# Thesis - Georgia Tech

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Thesis - Georgia Tech

Tail Buffet Alleviation of High Performance Twin Tail Aircraft using Offset Piezoceramic Stack Actuators and Acceleration Feedback Control A Thesis Presented to The Academic Faculty by Maxime P. Bayon de Noyer In Partial Fulfillment Of the Requirements for the Degree Doctor of Philosophy In the School of Aerospace Engineering Georgia Institute of Technology Atlanta, Georgia 30332 November 1999 Copyright 1999 by Maxime P. Bayon de Noyer Tail Buffet Alleviation of High Performance Twin Tail Aircraft using Offset Piezoceramic Stack Actuators and Acceleration Feedback Control Approved: _________________________ Sathya V. Hanagud _________________________ Erian Armanios _________________________ Wassim Haddad _________________________ Yves Berthelot _________________________ Steve Griffin Date Approved ____________ ii To my Parents, Florence and Philippe Bayon de Noyer my Sister, Astrid Bayon de Noyer for their support both material and spiritual. In memory of my grandfather, Xavier Baduel d’Oustrac. iii ACKNOWLEDGEMENTS I would like first to thank my advisor, Dr. Sathya Hanagud, for his time, his support and the knowledge that he shared with me during the course of this research work. I would also like to thank Dr. Erian Armanios and Dr. Wassim Haddad for the time they took to discuss with me some finer points on structures and control. I would further like to thank Dr. Steve Griffin for introducing me to the world of smart structures and imparting his knowledge to me at the early stage of this research work. I would like to acknowledge the technical support of Robert Englar and Robert Funk from the Georgia Tech Research Institute and thank them for their invaluable help during the wind tunnel tests. Finally, I would like to thank the machine shop staff of the School of Aerospace Engineering at the Georgia Institute of Technology, Harald Meyer, Harry Rudd and Wayne Springfield, who built the offset piezoceramic stack actuators and other specimens required for the experimental work of this research. I would also like to thank Chuck Albert for building some of the electronic equipment required for the experimental work. iv TABLE OF CONTENT ACKNOWLEDGEMENTS ...................................................................................iv TABLE OF CONTENT .......................................................................................... v LIST OF TABLES .................................................................................................. x LIST OF ILLUSTRATIONS ................................................................................xii SUMMARY .......................................................................................................xviii CHAPTER I. INTRODUCTION AND BACKGROUND ................................... 21 I.1. Tail Buffet of High Performance Twin Tail Aircraft.................................. 21 I.2. Tail Buffet Alleviation................................................................................ 23 I.3. Choice of Controller ................................................................................... 26 I.4. Choice of Actuator...................................................................................... 30 CHAPTER II. ACCELERATION FEEDBACK CONTROL .............................. 33 II.1. Acceleration Feedback Control for Single Degree of Freedom Systems.. 34 II.1.1. Generalized Single Degree of Freedom Acceleration Feedback Control....................................................................................................................... 35 v II.1.2. Design of the AFC Parameters for a SDOF System based on the Crossover Point ......................................................................................................... 37 II.1.3. Design of the AFC Parameters for a SDOF System based on the Optimization of the H2 Norm of the Closed Loop Modal Receptance ..................... 41 II.1.4. Design of the AFC Parameters for a SDOF System based on the Optimization of the H2 Norm of a Generalized Closed Loop Modal Transfer Function..................................................................................................................... 50 II.2. Acceleration Feedback Control for Multi-Degree of Freedom Systems... 58 II.2.1. Modal Space Representation of Single-Input Single-Output System under Acceleration Feedback Control....................................................................... 59 II.2.2. State Space Representation of a Multi-Input Single-Output System under Acceleration Feedback Control....................................................................... 64 II.2.3. AFC Crossover Design for the Control of Multi Degrees of Freedom Systems...................................................................................................................... 70 II.2.4. AFC H2 Norm Optimization of the Closed Loop Modal Response Design for Multi Degrees of Freedom Systems ........................................................ 75 CHAPTER III. OFFSET PIEZOCERAMIC STACK ACTUATOR.................... 80 III.1. Prototype Design of the Offset Piezoceramic Stack Actuator ................. 81 III.2. Modal Expansion Model for the OPSA acting on a Cantilever Beam..... 83 III.2.1. Development of the equations governing the piezoceramic stack .... 84 vi III.2.2. Development of the Equation Governing the Coupled System ........ 87 III.2.3. Solution of the Quasi-Static Problem................................................ 93 III.2.4. Solution of the homogeneous part of the dynamic problem: Mode Shapes and Natural Frequencies ............................................................................... 97 III.2.5. Low Frequency Approximation of the OPSA................................. 111 III.2.6. Transfer Function Model of the Low Frequency Approximation... 124 III.2.7. Numerical Validation of the Model Approximations ..................... 132 III.2.8. Experimental Validation of the Low Frequency Transfer Function Model ...................................................................................................................... 138 III.3. Optimization and Placement of the OPSA on the Experimental Benchmark .................................................................................................................. 145 III.3.1. Optimization of the Vertical Offset Distance for a Small Actuator 147 III.3.2. Optimal Placement of a Small Actuator.......................................... 149 CHAPTER IV. EXPERIMENTAL VALIDATIONS......................................... 153 IV.1. Control System Design Procedure based on Experimental Data........... 154 IV.1.1. Control System Objective ............................................................... 154 IV.1.2. Sensor Placement ............................................................................ 155 IV.1.3. Operating Vibration Magnitude for Control Authority Studies...... 156 IV.1.4. Actuator Placement ......................................................................... 158 vii IV.1.5. Actuator Authority Study................................................................ 162 IV.1.6. Plant Modeling................................................................................ 163 IV.1.7. Controller Design and Stability Assessment................................... 166 IV.1.8. Maximum Control Voltage Assessment ......................................... 168 IV.1.9. Controller Implementation.............................................................. 170 IV.1.10. Control System Design Procedure based on Experimental Data Pseudo Algorithm.................................................................................................... 171 IV.2. Wind Tunnel Tests for Active Tail Buffet Alleviation .......................... 172 IV.2.1. Wind Tunnel Facilities.................................................................... 172 IV.2.2. Aeroelastically Scaled Empennage Design..................................... 173 IV.2.3. Sensor Placement for Active Tail Buffet Alleviation ..................... 177 IV.2.4. Wind Tunnel Tail Buffet Response Study ...................................... 179 IV.2.5. Actuator Placement for Active Tail Buffet Alleviation .................. 183 IV.2.6. Actuators Authority Assessment..................................................... 185 IV.2.7. Plant Characterization ..................................................................... 188 IV.2.8. Controller Design and Stability Assessment................................... 192 IV.2.9. Maximum Control Voltage Assessment ......................................... 195 IV.2.10. Controller Implementation using a Digital Signal Processor........ 196 viii IV.2.11. Worse Buffet Conditions Control Experiment ............................. 199 IV.2.12. Angle of Attack Sweep Control Experiment (Robustness Issues) 200 IV.2.13. Free Stream Dynamic Pressure Sweep Control Experiment (Robustness Issues) ................................................................................................. 202 IV.3. Tests on a Full-Scale Vertical Tail......................................................... 205 IV.3.1. Full-Scale Control System Objective.............................................. 206 IV.3.2. Sensor and Actuator Placements..................................................... 207 IV.3.3. Plant Characterization ..................................................................... 212 IV.3.4. Controller Designs .......................................................................... 216 IV.3.5. Setup and Implementation of the Vibration Control System on the Full-Scale Sub-Assembly........................................................................................ 225 IV.3.6. Full-Scale Vibration Control Experimental Results ....................... 229 CHAPTER V. CONCLUSION AND RECOMMENDATIONS........................ 233 APPENDICES..................................................................................................... 238 REFERENCES.................................................................................................... 346 VITA ................................................................................................................... 353 ix LIST OF TABLES Table II-1 Optimal Controller Natural Frequency and Damping Ratio for an Open Loop System with ωs=100, ζs=0.01, a1=1, a2=1 and a Controller Gain of γ=0.05 ............. 58 Table III-1 Geometric and Material Properties of the Cantilever Beam......................... 132 Table III-2. Offset Distances of the OPSA ..................................................................... 133 Table III-3. Geometric and Material Properties of the Piezoceramic Stack ................... 133 Table III-4. Low Frequency Equivalent Characteristics of the Piezoceramic Stack ...... 134 Table III-5. Natural Frequencies (Hz) for the OPSA at Location 1................................ 135 Table III-6 Locations of Experimental Actuators ........................................................... 138 Table IV-1 Plant Parameters for the Bending Actuator Array........................................ 189 Table IV-2 Plant Parameters for the Torsion Actuator Array......................................... 190 Table IV-3 Controller Parameters ................................................................................... 193 Table IV-4 Modal Amplitudes associated with the Chordwise Study of the Actuator Placement ................................................................................................................ 209 Table IV-5 Modal Amplitudes associated with the Spanwise Study of the Actuator Placement ................................................................................................................ 210 x Table IV-6 Full-Scale Vertical Tail Plant Parameters .................................................... 215 Table IV-7 AFC Crossover Design Controller Parameters............................................. 217 Table IV-8 AFC H2 Optimal Design Controller Parameters........................................... 221 Table IV-9 Comparison of Functionals between SDOF Assumption and MDOF Optimization Designs.............................................................................................. 222 xi LIST OF ILLUSTRATIONS Figure II.1 Single Degree of Freedom System under AFC Root Locus Plot and Cross Over Point (o)............................................................................................................ 38 Figure III.1 Offset Piezoceramic Stack Actuator Structural Assembly ............................ 82 Figure III.2 Offset Piezoceramic Stack Actuator Mounted on a Cantilever Beam........... 84 Figure III.3 “Co-fired” Piezoceramic Stack Configuration .............................................. 85 Figure III.4 Model of the Offset Piezoceramic Stack Actuator Mounted on a Cantilever Beam.......................................................................................................................... 88 Figure III.5 Geometric Boundary Conditions Diagram .................................................... 91 Figure III.6 Natural Boundary Conditions Diagram ......................................................... 91 Figure III.7 Model of the Root OPSA Mounted on a Cantilever Beam.......................... 109 Figure III.8 Low Frequency Approximation of the OPSA acting on a Cantilever Beam ................................................................................................................................. 111 Figure III.9 Boundary Condition Diagram for the Approximation of the OPSA ........... 114 Figure III.10 Distribution of Forces generated by the OPSA and its Projection on the First ten Modes ................................................................................................................ 137 xii Figure III.11 Spanwise Distribution of the Residue of Normalized Force Distribution . 137 Figure III.12 Comparison of Cantilever Model and Experimental Transfer Functions for the First Actuator Location ..................................................................................... 140 Figure III.13 Comparison of Cantilever Model and Experimental Transfer Functions for the Second Actuator Location ................................................................................. 140 Figure III.14 Comparison of Torsional Spring Root Model and Experimental Transfer Functions for the First Actuator Location ............................................................... 143 Figure III.15 Comparison of Torsional Spring Root Model and Experimental Transfer Functions for the Second Actuator Location........................................................... 144 Figure III.16 Normalized Actuation Efficiency versus Vertical Offset Distance ........... 148 Figure III.17 Placement and Vertical Offset Optimization Curve for the First Bending Mode of the Benchmark Structure .......................................................................... 151 Figure III.18 Placement and Vertical Offset Optimization Curve for the Second Bending Mode of the Benchmark Structure .......................................................................... 151 Figure III.19 Placement and Vertical Offset Optimization Curve for the Third Bending Mode of the Benchmark Structure .......................................................................... 152 Figure IV.1 Principles of Actuator Placement ................................................................ 158 Figure IV.2 Pseudo-Algorithm of Control System Design Procedure............................ 171 Figure IV.3 Auto-Power Spectrum of the Buffet Load on the Outboard Trailing Edge Tip of the 1 to 1 Scale Model Left Vertical Tail at α=22° and q∞=7psf. ...................... 174 xiii Figure IV.4 1/16th Scale Model of the HPTTA with Aeroelastically Scale Empennage 175 Figure IV.5 First Four Modes of the Vertical Tail.......................................................... 178 Figure IV.6 Weighted (1, 2, -1, -1) Superposition of the Modes .................................... 179 Figure IV.7 Root Mean Square of the Trailing Edge Tip Acceleration versus Angle of Attack at q∞=9psf .................................................................................................... 181 Figure IV.8 Envelope of RMS of the Trailing Edge Tip Acceleration versus Angle of Attack and Free Stream Dynamic Pressure (o: Experimental Data Points)............ 182 Figure IV.9 Auto-Power Spectrum of Control Sensor under Operating Conditions ...... 183 Figure IV.10 Outboard(a) and Inboard(b) Views of the Experiment Vertical Tails....... 185 Figure IV.11 Actuator Authority Assessment................................................................. 187 Figure IV.12 Experimental and Modeled Transfer Functions for the Bending Array.... 191 Figure IV.13 Experimental and Modeled Transfer Functions for the Torsion Array..... 191 Figure IV.14 Bending and Torsion Controllers Transfer Functions ............................... 193 Figure IV.15 Generalized Root Locus Plot..................................................................... 194 Figure IV.16 Estimated Power Spectrums of Control Voltages ..................................... 195 Figure IV.17 Wiring for the Active Buffet Alleviation Experiment............................... 198 Figure IV.18 Experimental Setup for the Active Buffet Alleviation Experiment .......... 199 Figure IV.19 Comparison Between Open and Closed Loop Auto-Power Spectrum of Trailing Edge Tip Acceleration at α=20° and q∞=9psf........................................... 200 xiv Figure IV.20 Uncontrolled and Controlled Root Mean Square of the Trailing Edge Tip Acceleration versus Angle of Attack at q∞=9psf..................................................... 201 Figure IV.21 Uncontrolled and Controlled Root Mean Square of the Trailing Edge Tip Acceleration versus Free Steam Dynamic Pressure for Four Angles of Attack ..... 203 Figure IV.22 Percent RMS Reduction of the Trailing Edge Tip Acceleration of the Left Vertical Tail in the 0-300 Hz Frequency Band ....................................................... 203 Figure IV.23 Percent RMS Reduction of the Trailing Edge Tip Acceleration of the Left Vertical Tail in the 17.5-29.5 Hz Frequency Band around First Bending Mode.... 204 Figure IV.24 Percent RMS Reduction of the Trailing Edge Tip Acceleration of the Left Vertical Tail in the 65-99 Hz Frequency Band around First Torsion Mode ........... 204 Figure IV.25 Percent RMS Reduction of the Trailing Edge Tip Acceleration of the Left Vertical Tail in the 205-239 Hz Frequency Band around Second Torsion Mode... 205 Figure IV.26 Laboratory Sub-Assembly......................................................................... 206 Figure IV.27 Actuator Placement Optimization Plot for Chordwise Study.................... 210 Figure IV.28 Actuator Placement Optimization Plot for Spanwise Study...................... 211 Figure IV.29 Comparison between Modeled and Experimental Plant Transfer Functions ................................................................................................................................. 214 Figure IV.30 Comparison of SDOF Assumption and MDOF Optimization Designs AFC Crossover Controller Transfer Functions................................................................ 217 xv Figure IV.31 Comparisons of Open and Closed Loop Poles for SDOF Assumption and MDOF Optimization Designs of AFC Crossover Controller.................................. 218 Figure IV.32 Comparisons of Open and Closed Loop Transfer Functions for SDOF Assumption and MDOF Optimization Designs of AFC Crossover Controller ...... 219 Figure IV.33 Comparison of SDOF Assumption and MDOF Optimization Designs AFC H2 Optimal Controller Transfer Functions.............................................................. 222 Figure IV.34 Comparisons of Open and Closed Loop Poles for SDOF Assumption and MDOF Optimization Designs of AFC H2 Optimal Controller ............................... 223 Figure IV.35 Comparisons of Open and Closed Loop Transfer Functions for SDOF Assumption and MDOF Optimization Designs of AFC H2 Optimal Controller .... 224 Figure IV.36 Wiring for the Vibration Control Experiment of the Full-Scale Vertical Tail ................................................................................................................................. 227 Figure IV.37 Simulink Representation of the Controller................................................ 228 Figure IV.38 Experimental Comparison between Open and Closed Loop Accelerance at the Control Sensor Location.................................................................................... 230 Figure IV.39 Experimental Comparison between Open and Closed Loop Accelerance at the Secondary Performance Sensor Location.......................................................... 231 Figure IV.40 Zoom View of Experimental Comparison between Open and Closed Loop Accelerance at the Control Sensor Location ........................................................... 232 xvi Figure IV.41 Zoom View of Experimental Comparison between Open and Closed Loop Accelerance at the Secondary Performance Sensor Location................................. 232 xvii SUMMARY In high performance twin-tail aircraft (HPTTA), buffet induced tail vibrations occur when unsteady pressures associated with separated flow, or when vortices, excite the vibration modes of the vertical fin structural assemblies. At high angles of attack, flow separates at the leading edge of the wings, and vortices are generated at different locations such as the wing fuselage interface or the leading edge extensions. These vortices are convected by the geometry of the wing-fuselage interface toward the twin vertical tails. This phenomenon, along with the aeroelastic coupling of the tail structural assembly, results in vibrations that can shorten the fatigue life of the empennage assembly and limit the flight envelope due to the large amplitude of the fin vibrations. The main goal of this research work is to develop an active buffet alleviation system for high performance twin-tail aircraft using piezoceramic stack based actuators in combination with acceleration feedback control (AFC) theory. In order to complete this task, the work is divided into three main research areas. These areas are the formulation and the design of acceleration feedback controllers; the development and modeling of piezoceramic stack based actuator subassemblies; methods of implementing these controllers in structural systems control and the validation of the actuators and controllers for tail buffet alleviation. xviii For the acceleration feedback control work, new methods for the design of the controller parameters are presented for generalized single degree of freedom systems. These new designs are based on pure active damping and quadratic performance criteria, which are based on structural generalized coordinates, minimization. Then, noncollocated acceleration feedback multi-mode controller design methods are developed for a single sensor and a small number of actuator arrays (for Multi-Input Single-Output systems). Then, a new type of moment inducing actuator, the offset piezoceramic stack actuator, which is based on the use of piezoceramic stacks, is developed to provide the needed control authority for buffet alleviation. This actuator is also designed to satisfy high reliability and maintainability requirements. In addition, a technique is developed to analytically model the actuator on the basis of the modal expansion of the offset piezoceramic stack actuator driving a benchmark structure. The results of this analysis are used to create a low frequency approximation of the offset piezoceramic stack actuator as well as to optimize its offset distance and its placement. Because of the non-availability of reliable mathematical or numerical models for both the controlled structure and the buffet induced loads, a control system design method, which is based solely on the use of experimental data, is first developed. Then, two sets of experiments are conducted to show the feasibility of controlling the buffet induced vibrations during high angle of attack operations of a selected HPTTA. The first experiment validates both the effectiveness and the robustness of the developed active buffet alleviation system on an aeroelastically scaled model in wind tunnel studies. The xix second experiment shows that the combination of offset piezoceramic stack actuators and acceleration feedback control can suppress vibrations in a full-scale vertical tail subassembly. xx CHAPTER I. INTRODUCTION AND BACKGROUND I.1. Tail Buffet of High Performance Twin Tail Aircraft In high performance twin-tail aircraft (HPTTA), buffet induced tail vibrations were first noticed through their destructive effects of induced fatigue cracks in the vertical tail structural assembly. These fatigue cracks were noticed shortly (less than six months) after the aircraft was placed in service during the seventies[1], and many high angles of attack maneuvers were executed. After repeated temporary structural repairs, a thorough investigation of the conditions leading to the fatigue cracks confirmed that buffet induced tail vibrations were the cause of these fatigue cracks. In addition to the formation of fatigue cracks, buffet induced vibrations can restrict the flight maneuvering capability of the aircraft by restricting the angles of attack and speeds at which certain maneuvers can be executed. Furthermore, the formation of fatigue cracks may also lead to corrosion due to moisture absorption through these cracks. Because of these effects, a significant amount of maintenance efforts are spent on high performance twin-tail aircraft vertical tail assemblies in logistic centers. 21 The use of the term buffeting, as related to aircraft, appeared for the first time in the early 1930’s when the British Aeronautical Research Committee concluded that the cause of the accident of a commercial airplane was “buffeting” of the tail[2]. Buffeting was then defined as the irregular motion of a structure or parts of a structure excited by turbulence in the flow[3]. More recently, buffet was defined by Mabey[4] as the aerodynamic excitation provided by separated flows. Buffet induced tail vibrations occur when unsteady pressures associated with separated flow, or when vortices, excite the vibration modes of the vertical fin structural assemblies. At high angles of attack, flow separates at the leading edge of the wings, and vortices are generated at different locations such as the wing fuselage interface or the leading edge extensions. These turbulent flows are convected by the geometry of the wing-fuselage interface toward the vertical tails. This phenomenon, along with the aeroelastic coupling of the tail structural assembly, results in vibrations that can shorten the fatigue life of the empennage assembly and limit the flight envelope due to the large amplitude of the fin vibrations. The set of high performance twin-tail aircraft includes the F-14, F-15, F/A-18 and F-22. Most of these aircraft must maneuver at high angles of attack to meet their performance requirements. However, since the characteristics of the separated flow depend upon the geometry of the wing, the fuselage and the empennage, different kinds of tail buffet can exist. Vibrations of the vertical tails of the F/A-18 are attributed to broadband excitations resulting from the bursting of Leading-Edge Extension (LEX) vortices[5]. On the other hand, the tail buffet problem of the F-15 is associated with a 22 separated flow containing a narrow band of frequencies that engulf the tail assembly[6]. This narrow band of frequencies may contain fatigue critical frequencies. I.2. Tail Buffet Alleviation Many different approaches to tail buffet alleviation have been investigated. These approaches can be divided into two sets: the aerodynamic methods and structural dynamic methods. Aerodynamic approaches can be further divided into passive and active methods for reducing the strength, or for delaying the formation, of the vortices. The passive aerodynamic approach consists of adding fences on the wing or the fuselage. The leading edge extension (LEX) fences on the F/A-18 aircraft are examples of such devices. A LEX fence is a small trapezoidal plate attached to the upper surface of the LEX near the leading edge of the wing. The effects of the LEX fences on tail buffet have been studied extensively both experimentally[7-10] and numerically[11]. Other research on fences include the works of Klein et al.[12-13] on the F-15 aircraft as well as a generic twin-tail aircraft. The published active aerodynamic methods[14-19], which are active in the sense that power is required to drive the system, do not use any form of feedback control. One of the active aerodynamic methods is based on Tangential Leading Edge Blowing (TLEB). The effect of TLEB is to reduce the “effective angle of attack” of the vortex core. The application of TLEB for fin buffeting alleviation has been studied on single-fin aircraft[14-15] and twin-tail aircraft[16]. Another active aerodynamic approach is based on 23 forebody tangential slot blowing. Simulations carried out by Ken Gee et al.[17] have shown that such an apparatus reduces slightly the buffet induced vibrations by moving away the main frequencies of the vortices. The most recent active aerodynamic approach is a Flow Suction along the Vortex Cores (FSVC)[18-19] of the leading edges of a delta wing. Numerical studies of the method showed that it could delay the vortex breakdown flow upstream of the twin tail and modify the vortex core path. The structural dynamic methods for buffet alleviation can also be divided into passive and active approaches. The passive structural approach has been used extensively on F-15 aircraft[1]. This approach consists of reinforcing of the fin assembly with patches both to repair existing defects and to stiffen the assembly. An example of this approach is the “exoskin” doubler presented by Ferman et al.[1]. However, more recently, active structural control techniques have been investigated. The active structural dynamic approaches began with Ashley et al.[20] who numerically investigated the use of oscillating an aerodynamic control surface, namely the rudder for the F/A-18 and a tip vane for the F-15, to suppress tail buffet induced vibrations. Simple direct feedback controllers were designed to reduce the vibration of the vertical tails. The simulations showed the feasibility of using active vibration control for buffet alleviation. Later to obtain a buffet suppression system that operates independently of the flight control system, Lazarus et al.[21-22] numerically studied the feasibility of using an active piezoelectric buffet suppression system. This system called the Buffet Load Alleviation (BLA) consists of a large number of piezoceramic wafer actuators, strain 24 gages and accelerometers as sensors and Linear Quadratic Gaussian (LQG) design for the controller. Recently, Spangler and Jacques[23] have experimentally tested this Buffet Load Alleviation system on a full-scale F/A-18 empennage showing that their system could reduce RMS strain at the aft root of the vertical tail by up to 50%. Hauch et al.[24] developed an Active Vertical Tail (AVT) to study the feasibility of buffet alleviation using piezoceramic actuators, strain gage sensors and simple control techniques. The AVT was a 5% aeroelastically scaled structure with embedded piezoceramic wafers based on a generic aircraft. During wind tunnel experiments, the control techniques used to reduce buffet were based on Proportional, Integral and Differential (PID) feedback from each of the sensors. Nitzsche et al.[25] compared the two different approaches of using aerodynamic control surface or using piezoceramic actuators for tail buffet alleviation. Using LQG controllers, their simulations[25], showed that the strain actuation approach would appear to demonstrate a better performance. Then, Nitzsche et al.[26] tested their LQG controller on a full-scale F/A-18 empennage with strain actuation showing reduction of buffet vibrations by up to 58%. As part of the Actively Controlled Response Of Buffet Affected Tails (ACROBAT) program, Moses[27] performed wind tunnel tests on a 1/6-scale F/A-18 aircraft. The buffet alleviation controller, which was based on frequency domain compensation methods, was designed to suppress the response in the first bending mode of the tail using either the rudder or some piezoceramic actuators and an accelerometer as 25 sensor. The wind tunnel tests showed that using the rudder as the control actuator provided less alleviation than using piezoceramic actuators. During the SIDEKIC (Scaling Influences Derived from Experimentally-Known Impact of Controls) program[28], a hybrid actuator system using the rudder for the control of the first bending mode and piezoceramic wafers for the control of the first torsional mode was developed. Pado and Lichtenwalner[29] demonstrated, during wind tunnel tests, the ability of this hybrid actuator system in combination with Neural Predictive Control to alleviate buffet induced vibration in the vertical tails of the F/A-18 aircraft. An excellent summary on tail buffet alleviation advances can be found in the technical memorandum by Calarese and Turner[30]. More recently, Hopkins et al.[31] have summarized the different smart structure approaches for tail buffet alleviation of the F/A18 aircraft. Finally, Moses has presented, in Reference 28, the contributions of NASA to the field of active buffet alleviation. I.3. Choice of Controller Previous investigations, in the field of buffet alleviation, have used different types of controllers such as neural predictive control[29], LQG[21-23,25-26], PID[24], frequency domain compensation[27] or direct feedback[20]. These different methods have some advantages and some drawbacks. For example, one of the problems associated with the use of an LQG controller is that it requires an accurate model for both the structure and 26 the loads because the design of its observer depends on the external load influence matrix[32-33]. Further, LQG controllers do not provide guaranteed robustness properties[33]. In general, models of the structural systems are obtained in the form of a Finite Element Model (FEM). However, these models usually lack a good representation of the natural damping of the structure. Hence, before actually implementing any controller, the parameters of the structural systems are updated with experimentally obtained data resulting from system identification phase. Furthermore, for the problem of buffet alleviation, the loads have not, to date, been accurately modeled. As a result, some of the control designs have used methods of identifying the load profile by using wind tunnel data[20], while others have used a “linearized” concept which models the aeroelastic buffeting behavior of the tail as a superposition of two independent mechanisms (unsteady airloads induced by structural oscillations and the driving airloads due to buffeting)[25]. Another significant problem associated with most controllers is their spillover effect[32,34]. Spillover effects are the result of both sensing and actuating the unmodeled or uncontrolled modes or states of the controlled structure. These unmodeled modes arise from the fact that a continuous structure has an infinite number of degrees of freedom. As a result, the number of modes or states, considered for the control design, must be truncated to obtain a model with a finite number of degrees of freedom. Spillover effects will introduce changes between the modeled closed loop result and the actual behavior of the closed loop system. These changes may even result in instabilities[34]. To decrease the spillover effects, it is important that the magnitude of the controller transfer function 27 decrease rapidly as frequency increases which means that the controller has a fast roll-off at high frequencies. One method, currently used to achieve this objective, is the development of a weighted H∞ controller. Furthermore, for non-collocated sensors and actuators, the phase of the controller signal should be either 0 or 180 degrees, with respect to the sensor signal, at high frequencies such that the control forces do not drive the structure unstable by decreasing the closed loop damping. This last requirement results in controllers with even relative degrees between the denominator and numerator of each of the controller transfer functions. Another problem is the order of the controller, which is significant for its implementation. The order of the controller, when implemented digitally, is directly related to the speed at which the controller can be implemented and the size of the code which must be stored in the memory on the controller system. This means that for large order controllers, their implementation cannot be executed rapidly and will require a large amount of memory. As an example, LQG controllers usually result in high order controller even if the control objective is to control only a selected number of modes within a bandwidth because the controller order will be equal to the number of states within that bandwidth. In classical approaches to the control of flexible structures, system equations are usually rewritten in a state space domain. But these transformations to state space domain often lose insight into the physics of the problem from the point of view of a structural dynamicist. Since the work of Goh and Caughey[35], in 1985, and the introduction of the Positive Position Feedback (PPF) controller, the second order compensators enable 28 designers to keep the system of equations of motion in their second order form. But, control schemes such as PPF and Strain Rate Feedback (SRF) are not unconditionally stable. Juang and Phan[36] proposed a second order compensator using acceleration feedback, that was unconditionally stable. The unconditional stability of the scheme, applied to flexible structure, was later proved for multiple pairs of collocated sensors and actuators with their actuator dynamics[37]. In 1996, Goh and Yan[38] developed a method of assigning the damping ratio and scalar gain to these pairs of collocated sensors and actuators which was based on the use of critically damped compensators. As illustrated in Goh and Yan[38]’s design of Acceleration Feedback Control (AFC), the computation of the controller parameters does not require the external load influence matrix. As a result, the controller design does not require an accurate model of the loads. Further, AFC has a relative degree between the denominator and numerator of each of the controller transfer functions of two so that its roll-off at high frequencies is 40dB per decade and its phase at those frequencies is either 0 or 180 degrees, which is beneficial for non-collocated actuators and sensors. Furthermore, AFC enables the designer to control the vibration amplitude at selected frequencies within a given bandwidth without the additional order of the controller for uncontrolled states. Finally, the equations of motion of the closed loop system and of the controller can be written in a similar second order equation form. 29 I.4. Choice of Actuator The selection criteria for the choice of vibration control actuator involve mechanical properties, electrical properties and cost. For tail buffet induced vibration suppression, the primary concern is the control authority that can be generated by the actuator. Comparisons between the use of the rudder or induced strain actuators for buffet alleviation by both Nitzsche et al.[25] and Moses[27] indicate that piezoceramic actuators are more efficient. Furthermore, as noted by Lazarus et al.[21], using induced strain actuator instead of the rudder enable the buffet alleviation system to run independently of the flight controls and does not restrict in any way the maneuverability of the aircraft. For the purpose of buffet alleviation by induced strain actuation, every researcher[21-29], to date, has used Lead Zirconate Titanate (PZT) ceramic wafers as piezoelectric actuators. However, the control authority of PZT wafer actuators, unless used in large quantities, is usually not sufficient for most real world applications. Stack actuators can increase the control authority through a more efficient use of the piezoceramic material properties. This increase is obtained by the use of the longitudinal d33 coefficient instead of the transverse d31 or d32 coefficient generally used with wafers. Furthermore, the increased stack forces result from the addition of the effective piezoelectric reactions by using the accumulation of reaction from each PZT in series. Piezoceramic stack actuators have been successfully used for vibration control. This type of actuator has been used, as active elements for vibration suppression, in truss structures[39]. These actuators have also been used to generate point loads to control 30 vibrations in plates by placing them between the plate and a stiffener[40]. Furthermore, piezoceramic stack actuators have been implemented as moment inducing actuators by placing the stack within cutouts in stiff beams and plates[41] or mounting the stack in an external assembly for vibrations suppression in rapid fire guns[42]. These concepts have been the motivation for the choice of an actuator in this research program. Piezoceramic stack actuators can be used as “induced bending moment actuators”, which can generate much larger moments than PZT wafers. For this research program, an externally mounted actuator sub-assembly was designed, this actuator was called the “Offset Piezoceramic Stack Actuator” (OPSA). This OPSA was designed to have enhanced reliability and maintainability properties compared to previously designed piezoceramic stack based actuators. For active buffet alleviation, this piezoceramic stack actuator structural assembly would be bonded to the vertical tail skin and covered by aerodynamic shielding if necessary. In addition to the choice of the actuator, the next issue with the offset piezoceramic stack actuator is the modeling of its actuation. In general, piezoceramic stack actuators and their induced actuation on controlled structures are modeled either using Finite Element Models (FEM) or using the impedance method[43-46]. However, the FEM model, which is a numerical approach, does not provide the needed insight into the physics of the problem. The impedance approach was initially developed for piezoceramic wafers by Liang et al.[47] and was studied to provide the impedance between the input voltage to the actuator and the response of the controlled structure. However, the transfer function derived from the impedance method is in a compact form that cannot 31 be readily used to extract the plant modal parameters, such as the natural frequencies, needed for the control design and cannot be updated with experimental data. As a result, a technique is developed to analytically model the actuator on the basis of the modal expansion of the offset piezoceramic stack actuator driving a benchmark structure. The results of this analysis are then used to create a low frequency approximation of the offset piezoceramic stack actuator as well as to optimize its offset distance and its placement. 32 CHAPTER II. ACCELERATION FEEDBACK CONTROL As discussed in Section I.3, the only previously reported design procedure, to determine the parameters of acceleration feedback controller, was proposed by Goh and Yan[38]. In this method, damping ratios and gains were assigned, for pairs of collocated sensors and actuators, on the basis of the use of critically damped compensators. In practice, it is not always possible to collocate sensors and actuators. In particular, for a cantilevered structure, which is a close approximation for the case of the vertical tail of an aircraft, with acceleration sensors and bending moment inducing actuators, collocating sensors and actuators would result in a poor design. Either the signal to noise ratio of the sensor would be small or the actuator authority would be small. Hence, in this research work, two new designs for acceleration feedback controllers are developed for noncollocated multi-input, single-output (MISO) systems. For this purpose, different controller design strategies were considered. For pure active damping, the control designer would like to provide additional electronic damping to the controlled structure while avoiding perturbations to the natural frequencies or creations of new closed loop poles. As discussed in subsequent sections, this strategy leads to a crossover design for acceleration feedback controllers. On the other hand, for an optimal design of vibration controllers, the design is selected to minimize a chosen 33 performance criterion, which is possibly subjected to other constraints. The considered performance criterion, for this research work, is the H2 norm of a closed loop transfer function between a combination of the modal responses of selected modes and the disturbance load. This selected performance criterion does not directly involve the control signal magnitude, which is usually considered for a classical H2 optimization control such as the Linear Quadratic Regulator or the Linear Quadratic Gaussian controllers. Instead of using a controller cost to determine the gain matrix of the controller, the gains of the high authority acceleration feedback controllers can be chosen during an iterative process that insure that control signal will not be saturated. II.1. Acceleration Feedback Control for Single Degree of Freedom Systems First, to obtain close form solutions for the acceleration feedback controller parameters and to study the stability of non-collocated acceleration feedback control, the case of the control of a generalized Single Degree Of Freedom (SDOF) system is considered. This new representation for the SDOF system is generalized by introducing a sensor influence coefficient as well as an actuator influence coefficient. The addition of these coefficients enables the designer to consider actuation and sensing that can be in phase or out of phase. 34 II.1.1. Generalized Single Degree of Freedom Acceleration Feedback Control The generalized equations describing the closed loop behavior of a single degree of freedom system under Acceleration Feedback Control (AFC) consist of a structural equation with a feedback force due to the actuator and a disturbance force, and a compensator equation with acceleration sensing. These equations are generalized by introducing influence parameters for the actuator and for the sensor. These equations, in the modal space, are: ìξ + 2ζ s ωs ξ + ωs2 ξ = −a1γωc2 η + f í 2 îη + 2ζ c ωc η + ωc η = a 2 ξ (II.1.a-b) In these equations, ξ and η are the modal coordinates of the structure and of the compensator; respectively. Then, ωs, ωc, ζs and ζc are the natural frequencies and the damping ratios of the structure and the compensator, respectively. Further, γ is the controller gain applied to the feedback signal; and a1 and a2 are the influence parameters of the actuator and sensor, respectively. Finally, f is the external disturbance that drives the system. Without loss of generality, by taking the Laplace Transforms of Equations (II.1.ab), assuming zero initial conditions, and solving for η, the transfer functions of the closed loop system and of the controller, are obtained: 35 ( ) s 2 + 2ζ c ω c s + ω c2 ξ = 2 f s + 2ζ s ωs s + ωs2 s 2 + 2ζ c ω c s + ω c2 + a 1a 2 γωc2 s 2 í ω2 γ η ω c2 γ G c (s ) = c 2 = 2 a 2 s ξ s + 2ζ c ω c s + ωc2 î ì G s (s ) = ( ( )( ) (II.2.a-b) ) Equation (II.2.b) shows that the controller has a relative degree of two between its denominator and numerator. This result in a controller with 40 dB per decade roll-off and a phase of 0 or 180 degrees at high frequencies. The stability of such a system can be studied by applying the Routh-Hurwitz criterion[48] to the closed loop characteristic equation: (s 2 )( ) + 2ζ s ωs s + ωs2 s 2 + 2ζ c ω c s + ω c2 + a 1a 2 γωc2 s 2 = 0 (II.3) The Routh-Hurwitz criterion states that a necessary and sufficient condition for stability is that all the principal minors of the corresponding Routh-Hurwitz array be greater than zero. In this case the principal minors are: M1 = β s + β c β ω 2 + β c ω c2 + β s β c + a 1a 2 γωc2 (β s + β c ) M2 = s s βs + βc ( M3 = (( β s β c ωs2 − ω c2 M4 = ω ω 2 s 2 c ) ) + (β 2 ( )) ( + β c ) β s ω c2 + β c ωs2 + a 1a 2 γω c2 (β s + β c ) β s ω c2 + β c ωs2 β s ωs2 + β c ω c2 + β s β c + a 1a 2 γωc2 (β s + β c ) s ( ) ) (II.4.a-c) In the above equation, βs = 2ζsωs and βc = 2ζcωc. For the purpose of this research, the open loop system and the controller are assumed to be asymptotically stable. Hence, the modal coefficients, βs and βc, are positive. Then, if a1 and a2 have the same sign, meaning that the mode is observed in phase by the sensor with respect to the actuator 36 input, a sufficient condition for stability is that γ is positive. On the other hand, if a1 and a2 have opposite signs, meaning that the mode is observed out of phase by the sensor with respect to the actuator input, a sufficient condition for stability is that γ is negative. Since the only term depending on γ always appears in the form a1a2γ in Equations (II.4.a-c), a sufficient condition for stability is that a1a2γ be positive. II.1.2. Design of the AFC Parameters for a SDOF System based on the Crossover Point For the introduction of additional pure active damping, the control designer would like to provide additional electronic damping to the controlled structure while avoiding any change in its natural frequencies and avoiding the creation of new frequency peaks in the closed loop response. A particular point of interest in the root locus plot of a SDOF system controlled by AFC, as shown on Figure II.1, is the crossover point. At this point, the closed-loop poles of the structure and of the compensator coincide. This means that the frequency response of the closed-loop system will only exhibit a single peak. 37 Root Locus Plot 300 Imag 200 100 0 -100 -200 -300 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Real Figure II.1 Single Degree of Freedom System under AFC Root Locus Plot and Cross Over Point (o) The crossover point is obtained when the roots of the closed-loop characteristic equation are repeated complex conjugate pairs: (s 2 )( ) ( + 2ζ s ωs s + ωs2 s 2 + 2ζ c ωc s + ωc2 + a 1 a 2 γωc2 s 2 = s 2 + 2ζ f ωf s + ωf2 ) 2 (II.5) In the above equation, ωf and ζf are the natural frequency and the damping ratio of the closed-loop system, respectively. By expanding both sides of the equation and equating the terms of same power of s, four conditions are obtained to operate at the crossover point. These are: 38 ì2ζ f ωf = ζ s ωs + ζ c ωc , 2ωf2 + 4ζ f2 ωf2 = ωs2 + 4ζ s ωs ζ c ωc + ω2c + a1a 2 γωc2 , í 2ζ f ω3f = ωs2 ζ c ωc + ω2c ζ s ωs , (II.6.a-d) 4 2 2 îωf = ωs ωc . Using Equations (II.6.a), (II.6.c) and (II.6.d) to find a relation between the compensator natural frequency, the compensator damping ratio, the structural natural frequency and the structural damping ratio, we must have: ωs ωc (ζ s − ζ c )(ωs − ωc ) = 0 (II.7) By inspection of Equation (II.7), we obtain the following three possible conditions: ωc = 0, ζ c = ζs , ωc = ωs . (II.8.a-c) The condition (II.8.a) is not a useful condition since it would lead to ωf = 0. Further, condition (II.8.b) is not useful either since it leads to: ζ æω ω ö ζ f = s çç s + f 2 è ωf ωs (II.9) A design based on Equation (II.9) would mean that the final closed-loop natural frequency would be far away from the initial structural natural frequency to increase the damping. However, our objective is to avoid any changes in the natural frequencies of the system when the loop is closed. 39 The third condition, Equation (II.8.c), provides a practical design. It is taken to be the first crossover condition. Equation (II.6.d), under this condition, yields the desired condition that the closed-loop natural frequency will also be equal to the open-loop structural natural frequency. Using this result and Equations (II.6.a) and (II.6.b), another crossover design condition is: γ= (ζ s − ζ c )2 (II.10) a1a 2 Since the numerator of the right hand side of Equation (II.10) is quadratic, we obtain that, under cross over conditions, the product a1a2γ will always be positive so that the closed-loop system will be stable. Finally, the closed-loop damping ratio will be given from Equation (II.6.a) as: ζf = 1 (ζ s + ζ c ) 2 (II.11) From Equation (II.11), the closed loop damping ratio is the mean of the open loop and compensator damping ratios. Hence, to provide additional electronic damping to the system, the compensator damping ratio should be chosen larger than the open loop damping ratio. Thus, a design methodology will be to first obtain the initial structural parameters ωs and ζs as well as the influence parameters of the actuator and of the sensor, a1 and a2, either from a model of the system or by using experimental data. Different parameter designs for AFC to operate at the crossover point can be developed. From Equations 40 (II.8.c) and (II.6.d), the natural frequencies of the open loop system, controller and closed loop system are equal. Hence, the natural frequency of the compensator is not a design parameter. However, the gain and damping ratio of the controller as well as the closedloop damping ratio are valid design parameters. However, the crossover conditions must be satisfied which leaves only one free design parameter. Hence, one can choose a final closed loop value for the damping ratio, ζf, and solve Equation (II.11) for the necessary compensator damping ratio, ζc, and then use Equation (II.10) to compute the scalar gain, γ, to operate at the crossover point. Another design would use either the scalar gain, γ, or the damping ratio of the compensator, ζc, and use Equation (II.10) to compute the remaining parameter to operate at the crossover point. II.1.3. Design of the AFC Parameters for a SDOF System based on the Optimization of the H2 Norm of the Closed Loop Modal Receptance In the case of vibration suppression in a given structure, an optimization of the controller design parameters can be performed to meet a selected objective or a performance criterion. One way to design the single degree of freedom AFC compensator for vibration suppression is to minimize the H2 norm of the closed-loop receptance which is the system transfer function, G(jω), between the modal displacement and the external disturbance force. The performance criterion is defined to be: G 2 æ 1 =ç è 2π +∞ −∞ G ( jω) 2 F ö dω ÷ 1/ 2 = æç è +∞ 0 41 H (t ) 2 F dt ö 1/ 2 = H 2 (II.12) In this Equation, A F ( = tr A * A ) 1/ 2 is the Frobenius norm and H is the impulse- response matrix of the system, which is the inverse Laplace transform of the transfer function matrix. This design is equivalent to minimizing the L2 norm of the impulse modal response of the closed loop system, y(t), due to a unit impulse load, w(t), which is defined to be: y 2 = æç è +∞ 0 1/ 2 y T (t ) y(t ) dt ö (II.13) Or for a stochastic process, this design is equivalent to minimizing the covariance of the closed loop modal response, y(t), due to a unit white noise disturbance, w(t), which is defined to be: æ ì1 y 2 = çç lim E í è t →∞ î t t 0 üö y (τ) y(τ) dτý 1/ 2 T (II.14) It is to be noted that none of the definitions given by Equation (II.12) to (II.14) depend on the control signal directly. Hence, on the basis of this criterion, the controller will be a high authority controller. For this design, it will be proved that there does not exist an optimal controller gain. Therefore, instead of defining the controller gain by weighting the control signal, it can be defined by iterations in such a way that the control signal is not saturated. 42 To optimize the acceleration feedback control compensator for the norms defined in Equation (II.12) to (II.14), we define a functional J, which depends on the controller parameters: J (ωc , ζ c , γ ) = G 2 (II.15) 2 The closed loop system can be rewritten in a state space based on the modal states of the controlled system and the controller states: ~ ~ ìx (t ) = Ax (t ) + B w (t ) í ~ y(t ) = Cx (t ) (II.16) where: ìξ ü é 0 2 ïξ ï ê ï ï ~ ê − ωs x = í ý, A = ê 0 ïη ê 2 ïîη ë− a 2 ωs 1 − 2ζ s ωs 0 − 2a 2 ζ s ωs 0 − a 1 γω c 2 0 2 − ωc − a 1a 2 γωc 2 0 ù ì0ü ï ï 0 ~ ï1ï , B=í ý 1 ï0 ïîa 2 − 2 ζ c ωc ~ C= 1 0 0 0 (II.17) The transfer function between the modal response and the external disturbance force associated with the system described by Equation (II.16) is given by: ( ) ~ ~ ~ G (s ) = C sI − A B (II.18) ~ Since the matrix A represents to closed loop system parameters, to obtain a stable ~ closed loop system, the matrix A should be stable, meaning that the real parts of its 43 eigenvalues should be negative. Assuming that the closed loop system is stable, the modal impulse response due to an impulse load, w(t) = 2δ(t)v, where δ is the symmetric delta function, is given by: ~ ~ ~ y(t ) = C e At B v, t ≥ 0 (II.19) Hence, the functional defined in Equation (II.14) ∞ J (ωc , ζ c , γ ) = 0 ~ ~T ~ ~ ~ ~ ~ v T B T e A t C T C e At B v dt = tr é v v T B T êë +∞ 0 ~T ~ ~ ~ ~ e A t C T C e At dt B ù (II.20) In Equation (II.20), “tr” represent the trace operator. The set of all possible disturbance vectors v is spanned by the unit column vectors ei = coli(Im), i=1,…,m, where m is the length of the disturbance vector v and coli(Im) is the ith column of the identity matrix Im, hence by redefining the functional J for the sum of all possible unit disturbance vector ei, we have: m T ~ J (ωc , ζ c , γ ) = å tr é e i e i BT ê ë i =1 +∞ 0 ~T ~ ~ ~ ~ e A t C T C e At dt B ù ú +∞ ~ T ~ ~ ~ é m ~ù T ~ e A t C T C e At dt B ú = tr ê å e i e i B T 0 ë i=1 ~ +∞ A~T t ~ T ~ A~t ~ ù = tr é I m B T e C C e dt B êë 0 (II.21) Which can be rewritten in term of the observability Gramian of the closed loop system, P, which is given by: P= +∞ 0 ~T ~ ~ ~ e A t C T C e At dt (II.22) 44 Using the above definition into Equation (II.20), we obtain: ( ) ( ) ~ ~ ~~ J (ωc , ζ c , γ ) = tr B T PB = tr PBB T (II.23) ( ) ~ ~ Since the closed loop system is assumed to be stable and the pair A,C is assumed to be observable, the observability Gramian of the closed loop system, P, is positive definite and the solution of the Lyapunov equation[33]: ~ ~ ~ ~ PA + A T P + C T C = 0 (II.24) Since P is symmetric, its 10 independent elements are obtained by solving the linear system of equations given by Equation (II.24). Then by computing the trace of the product of matrices defined by Equation (II.23), the H2 norm of the transfer function between the modal response and the external disturbance is: ωs ζ c + ωc ζ c ( 1 + a 1a 2 γ ) + 4ωc ζ c ωs ζ s + 4 3 G 2 2 4 2 ( 2 ) 2 2 ( 3 2 2 ) ) )+ ωs ζ s ωc a 1a 2 γ + 4ζ c + 2ωs ωc ζ c 2ζ c + 2ζ s − 1 = 4ωs ( {a a γω ( ( ω 3 2 1 2 4 c 2 c ( 4 2 ) ( 2 + ωs ζ c ζ s + ωc ωs ζ c + ζ s 2 2 ) 2 2 ( 2 2 (II.25) ) )} 2 ζ c ζ s ωc + ωs + 4ωc ζ c ωs ζ s ωc + ωs + 2ωs ωc 2ζ c + 2ζ s − 1 Then, the derivative of the norm with respect to ωc, ζc and γ are set to zero: { ( ( 4ζ a 1a 2 γω c ζ c ω c ∂J = ∂ω c 2 c ( 2 ω s − a 1a 2 γω c 2 + 2ω s ζ c ω s − ω c 2 2 ) ((ω )(ω ζ s 2 2 s s ( )} + ω c ζ c ) − ζ c ωs − ω c 2 ) 2 + ω c ζ s + 2ω s ω c ζ c D 45 2 ) 2 2 ) = 0 (II.26.a) a 1a 2 γωc ∂J = ∂ζ c {( 4ζ 3 2 c 2 ωs − a 1a 2 γωc 2 ) (ω ζ s ( + ω c ζ c ) − ζ c ωs − ω c 2 s 2 2 D 2 − a 1a 2 ω c ζ c {(ζ ω 2 s s ( + ζ c ωc ) ωc − ωs 2 ∂J + 4ωs ωc ( ωs ζ c + ωc ζ s )( ωc ζ c + ωs ζ s ) = ∂γ D 2 )}= 0 2 2 (II.26.b) ) 2 2 } =0 (II.26.c) The denominator of Equations (II.26.a-c) is given by: [ (ω D = 16ωs ( 2 s − ωc ( )ζ 2 2 2 + a 1a 2 γωc ωc ζ c + ωs ωc ζ s ) ) ( − ωc 2 c 2 2 ( 2 2 ) + 4ω s ω c ζ c ω s ω c ζ s + ζ c + ω s + ω c ζ s ζ c ( a a γω 1 2 2 c )( ( 2 2 ) ( )] [ (ω 2 2 s 2 ) )ζζ )] 2 2 + 4ω s ω c ζ s ζ c ω s ω c ζ s + ζ c + ω s + ω c ζ s ζ c s c + (II.27) 3 First, it is to be noticed that, from inspection of the left hand side of Equation (II.26.c), the derivative with respect to γ is strictly negative and increases uniformly toward zero as γ tends to infinity. Hence, there is no optimal solution for the controller gain as a design parameter. However, for a particular controller gain, by setting Equations (II.26.b) to be identically zero, only the last term of Equation (II.26.a) remains. Hence, the following condition, to minimize the H2 norm, is obtained: æω2 öæ ω 2 ö ζ ω ζ c çç c 2 − 1÷÷ çç c 2 + 2 c c + 1 = 0 ζ s ωs è ωs è ωs (II.28) 46 In order to obtain a ratio between the natural frequency of the controller and the natural frequency of the open loop system that is real and positive, the only possible condition is: ωc = ωs (II.29) With this condition and Equation (II.26.a) or Equation (II.26.b), the second equation that should be satisfied to minimize the H2 norm of the closed loop transfer function is as follows: (ζ c + ζ s )2 ( a 1a 2 γ − 4ζ c 2 ) = 0 (II.30) For a stable controller, the compensator damping ratio should be positive. Hence, the only possible solution of Equation (II.30) is: ζc = 1 2 a 1a 2 γ (II.31) It is to be noted that for this design to meaningful, the controller damping ratio should be real. Hence, the product a1a2γ must be positive and the closed loop system will be stable. As a result, the design of an AFC controller that minimizes the H2 norm of the closed loop transfer function between modal displacement and external load of an SDOF system begins with the choice of a controller gain such that the product a1a2γ is positive. Then, Equation (II.31) is used to compute the optimal damping ratio for the controller. Finally, the natural frequency of the controller is set to be equal to the natural frequency 47 of the open loop system. In order to insure that the control signal will not be saturated, closed loop simulations can be investigated and an iteration process on the controller gain can be performed. The principal drawback associated with the use of optimal controllers arises from uncertainties in the model of the open loop system, which will create departure from optimality and may even lead to instabilities. As discussed in Section II.1.1, a sufficient condition for stability is that the product a1a2γ is positive. Hence, the only uncertainties that may lead to instability are uncertainties in the sign of the product of the actuator and sensor influence coefficients, a1a2. However, such type of uncertainty are extremely unlikely since the phase of the modeled transfer function is defined by this product and can be easily compared to the phase of the actual transfer function between the control sensor and the actuator input voltage. Hence, robust stability of the H2 optimal design for a single degree of freedom system is insured. To study the effect of uncertainty on the performance of the H2 optimal design, the departure from optimality due to uncertainties in the open loop system parameters is studied. Since the open loop parameters are scalars, any type of linear uncertainty can be modeled as: q s , true = q s ,identified (1 + ε q ) (II.32) The uncertainties are defined to be present in the natural frequency, εω, the damping ratio, εζ, and the product of the sensor and actuator influence coefficients, εa. 48 The departure from optimality due to uncertainties, Dopt, is defined as the difference between the H2 norms of the closed loop system using controller parameters computed with the identified uncertain modal parameters and the H2 norms of the closed loop system using controller parameters computed with the actual parameters normalized by the latest. Hence, the departure from optimality due to uncertainties is defined to be: D opt = = G cl,controller ( based on identified parameters ( G cl,controller 2 − G cl,controller based on true parameters based on true parameters 2 G cl ωs ( 1 + ε ω ), ζ s (1 + ε ζ ), ω c = ωs , ζ c = a 1a 2 γ 2, γ ) 2 (II.33) 2 G cl ωs ( 1 + ε ω ), ζ s (1 + ε ζ ), ωc = ωs ( 1 + ε ω ), ζ c = a 1 a 2 ( 1 + ε a )γ 2 , γ ) −1 2 By assuming small uncertainties, meaning all ε’s small with respect to one (ε<<1), and then a small open loop damping ratio, since typical damping ratios are less than 0.05, a Taylor series approximation, of order two, of the departure from optimality is given by: D opt ≈ 2 ù æ 5 7 öö 2 1æ 3 1 éæ 3 α2 æ ö ö 2 3 ê ç 1 − α ( 1 + ε ζ ) ÷ ε a + çç 1 − α + 2 ç 1 − α ÷ ÷÷ ε ω + ç α − 1÷ ε a + O ε 4 ëê è 2 2 2 4 2 ζ ø è ø ø è s ø è ( ) (II.34) In this equation, the non-dimensional parameter α is the ratio between the open loop damping ratio and the controller damping ratio computed for the model. It is given by α = 2ζ s a1a 2 γ . For most applications, the parameter α is chosen to be larger than the open loop damping ratio, ζs, to obtain a less than critically damped compensator. We can assess from Equation (II.34) that, for a large compensator gain (α << 1), the departure from optimality due to uncertainties in the model of the sensor and actuator 49 influence parameters behaves like εa. This means that a controller with actual influence parameters larger than the one modeled will likely perform better than expected. Simultaneously, the departure from optimality associated with uncertainties in the damping ratio of the open loop system decreases when the value of the controller gain is large. This also applies to the case of the uncertainties in natural frequency. Furthermore, to keep the departure from optimality due to uncertainties in the natural frequency small, the coefficient multiplying (α/ζs)2 in Equation (II.34) should be positive hence α should be kept below 2/7, this means that the minimum value of the gain that should be chosen for the controller is γ min = 49ζ s 2 a1a 2 . II.1.4. Design of the AFC Parameters for a SDOF System based on the Optimization of the H2 Norm of a Generalized Closed Loop Modal Transfer Function In general, the H2 norm of a closed loop transfer function between modal displacement and input disturbance force, that was used in the previous section, may not be the best criterion to achieve a particular goal. In fact, using a performance criterion based on a combination of modal displacement and modal velocity may yield a better design. Thus, instead of minimizing the H2 norm of the closed loop receptance, the minimization of the H2 norm of a generalized closed loop modal transfer function between a performance signal and the input disturbance force is considered. However, in a manner similar to the previous section, the control signal is not included directly in the performance criterion. 50 As discussed in the previous section, for a stochastic process, this design is equivalent to minimizing the covariance of the closed loop performance measurement, z(t), due to a unit white noise disturbance, w(t). Hence, the performance criterion is defined to be: æ ì1 z 2 = çç lim E í è t →∞ î t t 0 üö z (τ) z(τ)dτý 1/ 2 T (II.35) In Equation (II.35), E is the expectation function. The computation of the H2 norm of the generalized closed loop modal transfer function follows the same steps described in Section II.1.3. However, let us define the state space vector as a function of the generalized coordinate of the structure, ξ, and the coordinate of the controller, η, as follows: ì ωs ξ ü ï ξ ï ï ï x=í ý ïω c η ïî η (II.36) This choice of the state space vector is motivated by two reasons. First, the system state matrix associated with this state vector is better conditioned than the one given in Equation (II.17). This is due to the fact that the orders of magnitude of all the non-zero terms of the system state matrix, Equation (II.37.a), lie within a smaller interval than the ones in the system state matrix defined in the previous section. With this choice of state vector, the system state matrix and input disturbance force vector are given by: 51 é 0 ê −ω ~ s A=ê ê 0 ê ë − a 2 ωs ωs − 2ζ s ωs 0 − 2a 2 ζ s ωs 0 − a 1 γω c 0 − (1 + a 1a 2 γ ) ωc 0 ù ì0ü ï ï 0 ~ ï1ï , B=í ý ωc ï0 ïîa 2 − 2ζ c ωc (II.37.a-b) The second reason is that by defining the performance measurement, z(t), which is involved in the performance criterion, Equation (II.35), as: ~ z=Cx (II.38) There are three immediate choices for simple performance measurements. These choices result, for stochastic process, in the minimization of criteria corresponding to the potential energy, the kinetic energy or the sum of potential and kinetic energy. The ~ performance output matrices, C , associated with these three choices and the resulting performance criteria are: ~ C P = [1 0 0 0] , ~ C K = [0 1 0 0] , zP 2 zK é1 0 0 0ù ~ , zT CT = ê ë0 1 0 0 2 2 1 æ ì1 = çç lim E í m è t →∞ î t 2 0 1 æ ì1 çç lim E í = m è t →∞ î t 1 æ ì1 = çç lim E í m è t →∞ î t 1/ 2 üö k ξ (τ) dτý ÷÷ ø t t 0 t 0 üö m ξ 2 (τ) dτý (II.39.a) 1/ 2 (II.39.b) 1/ 2 üö k ξ (τ) + m ξ 2 (τ) dτý ÷÷ ø 2 (II.39.c) In Equations (II.39.a-c), m and k are the mass and stiffness of the single degree of freedom system respectively. 52 In order to generalize the result, let us define the following output measurement matrix: ~ ér C=ê1 ë0 0 r2 0 0ù 0 0 (II.40) In the previous section, the three controller parameters that were considered for the optimization process were the controller gain, γ, the controller natural frequency, ωc, and the controller damping ratio, ζc. Let us note however, that in the closed loop state matrix, given in Equation (II.37), the natural frequency of the controller, ωc, and the product ζcωc appears independently. As a result, for this section optimization process, we will consider the following three controller parameters instead, the controller gain, γ, the controller natural frequency, ωc, and a coefficient defined by the product of the controller natural frequency and damping ratio, βc = ζcωc. Hence, for all the following ~ computations, the (4,4) term of the closed loop state matrix A is replaced by -2βc. As discussed in the section II.1.3, the functional to be minimized, which is the H2 norm of the generalized modal transfer function, is given by: ( ~~ J (ωc , β c , γ ) = z 2 = tr PBB T ) (II.41) In the above equation, P is the positive definite solution of the Lyapunov equation: ~ ~ ~ ~ PA + A T P + C T C = 0 (II.42) 53 The computations in this section were performed with the Maple software and their details are given in Appendix I.1. First, the 10 independent elements of the matrix P are obtained by solving the linear system of equations given by Equation (II.42). Then by computing the trace of the product of matrices defined by Equation (II.41), the H2 norm of the transfer function between the performance measurement and the external disturbance is: (r )( a a γζ ω ω − 2ω ω β + ( ω + ω )β + 4ω β ( ω β + ζ ω )( β + ω ζ ) ) = 4ω ( ω ( 4ω ζ β + a a γω )( β + ζ ω ) + 4ω ζ β ( ω + ω ) + ω ζ β + ( a a γ − 2 )ω ζ ω β + ( a a γ + 1) ζ ω β ) 2 1 2 J = z 2 2 2 2 2 2 s ) 4 c c s s s c s c 1 2 c s c 2 c 4 1 2 s 1 2 c s s c 2 s 2 s c 2 4 s 2 c 2 c 2 s 2 4 s 2 2 ( 2 ωc + r2 ωs a 1a 2 γωc β c + r1 + r2 c 2 2 c s s c s c 1 2 2 s c (II.42) c 4 c s s 2 s s 2 2 c In order to find the values of the three controller parameters that minimizes the functional J, the derivatives of J with respect to γ, ωc and βc are computed. Then, assuming non-zero control parameters, the numerator of these three derivatives are set to zero resulting in the following three equations: [( r ω + r [ 4ω ( β + ω 2 0= 2 1 s 2 2 s ( 2 2 c 2 c ) ( ) ] ) + 4ζ ω ω β + 4ζ β ( ω 2 ζs 2 2 − r2 2 2 s 2 ) (( ω ζ s c c ( a a γω ( ω ζ + β ) − (r ω 4ζ ω β ( ω β + ζ ω ) (ω ζ 2 0 = r1 + r2 β c + r1 2 ωc ζ s + r1 + r2 ωs β c s 2 c 2 s s 2 c c s s c 3 ) ( 2 + ωs + ω c − ωs s c s 2 2 2 2 c 3 c 2 (II.43.a) )] 2 ) ) + r ω ) ω ζ β ( ( ω ζ + β ) 4β + ω ) + β )+ r ω ζ ω β (ω − ω ) 2 2 1 2 s c + β c ) ωc − 4ωs β c − ωs β c 4 s 4 1 2 s 4 2 2 c s s c s 2 s c 2 s c 2 s s 54 c c 2 c c 2 s s (II.43.b) 2 ( 4 4 2 ) 2 2 2 ( 2 2 ) 2 0 = β c a 1a 2 γωc − 4ωs ζ s r1 + ωc β c 4ωc ζ s − a 1a 2 γωs r2 ( )(ω β (ω ( 4β − ω ))+ ζ ( r 2 + β c r1 + r2 β c ωs 2 2 2 c c s − ωc + ωc ωs ( ωs β c − 2a 1a 2 γζ s ωc ) + 2 2 s 2 c s 2 1 ) 2 2 2 2 ωs + r2 ωc 2 )(8ω β s 3 c − a 1a 2 γζ s ωc 4 (II.43.c) ) First, it is to be noted that the left-hand-side of Equation (II.43.a) is strictly positive. Hence, the function J is monotonic with respect to the controller gain. Therefore, as in the previous section, the derivative with respect to the gain cannot be equal to zero. Further, the value of the function for zero and infinite gains are: lim(J ) = γ →0 2 1 r1 + r2 2 ωs ζ s 2 (II.44.a) 1 r1 ωc ( ωc ζ c + ωs ζ s ) + r2 ωS ( ωc ζ s + ωs ζ c ) lim(J ) = γ →∞ ( ωc ζ c + ωs ζ s )( ωc ζ s + ωs ζ c ) ωs 2 2 2 (II.44.b) The ratio of these two quantities is given by: lim(J ) γ →∞ lim(J ) γ →0 2 = (r 1 r1 ωc ζ s 2 + r2 2 )( ω ζ c s 2 + ωs ζ c ) + (r 1 r2 ωs ζ s 2 + r2 2 )( ω ζ c c + ωs ζ s ) <1 (II.45) Since the function J is monotonic with respect to the controller gain and since its value decreases with increasing gain values, we can conclude that there exists no optimal controller gain value and that the larger the gain the smaller the function J becomes with the minimum limit given by Equation (II.44.b). As a result, the gain of the controller is defined to be a control design parameter that can be selected on the basis of stability and control signal amplitude restrictions. 55 For a given controller gain, the optimal controller natural frequency and damping ratio are given by solving Equations (II.43.b) and (II.43.c) simultaneously. By combining these two equations, as illustrated in Appendix I.1, and assuming that the controller natural frequency is non-zero, the following equations is derived: t ω2 (β c , γ ) ωc − t ω0 (β c , γ ) = 0 2 (II.46) The coefficients tω2 and tω0 are functions of the open loop system, the controller gain, γ, and the opposite of the real part of controller pole, βc, given in Appendix I.1. Using Equations (II.46) into Equation (II.43.b), factoring the resulting relationship, and simplifying for the terms that cannot be equal to zero for a real positive βc, we obtain an equation for βc as a fifth order polynomial: t β5 (γ ) β c + t β 4 (γ ) β c + t β3 (γ ) β c + t β 2 (γ ) β c + t β1 (γ ) β c + t β0 (γ ) = 0 5 4 3 2 (II.47) The coefficients, tβi, are given in Appendix I.1 and are functions of the open loop system parameters as well as the controller gain. The design procedure associated with the computation of the acceleration feedback parameters for the minimization of a generalized modal transfer function is as follows. First, the two parameters of the output measurement matrix, r1 and r2, are chosen for a particular application. Then, a design controller gain, γd, is chosen is such a way that the product a1a2γd is positive to insure stability. To obtain the first controller parameter, the polynomial equation (II.47) is solved numerically for βc. The strictly positive real solution of Equation (II.47) is taken to be the optimal value for the opposite of the real 56 part of the controller pole, βc,o. This value is then used to compute the optimal controller natural frequency, ωc,o: ω c ,o = t ω0 (β c ,o , γ d ) (I.48) t ω2 (β c ,o , γ d ) Finally, the optimal controller damping ratio, ζc,o, is defined to be the ratio of the optimal value of the opposite of the real part of the controller pole, βc,o, divided by the optimal controller natural frequency, ωc,o. Since multiplying the output measurement matrix by any quantity will not change the optimal controller parameters, the coefficients r1 and r2 can be chosen on the [0,1] interval without any loss of generality. Further, as illustrated in Table II-1,for a given value of the controller gain, γ, the optimal natural frequency and optimal damping ratio for any pair {r1, r2} will be between the optimal values for the minimization of the H2 norm of the closed loop modal receptance and the optimal values for the minimization of the H2 norm of the closed loop modal mobility. The modal mobility is the transfer function between the first time derivative of the modal generalized coordinate and the input disturbance force. 57 Table II-1 Optimal Controller Natural Frequency and Damping Ratio for an Open Loop System with ωs=100, ζs=0.01, a1=1, a2=1 and a Controller Gain of γ=0.05 r1 r2 ωc,o ζc,o Case 1 (Receptance) 1 0 100 0.1118 Case 2 2/3 1/3 100.3 0.1121 Case 3 (Total Energy) 1/2 1/2 100.75 0.1124 Case 4 1/3 2/3 101.21 0.1126 Case 5 (Mobility) 0 1 101.52 0.1127 II.2. Acceleration Feedback Control for Multi-Degree of Freedom Systems In this section, a generalization of the single degree of freedom designs for the acceleration feedback controller parameters is developed for non-collocated multi-input single-output multi-mode acceleration feedback control. This generalization results in a new approach for the design of acceleration feedback controllers for multi-mode control. For this purpose, first, the modal equations governing a closed loop single-output singleinput system under acceleration feedback control are developed. These equations have a form familiar to the structural dynamicist and provide insights in the closed loop dynamics. Then, the modal state space equations of a loop multi-output single-input system under acceleration feedback control are developed. These equations are more useful for the design of the controller parameters than the modal equations. 58 II.2.1. Modal Space Representation of Single-Input Single-Output System under Acceleration Feedback Control For multi-mode acceleration feedback with a single pair of non-collocated sensor and actuator, the equations of motion of for the structure in the configuration space and the compensator are: ì[ M ]{x} + [C]{x } + [ K ]{x} = −{Γact } G [Ω c ]{η} + {f } í {η} + [ Λ c ]{η } + [Ω c ]{η} = {1p }ëΓacc {x} (II.49.a-b) Assuming that the structure is discretized by n locations, with n large enough such that, at least, all the modes necessary for the controller design are included, {x} is the vector of the n discretized spatial coordinates of the structure. [M], [C] and [K] are the inertia, damping and stiffness matrices, respectively. {Γact} is the sensitivity and location vector of the actuator, meaning that it is zero everywhere but at the location of the actuator where it is equal to a1. Γacc is the sensitivity and location row vector of the accelerometer, meaning that it is zero everywhere but at the location of the sensor where it is equal to a2. If we assume that we want to control p modes, p ≤ n, {η} are the p compensator coordinates, [Λc] = diag(2ζcωc) ; [Ωc] = diag(ωc2) are the compensator damping and natural frequency matrices respectively; ëG is the feedback gain row vector; and {1p} is a vector of length p with one for each entry such that all compensators are placed in parallel. In this research work, the general linear viscous damping matrix [C] is not used. Instead, a modal damping [Λs] is introduced after transformation of Equations (II.49.a-b) 59 into the modal space. Using the change of variable {x} = [Φ ]{ξ} , where [Φ] is the mass normalized mode shape matrix and {ξ} are the n modal coordinates of the structure, the modal closed-loop equations are obtained: {} { } {} {} ~ ~ ì ξ + [ Λ s ] ξ + [Ω s ]{ξ} = − Γ G [Ω c ]{η} + f act í ~ } + [ Λ c ]{η } + [Ω c ]{η} = {1p } Γacc ξ î{η {} (II.50.a-b) In Equations (II.50.a-b), [ Λ s ] = diag(2ζ s ωs ) , is the modal damping matrix for the structure; [Ω s ] = diag(ωs2 ) is the diagonal matrix of the structural natural frequencies; {Γ~ } = [Φ] T act {Γact } is the modal influence vector of the actuator; ~ Γacc = Γacc [Φ ] is the {} ~ T modal influence row vector of the sensor; and f = [Φ ] {f } is the modal force vector. By taking the Laplace Transform of Equation (II.50.b), assuming zero initial conditions, we solve for η. This value of η is substituted into the Laplace Transform of Equations (II.50.a): ( { } ) ([ ] ) −1 ~ ~ ì [ I ]s 2 + [ Λ ]s + [Ω ]+ Γ G [Ω c ] I p s 2 + [ Λ c ]s + [Ω c ] {1p } Γacc s 2 n s s act í −1 ~ 2 2 î{η} = I p s + [ Λ c ]s + [Ω c ] {1p }ëΓacc s {ξ} ([ ] ){ξ} = {~f } (II.51.a-b) Since the feedback term of Equation (II.51.a) is composed of a row vector that multiplies a diagonal matrix times a column vector, it is a scalar and can be rewritten as: ([ ] ë G [ Ω c ] I p s + [ Λ c ]s + [ Ω c ] 2 ) {1 }s −1 p p 2 = i =1 60 g i ωci2 s 2 s 2 + 2ζ ci ω ci s + ω ci2 (II.52) In this equation, gi’s are the entries of the gain row vector ëG . Then, the Equation (II.51.a), which describes the closed-loop behavior of the structure, becomes: æ ~ ~ ç [ I ]s 2 + [ Λ ]s + [ Ω ] + Γ n s s act Γacc ç è { } æ ç ç è p i =1 {} öö g i ωci2 s 2 ÷ {ξ} = ~ f 2 2 ÷ s + 2ζ ci ωci s + ωci (II.53) Without loss of generality, one can rearrange the row order of the above system of equations such that the first p rows correspond to the p modes that we want to control. Then, we can separate the system of equations as shown in Equations (II.54.a-b): ~ ~ æ 2 ö Γ Γ g ω2 s 2 ~ ç s + 2ζ s, k ωs ,k s + ωs2, k + act ,k acc,k k c,k + S1k ξ k + S 2 k ( ξ j ) = f k , k = 1,..., p 2 2 ç s + 2ζ c,k ωc ,k s + ωc ,k è (s 2 ) ~ + 2ζ s,k ωs, k s + ωs2, k + S 3k ξ k + S 2 k ( ξ j ) = f k , k = p + 1,..., n (II.54.a-b) The three terms S1, S2 and S3 are defined as follows: ~ ~ S1k = Γact ,k Γacc,k p i =1 i≠k ~ S 2 k (ξ j ) = Γact ,k n j=1 j≠ k S 3k ~ ~ = Γact ,k Γacc,k p i =1 g i ωc2,i s 2 (II.55.a) s 2 + 2ζ c ,i ωc ,i s + ωc2,i ~ æ Γacc, j ç ç è p s + 2ζ c , l ω c , l s + ω 2 l =1 ö g l ω c2,l s 2 g i ωc2,i s 2 2 c,l ξj (II.55.b) (II.55.c) s 2 + 2ζ c ,i ωc ,i s + ωc2,i 61 The first term, S1k, results from the fact that multiple modes are controlled simultaneously and that the compensators associated with each mode cannot, in general, be designed independently. The influence of this term can be minimized by performing multi-degrees of freedom system optimizations of the SDOF compensator designs presented in Sections II.1.2, II.1.3 and II.1.4. Such optimizations are presented in Sections II.2.3 and II.2.4. However, if the damping of each compensator is small enough, the magnitude and phase of this S1k term can be considered negligible with respect to the magnitude and phase of the controller associated with the kth mode. Such a design for multi-mode control is called a selective design since each compensator targets its associated mode under the assumption of negligible influence on the other modes. The second term S2k arises from the fact that the sensor signal contains some information on every mode. When the sensor signal is convoluted by the controller, these modes are coupled and fed back, coupled, to the controlled system through the actuator. This term can be considered as an additional driving force for each mode, which is dependent on a combination of every other mode, and as such will not result in any instability. However, coupling of the different modes will introduce variations of the closed loop pole behavior that are different from an uncoupled case. This fact will result in the loss of exact crossover or optimality conditions. Hence, to minimize the effect of the S2k term, to a certain extent, which is limited by the truncation of the actual dynamics, a multi-mode optimization of the SDOF design for acceleration feedback control should be performed. 62 The third term S3k represent the effects of the controller on uncontrolled or unmodeled modes. This term is the most important term for stability issues. If the phase of this term is close to –90 degrees, near any of the unmodeled or uncontrolled modes, and if its magnitude is greater that the damping coefficient of that mode, the controller will drive the system unstable. At high frequencies, the phase of the S3k term, using an acceleration feedback controller, will be 0 or 180 degrees hence the controller will not drive higher unmodeled modes unstable. However, the effect of the controller on the uncontrolled modes, which lie within the control bandwidth, must be checked and the controller tailored to guaranty the stability of the system. In the case where the control design is selective, which means that the S1k terms are considered negligible, and if the coupling S2k terms are considered negligible, the following set of uncoupled equation can is obtained from Equations (II.54): ~ ~ æ 2 Γ Γ g ω2 s 2 ö ~ ç s + 2ζ s ,k ωs ,k s + ωs2,k + act ,k acc,k k c ,k ξ k = f k , k = 1,..., p 2 2 ç s + 2ζ c , k ω c , k s + ω c , k è ~ s 2 + 2ζ s ,k ωs ,k s + ωs2,k + S 3k ξ k = f k , k = p + 1,..., n ( (II.56.a-b) ) In such a case, each compensator can be designed independently using a SDOF design procedure, such as the crossover design or the H2 optimal design. Then, all the independent compensators are implemented in parallel. If the terms S3k do not destabilize the unmodeled or uncontrolled modes, then the closed loop system will be stable. However, since the SDOF compensators are designed for the uncoupled set of Equations (II.56.a-b) while the actual closed-loop dynamics are given by Equations (II.54.a-b), the 63 crossover or optimality conditions will not be exactly satisfied when loop is closed on the actual system. II.2.2. State Space Representation of a Multi-Input Single-Output System under Acceleration Feedback Control For most real world applications, single-input single-output control systems are not the best solution. While a single sensor, positioned in such a way that all the modes in the control bandwidth are observable, is usually a sufficient solution, a single actuator will not, in most cases, be at an optimal location to control all the modes. In particular, for a beam like structure, bending and torsion modes do not have the same optimal orientation of the actuators. Hence, a method for the design of acceleration feedback control parameters for multi-input single-output systems should be developed. The first step in this development is the representation of the closed loop system in a state space form. For multi-mode acceleration feedback with multiple arrays of actuators and a single sensor, the equations of motion of for the structure in the configuration space and the compensator are: ì[ M ] {x} + [C] {x } + [ K ] {x} = −[Γact ] [G ] [Ω c ] {η} + {f } í {η} + [Λ c ] {η } + [Ω c ] {η} = {1} Γacc {x} (II.57.a-b) As in Section II.2.1, assuming that we only consider ns locations, with ns large enough such that all the necessary modes are included, {x} is the vector of the ns 64 discretized spatial coordinates of the structure; [M], [C] and [K] are the inertia, damping and stiffness matrices, respectively. Assuming that the control system uses nu actuator arrays, [Γact] is the ns x nu sensitivity and location matrix of the actuator arrays, meaning that it each column correspond to the influence vector of a single actuator array associated with one of the control signals. Γacc is the sensitivity and location row vector of the accelerometer. If we assume that we want to control nc modes, nc ≤ ns, {η} are the nc compensator coordinates, [Λc] = diag(2ζcωc) ; [Ωc] = diag(ωc2) are the compensator damping and natural frequency matrices respectively; since the control system is assumed to have nu actuators arrays driven by nu control signal, [G] is the nu x nc feedback gain matrix in which each row correspond to the gain vector of each controller signal; and {1} is a vector of length nc with one for each entry such that all compensators are placed in parallel. As in Section II.2.1, in the following equations, the general linear viscous damping matrix [C] is not used. Instead, a modal damping [Λs] is introduced after transformation of Equations (II.57.a-b) into the modal space. Using the change of variable {x} = [Φ ]{ξ} from the ns geometric locations to the first nm modes, where [Φ] is the ns x nm mass normalized mode shape matrix and {ξ} are the nm modal coordinates of the structure, the modal closed-loop equations are obtained: {} {} [ ] {} ~ ~ ì ξ + [ Λ s ] ξ + [Ω s ] {ξ} = − Γ act [G ][ Ω c ] {η} + f í {η}+ [Λ c ] {η }+ [Ω c ] {η} = {1} Γ~acc ξ {} 65 (II.58.a-b) In Equations (II.58.a-b), [ Λ s ] = diag(2ζ s ωs ) , is the nm x nm modal damping matrix for the structure; [Ω s ] = diag(ωs2 ) is the nm x nm diagonal matrix of the structural [ ] ~ T natural frequencies; Γact = [Φ ] [Γact ] is the nm x nu modal influence matrix of the ~ actuator arrays; Γacc = Γacc [Φ ] is the 1 x nm modal influence row vector of the sensor; {} ~ T and f = [Φ ] {f } is the nm x 1 modal force vector. To transform Equations (II.58.a-b) into the classical state space representation of a system under control and of the controller, let us introduce the following state space coordinate vectors based on Section II.1.4 considerations: {Ξ} = ìí [ωs ] ξü ξ î ì [ωc ] ηü ý , {Η} = í ý î η (II.59.a-b) In the above definitions, {Ξ} is the 2nm x 1 state space vector of the modal generalized coordinates of the controlled structure. Similarly, {H} is the 2nc x 1 state space vector of the controller coordinates. The two weighting matrices are defined to be [ωs ] = diag(ωs ) the nm x nm diagonal matrix of the open loop structure natural frequencies and [ωc ] = diag(ωc ) the nc x nc diagonal matrix of the open loop controller natural frequencies. The general equations governing the structure states, {Ξ}, sensor signal, y, controller states, {H}, and control signals, {u}, are given by: {Ξ } = [A ] {Ξ}+ [B ] {u}+ [D ] {~f } (II.60.a) {~f } (II.60.b) s y = Cs s 1 {Ξ}+ ëD s {u}+ ëD 2 66 {Η } = [A ] {Η} + {B } y (II.60.c) {u} = [C c ] {Η} (II.60.d) c c In the set of Equations (II.60.a-b), As is the 2nm x 2nm structural state matrix, Bs is the 2nm x nu control input matrix, D1 is the 2nm x ns disturbance input matrix, Cs is the 2nm system output vector, Ds is the nu control direct transmission vector and D2 is the nm disturbance direct transmission vector. All these vector and matrices are defined by the open loop structure as well as actuator arrays and sensor placements. On the other hand, the controller parameters define the matrices and vectors of the second set of Equations (II.60.c-d). In these equations, Ac is the 2nc x 2nc controller state matrix, Bc is the 2nc sensor input vector and Cc is the nu x 2nc overall gain matrix of the controller. Transforming Equations (II.58.a-b) in the form of Equations (II.60.a-d), using the state space vector defined in Equations (II.59.a-b), we obtain: {Ξ } = éê− [0ω ] −[[ωΛ] ]ùú {Ξ}+ éê[Γ~0 ]ùú {u}+ éê0I ù {~f } ë ë ë s s s y = ë− Γacc [ωs ] − ëΓacc (II.61.a) act [Λ s ] [Ξ] + ëΓacc [Γact ] {u}+ ëΓacc {f } (II.61.b) {Η } = éê− [0ω ] −[ω[Λ] ]ùú {Η} + ìí{01}üý y î ë û (II.61.c) {u} = [ − [G ][ωc ] (II.61.d) c c c 0 ] {Η} 67 From the two sets of Equations (II.60.a-d) and (II.61.a-d), all matrices and vectors governing the closed loop system can be identified. Further, a single combined state space equation can be developed to predict the dynamics of the closed loop system. This state space equation uses a global closed loop state vector defined as follows: {Ξ} {X cl } = ìí üý î{Η} (II.62) Using this global closed loop state space vector, the dynamic of the system can be written as a function of the 2(nm+nc) x 2(nm+nc) closed loop state matrix, Acl, and the 2(nm+nc) x nm closed loop disturbance input matrix, Bcl, as follows: {X } = [A ] {X }+ [B ] {~f } cl cl cl (II.63) cl Using the set of Equations (II.60.a-d), to obtain the single closed loop equation in term of the global closed loop state space vector, we have: {Ξ } = éê{B [A} C] ë s cl c s [B s ][C c ] é [D1 ] ù {Ξ cl } + ê ú [A c ] + {B c } ëD s [C c ] ë{B c } ëD 2 {} ù ~ f (II.64) Identifying the different components of the closed loop state matrix and the different component of the closed loop disturbance input matrix from the set of Equations (II.61.a-d), we obtain: 68 0 0 0 ù [ωs ] é ~ ê 0 − [ωs ] − [Λ s ] − Γact [G ][ωc ] (II.65.a) [A cl ] = ê ê [ωc ] 0 0 0 ê ~ ~ ~ ~ ë− {1} Γacc [ωs ] − {1} ëΓacc [Λ s ] − I + {1} ëΓacc Γact [G ] [ωc ] − [Λ c ] [ ] ( é 0 ù ê I [Bcl ] = ê ê 0 ~ ê ë{1} Γacc [ ] ) (II.65.b) It is to be noted at this point that the closed loop input disturbance matrix, Bcl, as given in Equation (II.65.b), depends on the sensor influence matrix only, which is independent of the external loads. Hence the closed loop input disturbance matrix does not in any way depends on the modal or geometric distribution of the disturbance loads. Therefore, the designs for multi-degrees of freedom systems discussed in the following sections do not require an accurate knowledge of the disturbance distribution. However, in the case where the spatial distribution of the disturbance, {Γf}, is known, the modal force vector can be written as: {~f }= [Φ] T {Γf } f (II.66) And, in this case, the closed loop input disturbance matrix can be reduced to a closed loop input disturbance vector given by: 0 ì ü T ï [Φ ] {Γf } ïï ï [Bcl ] = í ý 0 ï ~ T ïî{1} Γ acc [Φ ] {Γf } (II.67) 69 As a result, the knowledge of the spatial or modal distribution of the disturbance is not necessary for the designs presented in the following sections. However, its knowledge can insure further optimization of the controllers and a measurement of the needed control authority. II.2.3. AFC Crossover Design for the Control of Multi Degrees of Freedom Systems The concept of the crossover design, as presented in Section II.1.2, is based on the coincidence of the closed loop controller and structure poles in order to avoid the creation of new frequency peaks in the closed loop response. The coincidence of the poles means that the distance between the closed loop controller and structure poles is equal to zero when the closed loop system operates at the crossover point. In order to design an acceleration feedback controller for a multi degrees of freedom system to operate at as many crossover points as the number of modes that are controlled, a criterion for a numerical minimization is developed due to the fact that an exact solution for controller cannot be constructed. This criterion is a function of the three sets of control parameters, the natural frequencies, {ωc}, the damping ratios, {ζc}, and the controller gains, {γ}. As in the previous section, let us consider a controlled structure with nm modes and let us further consider nc modes, where nc is smaller than nm, that should be controlled. The criterion is based on the sum of the distances in the complex plane between each of the nc controlled poles of the closed loop structure and their associated closed loop controller poles. 70 J d ( { ωc }, { ζ c }, { γ} ) = nc k =1 σ c ,k − σ sc ,k 2 (II.68) In the above equation, σc,k denotes the kth closed loop controller pole and σsc,k its associated closed loop structural pole. In order to operate at nc crossover points, each of the distances should be equal to zero. Hence, the functional defined in Equation (II.68) should be identically zero. Then, a numerical scheme can be developed on the basis of the minimization of the criterion of Equation (II.68). However, it is to be noted that in practice, the distance between the closed loop controller and structure poles increases drastically with natural frequency and damping ratio. Hence, when the numerical scheme is applied to the design of a crossover based controller for the control of low and high natural frequencies, the program will favor the higher modes at the detriment of the lower ones. Furthermore, in practice, it is much more important to obtain close coincidence of poles with low damping or low frequencies in order to avoid the creation of new frequency peaks in the closed loop response. As a result, a more convenient functional for any numerical application is defined by normalizing each distance by the minimum of the absolute value of real part of the closed loop structure pole or of the closed loop controller pole: æ σ c ,k − σ sc,k ç J d ( { ωc }, { ζ c }, { γ} ) = ç Re( σ l,k k =1 ç min è l∈{c ,sc} nc ( )) ö 2 (II.69) As for the criterion defined in Equation (II.68), the criterion defined in Equation (II.69) is identically zero when crossover is reached for each of the controlled modes. 71 Since this criterion is strictly positive, if the minimum is attained, the value of the functional Jd will be zero and the controller will operate at crossover. The minimization scheme is based on finding the optimal controller parameter sets as a function of a predefined control design parameter set. Hence, the multi degree of freedom control design scheme is developed as follows. First, in general, the set of controller parameters chosen for design purposes should be chosen between the controller gains, {γ}, and the controller damping ratios {ζc}. However, as discussed in Section II.1.2, a third choice is possible by choosing the set of closed loop damping ratios for the controlled system, {ζf}. Or, more generally, the designer can choose the poles of the closed loop system, {σf}, with the restriction that the set of closed loop natural frequencies, {ωf}, should be very close to the set of open loop natural frequencies, {ωsc}. This last choice is essentially a pole placement method and the criterion should be somewhat altered to accommodate this design: J d ( { ωc }, { ζ c }, { γ} ) = nc k =1 2 σ f ,k − σ sc,k + σ f ,k − σ c ,k min l∈{c ,sc ,f ( Re( σ ) ) } 2 2 (II.70) l,k The second step of the controller design process is to define an initial set of controller parameters for the numerical optimization scheme. This step is performed by assuming that the closed loop system can be uncoupled into a set of controlled single degree of freedom systems and a set of uncoupled and uncontrolled single degree of freedom systems. Under this condition, the controller parameters can be computed for 72 each controlled mode independently using the equations of Section II.1.2. At this point, we have a triplet of control parameters, ({ωc0}, {ζc0}, {γ0}), which would satisfy the crossover conditions exactly if no closed loop coupling existed between the different modes. Since the coupling in practice is usually weak, this is a valid triplet as the initial guess of the optimization process. The third step in the controller design procedure is to find the control parameter sets that minimize the distance criterion, Jd. In this thesis work, the numerical minimization was performed using the fmins function of the Matlab software in which the minimization is performed using the Nelder-Mead simplex search[49,50] which is a direct search method that does not require gradients or other derivative information. The numerical computation of the distance criterion, Jd, is performed in four steps. First, the closed loop system state matrix is formed using Equation (II.65.a). Then, the closed loop pole diagonal matrix, Σcl, and its associated modal eigenvector matrix, Vcl, are extracted in the form: A cl Vcl = Vcl Σ cl (II.71) The third step in the computation of the distance criterion is the identification of the closed loop poles. This step is necessary to use Equation (II.69) or Equation (II.70). The structural closed loop poles and controller closed loop poles should be separated and each one matched to its open loop mode number. This identification is performed by analyzing the closed loop modal eigenvector matrix for the participation of each open loop modes in the closed loop system. The analysis is performed by row partitioning the 73 closed loop modal eigenvector matrix into the blocks of rows associated with the open loop modal displacements, VΦcl , and velocities, VΦ cl , and with the open loop controller coordinate, VHcl , and first time derivative, VH cl , as follows: éVΦcl ù êV Vcl = ê Φcl ê VHcl ê ë VH cl (II.72) In order to obtain the maximum participation of each open loop structure mode in the closed loop system, each row of VΦcl and VΦ cl is normalized by its maximum absolute value, then each column of the normalized VΦcl and VΦ cl is analyzed to obtain the open loop structure mode number associated with the maximum participation in the closed loop eigenvector. By combining these to sets of maximum participation, in general, each controlled structure open loop mode can be associated with two closed-loop poles and each uncontrolled structure open loop mode is associated with a single closed loop pole. Then VHcl is normalized in the same manner as VΦcl or VΦ cl , as described above. Finally, a combination of the normalized VHcl and VΦcl is used to separate each pair of closed loop poles into the structure pole and the controller pole. Once all the different poles of the closed loop system have been identified, the distance criterion is computed using either use Equation (II.69) or Equation (II.70) depending on which control design parameter set was chosen. 74 The result of the optimization can be evaluated by plotting the closed loop poles of the controlled system in the complex plane. However, it is to be noted at this point that the stability of the multi degree of freedom closed loop system is not insured by any step of the control design computation. Hence, the computed controller may result in an unstable system due to the coupling of the uncontrolled and controlled modes in the closed loop system. As a result, the stability of the closed loop system should be verified a posteriori. However, it should also be noted that the closed loop input disturbance matrix was not used in any step of the computation. Thus, the controller is completely independent of the geometric or modal disturbance distribution. The different programs that were developed for the Matlab software to implement the multi degrees of freedom crossover design of acceleration feedback controller are given in Appendix I.2. II.2.4. AFC H2 Norm Optimization of the Closed Loop Modal Response Design for Multi Degrees of Freedom Systems As for the multi degrees of freedom system crossover design, the generalization of the design of the AFC parameters based on the optimization of the H2 norm of a generalized closed loop modal transfer function is performed numerically. First, as in Section II.1.4, let us define a performance measurement vector, z, as a function of the closed loop system states, as defined in Equations (II.62) and (II.59.a-b): 75 z = C cl X cl = [R 1 ì [ ωs ]ξ ü ï ξ ï ï ï 0 0] í ý ï[ ωc ]η ïî η R2 (II.73) For a stochastic process, the H2 norm of the closed loop performance measurement vector, z, due to a unit white noise disturbance, w, is defined to be: æ ì1 z 2 = çç lim E í è t →∞ î t t 0 üö z (τ ) z(τ )dτý 1/ 2 T (II.74) The above equation can be written in terms of the generalized coordinates of the structure using Equation (II.73) as: æ ì1 z 2 = çç lim E í è t →∞ î t t 0 üö T ξ (τ )[ωs ]R 1 R 1 [ωs ]ξ(τ ) + ξ T (τ) R 2 R 2 ξ (τ ) dτý T T 1/ 2 (II.75) Further, as in the previous section, let us denote by the subscript “sc” the nc modes that will be controlled and define R1 and R2 to be diagonal matrices with only nonzero entries associated with the controlled modes. Then, we can rewrite Equation (II.75) as follows: æ ì1 z 2 = çç lim E í è t →∞ î t t 0 nc år k =1 2 1,k ω 2 sc , k ξ k (τ ) + r 2 2 2 ,k üö ξ k (τ ) dτý 1/ 2 (II.76) Hence, as discussed in Section II.1.4, the design method for acceleration feedback control based on the minimization of the H2 norm of the closed loop modal response can be associated with the minimization of some energy associated with some structural 76 modes. In fact, if all the weights r1 are unity and all the weights r2 are zeros, the design will minimize the potential energy associated with selected modes of the system. Similarly if all the weights r2 are unity and all the weights r1 are zeros, the design will minimize the kinetic energy associated with selected modes of the system. And finally, if all the weight r1 and r2 are unity, the design will minimize the sum of the kinetic and potential energy associated with selected modes of the system. This leads to three immediate types of designs for the minimization of the H2 norm of the closed loop modal response. As in Section II.2.3, the optimal controller design is divided into three steps. The first step consists of choosing the output measurement vector and the controller gain matrix. The choice of the output measurement vector, z, is equivalent to choosing which type of energy one wants to minimize for a particular application. As discussed in the previous paragraph, this first design step consist in the choice of the two weight matrices R1 and R2. When the output measurement vector is defined, the controller gain matrix should be defined in such a way that each entry of the 1 x nc vector given by the product [ ] ~ ~ Γacc Γact [G ] is positive. In the particular case where no mode is controlled by more than one actuator array, this condition is equivalent to insuring that the product a1a2γ is positive for every controlled mode. Hence, as discussed into Section II.1.1, this restriction on the gain matrix is given to insure that under the assumption that the structural closed loop system is uncoupled, the system is stable. 77 As in the previous section, the second step of the numerical optimization consists in the definition of some initial guess for the optimal controller natural frequencies and damping ratios. Again, this definition is performed under the assumption that the closed loop system can be uncoupled in a set of uncoupled structural modes under acceleration feedback control and a set of uncoupled and uncontrolled structural modes. Under this assumption, the parameters of each of the controller second order compensator can be computed independently based on the formulas given in Section II.1.3 or Section II.1.4 depending on the choice of weight matrices. The final step of the optimization procedure computes the optimal natural frequencies and damping ratios for the controller. In order to complete the step, the H2 norm criterion should be computed. This computation is completed in three steps. First, at each iteration of the optimization, the closed loop system state matrix, Acl, and the closed loop input disturbance matrix, Bcl, are computed using Equations (II.65.a) and (II.67), or (II.65.b), depending if the distribution of the disturbance is known or not. Further, the output performance measurement matrix, Ccl, is given by Equation (II.73). Then, as discussed in Section II.1.3, the optimization criterion, J, given by the H2 norm of the performance measurement, z, for a unit white noise disturbance, is defined to be, as a function of the observability Gramian, P, as follows: [( z 2 = J ( { ωc }, { ζ c } ) = tr B cl P B cl T )] 1/ 2 78 (II.77) Since, like in Section II.1.3, the closed loop system is assumed to be stable and the pair ( A cl ,C cl ) is assumed to be observable, the observability Gramian of the closed loop system, P, is positive definite and is the solution of the Lyapunov equation: T T PA cl + A cl P + C cl C cl = 0 (II.78) Hence, the second step of the computation of the optimization criterion consists in solving Equation (II.78) for the observability Gramian, P. If the computed observability Gramian is positive definite and the pair ( A cl ,C cl ) is observable, then the closed loop system will be stable. Then, the last step of the computation consists of evaluating Equation (II.77) for the optimization criterion. In contrast to the multi-degrees of freedom crossover design for acceleration feedback control, the minimization of the H2 norm of the closed loop modal response includes a guaranty of stability within the computation of the controller parameters. However, since in general, the pair ( A cl ,C cl ) is not necessarily observable, the controller stability should be checked a posteriori. In addition, this method makes use of the closed loop input disturbance matrix explicitly and hence even though this matrix can be written in such a way that the disturbance modal or geometric distribution is absent, better results could be obtained with knowledge of such a quantity. As for the previous section, the different programs that were developed for the Matlab software to implement the multi degrees of freedom crossover design of acceleration feedback controller are given in Appendix I.2. 79 CHAPTER III. OFFSET PIEZOCERAMIC STACK ACTUATOR To obtain a high authority actuator, a new design of an external actuator assembly was created to transform the longitudinal force generated by the piezoceramic stack into the induced actuation of local bending moments. In this new design, this transformation was obtained by placing the piezoceramic stack parallel to the controlled system in a mount bonded to the surface of the structure. Additional requirements for this design included reliability and maintainability issues arising from the fragility of the piezoceramic stack in tension or bending. To model the actuation in the modal space, the modal influence matrix of the actuators is required. To obtain such a matrix for a complicated structure, an FEM model of the actuation should be developed. However, to test the reliability of the finite element model of the actuation and to gain insight into the physics of the actuator, an analytical solution for a simple structure was initially studied. For this purpose, a new model of the offset piezoceramic stack actuator was developed using the modal expansion technique. This method was used to compute the transfer function between the response of an idealized structure and the input voltage to the actuator for a system made of an idealized offset piezoceramic stack actuator and an Euler-Bernoulli beam. 80 Finally, using the modal expansion model for the idealized actuator and the EulerBernoulli beam, an optimization of the actuator placement and of the actuator dimensions was investigated. The objective of the optimization was to maximize the authority of the offset piezoceramic stack actuator for the control of an ideal cantilever beam. The parameters that were used to optimize the actuator will include the distance between the mount and the root of the beam and the offset distance between the centerline of the stack and the centerline of the beam. III.1. Prototype Design of the Offset Piezoceramic Stack Actuator The main challenge, associated with the use of stacked piezoceramic actuators in tail buffet alleviation, is that a piezoceramic stack produces only longitudinal motion or forces. Hence, an assembly was designed to transform the longitudinal motion of the stack into moments that will produce the control actuation. Such a transformation was achieved by placing the piezoceramic stack parallel to the controlled structure at a distance from its neutral axis and at a selected orientation. This distance creates a lever arm so that the longitudinal forces results in local moments on the structure. This new high authority actuator assembly is made of two blocks bonded to the structure to provide an offset distance from the surface of the system to be controlled. A piezoceramic stack is placed between these two blocks and pre-compressed using a bolt. This pre-compression is necessary because the actuator can only provide control moments while extending from its rest position. By pre-compressing the active element, 81 an offset stress is created that will enable the piezoceramic stack to provide control moments over almost the full control cycle. Furthermore, before any control signal is applied, the active element is also electrically pre-compressed by applying a DC bias. This design is illustrated in Figure III.1. Nut and Bolt to provide mechanical precompression Piezoceramic Stack Rounded Contact Points to avoid tensile loads Controlled Structure Surface Figure III.1 Offset Piezoceramic Stack Actuator Structural Assembly Reliability benefits associated with the offset piezoceramic stack actuator are as follows. Since the piezoceramic stack is only compressed between the two blocks of the assembly, tensile loads are not transmitted to the active element. This fact reduces the possibility of actuator failure. Furthermore, in this design, the rounded point contacts between the piezoceramic stack and the mounts insure that bending loads are not transmitted to the active element as well, which reduces even further the possibility of stack failure in local tension. It is to be noted that, for maintainability, the bolt is used to permit an easy removal of the piezoceramic stack while the actuator assembly is bonded 82 to the system to be controlled. Hence, if failure were to occur, the active element could be replaced easily during regularly scheduled maintenance. For large amplitude vibration suppression, the primary concern is the control authority that can be generated by the actuator. To obtain a maximum control authority, the resultant forces that the actuator develops should be as large as possible. However, the power requirement of the actuator increases with the maximum authority that the active element can deliver. To produce the needed control moments, the active element of the Offset Piezoceramic Stack Actuator can be chosen from different types of piezoceramic stacks. Low-voltage (100V) piezotranslators can generate blocked force in the range of 180N to 3kN while high-voltage (1000V) piezotranslators can generate blocked force in the range of 1.5kN to 30kN. Once the type of the piezoceramic stack has been selected, the dimensions of the offset piezoceramic stack actuator assembly should be computed such that the chosen active element fits within the mount and optimal actuation energy transfer is obtained. III.2. Modal Expansion Model for the OPSA acting on a Cantilever Beam For controller design purposes, the modal influence matrix of the actuators is required to model the plant in the modal space. To obtain such a matrix for a complicated structure, a FEM model of the system should be developed. However, to test the reliability of the finite element model of the OPSA and to gain insight into the physics of the actuator, an analytical solution for a simple structure is now developed. 83 The simple system studied is a steel cantilever beam, which is 24-in long, 0.5-in thick and 2-in wide. An offset piezoceramic stack actuator (OPSA) is bonded on the upper surface of the beam close to its clamped end. The OPSA is made of two steel blocks bonded to the cantilever beam with a piezoceramic low-voltage translator, Physik Instrumente PI-830.10 (100V, 1kN), clamped between the blocks. This setup is illustrated in Figure III.2. Figure III.2 Offset Piezoceramic Stack Actuator Mounted on a Cantilever Beam III.2.1. Development of the equations governing the piezoceramic stack The selected active element of the offset piezoceramic stack actuator is a “cofired” type piezoceramic stack where the active ceramic material and the electrode material are “co-fired” in one step. A typical cross section of such a piezoceramic stack is illustrated in Figure III.3. 84 Piezoceramic Stack x, 3 Electrode Figure III.3 “Co-fired” Piezoceramic Stack Configuration For the purpose of modeling the piezoceramic stack, let us neglect the transverse effects in the active element. This assumption is made because the stresses and input electric fields in the transverse directions are zero. As a result, the strain distribution in the longitudinal direction, x, which is also the stack poling direction, 3, can be obtained as a function of the longitudinal stress distribution and the longitudinal applied electric field distribution. For this unidirectional model, the two constitutive equations governing the behavior of the piezoelectric material in its longitudinal direction are: σ + d 33 E YEs (III.1) D = ε 33 E + d 33 σ (III.2) S= In these equations, S is the longitudinal strain usually denoted εx, σ is the longitudinal stress, YEs is the longitudinal short-circuit Young’s modulus, d33 is longitudinal piezoelectric charge constant, E the electric field in the stack, D the electric displacement field and ε33 is the dielectric constant. 85 Using Newton’s second law, a relationship governing the dynamic longitudinal behavior of the stack as a function of the cross-sectional area, As, density, ρs, longitudinal stress, σ, and longitudinal displacement, u, is obtained: As ∂σ ∂2u = A s ρs 2 ∂x ∂t (III.3) Rewriting Equation (III.1), to obtain the stress field as a function of strain and electric field, we have: σ = Y E s S − d 33 Y E s E (III.1.b) Since the thickness of the electrodes are small compared to the thickness of the piezoceramic layers, let us assume that the Young’s modulus, YEs, is independent of the longitudinal coordinate, x. Further, let us assume that the product d33E is also independent of the longitudinal coordinate, x. This is because the sign of the product of the piezoelectric charge constant and the electric field, d33E, has to be the same over the entire stack to produce maximum actuation. It is further due to the fact that the thickness of each piezoceramic layer is small enough to assume that the d33 coefficient is homogeneous and that the electric field is constant within each layer. With these assumptions, the first spatial derivative of the constitutive Equation (III.1.b) yields: ∂σ ∂S ∂2u = YEs = YEs 2 ∂x ∂x ∂x (III.4) Using the above equation in the differential equation (III.3), a model of the dynamic behavior of the piezoceramic stack in the form of an active rod is obtained: 86 AsY E s ∂ 2u ∂ 2u = A sρs 2 ∂x 2 ∂t (III.5) There are two possible types of associated boundary conditions. The first type is geometric in which case the displacement at an end of the stack is prescribed: u (end, t ) = u prescribed (t ) (III.6) The second type is a natural boundary condition in which case the stress at an end of the stack is prescribed. In this case, using the constitutive Equation (III.1.b), we obtain: σ(end, t ) = Y E s ∂u (end, t ) + Y E s d 33 E(t ) = σ prescribed (t ) ∂x (III.7) III.2.2. Development of the Equation Governing the Coupled System In the following analysis, the beam is assumed to satisfy the Euler Bernoulli assumptions. The two mounts making the assembly of the OPSA are assumed to be massless rigid connections between the beam and the stack. Further, these connections are assumed to always remain at a right angle with respect to the elastic axis of the beam. The piezoceramic stack is hinged to the top of these connections in such a way that no bending moments are transmitted to the stack. Finally, the stack is modeled as an active rod as discussed in the previous section. This model is illustrated in Figure III.4. 87 w δ c b L x, u Figure III.4 Model of the Offset Piezoceramic Stack Actuator Mounted on a Cantilever Beam In order to solve the coupled actuator-beam problem, the beam is divided into three sections. The first segment spans from the root of the beam to the first connection ( 0 ≤ x ≤ b ). The second segment is the section of the beam directly below the stack ( b ≤ x ≤ b+c ). The third section spans from the second connection to the free end of the beam ( b+c ≤ x ≤ L ). The differential equations governing the transverse displacement of the beam, w(x,t), and the longitudinal motion of the stack, u(x,t) are given by: ∂4wi ∂2wi EbIb + A bρ b = 0 , i = 1, 2, 3 ∂x 4 ∂t 2 AsYEs (III.8) ∂2u ∂2u − A ρ =0 s s ∂x 2 ∂t 2 (III.9) Where Eb, Ib, Ab, ρb are the Young’s modulus, moment of inertia, cross sectional area and density of the beam, respectively. And As, YEs, ρs are the cross sectional area, the longitudinal short-circuit Young’s modulus and the density of the piezoceramic stack, respectively. 88 The geometric boundary conditions at the clamped end of the beam are given by: w 1 (0, t ) = 0 (III.10.a) ∂w 1 (0, t ) = 0 ∂x (III.10.b) Similarly, the natural boundary conditions free end of the beam are given by: ∂2w3 (L, t ) = 0 ∂x 2 (III.10.c) ∂3w 3 (L, t ) = 0 ∂x 3 (III.10.d) Under the small displacement assumptions, the continuity of the displacement, slope and shear force within the beam at the location of the connections between the beam and the rigid connections to the stack, imply the following boundary conditions: w 1 (b, t ) = w 2 (b, t ) (III.11.a) ∂w 1 (b, t ) = ∂w 2 (b, t ) ∂x ∂x (III.11.b) 3 E bIb ∂ 3 w1 (b, t ) = E b I b ∂w 23 (b, t ) 3 ∂x ∂x (III.11.c) w 2 (b + c, t ) = w 3 (b + c, t ) (III.12.a) 89 ∂w ∂w 2 (b + c, t ) = 3 (b + c, t ) ∂x ∂x (III.12.b) 3 E bIb ∂3w 2 (b + c, t ) = E b I b ∂w 33 (b + c, t ) 3 ∂x ∂x (III.12.c) The remaining boundary conditions yield the coupling terms between the actuator and the beam. First, the geometric boundary conditions, illustrated in Figure III.5, couple the longitudinal displacements of the piezoceramic stack to the slopes of the beam at the location of the connections through the lever arm dimension of the actuator, δ. These boundary conditions are of the form: u ( b, t ) = − δ ∂w 2 ( b, t ) ∂x u (b + c, t ) = −δ (III.13.a) ∂w 2 (b + c, t ) ∂x (III.13.b) Second, the natural boundary conditions couple the actuation forces of the piezoceramic stack, Fs, to the bending moments into the beam at the location of the connections via the lever arm distance of the actuator, δ. These boundary conditions, illustrated in Figure III.6, are of the form: EbIb ∂ 2 w1 ∂2w2 + δ = ( b , t ) F ( b , t ) E I (b, t ) s b b ∂x 2 ∂x 2 ∂2w2 ∂2w3 EbIb (b + c, t ) = E b I b 2 (b + c, t ) + δFs (b + c, t ) ∂x 2 ∂x 90 (III.14.a) (III.14.b) u x = b = −δ ∂w 2 ∂x x =b ∂w 2 ∂x ∂w 2 ∂x − x =b ∂w 2 ∂x b u x = b + c = −δ x =b ∂w 2 ∂x x = b + c − c x = b +c ∂w 2 ∂x δ x = b +c Figure III.5 Geometric Boundary Conditions Diagram Fs x =b Fs x =b +c u Fs x =b δFs x =b w2 w2 2 2 E bIb w1 ∂ 2 w1 E bIb ∂x 2 x =b ∂ w2 ∂x 2 δFs x =b+c E bIb E bIb ∂ w2 x =b ∂x 2 Fs x =b+ c x =b+c ∂x 2 w3 Figure III.6 Natural Boundary Conditions Diagram 91 ∂ 2w3 x =b+ c The actuation forces generated by the piezoceramic stack, Fs, as a function of the electric field within the stack, E(t), and the PZT longitudinal piezoelectric charge constant of the stack, d33, are given by: ∂u æ (x, t ) − Y E s d 33 E(t )ö Fs = A s σ = A s ç Y E s ∂x è (III.15) Using Equation (III.15) into the natural boundary conditions (III.14.a-b), we obtain: ∂ 2 w1 ∂2w 2 æ E ∂u ö E E bIb (b, t ) + δA s ç Y s (b, t ) − Y s d 33 E(t ) = E b I b 2 (b, t ) ∂x ∂x 2 ∂x è E bIb ∂2w3 ∂2w 2 ( ) (b + c, t ) + δA s æç Y E s ∂u (b + c, t ) − Y E s d 33 E(t )ö b + c , t = E I b b 2 2 ∂x ∂x ∂x è (III.16.a) (III.16.b) It is to be noted at this point that the boundary conditions (III.16.a) and (III.16.b) are nonhomogeneous. In order to solve this problem using modal analysis, we need to transform the problem into a nonhomogeneous differential equation form with homogeneous boundary conditions[51]. This task is achieved by dividing the deformation into dynamic, ŵ i , û , and quasi-static, wi0, u0, deflections: w i (x, t ) = ŵ i (x, t ) + w i 0 (x, t ) , i = 1, 2, 3 (III.17.a) u (x, t ) = û (x, t ) + u 0 (x, t ) (III.17.b) 92 III.2.3. Solution of the Quasi-Static Problem In order to obtain a nonhomogeneous differential equation problem with homogeneous boundary conditions for the dynamic deflections, ŵ i , û , the following relations for the quasi-static problem should be identically satisfied. From Equations (III.10.a-b), at the root of the beam, x = 0, we must have: w 10 (0, t ) = 0 (III.18.a) ∂w 10 (0, t ) = 0 ∂x (III.18.b) From Equations (III.11.a-c), (III.13.a) and (III.16.a), at the first connection between the beam and the stack, x = b, we must have: u 0 ( b, t ) = − δ ∂w 20 ( b, t ) ∂x (III.19.a) w 10 (b, t ) = w 20 (b, t ) (III.19.b) ∂w 10 (b, t ) = ∂w 20 (b, t ) ∂x ∂x (III.19.c) E bIb ∂ 2 w 10 ∂ 2 w 20 æ E ∂u 0 ö E + δ − = ( b , t ) A Y ( b , t ) d Y E ( t ) E I (b, t ) s s ç s 33 b b ∂x ∂x 2 ∂x 2 è (III.19.d) 3 ∂ 3 w 10 (b, t ) = ∂w 203 (b, t ) 3 ∂x ∂x (III.19.e) 93 From Equations (III.12.a-c), (III.13.b) and (III.16.b), at the second connection between the beam and the stack, x = b+c, we must have: u 0 (b + c, t ) = −δ EbIb ∂w 20 (b + c, t ) ∂x (III.20.a) w 20 (b + c, t ) = w 30 (b + c, t ) (III.20.b) ∂w 20 (b + c, t ) = ∂w 30 (b + c, t ) ∂x ∂x (III.20.c) ∂ 2 w 20 ∂ 2 w 30 ( ) (b + c, t ) + δA s æç Y E s ∂u 0 (b + c, t ) − d 33 Y E s E(t )ö (III.20.d) b + c , t = E I b b 2 2 ∂x ∂x ∂x è 3 ∂ 3 w 20 (b + c, t ) = ∂w 303 (b + c, t ) 3 ∂x ∂x (III.20.e) From Equations (III.10.c-d), at the free end of the beam, x = L, we must have: ∂ 2 w 30 (L, t ) = 0 ∂x 2 (III.21.a) ∂ 3 w 30 (L, t ) = 0 ∂x 3 (III.21.b) Since the only prescribed time dependant variable is the electric field within the stack and since there is no time differentiation in Equations (III.18.a) to (III.21.b), the quasi-static deflections can be assumed as polynomial function of the longitudinal coordinate, x, multiplied by the electric field, E(t). Hence, we have: 94 ( ) w i 0 (x , t ) = e i 3 x 3 + e i 2 x 2 + e i1 x + e i 0 E(t ) , i = 1,2,3 (III.22.a) u 0 (x, t ) = (e 41 x + e 40 ) E(t ) (III.22.b) Using the above forms in the fourteen equations (Equations (III.18.a) to (III.21.b)) that the quasi-static displacements must satisfy. The solution of the fourteen equations is obtained using the Maple software and the derivation of this quasi-static problem is presented in the first part of Appendix II.1.a. The following 14 coefficients are identified: e10 = e11 = e12 = e13 = 0 e 20 = − e 21 = (III.23) δ2AsY Es 1 b2 d 33 2 E bIb + δ2AsY Es δ (III.24.a) δ2AsY Es b d 33 2 E E b Ib + δ AsY s δ e 22 = − (III.24.b) δ2AsY Es 1 1 d 33 2 E 2 EbIb + δ AsY s δ (III.24.c) e 23 = 0 e 30 = (III.24.d) δ2As Y E s cæ cö ç b + d 33 2 E E b Ib + δ AsY s δ è 2 (III.25.a) δ2As Y E s c e 31 = − d 33 2 E EbIb + δ AsY s δ (III.25.b) 95 e 32 = e 33 = 0 (III.25.c-d) δ2As Y E s e 40 = − b d 33 E b Ib + δ2As Y E s e 41 = (III.26.a) δ2As Y E s d 33 E bIb + δ2AsY Es (III.26.b) These coefficients give the following quasi-static displacement: w 10 (x, t ) = 0 w 20 (x , t ) = −R s (III.27.a) (x − b )2 d 2δ 33 E (t ) (III.27.b) cæ cùö é w 30 (x, t ) = − R s çç x − ê b + d 33 E(t ) δè 2 ë (III.27.c) u 0 (x, t ) = R s (x − b ) d 33 E(t ) (III.27.d) In the above equations, Rs is a non-dimensional stiffness ratio. This ratio is the bending stiffness due to the stack, which is the longitudinal stiffness of the stack, AsYEs, multiplied by the square of the distance from the neutral axis of the stack to the beam neutral axis, δ, divided by the total bending stiffness of the beam in parallel with the stack. This ratio is then given by: δ2AsYEs Rs = EbIb + δ2AsYEs (III.28) 96 Using the definition of the dynamic and quasi-static deflections into the original differential equations, the set of nonhomogeneous differential equations governing the dynamic behavior of the coupled actuator-beam system are: E b I b ∂ 4 ŵ 1 ∂ 2 ŵ 1 + =0 A b ρ b ∂x 4 ∂t 2 (III.29.a) E b I b ∂ 4 ŵ 2 ∂ 2 ŵ 2 (x − b ) d E (t ) = Rs + 33 4 2 2δ A b ρ b ∂x ∂t (III.29.b) E b I b ∂ 4 ŵ 3 ∂ 2 ŵ 3 + = Rs A b ρ b ∂x 4 ∂t 2 (III.29.c) 2 cæ c ùö é (t ) çç x − ê b + d 33 E δè 2 ë A s Y E s ∂ 2 û ∂ 2 û (t ) − = R s (x − b )d 33 E A s ρ s ∂x 2 ∂t 2 (III.29.d) The boundary conditions associated with the dynamic behavior of the coupled system are given by Equations (III.10.a) through (III.14.b) with the electric field, E(t), set to be identically zero. III.2.4. Solution of the homogeneous part of the dynamic problem: Mode Shapes and Natural Frequencies To solve for the mode shapes and natural frequencies of the coupled system, let us first use separation of variable. Hence we assume dynamic deflections of the form: ŵ i (x, t ) = Wi (x ) T(t ) , i = 1,2,3 (III.30.a) 97 û (x , t ) = U(x ) T(t ) (III.30.b) Using the above form for the dynamic deflections into the homogeneous part of the differential equations (III.29), we have: E b I b d 4 Wi d 2T T + W = 0 , i = 1,2,3 i A b ρ b dx 4 dt 2 (III.31.a) AsYEs d 2U d 2T T − U =0 A s ρ s dx 2 dt 2 (III.31.b) In order for the above system to be satisfied, the spatial part of the equations should be equal to the temporal part of the equations and both should be equal to a constant. Let us denote this constant ω2, the spatial parts of Equations (III.31a-b) yield: d 4 Wi A ρ − ω2 b b Wi = 0 , i = 1,2,3 4 dx E bIb (III.32.a) Aρ d2U + ω2 s Es U = 0 2 dx AsY s (III.32.b) Let us define the following two dimensionless coefficients: β 4 = L4 ω2 A bρb E bIb (III.33.a) α 2 = c 2 ω2 A s ρs AsY Es (III.33.b) 98 Based on the two non-dimensional parameters α and β, the general solutions of the differential equations (III.32) have the form: æ xö æ xö æ xö æ xö , i = 1, 2, 3 Wi (x ) = C i1 sin ç β ÷ + C i 2 cosç β ÷ + C i 3 sinh ç β ÷ + C i 4 cosh ç β è L è L è L è L æ xö æ xö U(x ) = C 41 sin ç α ÷ + C 42 cosç α è c è c (III.34.a) (III.34.b) In order to obtain the mode shapes and the characteristic equation, we should solve for the coefficients of the assumed solutions using the boundary conditions. The solution of this problem is obtained using the Maple software and is presented in the second part of Appendix II.1.a. The main steps of the derivation are also reported in this section of the Thesis. First, using the definitions (III.34.a-b) into the boundary conditions (III.10.a-d), we have: C12 + C14 = 0 (III.35.a) C11 + C13 = 0 (III.35.b) − C 31 sin (β) − C 32 cos(β ) + C 33 sinh (β ) + C 34 cosh (β) = 0 (III.35.c) − C 31 cos(β ) + C 32 sin (β ) + C 33 cosh (β ) + C 34 sinh (β ) = 0 (III.35.d) Similarly, using the definitions (III.34.a-b) into equations (III.11.a-c) and the homogeneous part of (III.14.a), we have: 99 (C11 − C 21 )sin æç β b ö÷ + (C12 − C 22 )cosæç β b ö÷ è L è L æ bö æ bö + (C13 − C 23 )sinh ç β ÷ + (C14 − C 24 ) cosh ç β =0 è L è L (III.36.a) (C11 − C 21 ) cosæç β b ö÷ − (C12 − C 22 )sin æç β b ö÷ è L è L æ bö æ bö + (C13 − C 23 ) coshç β ÷ + (C14 − C 24 ) sinh ç β =0 è L è L æ bö æ bö − (C11 − C 21 ) cosç β ÷ + (C12 − C 22 )sin ç β ÷ è L è L æ bö æ bö + (C13 − C 23 ) cosh ç β ÷ + (C14 − C 24 )sinh ç β =0 è L è L æ bö æ bö æ bö − (C11 − C 21 ) sin ç β ÷ − (C12 − C 22 ) cosç β ÷ + (C13 − C 23 ) sinhç β ÷ è Lø è Lø è Lø E 2 δA Y s L α é æ b öù æ bö æ bö + (C14 − C 24 ) coshç β ÷ = − s C 41 cosç α ÷ − C 42 sinç α ÷ ê 2 EbIb β c ë è cø è Lø è cø (III.36.b) (III.36.c) (III.36.d) Again, using the definitions (III.34.a-b) into equations (III.12.a-c) and the homogeneous part of (III.14.b), we have: (C 31 − C 21 )sinæç β b + c ö÷ + (C 32 − C 22 ) cosæç β b + c ö÷ è L è L æ b +cö æ b +cö =0 + (C 33 − C 23 ) sinh ç β ÷ + (C 34 − C 24 ) cosh ç β è L è L (III.37.a) (C 31 − C 21 ) cosæç β b + c ö÷ − (C 32 − C 22 )sinæç β b + c ö÷ è L è L æ b + cö æ b+cö =0 + (C 33 − C 23 ) coshç β ÷ + (C 34 − C 24 )sinh ç β è L è L 100 (III.37.b) æ b+cö æ b+cö − (C 31 − C 21 ) cosç β ÷ + (C 32 − C 22 )sin ç β ÷ è L è L æ b+cö æ b+cö =0 + (C 33 − C 23 ) coshç β ÷ + (C 34 − C 24 )sinh ç β è L è L (III.37.c) æ b + cö æ b + cö æ b + cö − (C 31 − C 21 ) sin ç β ÷ ÷ + (C 33 − C 23 ) sinh ç β ÷ − (C 32 − C 22 ) cosç β L ø L ø L ø è è è (III.37.d) δA s Y E s L2 α é æ b + cö æ b + cö æ b + c öù + (C 34 − C 24 ) cosh ç β C 41 cosç α ÷=− ÷ − C 42 sin ç α ÷ L ø E b I b β 2 c êë c ø c ø è è è Finally, using the definitions (III.34.a-b) into boundary conditions (III.13.a-b), we have: æ bö æ bö C 41 sinç α ÷ + C 42 cosç α ÷ è cø è cø δβ ì æ bö æ bö æ bö æ b öü = − íC 21 cosç β ÷ − C 22 sin ç β ÷ + C 23 coshç β ÷ + C 24 sinh ç β ÷ý L î è Lø è Lø è Lø è Lø æ b + cö æ b + cö C 41 sin ç α ÷ + C 42 cosç α ÷ c ø c ø è è δβ ì æ b + cö æ b + cö æ b + cö æ b + c öü = − íC 21 cosç β ÷ − C 22 sin ç β ÷ + C 23 coshç β ÷ + C 24 sinh ç β ÷ý L î L ø L ø L ø L ø è è è è (III.38.a) (III.38.b) From the sum and subtraction of Equations (III.36.b) and (III.36.c), we have: (C11 − C 21 ) cosæç β b ö÷ − (C12 − C 22 ) sinæç β b ö = 0 (III.39.a) (C13 − C 23 ) coshæç β b ö÷ + (C14 − C 24 )sinhæç β b ö = 0 (III.39.b) è L è L è L è L 101 Similarly, for the pair of Equations (III.36.a)-(III.36.d), (III.37.b)-(III.37.c) and (III.37.a)-(III.37.d), we obtain: (C11 − C 21 )sinæç β b ö÷ + (C12 − C 22 )cosæç β b ö÷ = R 1 éêC 41 cosæç α b ö÷ − C 42 sinæç α b ö÷ù è Lø è Lø è cø ë è cø (III.39.c) (C13 − C 23 )sinhæç β b ö÷ + (C14 − C 24 )coshæç β b ö÷ = −R 1 éêC 41 cosæç α b ö÷ − C 42 sinæç α b ö÷ù (III.39.d) è Lø è Lø è cø ë è cø (C 31 − C 21 )cosæç β b + c ö÷ − (C 32 − C 22 )sin æç β b + c ö = 0 (III.40.a) (C 33 − C 23 )cosh æç β b + c ö÷ + (C 34 − C 24 )sinh æç β b + c ö = 0 (III.40.b) L è è L è L è L (C31 − C21 )sinæç β b + c ö÷ + (C32 − C22 ) cosæç β b + c ö÷ L ø è è L ø é æ b + c öù æ b+cö = R1 êC41 cosç α ÷ ÷ − C42 sinç α c ø c ø è è ë (III.40.c) (C33 − C 23 )sinhæç β b + c ö÷ + (C34 − C24 )coshæç β b + c ö÷ è L ø è é æ b+cö æ b + c öù = −R 1 êC 41 cosç α ÷ − C 42 sinç α ÷ c ø c ø è è ë L ø (III.40.d) In Equations (III.39.c-d) and (III.40.c-d), R1 is a non-dimensional coefficient given by: 1 δ 2 A s Y E s L2 α R1 = 2 E b I b cδ β 2 (III.41) 102 From Equations (III.39.a) and (III.39.c), we obtain: é æ b öù æ b ö æ bö C11 = C 21 + R 1 êC 41 cosç α ÷ − C 42 sinç α ÷ sinç β è cø è L è cø ë (III.42.a) é æ bö æ b öù æ b ö C12 = C 22 + R 1 êC 41 cosç α ÷ − C 42 sinç α ÷ cosç β è cø è cø è L ë (III.42.b) From Equations (III.39.b) and (III.39.d), we obtain: é æ bö æ xö æ b öù C13 = C 23 + R 1 êC 41 cosç α ÷ − C 42 sinç α ÷ú sinhç β ÷ è cø è cø è Lø ë (III.42.c) é æ bö æ b öù æ bö C14 = C 24 − R 1 êC 41 cosç α ÷ − C 42 sinç α ÷ coshç β è L è cø è cø ë (III.42.d) From Equations (III.40.a) and (III.40.c), we obtain: é æ b+cö æ b + c öù æ b + c ö C 31 = C 21 + R 1 êC 41 cosç α ÷ − C 42 sinç α ÷ ÷ sinç β c ø c øú è L ø è è ë (III.43.a) é æ b+cö æ b + c öù æ b + c ö C 32 = C 22 + R 1 êC 41 cosç α ÷ − C 42 sinç α ÷ cosç β c ø c ø è è L è ë (III.43.b) From Equations (III.40.b) and (III.40.d), we obtain: é æ b+cö æ b + c öù æ b+cö C 33 = C 23 + R 1 êC 41 cosç α ÷ − C 42 sinç α ÷ ú sinhç β ÷ c ø c ø è è è L ø ë (III.43.c) é æ b + c öù æ b+cö æ b+cö C34 = C 24 − R 1 êC 41 cosç α ÷ú coshç β ÷ − C 42 sin ç α ÷ c ø c ø è è è L ø ë (III.43.d) 103 From Equations (III.42.a-c) and (III.43.a-c), the coefficients C11, C12, C13, C13, C31, C32, C33 and C34 can be obtained recursively from the coefficients C21, C22, C23, C24, C41 and C42. Using Equations (III.42.a-c) into Equations (III.35.b) and (III.35.a), we obtain the C21 and C22 coefficients respectively: é æ bö æ b öù é æ b ö æ b öù C 21 = −C 23 − R 1 êC 41 cosç α ÷ − C 42 sinç α ÷ú êsinç β ÷ + sinhç β ÷ è cø è c ø ë è Lø è Lø ë (III.44.a) é æ bö æ b öù æ b öù é æ b ö C 22 = −C 24 − R 1 êC 41 cosç α ÷ − C 42 sinç α ÷ú êcosç β ÷ − coshç β ÷ è cø è cø ë è Lø è Lø ë (III.44.b) Then, using Equations (III.43.a-c) and Equations (III.4.a-b) into Equations (III.35.c) and (III.35.d), the following two Equations results: [sin(β) + sinh(β)] C 23 + [cos(β) + cosh(β)] C 24 − L11C 41 + L12 C 42 = 0 (III.45.a) [cos(β) + cosh(β)] C 23 − [sin (β) − sinh (β)] C 24 + L 21C 41 − L 22 C 42 = 0 (III.45.b) In the above equations, the coefficients L11, L12, L21 and L22 are given by: é æ b+c−Lö æ b + c − L öù æ b + c ö L11 = R 1 êcosç β ÷ú cosç α ÷ + coshç β ÷ L L c ø è ø ø è ë è é æ b − L öù æ b ö æ bö æ bö + R 1 êcos(β) coshç β ÷ − sin (β)sinhç β ÷ − cosç β ÷ cosç α L ø è è Lø è Lø è c ë (III.46.a) é æ b+c−Lö æ b + c − L öù æ b + c ö L12 = R 1 êcosç β ÷ú sinç α ÷ + coshç β ÷ L L c ø è ø ø è ë è é æ b − L öù æ b ö æ bö æ bö + R 1 êcos(β)coshç β ÷ − sin (β)sinhç β ÷ − cosç β ÷ sinç α L ø è è Lø è Lø è c ë (III.46.b) 104 é æ b+c−Lö æ b + c − L öù æ b + c ö L 21 = R 1 ê− sin ç β ÷ + sinh ç β ÷ cosç α ÷ L L c è ø è ø è ë é æ b − L öù æ b ö æ bö æ bö + R 1 êcos(β )sinh ç β + sin (β )coshç β ÷ + sin ç β ÷ cosç α ÷ Lø L øú è c ø è è L ë é æ b+c−Lö æ b + c − L öù æ b + c ö L 22 = R 1 ê− sin ç β ÷ + sinh ç β ÷ú sin ç α ÷ L L c ø è ø è ø è ë é æ bö æ bö æ b − L öù æ b ö + R 1 êcos(β )sinh ç β ÷ + sin (β) coshç β ÷ + sin ç β ÷ sin ç α L ø è Lø è Lø è è c ë (III.46.c) (III.46.d) Solving Equations (III.45.a) and (III.45.b) simultaneously for C23 and C24, we obtain: C 23 = [cos(β) + cosh(β)](− L 21C 41 + L 22 C 42 ) + [sin (β) − sinh (β)](L11C 41 − L12 C 42 ) 2 (1 + cos(β )cosh (β )) (III.47.a) C 24 = [cos(β) + cosh (β)](L11C 41 − L12 C 42 ) + [sin (β) + sinh (β)](L 21C 41 − L 22 C 42 ) 2 (1 + cos(β )cosh (β )) (III.47.b) Finally, using Equations (III.47.a-b) and (III.45.a-b) into Equations (III.38.a-b), we obtain two equations of the form: é α α æ b + cö æ cö æ c öù ê− R d M 11 β cosç α c ÷ − R d M12 β cosç α c ÷ + 2(1 + cos(β) cosh(β))sin ç α c ÷ú C 41 + è ø è ø è ø ë é α æ b + cö α æ c öù æ cö êR d M 11 β sin ç α c ÷ + 2(1 + cos(β) cosh(β)) cosç α c ÷ + R d M12 β sin ç α c ÷ C 42 = 0 è ø è ø è ø ë é α α æ c öù æ b + cö æ b + cö êR d M 21 β cosç α c ÷ + 2(1 + cos(β) cosh(β))sinç α c ÷ − R d M 22 β cosç α c ÷ú C 41 + è ø ø è è ø ë é α æ b + cö α æ c öù æ b+ cö ê2(1 + cos(β) cosh(β )) cosç α c ÷ − R d M 21 β sinç α c ÷ + R d M 22 β sinç α c ÷ C 42 = 0 è ø è ø è ø ë 105 (III.48.a) (III.48.b) In the above equations, the coefficients M11, M12, M21 and M22 are given by: æ cö æ cö æ b + c − L öé æ b − L ö æ b − L öù M11 = sinh ç β ÷ + sinç β ÷ − cosç β ÷ êsin ç β ÷ + sinh ç β ÷ L ø L L L L è ø è ø è øë è ø è æ bö æ b+ cö æ b ö æ b+cö æ bö æ b+cö æ b ö æ b +cö − sin ç β ÷ coshç β ÷ − sinh ç β ÷ cosç β ÷ − cosç β ÷ sinh ç β ÷ − coshç β ÷ sinç β ÷ è L è L è L è L è L è L è L è L æ 2b + c − L ö æ 2b + c − L ö æ c−Lö æ c−Lö − sin (β) coshç β ÷ − sinh (β)cosç β ÷ + cos(β)sinh ç β ÷ + cosh (β)sin ç β ÷ L L è è è L è L (III.49.a) æ b+c−Lö æ b−Lö æ b+c−Lö æ b−Lö − sin ç β ÷ coshç β ÷ − sinh ç β ÷ cosç β L L L L è è è è æ b−Lö æ b−Lö æ b−Lö æ b−Lö M12 = 2 sinhç β ÷ cosç β ÷ + 2 coshç β ÷ sinç β ÷ + sinh (β)cos(β) + cosh(β)sin (β) L L L L è è è è æ bö æ bö æ bö æ bö æ 2b − L ö æ 2b − L ö + 2 sinhç β ÷ cosç β ÷ + 2 coshç β ÷ sinç β ÷ + sin (β)coshç β ÷ + sinh (β)cosç β L L è L è L è L è L è è æ b + cö æ b + cö æ b + cö æ b + cö M 21 = 2 cosh ç β ÷ sinh ç β ÷ + cosh (β ) sin (β) + sinh (β ) cos(β) ÷ sin ç β ÷ + 2 cosç β L L L L è è è è æ b + c − Lö æ b + c − Lö æ b + c − Lö æ b + c − Lö 2 sinh ç β ÷ sin ç β ÷ ÷ cosç β ÷ + 2 cosh ç β L L L L è è è è (III.49.b) (III.49.c) æ 2 b + 2c − L ö æ 2 b + 2c − L ö + sinh (β) cosç β ÷ + sin (β) cosh ç β L L è è æ cö æ cö æ b ö æ b + cö æ b ö æ b + cö M 22 = − sinç β ÷ − sinh ç β ÷ + cosh ç β ÷ sin ç β ÷ + sinh ç β ÷ cosç β ÷ L L è L è L è L è è L è æ bö æ b + cö æ bö æ b + cö æ c − Lö æ 2b + c − L ö + cosç β ÷ sinh ç β ÷ + sin ç β ÷ cosh ç β ÷ − cosh (β) sin ç β ÷ + sinh (β) cosç β ÷ L L è L è L è L è L è è (III.49.d) æ c − Lö æ 2b + c − L ö æ b − Lö æ b + c − Lö − cos(β ) sinh ç β ÷ + sin (β ) cosh ç β ÷ + sinh ç β ÷ cosç β ÷ L L L L è è è è æ b − Lö æ b + c − Lö æ b − Lö æ b +c − Lö æ b − Lö æ b + c − Lö + cosç β ÷ sinh ç β ÷ + sin ç β ÷ cosh ç β ÷ + cosh ç β ÷ sin ç β L L L L L L è è è è è è And the dynamic non-dimensional ratio Rd is given by: Rd = 1 δ2AsY Es L 2 c E bIb (III.50) From Equation (III.48.a), we obtain C41 as a function of C42: 106 αé æ cö æ b+cö æ c öù 2(1 + cos(β ) cosh (β )) cosç α ÷ + R d ê M11 sin ç α ÷ + M12 sin ç α ÷ ú c ø βë è è cø è cø C 41 = C 42 (III.51) αé æ c öù æ b+cö æ cö 2(1 + cos(β) cosh (β ))sin ç α ÷ − R d ê M11 cosç α ÷ − M12 cosç α ÷ c ø βë è cø è è cø Using the above result in Equation (III.48.b), and dividing by C42 in order to obtain a non-trivial solution for the coefficients Cij, we obtain the characteristic equation of the coupled system. After collecting the terms of same power of the dynamic nondimensional ratio Rd and then separating the different terms as functions of α, the characteristic equation has the following form: 2 α α 2 α 1 + cos(β ) cosh (β ) + R d F1 + R d F2 + R d 2 F3 = 0 β tan (α ) β sin (α ) β (III.52) In the above Equation, the coefficients F1, F2 and F3 are independent of the nondimensional coefficient α and are given by: æ bö æ bö æ bö æ bö F1 = sinh (β) cos(β) + cosh (β ) sin (β) + sinh ç β ÷ cosç β ÷ + coshç β ÷ sinç β ÷ è Lø è Lø è Lø è Lø b c b c b c b c + + + + æ b − Lö æ b − Lö ö æ ö æ ö æ ö æ + sinh ç β ÷ ÷ cosç β ÷ + sinh ç β ÷ sinç β ÷ + coshç β ÷ cosç β L ø è L ø L ø è L ø L ø è L ø è è è (III.53.a) æ b − Lö æ b − Lö æ b + c − Lö æ b + c − Lö æ b + c − Lö æ b + c − Lö + coshç β ÷ sin ç β ÷ + sinh ç β ÷ cosç β ÷ + cosh ç β ÷ sin ç β ÷ L L L L L L è ø è ø è ø è ø è ø è ø é æ 2b − L ö é 1 æ 2b − L ö æ 2b + 2c − L öù æ 2 b + 2c − L ö ù 1 + sinh (β) êcosç β ÷ + cosç β ÷ú + sin (β ) êcoshç β ÷ + coshç β ÷ 2 L ø L 2 L ø L è ø è ø è ë è ë æ cö æ cö æ 2b + c − L ö æ 2b + c − L ö F2 = sinh ç β ÷ + sin ç β ÷ − sinh (β) cosç β ÷ − sin (β ) coshç β ÷ L L è L è L è è æ c − Lö æ c −Lö æ bö æ b + cö æ bö æ b + cö + cosh (β) sin ç β ÷ + cos(β ) sinh ç β ÷ − sinh ç β ÷ cosç β ÷ − sinç β ÷ cosh ç β ÷ L L L L L L è è è è è è æ b − Lö æ b + c − Lö æ bö æ b + cö æ b ö æ b + cö − coshç β ÷ sin ç β ÷ − sinh ç β ÷ cosç β ÷ ÷ − cosç β ÷ sinh ç β L L L L L L è è è è è è æ b − Lö æ b + c − Lö æ b − Lö æ b + c − Lö æ b − Lö æ b + c − Lö − sin ç β ÷ coshç β ÷ − coshç β ÷ sin ç β ÷ − cosç β ÷ sinh ç β L L L L L L è è è è è è 107 (III.53.b) 1 æ cö æ b + cö æ b + cö æ cö F3 = − sinh (β) sin (β) + 2 sin ç β ÷ sinh ç β ÷ − sin ç β ÷ sinhç β ÷ 2 L L è L è è è L + 1 æ c − Lö æ c − Lö æ b − Lö æ b − L ö 1 æ 2b + c − L ö æ 2b + c − L ö sinhç β ÷ sin ç β ÷ + sin ç β ÷ sinhç β ÷ − sin ç β ÷ sinhç β ÷ 2 L L L L 2 è L L è è è è è 1 æ b + cö æ b − cö 1 æ b + cö æ b − cö 1 æ b + cö æ b − cö sinh ç β ÷ sin ç β ÷ + sin ç β ÷ sinh ç β ÷ − coshç β ÷ cosç β ÷ 2 L L 2 L L 2 L L è è è è è è 1 æ b + cö æ b − cö 1 æ c − L ö æ 2b + c − L ö 1 æ c − Lö æ 2b + c − L ö + cosç β ÷ coshç β ÷ + coshç β ÷ cosç β ÷ − cosç β ÷ coshç β ÷ 2 L L 2 L L 2 L L è è è è è è − − 1 æ 2 b + 2c − L ö 1 æ 2 b + 2c − L ö 1 æ 2b + 2c − L ö sinh (β ) sin ç β ÷ − sin (β) sinhç β ÷ − cosh (β) cosç β ÷ 4 L 4 L 4 L è è è + 1 æ 2b + 2c − L ö 1 æ 2c − L ö 1 æ 2c − L ö cos(β) coshç β ÷ − cosh (β) cosç β ÷ + cos(β ) coshç β ÷ 4 L 4 L 4 L è è è + 1 æ 2b − L ö 1 æ 2b − L ö æ 2b − L ö 1 æ 2b − L ö 1 sinh (β) sin ç β ÷ + cos(β) cosh ç β ÷ ÷ + sin (β) sinh ç β ÷ − cosh (β) cosç β 4 L 4 L 4 L 4 L è è è è 1 æ b − L ö æ b + 2c − L ö 1 æ b − L ö æ b + 2c − L ö sinh ç β ÷ sin ç β ÷ + coshç β ÷ cosç β ÷ 2 L L 2 L L è è è è 1 æ b − Lö æ b + 2c − L ö 1 æ b − Lö æ b + 2c − L ö − sin ç β ÷ sinh ç β ÷ − cosç β ÷ coshç β ÷ 2 è L L 2 L L è è è − + (III.53.c) 1 æ 2b − L ö æ 2b + 2c − L ö 1 æ 2b − L ö æ 2b + 2c − L ö sin ç β ÷ sinh ç β ÷ + sinh ç β ÷ sin ç β 4 è L L 4 L L è è è From Equations (III.50), if the stiffness of the stack is set to zero, meaning that there is no stack attached between the two rigid connections, the non-dimensional dynamic ratio Rd becomes identically zero. In this case, the characteristic equation (III.53) reduces to the characteristic equation for a cantilever beam: cos(β)cosh (β) = −1 (III.55) Similarly, if the length of the actuator, c, is set to zero, from Equation (III.33.b), the coefficient α becomes zero and the characteristic equation (III.53) reduces to the characteristic equation for a cantilever beam (III.55). Under the assumption that the first segment of the beam still satisfies EulerBernoulli conditions, as its length decreases to zero. Hence, neglecting any shear effects, if the distance from the root of the beam to the actuator, b, is set to zero, the system is 108 equivalent to hinging one side of the stack to the cantilever beam mount, as illustrated in Figure III.7. w δ c L x, u Figure III.7 Model of the Root OPSA Mounted on a Cantilever Beam This problem is solved in a manner similar to the one presented in Sections III.2.2 to III.2.4 with the first section of the cantilever beam being absent. The derivation of the solution is obtained using the Maple software and is presented in Appendix II.1.b. For this problem, the characteristic equation has the form: 1 + cos(β)cosh (β) + R d α F1b = 0 β tan (α ) (III.56) Where the coefficient F1b is given by: 1 1 æ cö æ cö æ cö æ cö F1b = sinh (β)cos(β) + cosh(β)sin (β) + sinhç β ÷ cosç β ÷ + coshç β ÷ sinç β ÷ 2 2 è L è L è L è L æ c−Lö æ c−Lö æ c−Lö æ c−Lö + sinhç β ÷ cosç β ÷ + coshç β ÷ sinç β ÷ è L è L è L è L 1 æ 2c − L ö 1 æ 2c − L ö + sinh (β)cosç β ÷ + sin (β)coshç β 2 L 2 L è è 109 (III.57) This solution for the root OPSA problem, given by Equations (III.56) and (III.57), is identical to setting the length b to zero in equations (III.52) and (III.53.a-c). At frequencies well below the first characteristic frequency of the piezoceramic stack, which is the first unloaded resonant frequency given by the solution of Equation (III.33.b) for α = 1, the non-dimensional parameter α is small with respect to 1 (α << 1). Further, if the first few frequencies of the cantilever beam are also lower than the first frequency of the stack, the ratio α/β is also small with respect to 1 (α << β). Since the first unloaded natural frequency of a piezoceramic stack is usually of the order of 10kHz, it is a reasonable assumption to set the parameter α to zero when studying the first few frequencies of the system. In such a case, Equation (III.53) becomes: 1 + cos(β)cosh (β) + Rd (F1 + F2 ) = 0 β (III.58) Since in Equation (III.58) the only parameter that depends on the stack material properties is Rd. Further, since, from its definition in Equation (III.50), this ratio only depends on the stiffness property of the stack. It is to be noted that Equation (III.58) does not depend on the inertia of the stack. Further, for a zero non-dimensional coefficient α, both Equations (III.48.a) and (III.48.b) reduce to: [1 + cos(β)cosh(β)] C 42 = 0 (III.59) Since, in general, the sum of the functions F1 and F2 is non-zero, from Equation (III.58), we have that the first term of Equation (III.59), 1+cos(β)cosh(β), must be nonzero. Hence, we must have that the mode shape coefficient C42 is identically zero. In such 110 a case, with a zero non-dimensional coefficient α, from Equation (III.34.b), the mode shape associated with the dynamic displacement of the stack, U, is identically zero. Therefore from Equations (III.17.b) and (III.27.d), the displacement field in the stack becomes: u (x, t ) = R s (x − b ) d 33 E(t ) (III.60) Taking into account both this result and the fact that the characteristic equation is independent of the stack inertia, for low frequency operation, the behavior of the piezoceramic stack can be modeled as a spring and two induced forces at its ends. III.2.5. Low Frequency Approximation of the OPSA As discussed above, for low frequency operation, the piezoceramic stack that is the active element of the Offset Piezoceramic Stack Actuator can be modeled as a spring of stiffness, ks, with its actuation modeled as two driving forces, Fs, at the ends of the spring. This model is illustrated in Figure III.8. ks Fs δ b c L Figure III.8 Low Frequency Approximation of the OPSA acting on a Cantilever Beam 111 In order to be able to identify the equivalent spring stiffness and driving forces of the low frequency approximation of the OPSA, a modal analysis of the system illustrated in Figure III.8 is performed. As for the previous sections, the beam is segmented into three sections, which are the same as the ones described in Section III.2.2. Again, the cantilever beam is assumed to satisfy the Euler-Bernoulli assumptions and the mass-less rigid connections remains orthogonal to the elastic axis of the beam. Finally, the spring is hinged to the top of these connections. The differential equations governing the transverse displacement of each section of the beam, w(x,t), are given by: EbIb ∂2wi ∂4wi = 0 , i = 1, 2, 3 + A ρ b b ∂t 2 ∂x 4 (III.61) The geometric boundary conditions at the clamped end of the beam are given by: w 1 (0, t ) = 0 (III.62.a) ∂w 1 (0, t ) = 0 ∂x (III.62.b) Similarly, the natural boundary conditions free end of the beam are given by: ∂2w3 (L, t ) = 0 ∂x 2 (III.62.c) ∂3w 3 (L, t ) = 0 ∂x 3 (III.62.d) 112 Under small displacement assumptions, the continuity of the displacement, slope and force at the connections between the beam and the rigid connections to the spring, yield the following boundary conditions: w 1 (b, t ) = w 2 (b, t ) (III.63.a) ∂w 1 (b, t ) = ∂w 2 (b, t ) ∂x ∂x (III.63.b) 3 ∂3w1 (b, t ) = ∂w 23 (b, t ) 3 ∂x ∂x (III.63.c) w 2 (b + c, t ) = w 3 (b + c, t ) (III.64.a) ∂w ∂w 2 (b + c, t ) = 3 (b + c, t ) ∂x ∂x (III.64.b) 3 ∂w ∂3w 2 (b + c, t ) = 33 (b + c, t ) 3 ∂x ∂x (III.64.c) The remaining boundary conditions yield the coupling terms between the driving forces, the spring and the beam. The natural boundary conditions couple the driving forces, Fs, to the bending moments into the beam at the location of the connections via the lever arm distance of the actuator, δ, and the slope in the beam at the location of the connections through the spring stiffness, ks, and the lever arm distance of the actuator, δ. These boundary conditions, illustrated in Figure III.9, are of the form: 113 ∂ 2 w1 ∂2w 2 EbIb (b, t ) = E b I b 2 (b, t ) + δFs (t ) + k s δ 2 æç ∂w 2 (b + c, t ) − ∂w 2 (b, t ) ö 2 ∂x ∂x ∂x è ∂x E b Ib (III.65.a) ∂2w3 ∂w 2 ∂2w 2 ö 2 æ ∂w 2 ( ) ( ) (b + c, t ) (III.65.b) ( b , t ) = E I ( b c , t ) b c , t F t k + − + + δ + δ ç s s b b ∂x 2 ∂x ∂x 2 è ∂x Fs æ ∂w δFs + k s δ 2 çç 2 è ∂x − x = b+c æ ∂w k s δçç 2 è ∂x ∂w 2 ∂x − x =b æ ∂w Fs + k s δçç 2 è ∂x ö x =b+ c − x =b+c 2 EbIb w1 2 EbIb ∂ w1 ∂x 2 ∂x ∂w 2 ∂x ö x =b EbIb 2 ∂ w2 2 Fs ö w2 w2 x =b ∂w 2 ∂x E bIb x =b ∂ w2 ∂x 2 x =b+c ∂2w3 ∂x 2 x = b +c w3 x =b Figure III.9 Boundary Condition Diagram for the Approximation of the OPSA As in Section III.2.2, it is to be noted at this point that the boundary conditions (III.65.a) and (III.65.b) are nonhomogeneous. In order to solve this problem using modal analysis, we need to transform the problem into a nonhomogeneous differential equation form with homogeneous boundary conditions. This task is achieved by dividing the beam deflection into dynamic, ŵ i , and quasi-static, wi0, deflections: w i (x, t ) = ŵ i (x, t ) + w i 0 (x, t ) , i = 1, 2, 3 114 (III.66) Using the procedure described in Section III.2.3, the pseudo static deflections are assumed to be the product of polynomial functions of the spatial coordinate x with the driving force Fs(t). Solving for the coefficients of the polynomial functions such that we obtain an homogeneous boundary condition problem for the dynamic deflections, ŵ i , we obtain the following quasi-static deflections: w 10 (x, t ) = 0 w 20 (x , t ) = − (III.67.a) (x − b ) F δ2 s 2 δ k s c + E b I b 2δ 2 (III.67.b) δ2 cæ cùö é w 30 (x, t ) = − 2 çç x − ê b + Fs 2 δ k sc + E bIb δ è ë (III.67.c) These results are obtained by solving the equations with the Maple software and the process is shown in the first part of Appendix II.1.c. As expected, the pseudo-static displacement is identically zero in the section spanning from the root of the beam to the actuator. Further, The pseudo-static deflection distribution is the same for the low frequency model as the one given by Equations (III.27.b-c). Hence, the equivalent spring stiffness of the stack and the driving forces generated by the stack which are the coefficients of the low frequency model can be identified using Equations (III.67.b-c) with Equations (III.67.b-c). From these two sets of equations, we obtain that the equivalent spring stiffness of the stack and its driving force are given by: AsYEs ks = c (III.68.a) 115 Fs = A s Y E s d 33 E (t ) (III.68.b) In this case, Equations (III.67.a-c) are identical to Equations (III.27.a-c). Further, the nonhomogeneous differential equations governing the dynamic behavior of each section are given by: E b I b ∂ 4 ŵ 1 ∂ 2 ŵ 1 =0 + A b ρ b ∂x 4 ∂t 2 (III.69.a) E b I b ∂ 4 ŵ 2 ∂ 2 ŵ 2 (x − b ) d E (t ) = Rs + 33 4 2 A b ρ b ∂x 2δ ∂t (III.69.b) E b I b ∂ 4 ŵ 3 ∂ 2 ŵ 3 = Rs + A b ρ b ∂x 4 ∂t 2 (III.69.c) 2 cæ c ùö é (t ) çç x − êb + d 33 E δè 2 ë As expected these three differential equations are the same as the ones for the continuous model for the piezoceramic stack, Equations (III.29.a-c). As in Section III.2.4, the homogeneous part of Equations (III.69.a-c) is used to solve for the mode shapes and characteristic equation of the system. Based on separation of the variables, we obtain the following spatial differential equations for the system: d 4 Wi β 4 − Wi = 0 , i = 1,2,3 dx 4 L4 (III.70) Where the non-dimensional coefficient β is the same as in Section III.2.4 and is given by: 116 β 4 = L4 ω2 A bρb E bIb (III.71) As in Section III.2.4, the general solutions of the differential equations (III.70) have the form: æ xö æ xö æ xö æ xö , i = 1, 2, 3 Wi (x ) = C i1 sin ç β ÷ + C i 2 cosç β ÷ + C i 3 sinh ç β ÷ + C i 4 cosh ç β è L è L è L è L (III.72) As for the previous section, the mode shape coefficients and the characteristic equation are computed using the Maple software and the computation are illustrated in the second part of Appendix II.1.c. Using the homogeneous parts of the boundary conditions (III.62.a) to (III.65.b), the eleven relationships between the mode shape coefficients and the characteristic equation are obtained. The procedure described in Section III.2.4 is used to obtain the following equations. From the boundary conditions (III.63.a-c) and the homogeneous part of boundary condition (III.65.a), we obtain: é R ì æ cö æ 2b + c ö üù æ 2b ö C11 = ê1 + 2 ísin ç β ÷ + sin ç β ÷ − sin ç β ÷ ýú C 21 2 î è Lø L ø û è è Lø ë R ì æ 2b + c ö ü æ 2b ö æ cö + 2 í− 1 + cosç β ÷ + cosç β ÷ − cosç β ÷ ýC 22 2 î L ø è è Lø è Lø ì æ b + c öü ù æ b + c öü æ bö æ bö æ b ö éì + R 2 sin ç β ÷ êícoshç β ÷ − coshç β ÷ýC 24 ÷ ýC 23 + ísinh ç β ÷ − sinhç β è L ø è L ø è Lø è Lø è L ø ëî î 117 (III.73.a) C12 = R2 ì æ 2b + c ö ü æ 2b ö æ cö ÷ýC 21 í1 − cosç β ÷ + cosç β ÷ − cosç β 2 î L ø è è Lø è Lø é R ì æ cö æ 2b + c öüù æ 2b ö + ê1 + 2 ísin ç β ÷ − sin ç β ÷ + sin ç β ÷ýú C 22 2 î è Lø L ø û è è Lø ë ì æ b + c öü ù æ b + c öü æ bö æ bö æ b ö éì + R 2 cosç β ÷ êícoshç β ÷ − coshç β ÷ ýC 24 ú ÷ ýC 23 + ísinh ç β ÷ − sinh ç β è L ø è L ø è Lø è Lø è L ø ëî î û é R ì æ cö æ 2b ö æ 2 b + c ö üù C13 = ê1 + 2 ísinh ç β ÷ + sinh ç β ÷ − sinh ç β ÷ ýú C 23 L ø û 2 î è Lø è Lø è ë R ì æ cö æ 2b ö æ 2b + c ö ü + 2 í− 1 + coshç β ÷ + coshç β ÷ − coshç β ÷ ýC 24 2 î L ø è Lø è Lø è (III.73.b) (III.73.c) ì æ bö æ b ö éì æ b ö æ b + c öü æ b + c öü ù + R 2 sinh ç β ÷ êícosç β ÷ − cosç β ÷ ýC 21 − ísin ç β ÷ − sin ç β ÷ ýC 22 è L ø ëî è L ø è L ø è L ø î è Lø R2 ì æ cö æ 2b ö æ 2b + c ö ü ÷ýC 23 í1 − coshç β ÷ + coshç β ÷ − coshç β 2 î L ø è Lø è Lø è é R ì æ 2 b + c ö üù æ 2b ö æ cö + ê1 + 2 ísinh ç β ÷ − sinh ç β ÷ + sinh ç β ÷ýú C 24 L ø û 2 î è è Lø è Lø ë C14 = − (III.73.d) ì æ bö æ b + c öü ù æ b + c öü æ b ö éì æ b ö − R 2 coshç β ÷ ê ícosç β ÷ − cosç β ÷ýC 22 ÷ýC 21 − ísinç β ÷ − sinç β è L ø è L ø è L ø ëî è L ø î è Lø In the above equations, the non-dimensional ratio R2 is given by: R2 = 1 δ2AsY Es L R d = 2 E b I b cβ β (III.74) Similarly, from the boundary conditions (III.64.a-c) and the homogeneous part of boundary condition (III.65.b), we obtain: 118 é R ì æ cö æ 2 b + 2c ö ü ù æ 2b + c ö C 31 = ê1 + 2 ísin ç β ÷ + sin ç β ÷ýú C 21 ÷ − sin ç β 2 î è Lø L ø L ø û è è ë R ì æ cö æ 2b + c ö æ 2 b + 2c ö ü + 2 í1 − cosç β ÷ + cosç β ÷ − cosç β ÷ýC 22 2 î L ø L ø è Lø è è (III.75.a) ì æ b + c öü ù æ bö æ b + c öü æ bö æ b + c ö éì + R 2 sin ç β ÷ýC 24 ÷ýC 23 + ísinh ç β ÷ − sinh ç β ÷ êícoshç β ÷ − coshç β è L ø è Lø è L ø è Lø è L ø ëî î C 32 = R2 2 é R + ê1 + 2 2 ë ì æ cö æ 2b + c ö æ 2 b + 2c ö ü ÷ − cosç β ÷ýC 21 í− 1 + cosç β ÷ + cosç β L ø L L è ø è ø è î ì æ cö æ 2b + c ö æ 2b + 2c öüù ÷ + sin ç β ÷ýú C 22 ísin ç β ÷ − sin ç β L ø L ø û è è î è Lø (III.75.b) ì æ b + c öü æ bö æ b + c öü ù æ b + c ö éì æ bö + R 2 cosç β ÷ýC 23 + ísinh ç β ÷ − sinh ç β ÷ýC 24 ÷ êícoshç β ÷ − cosh ç β è L ø ëî è Lø è L ø è Lø è L ø î é R ì æ 2b + c ö æ 2b + 2c öüù æ cö C 33 = ê1 + 2 ísinh ç β ÷ + sinh ç β ÷ − sinh ç β ÷ýú C 23 L ø û 2 î L ø è è Lø è ë R ì æ cö æ 2b + c ö æ 2b + 2c öü + 2 í1 − cosh ç β ÷ + coshç β ÷ − cosh ç β ÷ýC 24 2 î L ø L ø è Lø è è (III.75.c) ì æ bö æ b + c öü ù æ b + c öü æ b + c ö éì æ b ö + R 2 sinh ç β ÷ýC 22 ÷ýC 21 − ísin ç β ÷ − sin ç β ÷ êícosç β ÷ − cosç β è L ø è L ø è L ø ëî è L ø î è Lø R2 ì æ cö æ 2b + c ö æ 2b + 2c öü ÷ − coshç β ÷ýC 23 í− 1 + coshç β ÷ + coshç β 2 î L ø L ø è Lø è è é R ì æ 2b + 2c öüù æ 2b + c ö æ cö + ê1 + 2 ísinh ç β ÷ − sinh ç β ÷ýú C 24 ÷ + sinh ç β 2 î L ø L ø û è è è Lø ë ì æ bö æ b + c ö éì æ b ö æ b + c öü æ b + c öü ù − R 2 coshç β ÷ êícosç β ÷ − cosç β ÷ýC 21 − ísin ç β ÷ − sin ç β ÷ýC 22 è L ø ëî è L ø è L ø è L ø î è Lø C 34 = − (III.75.d) From Equations (III.73.a-d) and (III.75.a-d), the coefficients C11, C12, C13, C13, C31, C32, C33 and C34 can be obtained recursively from the coefficients C21, C22, C23, C24. 119 As in Section III.2.4, using Equations (III.73.a-d) into Equations (III.62.a-b), we obtain the C21 and C22 coefficients respectively: C 21 = I13 C 23 + I14 C 24 J (III.76.a) C 22 = I 23 C 23 + I 24 C 24 J (III.76.b) The coefficients of Equations (III.76.a-b) are given by: I13 = R2 2 é æ 2b ö æ 2b ö æ cö æ cö êsinhç β L ÷ + sin ç β L ÷ + sinh ç β L ÷ − sin ç β L ÷ ø ø è ø è ø è è ë æ 2b + c ö æ 2b + c ö ù æ b öé æ b ö æ b + c öù − sinh ç β ÷ + sin ç β ÷ú + R 2 coshç β ÷ êsin ç β ÷ − sin ç β ÷ú L ø L ø è è è L øë è L ø è L ø (III.77.a) æ b öé æ bö æ b + c öù + R 2 sin ç β ÷ êcoshç β ÷ − coshç β ÷ +1 è L øë è Lø è L ø I14 = R2 2 é æ 2b ö æ 2b ö æ cö æ cö êcoshç β L ÷ − cosç β L ÷ + coshç β L ÷ − cosç β L ÷ è ø ø è è ø è ø ë æ b + c öù æ b öé æ b ö æ 2b + c ö æ 2b + c ö ù − cosh ç β ÷ú ÷ + cosç β ÷ú + R 2 sinh ç β ÷ êsin ç β ÷ − sin ç β L ø L ø è L ø è è è L øë è L ø (III.77.b) æ bö æ b + c öù æ b öé + R 2 sin ç β ÷ êsinh ç β ÷ − sinh ç β ÷ è Lø è L ø è L øë é æ 2b ö æ cö æ cö æ 2b ö êcoshç β L ÷ + cosç β L ÷ − coshç β L ÷ − cosç β L ÷ ø ø ø ø è è è è ë æ 2b + c ö ù æ b + c öù æ 2b + c ö æ b öé æ b ö + coshç β ÷ ú + R 2 coshç β ÷ êcosç β ÷ − cosç β ÷ú ÷ + cosç β L ø L ø è è L ø è è L øë è L ø I 23 = R2 2 æ b öé æ bö æ b + c öù + R 2 cosç β ÷ êcoshç β ÷ − coshç β ÷ − R2 è L øë è Lø è L ø 120 (III.77.c) é æ 2b ö æ cö æ cö æ 2b ö êsinh ç β L ÷ + sin ç β L ÷ − sinh ç β L ÷ + sin ç β L ÷ ø ø ø ø è è è è ë æ 2b + c ö ù æ b + c öù æ 2b + c ö æ b öé æ b ö + sinh ç β ÷ ú + R 2 sinh ç β ÷ êcosç β ÷ − cosç β ÷ú ÷ − sin ç β L ø L ø è è L ø è è L øë è L ø I 24 = R2 2 (III.77.d) æ b öé æ bö æ b + c öù + R 2 cosç β ÷ êsinh ç β ÷ − sinh ç β ÷ +1 è L øë è Lø è L ø é æ cö æ b öù æ bö æ bö æ bö J = −1 − R 2 êsinç β ÷ + sinç β ÷ coshç β ÷ + cosç β ÷ sinhç β ÷ú è Lø è Lø è Lø è Lø ë è Lø é æ b ö æ b+cö æ b ö æ b + c öù + R 2 êsinhç β ÷ cosç β ÷ ÷ + coshç β ÷ sin ç β è Lø è L ø è Lø è L ø ë (III.78) Using Equations (III.75.a-c) and (III.76.a-b) into the boundary condition (III.62.c), we obtain a relationship between the remaining two mode shape coefficients C23 and C24 in the following form: C 23 = L4 C 24 L3 (III.79) In the above equation, the coefficients L3 and L4 are given by: 121 é æ c − Lö æ c − Lö æ 2b − L ö L 3 = êcosh (β ) − 3 cos(β) − coshç β ÷ + 2 cosç β ÷ − cosç β ÷ L ø L ø L ø è è è ë æ 2 b + 2c − L ö æ 2 b + 2c − L ö æ 2b + c − L ö æ 2b + c − L ö '− coshç β ÷ ÷ − cosç β ÷ + 2 cosç β ÷ + coshç β L L L L ø ø è ø è ø è è æ cö æ 2b ö æ cö æ cö + 2 sinh (β ) sin ç β ÷ + sin (β) sinh ç β ÷ + cos(β ) coshç β ÷ + sin (β) sinh ç β ÷ è Lø è Lø è Lø è Lø æ 2b + c ö æ 2b + c ö æ 2b ö − cos(β ) coshç β ÷ − sin (β) sinh ç β ÷ ÷ + cos(β ) coshç β L ø L ø è è Lø è æ b ö æ b − Lö æ bö æ b − Lö æ bö æ b −Lö + 4 cosh ç β ÷ cosç β ÷ − sin ç β ÷ sinh ç β ÷ − cosç β ÷ cosh ç β ÷ L ø L ø L ø è Lø è è Lø è è Lø è æ bö æ b + Lö æ bö æ b + Lö æ b ö æ b + c − Lö + sin ç β ÷ sinh ç β ÷ + cosç β ÷ coshç β ÷ − 4 coshç β ÷ cosç β ÷ L L L L L è ø è ø è ø è ø è Lø è ø æ bö æ b + c − Lö æ b + cö æ b − Lö + 2 cosç β ÷ cosh ç β ÷ − 2 coshç β ÷ cosç β ÷ L L ø è L ø è Lø è ø è æ b + cö æ b − Lö æ b + cö æ b − Lö æ b + cö æ b + Lö + sin ç β ÷ sinh ç β ÷ + cosç β ÷ coshç β ÷ − sin ç β ÷ sinh ç β ÷ L ø L ø L ø L ø L ø L ø è è è è è è æ b + cö æ b + c − Lö æ b + cö æ b + Lö − cosç β ÷ coshç β ÷ + 2 coshç β ÷ cosç β ÷ L ø L ø L ø è L è è è ø (III.80.a) æ b + cö æ b + c − L öù − 2 cosç β ÷ coshç β ÷ R 2 + 2 sin (β ) + 2 sinh (β) L ø L è è ø æ 2b + c − L ö æ c − Lö æ c − Lö L 4 = − sinh (β) − sin (β ) − sinh ç β ÷ ÷ + sinh ç β ÷ − 2 sin ç β L L L è ø è ø è ø æ 2b − L ö æ 2 b + 2c − L ö æ 2 b + 2c − L ö æ 2b + c − L ö + 2 sin ç β ÷ ÷ − sin ç β ÷ − sin ç β ÷ − sinh ç β L L L L ø è è ø è ø è ø æ cö æ cö æ cö æ 2b ö − cosh (β) sin ç β ÷ − sin (β ) coshç β ÷ − cos(β) sinh ç β ÷ − sin (β) cosh ç β ÷ è Lø è Lø è Lø è Lø æ 2b + c ö æ b ö æ b − Lö æ 2b + c ö æ 2b ö + cos(β) sinh ç β ÷ + sin (β ) coshç β ÷ ÷ − 4 sinh ç β ÷ cosç β ÷ − cos(β) sinh ç β L L L L ø è ø è Lø è è ø è ø æ b + Lö æ b − Lö æ bö æ b − Lö æ bö æ bö − sin ç β ÷ coshç β ÷ ÷ − sin ç β ÷ coshç β ÷ − cosç β ÷ sinh ç β L L L L L L ø è è ø è ø è ø è ø è ø æ bö æ b + Lö æ b ö æ b + c − Lö æ bö æ b + c − Lö − cosç β ÷ sinh ç β ÷ + 4 sinh ç β ÷ cosç β ÷ + 2 sin ç β ÷ coshç β ÷ L ø L L è Lø è è Lø è ø è Lø è ø æ b − Lö æ b + cö æ b − Lö æ b + cö æ b + cö æ b − Lö + 2 sinh ç β ÷ ÷ sinh ç β ÷ + cosç β ÷ coshç β ÷ + sin ç β ÷ cosç β L ø è L ø L ø L ø L ø L ø è è è è è æ b + cö æ b + c − Lö æ b + Lö æ b + cö æ b + Lö æ b + cö + cosç β ÷ ÷ cosç β ÷ − 2 sinh ç β ÷ coshç β ÷ + sin ç β ÷ sinh ç β L L L L L ø è L ø è ø è ø è ø è ø è æ b + cö æ b + c − L öù − 2 sin ç β ÷ coshç β ÷ R 2 − 2 cos(β ) − 2 cosh (β) L ø L è è ø 122 (III.80.b) Finally, using Equations (III.75.a-c), (III.76.a-b) and (III.79) into the boundary condition (III.62.d), and dividing by the remaining mode shape coefficient C24 in order to eliminate the trivial solution for the mode shapes, we obtain the characteristic equation for the low frequency approximation of the OPSA acting on a cantilever beam in the following form: 1 + cos(β) cosh (β) + Rd F4 = 0 β (III.81) In the above equation, the term F4 is given by: æ cö æ cö æ c − Lö F4 = sinh ç β ÷ + sin ç β ÷ + sinh (β) cos(β) + sin (β) cosh (β) + cosh (β ) sin ç β ÷ L L L è è è æ 2b − L ö æ 2b − L ö 1 æ c − Lö 1 + cos(β) sinh ç β ÷ ÷ + sin (β) coshç β ÷ + sinh (β) cosç β L L 2 L 2 è è è æ 2b + c − L ö æ 2b + c − L ö 1 æ 2b + 2c − L ö − sinh (β) cosç β ÷ − sin (β) coshç β ÷ + sinh (β ) cosç β ÷ L L 2 L è è è 1 æ 2b + 2c − L ö æ bö æ bö æ bö æ bö + sin (β) coshç β ÷ + sinhç β ÷ cosç β ÷ + coshç β ÷ sin ç β ÷ 2 L è è L è L è L è L æ b ö æ b + cö æ b ö æ b + cö æ bö æ b + cö − sinh ç β ÷ cosç β ÷ − coshç β ÷ sin ç β ÷ − sin ç β ÷ coshç β ÷ L L L è L è è L è è L è æ bö æ b + cö æ b − Lö æ b + c − Lö æ b − Lö æ b + c − Lö − cosç β ÷ sinh ç β ÷ − sinhç β ÷ cosç β ÷ − coshç β ÷ sin ç β ÷ L L L L L L è è è è è è æ b − Lö æ b + c − Lö æ b − Lö æ b + c − Lö æ b + cö æ b + cö − sin ç β ÷ coshç β ÷ − cosç β ÷ sinh ç β ÷ + sin ç β ÷ coshç β ÷ L L L L L L è è è è è è æ b + cö æ b + cö æ b − Lö æ b − Lö æ b − Lö æ b − Lö + cosç β ÷ sinh ç β ÷ + sin ç β ÷ coshç β ÷ + cosç β ÷ sinh ç β ÷ L L L L L L è è è è è è æ b + c − Lö æ b + c − Lö æ b + c − Lö æ b + c − Lö + sin ç β ÷ coshç β ÷ + cosç β ÷ sinh ç β L L L L è è è è (III.82) By comparing the coefficient F4 with the sum of the coefficients F1 and F2 defined in Section III.2.4, it is found that Equations (III.81) and (III.58) are identical. This result 123 validates both the low frequency approximation and the identification of the spring constant and the driving force for the model of the piezoceramic stack. III.2.6. Transfer Function Model of the Low Frequency Approximation The natural frequencies of the low frequency model, ωj, are the roots of the characteristic equation (III.81). Similarly, the mode shapes associated with the dynamic part of the model can be written in a compact form as: φˆ j (x ) = φˆ 1 j (x )H(b − x ) + φˆ 2 j (x )H(x − b )H(b + c − x ) + φˆ 3 j (x )H(x − b − c ) (III.83) In the above equation, H represent the Heaviside or Step function which is one if its coefficient is positive and zero elsewhere. Further, the φ̂ ij ’s are the mode shapes of the dynamic part of the model on the ith section. Hence, these parts of the mode shapes are given by: æ xö æ xö æ xö æ xö , i = 1, 2, 3 (III.84) φˆ ij (x ) = C i1 sin ç β j ÷ + C i 2 cosç β j ÷ + C i 3 sinh ç β j ÷ + C i 4 coshç β j è L è L è L è L In the above equations, the non-dimensional coefficients, βj, are functions of the natural frequencies, ωj, and these coefficients are given by: 4 β j = L4 ω j 2 A bρb E bIb (III.85) Similarly, the coefficients C11 to C34 are the mode shapes coefficients and are given by first choosing C24 and then using recursively Equations (III.79), (III.76.a-b), 124 (III.75.a-d) and (III.73.a-d). Since the mode shapes are defined within a multiplicative constant near, for the purpose of this analysis, C24 is chosen such that the mode shapes form an orthonormal basis for the response of the system. In a manner similar to the Expression (III.83) representing a compact mode shape, the differential equation governing the dynamic part of the model can also be written compactly as: E b I b ∂ 4 ŵ ∂ 2 ŵ (t ) + 2 = Γf (x ) R d 33 c E A b ρ b ∂x 4 ∂t (III.86) In the above equation, ŵ is the dynamic part of the global displacement field of the actuated cantilever beam, which means the dynamic part of the displacement field of the cantilever beam due to the actuation of the low frequency model of the OPSA, and is given by: ŵ (x ) = ŵ 1 (x ) H(b − x ) + ŵ 2 (x ) H(x − b ) H(b + c − x ) + ŵ 3 (x ) H(x − b − c ) (III.87) Similarly, Γf (x) is the normalized spatial distribution of the actuator equivalent normal forces. This distribution is given by: cù é x − êb + ú (x − b ) 2 ë Γf (x ) = H (x − b − c ) H(x − b ) H(b + c − x ) + cù æ é c ùö é L − êb + 2c çç L − êb + ú ÷÷ 2 2 ë ë è ø 2 Finally, the non-dimensional ratio R is defined to be: 125 (III.88) R= Rs δ δ AsY s æ c ùö é çç L − êb + ÷÷ = 2 E bIb + δ2AsY Es ë è 2 E cö æ L −çb + 2 è δ (III.89) Using a modal expansion procedure, we can write the dynamic part of the global displacement field of the actuated cantilever beam, ŵ , as a function of its associated mode shapes, φ̂ j , and generalized coordinates, η̂ j : ŵ = ∞ j=1 φˆ j ηˆ j (III.90) Using the above definition into the global differential equation (III.86), we obtain ∞ E bIb A bρb j=1 d 4 φˆ j dx 4 ηˆ j + ∞ j=1 ˆ = Γ (x ) R d c E (t ) φˆ j η j f 33 (III.91) Since the mode shapes satisfy the homogeneous spatial differential equation, we have: 4 E b I b d φˆ j 2 = ω j φˆ j A b ρ b dx 4 (III.92) Hence, the differential equation (III.91) can be rewritten as: ∞ j=1 ( ) 2 (t ) φˆ j ηˆ j + ω j ηˆ j = Γf (x ) R d 33 c E (III.93) 126 Since the mode shapes have been chosen to form an orthonormal basis for the dynamic displacement field of the actuated beam, the differential equation (III.93) can be decoupled into an infinite set of single degree of freedom ordinary differential equations: ˆ + ω 2 ηˆ = Γ R d c E (t ) , j = 1, …, ∞ η j j j fj 33 (III.94) In the above equation, the Γf j ’s are the normalized modal influence coefficients of the actuation provided by the low frequency model of the offset piezoceramic stack actuator. These influence coefficients are given by: L Γf j = Γf (x )φˆ j (x )dx (III.95) 0 For the purpose of this analysis and without loss of generality, let us assume that at the initial time t = 0, both the driving electric field, E(t), and the generalized coordinates, η̂ j , are zero. Taking the Laplace transform of Equation (III.94) and substituting jω for the Laplace coordinate, we obtain: (− ω 2 ) 2 + ω j ηˆ j = −ω2 Γf j R d 33 c E ( jω) , j = 1, …, ∞ (III.96) Then, the modal coordinates η̂j are given by: ηˆ j = − ω 2 Γf j − ω2 + ω j 2 R d 33 c E ( jω) (III.97) 127 Using this result in the modal expansion definition (III.90), the frequency response function between the dynamic part of the displacement field of the actuated beam and the input electric field to the piezoceramic stack is given by: ŵ (x, jω) = E ∞ j=1 − ω2 φˆ j (x ) Γf j − ω2 + ω j 2 (III.98) R d 33 c As for the mode shapes and the dynamic part of the displacement field, a global formulation of the quasi-static deflection of the actuated beam can be written as: w 0 (x ) = w 10 (x ) H(b − x ) + w 20 (x ) H(x − b ) H(b + c − x ) + w 30 (x ) H(x − b − c ) (III.99) Using the definition of the quasi-static deflection on each of the section of the beam from Equations (III.67.a-c), we have: w 0 (x , t ) = −Γf (x ) R d 33 c E(t ) (III.100) In Equation (III.100), the normalized spatial distribution Γf (x) and the nondimensional ratio R are defined in Equations (III.88) and (III.89), respectively. In order to obtain a partial expansion form for the total displacement of the actuated beam, the quasi-static displacement field is projected on the mode shapes associated with the dynamic part of the displacement field. However, the mode shapes φ̂ j form an orthonormal basis for the dynamic part of the displacement of the beam. Since the boundary conditions for the dynamic and quasi-static displacement field are not the 128 same, the difference between the projection and the actual static displacement fields will result in a residue, H0(x)E(t). Hence, we can write the quasi-static displacement as: w0 = ∞ j=1 φˆ jη0 j + H 0 (x ) E(t ) = −Γf (x ) R d 33 c E(t ) (III.101) In the above equation, η0j are the quasi-static modal displacements and are given by: η0 j = − Γf j R d 33 c E(t ) (III.102) Again, the Γf j ’s are the normalized modal influence coefficients of the actuation provided by the low frequency model of the offset piezoceramic stack actuator and are given in Equation (III.95). Hence, the frequency response function between the quasi-static part of the displacement field of the actuated beam and the input electric field to the piezoceramic stack is given by: w0 ( x , jω ) = − E ∞ j=1 φˆ j (x ) Γf j R d 33 c + H 0 (x ) (III.103) Since the total displacement field of the actuated cantilever beam is the sum of its quasi-static and dynamic parts, the frequency response function between the displacement at any point within the beam and the input electric field to the piezoceramic stack is given by: 129 é ù w (x, jω) = ê ŵ + w 0 (x, jω) = E ëE E ∞ j=1 2 − ω j φˆ j (x ) Γf j − ω2 + ω j 2 R d 33 c + H 0 (x ) (III.104) The electric field within the stack, E(t), is given by the ratio of the input voltage to the stack, V(t), and the typical thickness of each layer within the stack, ts,: E(t ) = V(t ) ts (III.105) Furthermore, the length of the stack, c, divided by the typical thickness of each layer, ts, is equal to the number of layers within the stack, ns,: ns = c ts (III.106) Using Equations (III.105) and (III.106), a frequency response function, which is more useful for control application and the transverse displacement field of the actuated cantilever beam to the input voltage to the stack is obtained as follows: w (x, jω) = V ∞ j=1 2 − ω j φˆ j (x ) Γf j − ω + ωj 2 2 ~ R d 33 n s + H 0 (x ) (III.107) In order to update this model with damping, modal damping can be extracted from experimental data. In this case, the equations governing the behavior of the generalized coordinates of the dynamic part of the displacement field of the actuated beam becomes: (− ω 2 ) 2 + 2 ζ j ω j ω j + ω j ηˆ j = −ω2 Γf j R d 33 c E ( jω) , j = 1, …, ∞ 130 (III.108) Hence the transfer function between the total displacement of the actuated beam, at any coordinate x, and the input voltage to the piezoceramic stack actuator becomes: w (x , jω) = V 2 ∞ − 2ζ j ωj ω j − ωj j=1 − ω + 2ζ j ωj ω j + ωj 2 2 ~ φˆ j (x ) Γf j R d 33 n s + H 0 (x ) (III.109) Since the boundary conditions of the quasi-static part and dynamic part of the problem are only different at the location of the two connections between the beam and ~ the piezoceramic stack, the residue, H 0 , will be negligible for almost every sensor location but in the immediate vicinity of the actuator. As indicated by the boundary conditions, the actuator induces moments into the structure hence is best located at maximum curvature locations. Further, in this thesis research, the sensor is an accelerometer, which should be located at maximum displacement locations. Therefore, for a cantilever beam, the sensor should be located at the tip of the beam, where the boundary conditions of the quasi-static and dynamic parts of the problem agree. Hence, without loss of generality, for a cantilever beam, the residue arising form the difference between the projection of the quasi-static displacement onto the dynamic mode shape and the actual quasi-static displacement can be neglected. Hence, the transfer function between an accelerometer sensor located at the tip of the beam and the low frequency approximation of the OPSA is: a (L, jω) = V ∞ j=1 ( ω2 2 ζ j ω j ω j + ω j 2 ) − ω2 + 2 ζ j ω j ω j + ω j 2 φˆ j (L ) Γf j R d 33 n s 131 (III.110) III.2.7. Numerical Validation of the Model Approximations To validate both the low frequency approximation and the approximation made by neglecting the residue in the transfer function model for a sensor located at the free tip of the beam, a numerical analysis of an actuated steel cantilever beam was developed. The geometric and material properties of the beam are given in Table III-1. Then, it was assumed than an actuator was bonded to the upper surface of the cantilever beam. The vertical and horizontal offset distances for the actuator are given in Table III-2. Table III-1 Geometric and Material Properties of the Cantilever Beam Length L 609.6 mm Width bb 50.8 mm Thickness hb 12.7 mm Cross Sectional Area Ab 6.45 10-4 m2 Moment of Inertia Ib 8.67 10-9 m4 Modulus of Elasticity Eb 200 GPa Density ρb 7860 kg/m3 132 Table III-2. Offset Distances of the OPSA Vertical Offset δ 20.5 mm Horizontal Offset b 30 mm The active element for the actuator was selected to be a piezoceramic low-voltage translator, Physik Instrumente PI-830.10 (100-V, 1-kN), which geometric and material properties are given in Table III-3. For this piezoceramic stack, its first characteristic frequency, equivalent spring stiffness and force per unit input voltage are given in Table III-4. Table III-3. Geometric and Material Properties of the Piezoceramic Stack Length c 18 mm Cross Sectional Area As 25 10-6 m2 Typical Layer Thickness ts 120 µm Number of Layers ns 150 Modulus of Elasticity YEs 55.25 GPa Density ρs 8000 kg/m3 Piezoelectric Charge Constant d33 6.35 10-10 m/V 133 Table III-4. Low Frequency Equivalent Characteristics of the Piezoceramic Stack First Characteristic Frequency fs 23.25 kHz Equivalent Spring Stiffness ks 76.75 106 N/m Induced Force per Input Voltage Fs 7.3 N/V The numerical analysis for the validation of the approximations of the actuated cantilever beam was performed using the Matlab software. The different functions that were written to perform this analysis are given in Appendix II.2. The first part of the analysis was to compare the natural frequencies derived with the continuous model obtained in Section III.2.4 with the natural frequencies derived with the low frequency spring model for the active element obtained in Section III.2.5. These frequencies are also compared to the natural frequencies of the cantilever beam in the absence of any actuator. The first seven natural frequencies are given in Table III-5. 134 Table III-5. Natural Frequencies (Hz) for the OPSA at Location 1 Continuous Model Spring Model No Actuator 28.19 28.19 27.85 175.80 175.80 174.52 490.53 490.54 488.67 958.91 958.97 957.59 1583.13 1583.32 1582.97 2364.36 2364.78 2364.68 3304.22 3304.93 3302.73 As expected, the natural frequencies of the continuous model and the low frequency approximation match almost exactly. Further, the difference between the two models tends to increase with frequency but is still of the order of 0.025% for the seventh frequency. It is also to be noted that the low frequency approximation tends to overpredict the natural frequencies. This last result is due to the fact that the inertia associated with the actuator is neglected in the low frequency model. By comparing the frequencies of the actuated beam with the frequencies of the beam by itself, it can be noticed that in general, the addition of the actuator increases slightly the natural frequencies, which means that the actuator stiffens the beam. However, for some higher modes, such as the sixth mode, the continuous model shows 135 that the addition of the actuator adds more inertia than stiffness to the beam resulting in a lower frequency. However, this result is not duplicated with the low frequency model in which the actuator has no inertia. The second approximation, which was effectuated in the model and needs validation, is the assumption that the residue arising form the difference between the projection of the quasi-static displacement onto the dynamic mode shape and the actual quasi-static displacement can be neglected for a sensor located at the free tip of the beam. Figure III.10 illustrates the normalized spatial distribution of the actuator equivalent normal forces, Γf (x), and its projection onto the dynamic mode shapes. Further, Figure III.11 shows the spanwise distribution of the normalized residue, which is the difference between the two curves illustrated in Figure III.10. As is illustrated in Figure III.10, the projection approximates the force distribution closely over most of the span but in the immediate vicinity of the actuator. As explained in Section III.2.6, this is due to the fact the out of the twelve boundary conditions for the low frequency model, only two are different between the quasi-static and dynamic part of the problem. Further, from Figure III.11, the residue magnitude tends to decrease as the longitudinal coordinate increases toward the free tip of the beam. From the same figure, the approximation made to derive Equation (III.110) is only 0.01% of the quasi-static displacement at the tip of the beam. Hence, the approximation, which consists in neglecting the residue for a sensor located at the free tip of the beam, is valid. 136 Distribution of Forces generated by the OPSA and its Projection on the First Modes 1 Initial Distribution Projection of Distribution 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 Spanwise Direction (m) 0.5 0.6 Figure III.10 Distribution of Forces generated by the OPSA and its Projection on the First ten Modes -3 6 Residue of the Projection x 10 4 2 0 -2 -4 -6 -8 -10 -12 0 0.1 0.2 0.3 0.4 Spanwise Direction (m) 0.5 0.6 0.7 Figure III.11 Spanwise Distribution of the Residue of Normalized Force Distribution 137 III.2.8. Experimental Validation of the Low Frequency Transfer Function Model In order to validate the low frequency transfer function model, an experiment was designed and carried out. The specimen for the experiment was the cantilever beam described in Table III-1. For this experiment, two actuators were bonded to the upper surface of the cantilever beam. The distance from the cantilever end of the beam to the first connection of each actuator is given in Table III-6 and the vertical offset distance for both actuators is the one given in Table III-2. Table III-6 Locations of Experimental Actuators Location of Actuator 1 b1 30 mm Location of Actuator 2 b2 122.5 mm As in Section III.2.7, the active element of each actuator is a piezoceramic lowvoltage translator, Physik Instrumente PI-830.10, which geometric and material properties are given in Table III-3 and which low frequency characteristics are given in Table III-4. For the purpose of this validation, the maximum frequency that was selected for the experiment was defined such that the first three modes of the coupled system could be identified. Since the third bending mode has a computed frequency of about 490 Hz, the Nyquist frequency was chosen to be around 610 Hz. 138 In this experiment, in order to validate Equation (III.110), a frequency response function was obtained between acceleration at the free tip of the beam and input voltage to the piezoceramic stack. The sensor was a PCB 303A03 accelerometer, located in the middle of the free tip of the cantilever beam, its signal was amplified through a PCB Signal Conditioner and filtered at the Nyquist frequency for anti-aliasing purpose before post processing. A Wavetek Signal Generator generated a random driving signal. This signal was both fed to a low voltage piezo driver and to a low pass filter set at the Nyquist frequency, again for anti-aliasing purpose before post processing. By applying the sensitivity and amplification factors to the measured anti-aliased signals, a frequency response function was obtained in g per unit input voltage to the piezoceramic stack. The experimental and modeled transfer functions are shown in Figure III.12 for the actuator placed at the first location and in Figure III.13 for the second actuator. However, these two figures display a relatively poor agreement between the low frequency model for the actuated cantilever beam and the actual experimental data. 139 Comparison of FRF between free tip acceleration and input voltage to stack 20 10 Magnitude (dB re 1g/V) 0 -10 -20 -30 -40 -50 -60 Experimental FRF Spring Model FRF -70 0 100 200 0 100 200 300 400 500 600 400 500 600 Phase (rad) 2 0 -2 300 Frequency (Hz) Figure III.12 Comparison of Cantilever Model and Experimental Transfer Functions for the First Actuator Location Comparison of FRF between free tip acceleration and input voltage to stack 20 10 Magnitude (dB re 1g/V) 0 -10 -20 -30 -40 -50 -60 -70 Experimental FRF Spring Model FRF 0 100 200 0 100 200 300 400 500 600 400 500 600 Phase (rad) 2 0 -2 300 Frequency (Hz) Figure III.13 Comparison of Cantilever Model and Experimental Transfer Functions for the Second Actuator Location 140 The main reason for the poor correlation between experimental and modeled data is due to the fact that the experimental cantilever beam is not perfectly clamped at its root. In fact, the boundary conditions associated with the root of the beam are closer to a hinge support with a stiff torsional spring. These boundary conditions should have the following form: w 1 (0, t ) = 0 EbIb (III.111.a) ∂ 2 w1 (0, t ) = C a ∂w 1 (0, t ) 2 ∂x ∂x (III.111.b) In Equation (III.111.b), Ca is the torsional stiffness of the root spring. In order to obtain a better agreement between the model and the experimental data, the problem that is defined by the differential equations (III.61) with the boundary conditions (III.111), (III.63), (III.64) and (III.65) was solved using the Maple software and the method described in Section III.2.5. This Maple solution is given in Appendix II.1.d. Based on the results of this analysis, an equation similar to Equation (III.110) is obtained with the only difference that the mode shape coefficient and natural frequencies depend on the torsional stiffness constant of the spring support of the beam. Based on these equations, a numerical analysis of the beam with the torsional spring support was performed using the Matlab software. The files associated with this analysis are given in Appendix II.2. Using this analysis, the spring stiffness of the torsional spring support was identified in such a way that the first three modeled frequencies match the experimental resonant frequencies. The torsional spring stiffness 141 was chosen to be 7 104 N.m/rad. The order of magnitude for the torsional spring stiffness is reasonable for the modeling of an imperfect cantilever beam support. A comparison of resulting modeled transfer functions and experimental frequency response functions is shown in Figure III.12 for the actuator placed at the first location and in Figure III.13 for the second actuator. With the addition of the torsional spring support at the root of the beam, the correlation between experimental and modeled frequency response functions is almost complete for the first actuator location. The remaining discrepancies between modeled and experimental frequencies can be attributed to the finite dimensions of the connections between the piezoceramic stack and the beam. The steel blocks, which are bonded on the upper surface of the beam, will change the local stiffness and inertia properties of the beam. Another factor, which will contribute to the difference, is the fact that the cantilever beam was assumed to satisfy Euler Bernoulli assumptions so that its shear and rotary inertia effects were neglected. However, these effects should not contribute to large discrepancies since the dimensions of the beam are compatible with the assumptions. 142 Comparison of FRF between free tip acceleration and input voltage to stack 20 10 Magnitude (dB re 1g/V) 0 -10 -20 -30 -40 -50 -60 -70 Experimental FRF Spring Model FRF 0 100 200 0 100 200 300 400 500 600 400 500 600 Phase (rad) 2 0 -2 300 Frequency (Hz) Figure III.14 Comparison of Torsional Spring Root Model and Experimental Transfer Functions for the First Actuator Location The peaks in the experimental transfer function which are located to the right and left of the second and third natural frequencies are most probably due to a nonlinear contact-noncontact behavior. These contact-noncontact interactions occur because, for reliability issues, the stack is only clamped along the longitudinal axis in the mounts. When the beam deflects downward, its own inertia may deflect it more than the actuation provided by the OPSA. In this case, a gap between the end of the stack and the inside of the mount sockets is created. Then, as the beam returns to its neutral position, the stack comes into hard contact with the mount. 143 Comparison of FRF between free tip acceleration and input voltage to stack 20 10 Magnitude (dB re 1g/V) 0 -10 -20 -30 -40 -50 -60 -70 Experimental FRF Spring Model FRF 0 100 200 0 100 200 300 400 500 600 400 500 600 Phase (rad) 2 0 -2 300 Frequency (Hz) Figure III.15 Comparison of Torsional Spring Root Model and Experimental Transfer Functions for the Second Actuator Location As illustrated in Figure III.15, the experimental frequency response function for the second actuator location does not match the modeled transfer function as well as for the first location. In addition to the reasons given for the first location, the differences are most probably due to added stiffness to the beam resulting from the presence of the mount of the first actuator, which was bounded on the beam when the experimental data was obtained. The presence of the mount mostly results in a local stiffening of the root of the beam which would explained why the predicted magnitude is somewhat higher than the one obtained experimentally. 144 III.3. Optimization and Placement of the OPSA on the Experimental Benchmark In this thesis research, the experimental benchmark for the validation of the modal expansion based model is an Euler-Bernoulli beam with a torsional spring support on one end and free at the other end. The dimensions and material properties of the beam are given in Table III-1. Since, the modal expansion model for the low frequency actuator model acting on an the experimental benchmark has been validated experimentally, an optimization of the actuator placement and of the vertical offset distance can be derived based on this model. Given the active element of the OPSA, namely the piezoceramic stack, the objective of the optimization is to maximize the control authority of the offset piezoceramic stack actuator for the control of the experimental benchmark. From Section III.2.6, the general form of the transfer function between an accelerometer located on the beam, far from the actuator, and the input voltage to the piezoceramic stack can be written as: a (x acc , jω) = V ∞ ω 2 Γacc, i (x acc ) ΓOPSA , i i =1 − ω 2 + 2 ζ i ω i ω j + ωi (III.112) 2 In the above equation, Γacc and ΓOPSA are the modal influence coefficients of the accelerometer and offset piezoceramic actuator respectively. From Equation (III.110), we can identify both influence coefficient: Γacc, i (x acc ) = φˆ i (x acc ) ( (III.113.a) 2 ) ΓOPSA , i = 2 ζ i ωi ω j + ωi Γf i R d 33 n s (III.113.b) 145 For the accelerometer, the optimal location on a cantilever beam is the free tip. In such a case all of the Γacc, i’s are maximum. Hence, for the optimum sensor location, Equation (III.112) is exactly Equation (III.110). In order to maximize the control authority of the control system, we need to optimize the actuator authority of the OPSA. This means that we should maximize ΓOPSA for the modes that have been selected for control. If the length of the actuator is small compared to the beam length (c << L), which is the case for the experimental benchmark (c/L = 0.03), the characteristic equation can be approximated by: 1 + cos(β ) cosh (β ) + E bIb β(cos(β )sinh (β ) − sin (β )cosh (β )) = 0 Ca L (III.114) Equation (III.114) is the characteristic equation of a beam, without any actuator, with a torsional spring support on one end and free at the other end. Hence, if the length of the actuator is small compared to the beam length, we can make the following two approximations. First, the natural frequencies of the actuated beam are independent of the vertical offset distance and actuator position. Second, the mode shapes of the actuated beam are also independent of the vertical offset distance and actuator position. These two approximations combined insure that the approximation of the optimal vertical offset distance is the same for every mode and that the placement analysis can be performed on the structure without any actuator mounted. 146 III.3.1. Optimization of the Vertical Offset Distance for a Small Actuator If the length of the actuator is small compared to the beam length, the only term, which remains in the influence coefficient, ΓOPSA, that depends on the vertical offset distance, δ, is the non-dimensional ratio, R . By taking the derivative of this nondimensional ratio with respect to the vertical offset distance and setting this derivative to zero, we obtain: 2 δo A s Y E s = E b I b (III.115) The above equation means that the offset distance is optimal when the added bending stiffness due to the piezoceramic stack matches the bending stiffness of the beam alone. In order to validate the approximations, a numerical analysis of the experimental benchmark, the beam supported by a torsional spring at one end and free at the other, is performed for the first five modes. Let us define the relative efficiency of the actuator by the real part of its influence coefficients normalized for each mode by its maximum. The relative efficiency of the actuator, described in Section III.2.7, for the first five modes of the actuated benchmark structure and for different values of the vertical offset distance is presented in Figure III.16. 147 Normalized Modal Actuation versus Vertical Offset for the First Five Modes 1 0.9 Normalized Modal Actuation 0.8 0.7 0.6 0.5 0.4 <-- Approximate Optimal Vertical Offset 0.3 0.2 <-- Beam Surface 0.1 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Vertical Offset (m) 0.07 0.08 0.09 0.1 Figure III.16 Normalized Actuation Efficiency versus Vertical Offset Distance As assumed, the optimal vertical offset distance depends on the mode number only very slightly. In fact, the difference between the optimum distance between the first and fifth mode is only 0.7 mm. However, the approximate optimal vertical offset distance, given by Equation (III.115), is about 1.75 mm lower than the optimal distance for the first bending mode. This means that the approximate optimal offset will produce a slightly smaller actuation than a fully optimized value. Still, for this particular application, if an actuator were built at the computed approximate offset distance, the efficiency of the actuator would still be about 99.8%. Hence, for most applications, the approximate optimal vertical offset distance given by Equation (III.115) is within reasonable bounds. Further, if a full optimization was performed for the control of 148 selected modes, the approximate optimal value is a valid initial condition for the procedure. III.3.2. Optimal Placement of a Small Actuator The placement of the actuator is based on reciprocity theory. The principal mode of actuation, which is induced by the offset piezoceramic stack actuator, is the input of a pair of bending moments in opposite directions located at a distance b and b+c from the root of the beam. The objective of the placement optimization is to obtain maximum transverse acceleration at the sensor location, the free tip of the benchmark structure, due to the pair of moments. By reciprocity theory, this optimization is equivalent to finding the locations with maximum curvature due to an input of transverse disturbance force located at the sensor position. As discussed before, if the length of the actuator is small compared to the beam length, the placement analysis can be performed on the structure without any actuator mounted. In this case, using modal expansion, the curvature response of the beam, Κ(x,t), can be written in terms of its curvature mode shapes, κi(x), and generalized coordinates, ηi(t), as follows: Κ (x , t ) = ∞ i =1 κ i (x ) ηi (t ) (III.116) One method to obtain the optimal actuator location is to replace the generalized coordinates by weights on the curvature mode shapes, Wi, which are chosen such that the 149 modes to be controlled by a given actuator are the principal component of the sum. Then, the objective is to obtain the location, xo, of maximal total weighted curvature: ∞ ì ü ( ) x o = íx : K x o = max κ i (x ) Wi ý x i =1 î (III.117) In order to validate the reciprocity approach for the optimal position of the offset piezoceramic stack actuator placement, a numerical analysis of the benchmark structure is performed. This analysis is designed to validate the small actuator approximation and the fact that the optimal vertical offset distance does not depend on actuator location. In this analysis, the influence coefficient of the OPSA is computed at different location on the beam and for different values of the vertical offset. Then, these coefficients are normalized with respect to the maximum value on the set. The comparison of these normalized efficiency coefficients, in percent, with the curvature modes of the first three modes of the benchmark structure without actuator mounted are illustrated in Figure III.17, Figure III.18 and Figure III.19. 150 Placement and Vertical Offset for 1st Mode 0.055 50 80 70 0.045 0.04 90 Vertical Offset (m) 30 0.05 0.035 0.03 0.025 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Span 0.7 0.8 0.9 1 Curvature 1 0.5 0 Figure III.17 Placement and Vertical Offset Optimization Curve for the First Bending Mode of the Benchmark Structure Placement and Vertical Offset for 2nd Mode 0.055 50 70 30 0 80 50 80 0.04 70 Vertical Offset (m) 30 -95 0.045 -30 -50 -70 -80 -90 0.05 0.035 0.03 0.025 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Span 0.7 0.8 0.9 1 Curvature 1 0 -1 Figure III.18 Placement and Vertical Offset Optimization Curve for the Second Bending Mode of the Benchmark Structure 151 30 50 70 80 90 95 -80 -70 -80 -50 Vertical Offset (m) 30 50 90 95 -80 0.035 -30 -50 0.04 70 0 0.045 80 50 80 70 0.05 -70 30 0 -30 Placement and Vertical Offset for 3rd Mode 0.055 0.03 0.025 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Span 0.7 0.8 0.9 1 Curvature 1 0 -1 Figure III.19 Placement and Vertical Offset Optimization Curve for the Third Bending Mode of the Benchmark Structure All three figures indicate that the optimal locations for the offset piezoceramic stack actuator are associated with maximum curvature of the benchmark structure in the absence of actuator. Hence, both the reciprocity approach and the small actuator assumption are validated. Further, the horizontal line representing the approximate optimal vertical offset distance for the actuator is very close to the actual optimal offset for all three modes at every location. 152 CHAPTER IV. EXPERIMENTAL VALIDATIONS In this chapter, the developed buffet alleviation system is validated by two different experiments. The first experiment validates the proposed active buffet alleviation system during wind tunnel tests. For these wind tunnel experiments, a 1/16thscale model of the F-15 aircraft is used. The empennage of this model is aeroelastically scaled in such a way that the buffet flow characteristic reduced frequency is conserved and the structural reduced frequencies match the ones of the full-scale aircraft. The control system is tested at the worse angle of attack and at a wind tunnel speed that is approximately scaled to represent a full-scale Mach number of 0.6 at an altitude of 20,000-ft. Further tests are conducted at free stream velocities and angle of attacks in the neighborhood of the worse operating conditions. With these tests, both effectiveness and robustness of the control system are tested. The second experiment is performed to study if a combination of offset piezoceramic stack actuators and acceleration feedback control can suppress vibrations in a full-scale vertical tail sub-assembly that is supported in such a way that the dynamics of the complete aircraft are reasonably simulated. This subassembly consists of the left half empennage of an HPTTA which includes the vertical tail, the horizontal stabilator and the rear engine mount. 153 IV.1. Control System Design Procedure based on Experimental Data Because of the non-availability of reliable mathematical or numerical models for both the controlled structure and the buffet induced loads, the first task of the experimental validation study was to derive a control system design methodology that is based on the use of experimental data only. This design procedure includes the development of methods for the placement of the sensor and the actuators; for the characterization of the plant; for the design of the acceleration feedback controller parameters in such a way that the closed loop control signal is not saturated; and, finally, for the controller implementation using a Digital Signal Processor (DSP). IV.1.1. Control System Objective The first step of any control system design procedure is to identify its objectives. These control system objectives usually arise from a previously defined problem. In the particular case of the buffet induced vibrations of the vertical tails of an HPTTA, wind tunnel tests[52] have shown that only the first few modes of the vertical tails are excited by the buffet loads and should be suppressed in order to increase the fatigue life of these assemblies. Therefore, on the basis of observed fatigue failures in the structure, the control system objective is to suppress the major modes that lie below 100 hertz for the full-scale aircraft. 154 IV.1.2. Sensor Placement The second step of the active vibration control system design methodology is the choice of the control sensor location. From our choice of acceleration feedback controller, the type of sensor is determined to be an accelerometer. The location of the control sensor is dictated by the fact that all modes within the control bandwidth should be observable. These modes should include the controlled modes as well as the uncontrolled modes so that when the stability is checked using data from the sensor response, the stability is guaranteed for the entire control bandwidth. An additional constraint on the sensor location is given by the fact that the sensor should be placed such that its signal to noise ratio is as large as possible. Assuming that the measurement noise is added to the true sensor signal and that this noise spectrum is flat in the control bandwidth, maximizing the signal to noise ratio is equivalent to maximize the true sensor signal. In the particular case where closely spaced modes are present in the system, these modes should be observed in phase by the sensor when actuated by the control system actuators. Hence, the sensor location may be slightly altered after the actuator locations have been chosen. To experimentally determine the sensor location, an experimental modal analysis of the system to be controlled should first be performed using accelerometers. The result of this experimental modal analysis will be the natural frequencies, ωi, the damping ratios, ζi, and the orthonormal mode shapes, ϕi(pj), of the system to be controlled. If a mass matrix of the system is available, the mode shapes should be mass normalized for 155 later use in the plant model derivation. Using modal expansion theory, the acceleration response of the system at any location pj can be written as: a( p j , t ) = ∞ i ( t ) ϕ i ( p j )η (IV.1) i =1 In Equation (IV.1), the ηi’s are the modal, or generalized, coordinates of the structural system. To select the sensor location, the modal coordinates can be replaced by scalar factors, or weights, which are defined to be zero for modes outside of the selected control bandwidth. The magnitudes and signs of these weights should be chosen such that the control system objectives are satisfied. Hence, if the control bandwidth includes N modes, N weights are chosen, A1,…,AN, and the sensor location ps can be chosen such that the weighted modal response is maximum: é a ( p k ) = max ê pj ëê N ϕ i (p j )A i ù (IV.2) i =1 At this defined pk location, an accelerometer is mounted on the system. IV.1.3. Operating Vibration Magnitude for Control Authority Studies The choice of using acceleration feedback control with either the crossover design or the H2 optimal design, as described in Sections II.2.3 and II.2.4, enable us to design controllers without an accurate model of the disturbance forces. For the crossover design of acceleration feedback controller this is due to the fact that, as discussed in Section 156 II.2.3, none of the equations used to compute the controller parameters depends on the load influence matrix. For the H2 optimal design of acceleration feedback controllers this is due to the fact that, as discussed in Section II.2.2 and II.2.4, the input disturbance matrix used for the controller parameter computations can be chosen to be independent of the load distribution. However, to size the number of actuators needed to satisfy the control system objectives and to define the gains of the controllers, a representation of the disturbance is needed. In cases where the operating external loads of the system cannot be modeled accurately, a different procedure to obtain a representation of the disturbance should be used. For control authority computations, the auto-power spectrum of the acceleration, at the sensor location, during operating conditions can be used to assess the magnitude and frequency content of the external loads. Hence, an experimental representation of the acceleration at the sensor location, af(t), due to the disturbance at the operating conditions, f(t), is obtained in the form of an auto-power spectrum, PAfAf : [ * ] PA f A f ( ω) = E A f ( ω) A f ( ω) = lim n d →∞ 1 nd nd p =1 * A f p ( ω) A f p ( ω) (IV.3) In Equation (IV.3), Af(ω) is the Fourier transform, for the computation the Fast Fourier Transform (FFT), of the measured acceleration af(t). 157 IV.1.4. Actuator Placement The fourth step of the control system design procedure is the placement of the actuator arrays. An actuator array is defined to be a series of actuators that will be driven by the same control signal. The number of actuators making an actuator array is defined by the required control authority to satisfy the control system objectives. Depending on the types of mode to be controlled a single or multiple actuator arrays may be needed. The principle of the actuator placement procedure is based on reciprocity theory as illustrated in Figure IV.1. Reciprocal Theorem Curvature Modal Survey Control Actuation Figure IV.1 Principles of Actuator Placement The optimization is performed such that local moments, which are induced by the offset piezoceramic stack actuators, produce maximum response from the control sensor. 158 As discussed in Section III.3.2, using reciprocity theory, the actuators should be placed at the locations and in the directions where the curvature responses due to a disturbance force at the sensor location, in the measured direction, are maximum. The initial choices for the placement of the actuator arrays result directly from the type of actuators as well as from the type of modes to be controlled. These initial guesses for the locations of the actuators can be obtained from the results of the experimental modal analysis performed for the sensor placement in Section IV.1.2. For moment inducing actuators, the optimal actuator locations will be associated with the locations that have maximum curvature for the modes that these actuators will control. Since the actuators will be bonded on the surface of the controlled structure, a local two-dimensional approximation of the system is assumed. This approximation means that, at each actuator location, the structure is supposed to behave like a twodimensional system. Using the mode shapes defined at the points pj, ϕi(pj), obtained for the sensor placement and assuming that the two-dimensional approximation of the structure lies in the xz-plane, we can numerically obtain the curvature modes, ϕxx,i(pj), ϕzz,i(pj) and ϕxz,i(pj). Then, the initial locations and directions can be identified such that the weighted modal curvature response is large. Hence, for a specified set of modal weights, Ci, we want to obtain the point pk and direction angle θk with respect to the elastic axis such that: é κ( p k , θ k ) = max êê p j ,θm ê ë N i =1 [ cos 2 (θ m ) sin ( θ m ) 2 ì ϕ xx ,i ( p j ) ü ù ï ï 2 cos( θ m ) sin ( θ m ) í ϕ zz,i ( p j ) ý C i ï ϕ (p ) ï î xz ,i j ] 159 (IV.4) Once initial placements are identified, a curvature modal survey should be performed in the neighborhood of these locations to refine the placement. The curvature modal survey is an experimental modal analysis in which, instead of measuring acceleration, curvature is measured in three directions, due to the local two-dimensional approximation of the structure, at each point. This analysis is performed by disturbing the structure at the control sensor location and in the measurement direction. Then, the response from a curvature sensor, such as a PVDF film, is measured in the immediate area of each candidate actuator location in three different directions. The thickness of the PVDF film transducers, which are used in this research work, range from 20 to 100 µm. Thus, in practice, the sensor thickness can be neglected with respect to the thickness of the structure without introducing significant amount of error. With the assumptions that all in-plane strains are negligible, we have a general chargedeformation relation for a two dimensional structure as[53]: æ ∂2w ∂2w ∂2w ö ç q(t ) = −Gh ç e 31 2 + e 32 2 + 2e 36 dxdz ∂x∂z ∂x ∂z è S (IV.5) In equation (IV.5), q is the sensor signal input to the A/D converter; G is the gain associated with the electronic circuit, such as a charge amplifier; h is the distance form the sensor surface to the structure local neutral plane; S is the sensor area; w is the flexural displacement of the structure; eij’s are the piezoelectric voltage coefficients of the PVDF film. 160 For PVDF, e36 is usually negligibly small. Further, in most cases, the PVDF piezoelectric voltage coefficients are tailored such that e31>>e32. The dimensions of the PVDF film transducer, which shows a large aspect ratio, reinforce this effect. Hence, the sensitivity of the PVDF film sensor is predominant in the 1-direction. Thus, equation (IV.5) can be simplified as follows: q(t ) = −GhSe 31 κ (IV.6) In the above equation, κ is the mean local curvature in the sensor principal direction, the 1-direction. This makes the PVDF film transducer a uniaxial sensor. As discussed at the earlier in the section, based on the local two-dimensional approximation of the controlled structure at the actuator locations, the curvature modal survey should be performed at three different angles with respect to the elastic axis of the system. As for Equation (IV.4), for the optimal actuator location, pk, and direction, θk, we have: é κ( p k , θ k ) = max êê p j ,θm ê ë N [ cos 2 (θ m ) sin ( θ m ) 2 i =1 ì ϕ xx ,i (p j ) ü ù ï ï 2 cos( θ m ) sin ( θ m ) í ϕ zz,i (p j ) ý C i (IV.7) ï ϕ (p ) ï î xz ,i j ] Where the three identified curvatures, ϕxx, ϕzz and ϕxz, result from the following relation: ì ϕ xx ,i (p j ) ü é cos 2 ( θ1 ) sin 2 ( θ1 ) 2 cos( θ1 )sin ( θ1 ) ï ê ï 2 2 í ϕ zz ,i (p j ) ý = ê cos ( θ 2 ) sin ( θ 2 ) 2 cos( θ 2 )sin ( θ 2 ) ï ϕ (p ) ï ê cos 2 ( θ ) sin 2 ( θ ) 2 cos( θ )sin ( θ ) 3 3 3 3 î xz ,i j ë 161 ù ú ú ú û −1 ì κ i (p j , θ1 ) ü ï ï í κ i (p j , θ 2 ) ý ï κ (p , θ ) î i j 3 (IV.8) Once the actuator arrays locations are selected, the actuator assemblies are bonded to the structure using structural adhesives. IV.1.5. Actuator Authority Study The next step of the control system design process is to assess if the actuators can provide enough authority to suppress the vibrations of the controlled modes. This assessment can be performed by obtaining experimental frequency response functions, HAvVi between the input voltage to the actuators, vi(t), and the acceleration response at the sensor location, avi(t). This frequency response function is obtained using the formula developed by Vold et al.[54]: H A v Vi = PA v Vi PA v A v PA v Vi PVi Vi (IV.9) In the above equation, PAiAi and PViVi are the auto-power spectrums of the acceleration response and input voltage to the ith actuator respectively while PAiVi is the cross-power spectrum of these two signals. It is to be noted that these frequency response functions include the dynamics of the actuators if there are any in the control bandwidth. Then, by assuming an input voltage with a constant spectrum, PVmax, close to the maximum possible voltage input to the actuators, an equivalent auto-power spectrum is computed for the acceleration, at the ~ sensor location, PA v A v max , due to each actuator arrays: 162 ~ PA v A v max = H A v Vi 2 (IV.10) PV max For each actuator array, the magnitudes of the modes, that will be controlled with this array, are compared with the magnitudes of the equivalent modes from the autopower spectrum obtained in Section IV.1.3, PAfAf. If the magnitudes are of the same order, the control procedure can continue, else additional actuators should be bonded to the structure. The number of additional actuators for the kth actuator array can be estimated by inspection of the following ratio: N act ,k = max ωi ∈ Ω k PAf A f ( ωi ) H A v Vi ( ωi ) PV max 2 , Ω k = {ωi controlled by actuator array k} (IV.11) IV.1.6. Plant Modeling Once the sensor and actuators have been placed, a “plant” characterization step should be performed to extract the system parameters needed for the design of the controller. The “plant” is defined to be the mathematical representation of the controlled system. For acceleration feedback control the plant parameters can be derived from the transfer function matrix between the voltage response of the sensor, vs(t), and the voltage input to each actuator array, vi(t). These modeled plant parameters can be obtained by fitting the frequency response functions, HVsVi, obtained experimentally during the control authority assessment step in a partial fraction expansion form: 163 N H VsVi (ω) = k =1 − A ik ω2 − ω 2 + 2ζ ik ωik jω + ωik (IV.12) 2 In this equation, Aik is the modal amplitude, ζik is the modal damping ratio and ωik is the natural frequency of mode k when the system is excited by the ith actuator array. As a first step, the fitting of each frequency response function is performed using a least square single degree of freedom fit around each of the modal frequency ωk: H Vs Vi ({ω}k ) = B ik {ω}ik j {ω}ik − σ ik 2 (IV.13) In the above equation, {ω}ik is a set of frequencies centered around the modal frequency ωik, Bik is the amplitude coefficient and σik is the pole associated with mode k. Hence, the natural frequency, ωik modal damping ratio, ζik, and modal amplitude, Aik, of mode k, can be extracted as: ωik = σ ik ζ ik = − (IV.14.a) Re( σ ik ) ωik ( A ik = 2 Re B ik σ ik (IV.14.b) * ) (IV.14.c) During the early stage of this research, it was found that this method yields relatively accurate results for ζik and ωik. However, it was also found that the Aik’s are 164 usually not accurate enough. As a result, the Aik’s are obtained by circle fitting data in the Nyquist plane around each natural frequency ωik. ( A ik = 2d ik ζ ik sign ∠ H Vs Vi (ω k ) ) (IV.15) In Equation (IV.15), dik is the diameter of the Nyquist circle associated with the kth mode. As described in Sections II.2.3 and II.2.4, for multi-degree of freedom optimization, a state space model of the plant is necessary. This state space model, which correspond to Equations (II.61.a-b) with zero disturbance force, has the following form: ì ωs ù é 0 ù é 0 {X}+ ê ~ {v i } X =ê ú ëΓact ë − ωs − Λ s í ~ ~ ~ ~ îv s = − Γacc ωs − Γacc Λ s {X} + Γacc Γact {v i } {} [ ] [ (IV.16) ] In the above equations, the modal influence row vector of the accelerometer can computed by multiplying the row of the mass normalized mode shape matrix, obtained in Section IV.1.2, associated with the sensor location by the sensitivity coefficient of the accelerometer a2. On the other hand, the modal influence vector of the actuator can be identified from the modal residues computed from Equation (IV.15) by computing each coefficient ~ ~ independently using Γact ,ik = A ik / Γacc,i . 165 IV.1.7. Controller Design and Stability Assessment Once the plant parameters are identified, a controller is designed for each actuator array. As a first step, each controller is designed by implementing in parallel single degrees of freedom acceleration feedback control compensators. Each single degree of freedom compensators is designed following either the crossover point approach, described in Section II.1.2, or an H2 optimal approach, described in Section II.1.3 and II.1.4. Regardless of the choice of the design approach, each single degree of freedom compensator is defined by the choice of a single design parameter. Usually, this parameter is chosen to be the gain of the compensator. For stability, this gain is chosen such that the product γikAik is positive since the modal amplitude, Aik, is the product of the modal influence coefficient of the kth actuator array and sensor ~ ~ respectively, A ik = Γact ,ik Γacc,i . The second step of the control design consists in optimizing the controller for multi-degree of freedom systems using the algorithms described in Sections II.2.3 and II.2.4 for the crossover and H2 optimal designs respectively. For the crossover design approach to acceleration feedback control, this optimization step is performed to insure that the controller operates at crossover for multi-degree of freedom environment. And, for the H2 optimal approach to the design of acceleration feedback controller, this optimization insure that the multi-degree of freedom system criterion is minimum. Then, for each controller, the compensators are implemented in parallel. Hence, the controller 166 driving the kth actuator array is given by the transfer function between the input voltage to that actuator array, vk(t), and the voltage from the control sensor, vs(t): − γ cik ωcik G ck = i 2 − ω2 + 2ζ cik ωcik jω + ωcik (IV.17) 2 In this work, all the considered control schemes are based on multi-input singleoutput (MISO) systems. In this case, let us consider a performance sensor response, yp(t), located at point p and an excitation force, fk(t), driving the system at point k. For the MISO system, the closed loop transfer function between yp and fk is given by: ( G pk + H ypf k = G ci G pu i G sk − G su i G pk ) i 1− (IV.18) G ci G su i i Where Gpk and Gpui are the open loop transfer functions between the performance sensor and the excitation force and the input voltage to the ith actuator array respectively. Gsk and Gsui are the open loop transfer functions between the control sensor and the excitation force and the input voltage to the ith actuator array respectively. Finally, Gci is the controller driving the ith actuator array. A sufficient condition for the stability of the closed loop system is that the poles of the closed loop transfer function between yp and fk have negative real parts for any performance sensor and disturbance locations on the system. The poles of the closed loop transfer function are the roots of the denominator of Equation (IV.18): 167 ( ) æ den H y p f k = denç G pk + ç è ö æ G ci G pu i G sk − G su i G pk ÷ numç 1 − ÷ ç è ( i ) G ci G su i ö (IV.19) i The first term of the right-hand side of Equation (IV.19) does not depend on the performance sensor location, p. Hence the equation can be simplified by collocating the performance and control sensors. The characteristic equation is then given by: æ 0 = den ( G sk ) numç 1 − ç è G ci G su i ö (IV.20) i The above equation is also independent of the disturbance excitation location, k, so that we finally obtain, using the kth actuator array for disturbance: ìï 0 = den G su k í ïî ( ) ∏ ( [ num( G ) den G ci G su i − i ci G su i )∏ den( G cj G su j ü )] ïý (IV.21) j≠ i i The closed loop stability is assessed by computing the roots of Equation (IV.21) before implementing the controllers. IV.1.8. Maximum Control Voltage Assessment Before implementing the controller, it is necessary to check if any saturation of the control signal will occur. In fact, it is important to insure that the control voltage is not larger than either the maximum allowable voltage for driving the actuators or the maximum that can be generated by the digital to analog converter of the controller, whichever is smaller. 168 The first bound for the control voltage is related with the reliability of the control scheme. If the control voltage was larger than the largest allowable voltage input to the piezoceramic stack, depoling of the actuator could occur. However, specialized equipment such as the PI piezo driver will saturate below the maximum voltage allowable for the PI PZT stacks. Hence, the equipment will insure the reliability of the control scheme. The second bound is associated with both the saturation voltages for the D/A converter and for the piezo driver. If the control signal exceeds either of the saturation voltages, the signal will be clipped and hence new unwanted frequencies will be added to the control signal. These frequencies may cause instabilities and the controller will not exactly perform as designed. The maximum control voltage assessment is performed by generating auto-power spectrum of the control signals and computing an upper bound for the Root Mean Square (RMS) value of each control voltage. Since the maximum value of the control signal will be attained during the first instant of the control, when the open loop sensor signal drives the controller, the auto-power spectrum of the voltage of the accelerometer during operating conditions, PVsfVsf, can be used to drive the controller: RMS Vk = 1 PV V = N k k 1 G ck N 2 PVsf Vsf (IV.22) In Equation (IV.22), N is the number of spectral rays that are present in the power spectrum. The expected RMS voltage for each controller should lie below the maximum 169 voltage allowable. It is to be noted that this estimate is very conservative because the closed loop magnitude of the peaks of the sensor voltage spectrum will be lower than their open loop values once the controlled system reaches steady state. Thus, if the computed RMS voltages are somewhat higher than the maximum allowable voltage, the control signal might not be clipped when the steady state of the closed loop system is reached. IV.1.9. Controller Implementation Once the controller has been designed and has been checked for both stability and maximum control voltage, it is implemented using a digital signal processor (DSP). The computer program for the controller is written using the block programming capability of Simulink. Then, using the Real Time Workshop Toolbox of Matlab with the Real Time Interface of dSPACE, the code is converted to C language and then to the DSP language. In order to perform the time integration required by the controller, several methods are available in the Real Time Workshop Toolbox. It has been found that any algorithm with accuracy lower than the one offered by a fifth order Runge-Kutta algorithm will not produce satisfactory results and may even lead to instabilities. Hence, this algorithm is chosen to perform the time integration. To avoid any aliasing problem with the controller and to avoid any phase shift that would be introduced by an anti-aliasing filter, the controller is set to run at a frequency much higher than the highest frequency in the control bandwidth. Typically, 170 setting the controller to run at 10 times the highest frequency of interest will result in a good performance. IV.1.10. Control System Design Procedure based on Experimental Data Pseudo Algorithm In summary, the control design procedure can be reduced to the pseudo algorithm illustrated in Figure IV.2. Identify Modes to be Suppressed Sensor Placement Operating Vibrations Magnitude at Sensor Location Actuator Arrays Placement Actuators Authority Assessment no yes Plant Characterization Controller Design and Stability Assessment Maximum Control Voltage Assessment no yes Controller Implementation Control Validation Figure IV.2 Pseudo-Algorithm of Control System Design Procedure 171 IV.2. Wind Tunnel Tests for Active Tail Buffet Alleviation The main objective of the wind tunnel tests is to demonstrate our ability to suppress all the principal modes of the HPTTA vertical tails that participate in the buffet response at high angles of attack. In addition, the control system should not only perform well at the angle of attack corresponding to the worst buffet conditions but also over a wide range of angles of attack and free stream dynamic pressure. Hence, to validate the controller, three different experiments are conducted. First, a control experiment is run at the predetermined worse buffet condition of 20 degrees angle of attack and 9 psf of free stream dynamic pressure. Then, for a free stream dynamic pressure of 9 psf, the angle of attack is varied from 0 to 23 degrees. Finally, four different angles of attack are selected and the free stream dynamic pressure is varied from 5 to 13 psf. IV.2.1. Wind Tunnel Facilities The wind tunnel tests are performed at the Georgia Tech Research Institute Model Test Facility (GTRI-MTF). The GTRI-MTF wind tunnel is a closed-return, atmospheric, low-speed wind tunnel that has a rectangular test section 30 inches high and 43 inches wide with a usable length of 90 inches. This facility is capable of empty tunnel speeds of up to 200 ft/sec and corrected maximum dynamic pressures of 50 psf. During the tests wall corrections are not considered for the following reasons. The vortices, which are responsible for the buffet, are formed near the junction of the fuselage, the engine inlets 172 and wing leading edges. Furthermore, the tail sub-assembly, where the measurement of buffet loads are obtained, are near the center of the tunnel cross section. Because blockage correction factors for separated flow are not available, the free stream dynamic pressure is corrected by using approximate correction factors that are obtained from flows at low angles of attack. IV.2.2. Aeroelastically Scaled Empennage Design Some initial tests in the wind tunnel showed that there exists a characteristic frequency associated with the maximum buffet load. As illustrated in Figure IV.3, for our 1/16th scaled model at α=22° and q∞=7psf, this frequency is around 53 Hz. Based on a characteristic length given by the full-scale fin mean aerodynamic chord (81 inches) scaled to 1/16th and a free stream velocity of 23.75 m/s (equivalent to a dynamic pressure of 7 psf at 1000 ft altitude), the reduced frequency associated with that maximum buffet load is 0.29. Where the reduced frequency, r, is defined to be as a function of the fin mean aerodynamic chord, c, the buffet frequency, f, and the free stream velocity, U∞: r= c×f U∞ (IV.23) The result of the MacAir experiments[52] reported by Triplett on a 13% scaled model shows a frequency associated with the maximum buffet load around 30 Hz at a free stream dynamic pressure of 12 psf. By scaling the full-scale fin mean aerodynamic chord and using a free stream velocity of 30.94 m/s (equivalent to a free stream dynamic 173 pressure of 7 psf at 600 ft altitude), we obtain a reduced frequency associated with the maximum buffet load of about 0.26. Furthermore, the results of some earlier Georgia Institute of Technology tests reported by Komerath et al.[6] indicated a frequency associated with the maximum buffet load of about 150 Hz for a 1/32th scale model at a free stream velocity of 30.48 m/s and about 13 Hz for a 1/7th scale model at a free stream velocity of 12.19 m/s. By scaling the full-scale fin mean aerodynamic chord, the reduced frequency associated with the maximum buffet load for both the experiments is about 0.31. Pressure on Outboard Trailing Edge Tip of the 1 to 1 Model Left Vertical Tail at α=22° and q∞=7psf -30 Auto-Power Spectrum of Buffet Load (dB) -35 -40 -45 -50 -55 -60 0 20 40 60 80 100 120 Frequency (Hz) 140 160 180 200 Figure IV.3 Auto-Power Spectrum of the Buffet Load on the Outboard Trailing Edge Tip of the 1 to 1 Scale Model Left Vertical Tail at α=22° and q∞=7psf. 174 Since the reduced frequencies associated with the maximum buffet load agree within a few percent for all the experiments, we determined that it was very important for the active tail buffet alleviation tests to scale the model such that both the aerodynamic and structural reduced frequencies are conserved. It is also important to note that these computed reduced frequencies would mean that the frequency associated with the maximum buffet load for a full-scale aircraft flying at 20,000 ft at Mach 0.6 would be around 26 Hz. This frequency does not coincide with any natural frequency of the fullscale empennage. In fact, it is located between the first bending mode around 10 Hz and the first torsion mode around 37 Hz. Figure IV.4 1/16th Scale Model of the HPTTA with Aeroelastically Scale Empennage Due to blockage effects, the Georgia Tech Research Institute low speed wind tunnel performs optimally with our 1/16th scaled model between free stream dynamic 175 pressures of 5 and 13 psf. In order to obtain the same reduced frequencies for the wind tunnel tests and the full-scale aircraft, a frequency ratio between the scaled model and the full scale aircraft is defined to be: f scaled mod el f full scale = c full scale U ∞ , scaled mod el (IV.24) c scaled mod el U ∞ , full scale Based on Equation (IV.24), to operate in this optimal range, the frequencies of the model should be between 2 and 2.5 time larger than the full-scale, which would be equivalent to running the wind tunnel with a free stream dynamic pressures in the range of 7 and 11 psf. To operate in the middle of the optimal range for the wind tunnel, the scale model of the empennage was designed to have natural frequencies 2.25 times larger than the full-scale tail subassembly. This model would then operate at a free stream dynamic pressure of 9 psf to conserve the reduced frequencies of the flow and structure. This dynamic pressure was equivalent to a free stream velocity of 26.9 m/s that translates approximately (because of compressibility effects) to a free stream velocity of 191.5 m/s for the full-scale aircraft or about Mach 0.6 at 20,000 ft. Using a finite element model with specified span-wise and chord-wise dimensions, corresponding to 1/16th of the full-scale vertical tails, we iterated over the thickness dimensions to obtain natural frequencies at a ratio of 2.25 to 1 with respect to the full-scale aircraft. The resulting scaled model of the vertical tails was built from a wet lay-up of Bondo plain weave fiberglass fabric and Bondo epoxy resin. Each vertical tail has 8 layers of fiberglass at the root and 5 layers at the tip. In addition, some carbon fiber 176 strips were added to represent the two main spars of the vertical tails. Finally, brass tubes of appropriate diameters were bonded to the tip to simulate the tip pods. This 1/16th scaled model is illustrated in Figure IV.4. IV.2.3. Sensor Placement for Active Tail Buffet Alleviation The first task of the wind tunnel tests is to place the sensor for the control experiments. The procedure developed in Section IV.1.2 is used for this task. An additional requirement for the placement of the sensor for the wind tunnel tests is that, to maintain the vortex cohesion, minimum flow disturbance due to the sensor should be obtained. Further, instead of using an experimental modal analysis, the first four modes of the aeroelastically scaled vertical tail are computed using the finite element model developed in Section IV.2.2. These modes are illustrated in Figure IV.5. From inspection of the mode shapes, the upper end of the tail seems to be the best location for our control sensor. Furthermore, to disturb the flow as little as possible, the trailing edge of the tail makes a better location. Since the first torsion mode is the most important to control and to have all the modes in phase at the trailing edge tip of the tail, we assign the weights 1, 2, -1 and -1 to the first four modes respectively. Then, as illustrated in Figure IV.6, we superpose all four modes to obtain the best candidate for the sensor location. The analysis of the weighted modal superposition shows that, as we guessed initially, the trailing edge tip of the vertical tail will maximize the signal to noise ratio for 177 the control accelerometer. Furthermore, from the mode shape analysis, we can see that all modes in the control range will be observable. As a result of this analysis, a PCB 303-A02 accelerometer was mounted on the inboard trailing edge tip of each vertical tail. 2st Mode 0.2 0.2 0.15 0.15 z (m) z (m) 1st Mode 0.1 0.1 0.05 0.05 0 0.1 0 0.1 0.2 0 y (m) 0.2 0 0.1 -0.1 0 y (m) x (m) -0.1 0 x (m) 4st Mode 0.2 0.2 0.15 0.15 z (m) z (m) 3st Mode 0.1 0.1 0.05 0.1 0.05 0 0.1 0 0.1 0.2 0 y (m) 0.2 0 0.1 -0.1 0 y (m) x (m) 0.1 -0.1 0 x (m) Figure IV.5 First Four Modes of the Vertical Tail 178 Weighted (1,2,-1,-1) Superposition of the Modes 0.2 0.18 0.16 0.14 z (m) 0.12 0.1 0.08 0.06 0.04 0.02 0 0.1 0.05 0.2 0.15 0 0.1 -0.05 y (m) 0.05 -0.1 0 x (m) Figure IV.6 Weighted (1, 2, -1, -1) Superposition of the Modes IV.2.4. Wind Tunnel Tail Buffet Response Study The goal of this phase of the tests is to determine at which attitude the worse disturbances are encountered. For this study, it is assumed that the dynamic response of the structure provides a measure of the buffet pressure on the vertical tail. This set of experiments is divided into two parts. First, a survey of the trailing edge tip acceleration is conducted for angles of attack ranging from 0 to 23 degrees. Then, a second survey of the tail response is conducted for different angles of attack and free stream dynamic pressures in the neighborhood of the worse buffet conditions. 179 The first test is performed to determine the angle of attack at which the worse disturbances are encountered. As discussed in Section IV.2.2, the experiment is run at a free stream dynamic pressure of 9 psf. The results of this experiment are illustrated in Figure IV.7. The experimental observations indicate that the angle of attack that displays the maximum tip response is about 20 degrees. This agrees with the results reported by Komerath et al.[6] during their tests. However, it does not agree exactly with the results of Triplett[52]. He reported a worse case at an angle of attack of 22 degrees. The decrease in vibrations above 20 degrees is due to the fact that the flow on the wing is completely separated and the separated flow destroys the coherence of the buffet vortices. It is also interesting to notice that the vibrations below 12 degrees of angle of attack increase slowly and almost linearly with respect to the angles of attack. However from 12 to 20 degrees the amplitude of vibrations increases by a factor of 10 and very rapidly, as shown in Figure IV.7. This is due to the buffet induced vortices that increase in strength and immerse increasing area of the vertical tail. 180 RMS Tip Acceleration versus Angle of Attack 7 Experimental Data Points Polynomial Fit of Data RMS Acceleration (g/ √Hz) 6 5 4 3 2 1 0 0 5 10 15 Angle of Attack (deg) 20 25 Figure IV.7 Root Mean Square of the Trailing Edge Tip Acceleration versus Angle of Attack at q∞=9psf The second test was performed to determine the behavior of the buffet load and buffet response in the neighborhood of the worse buffet conditions, i.e. at 20 degrees angle of attack and at a free stream dynamic pressure of 9 psf. This experiment used acceleration response data from the sensor as a measure of the exciting buffet loads. The model was set at five different angles of attack, 14, 17, 20, 21 and 23 degrees. For each angle of attack, the wind tunnel test was run at five different free stream dynamic pressures, 5, 7, 9, 11 and 13 psf, which correspond to the defined free stream velocity of 26.9 m/s with additional points in the +25% to –25% of operational free stream velocity range. From these experimental data, an envelope of the effect of buffet loads was extrapolated and is illustrated in Figure IV.8. 181 The results, illustrated in Figure IV.8, confirm that at all free stream velocities the maximum disturbance (buffet load) occurs at an angle of attack close to 20 degrees. Furthermore, the rate at which the disturbance increases is also maximum along the 20 degrees line for the different dynamic pressures. RMS Tip Acceleration versus Angle of Attack and Free Stream Dynamic Pressure 12 RMS Acceleration (g/ √Hz) 10 8 6 4 2 0 14 12 24 10 22 20 8 18 6 Free Stream Dynamic Pressure (psf) 16 4 14 Angle of Attack (deg) Figure IV.8 Envelope of RMS of the Trailing Edge Tip Acceleration versus Angle of Attack and Free Stream Dynamic Pressure (o: Experimental Data Points). As discussed in Section IV.1.3, for the actuator and control authority studies, the acceleration data of the control sensor are recorded to obtain the sensor data at the operating conditions, which for these wind tunnel tests are the worse buffet conditions, namely at an angle of attack of 20 degrees and at a free stream dynamic pressure of 9 psf. 182 The auto-power spectrum of the control sensor acceleration was then computed and is illustrated in Figure IV.9. Tip Acceleration due to Wind at α=20° and q =9psf ∞ 40 2 Auto-Power Spectrum of Acceleration (dB re 1g /Hz) 60 20 0 -20 -40 -60 0 50 100 150 Frequency (Hz) 200 250 Figure IV.9 Auto-Power Spectrum of Control Sensor under Operating Conditions IV.2.5. Actuator Placement for Active Tail Buffet Alleviation At this point, it was decided to instrument the left vertical tail with offset piezoceramic stack actuators based on PI-810.10 (180N) piezoceramic stacks. The placement of these actuators is based on the procedure developed in Section IV.1.4. From the worse buffet condition response, shown in Figure IV.9, and the finite element model developed for Section IV.2.2, the modes to be suppressed can be identified as the first and second bending modes and the first and second torsion modes of the vertical tail. To be able to optimally control both types of modes, we require two actuator arrays. 183 The first pair of actuators is located to obtain large bending actuation authority. Since the vertical tail is a tapered cantilevered structure with elastic root restraints, we can assume that maximum bending effectiveness will be obtained by placing the actuator near the root of the tail along its elastic axis. This assumption is refined by performing a local curvature modal survey, as discussed in Section IV.1.4, of the vertical tail along the elastic axis both aft and for of it. The curvature modal survey is performed by impacting the tail structure with an impact hammer, PCB Impulse Hammer, on the outboard trailing edge tip (sensor location) and recording the response of a PVDF sensor, Amp DT1-052K, located at different candidate actuator locations. The analysis of the local curvature mode shows a maximum effectiveness for the actuator along the 50% chord line at the root of the tail. The second pair of piezoceramic stack actuators was placed for the control of torsion modes. For a cantilevered structure, the torsional actuation is usually optimal when the array is centered on the elastic axis with an angle of 45 degrees with respect to it. Again, this assumption was refined by performing a local curvature modal survey of the tail above the location of the bending control array and at angles varying from 25 to 65 degrees with respect to the 50% chord line. The analysis of the results of the survey shows that a location above the first array and at an angle of 35 degrees with respect to the 50% chord line provides the best effectiveness. 184 (a) (b) Figure IV.10 Outboard(a) and Inboard(b) Views of the Experiment Vertical Tails Two offset piezoceramic stack actuators were mounted per array, one on each side of the vertical tail, to provide maximum moment actuation to the tail. As illustrated in Figure IV.10, during the wind tunnel experiments the stack based actuators were covered with masking tape both for safety during tests and to reduce the flow perturbation generated by the actuators. IV.2.6. Actuators Authority Assessment Once the actuators had been bonded to the structure, their authority was checked. It is often a good idea to perform the operating condition test, which is the worse buffet condition test, again to check that the dynamic behavior of the structure has not been 185 modified significantly by the addition of the actuator arrays. For this purpose, the model was reintroduced in the wind tunnel and the worse buffet conditions were set, as define in Section IV.2.4, to acquire new acceleration data from the control sensor. These time data are then transformed to an auto-power spectrum. Afterwards, experimental frequency response functions between the acceleration response at the trailing edge tip and the input voltage to the actuator arrays were collected. This operation was obtained by driving one actuator array with a random signal, while the other was in an open circuit state, and recording the sensor response. Then both driving random signal and sensor response were converted to the frequency domain and transformed to a frequency response function. Once this operation had been performed with the bending array, it was repeated with the torsion array. As discussed in Section IV.1.5, the frequency response functions were then converted to auto-power spectrums for a flat maximum input voltage of 7V, which correspond to the maximum input voltage to the piezo-drivers. Finally, the experimental operating condition auto-spectrum was compared to the computed auto-power spectrum for each array. The results are illustrated in Figure IV.11. 186 Actuator Athority Assessment 60 Auto-Power Spectrum of Acceleration (dB re 1g 2/Hz) 40 20 0 -20 -40 due to Bending Array at Full Input due to Torsion Array at Full Input due to Wind at α =20° and q =9psf ∞ -60 0 50 100 150 Frequency (Hz) 200 250 Figure IV.11 Actuator Authority Assessment First, it is to be noted at this point that the addition of the two pairs of actuators has strongly reduced the modal amplitude of the second bending mode as seen by the control sensor. This reduction could be attributed to two factors. First, the mode shape has changed and it is not well observable by the sensor anymore. To check this hypothesis other candidate sensor locations should be checked such as the leading edge tip or even the 50% chord tip acceleration. However, this first hypothesis proved to be incorrect, the second possibility remains. The addition of the actuator arrays has stiffened the structure, which in turn passively suppressed the second bending mode. Hence, for the following control experiments, we will concentrate on suppressing the first bending mode with the bending array and the first and second torsion modes with the torsion array. 187 Since the auto-power spectrums of the acceleration response of the control sensor due to the wind and due to the bending array are about the same, the bending array has enough control authority to suppress this particular mode. For the first torsion mode, the torsion array produces a slightly smaller response than the wind, however this differences is not large enough to require an extra actuator. Finally, for the second torsion mode, both of the actuator arrays produce a larger response than the wind hence there will not be any difficulty to suppress this mode using either of the actuator arrays. Since the torsion array has a larger authority on the second torsion mode, we will use these actuators to suppress this mode. IV.2.7. Plant Characterization The next task associated with designing a control system is to obtain a mathematical model of the system to be controlled. This model is usually referred to as a “plant”. To obtain the plant model, experimental transfer functions were obtained between the input voltage to each actuator array and the sensor response voltage. Then, as discussed in Section IV.1.6, using a combination of system identification techniques such as single pole fitting and complex circle fitting around the poles, the parameters of each of the transfer functions were extracted. Finally, each experimental frequency response function is compared to the modeled transfer function of the form: H VsVi (ω) = N k =1 − A ik ω2 B + i 2 2 jω − ω + 2ζ ik ωik jω + ωik 188 (IV.25) The second term of Equation (IV.25), Bi/jω, is added to obtain a better fit at low frequencies. Using the frequency response function for the bending array obtained previously and using the procedure described above, we get the results shown in Table IV-1 and Figure IV.12. Similarly, for the torsion array, we get the results shown Table IV-2 in and Figure IV.13. As shown by both figures, we have a good agreement between the modeled and experimental transfer functions. Hence, the plant parameters are considered to be acceptable. Table IV-1 Plant Parameters for the Bending Actuator Array Mode VT: Vert. Tail HT: Horiz. Tail Initial guess length of the for natural segments frequency (Hz) {ω}ik Computed ωik (Hz) Computed ζik (%) Computed Aik (*10-4) Sting Mode 14 5 13.49 0.73 -0.1938 VT 1st Bending 25 17 25.02 1.92 -51.2332 HT 1st Bending 36 7 36.41 2.42 -8.0237 VT 1st Torsion 85.5 7 85.13 2.07 7.4392 VT 2nd Bending 120 11 119.04 1.03 2.0736 VT 2nd Torsion 223.5 17 224.11 1.74 94.5243 VT 3rd Bending 302.5 9 301.04 1.20 -7.8650 And the low frequency parameter was Bi = 29.10-3 189 Table IV-2 Plant Parameters for the Torsion Actuator Array Mode VT: Vert. Tail HT: Horiz. Tail Initial guess length of the for natural segments frequency (Hz) {ω}ik Computed ωik (Hz) Computed ζik (%) Computed Aik (*10-4) Sting Mode 14 5 12.97 0.42 -0.0404 VT 1st Bending 25 17 25.01 1.90 -25.9468 HT 1st Bending 36 7 36.11 1.84 -3.7122 VT 1st Torsion 85.5 7 85.63 1.71 -118.7604 VT 2nd Bending 120 11 99.32 6.26 3.7396 VT 2nd Torsion 223.5 17 223.34 1.77 127.8960 VT 3rd Bending 302.5 9 302.15 1.19 -12.1224 And the low frequency parameter was Bi = 29.10-3 190 Control Plant Model Transfer Function for Bending Actuation -10 -15 -20 Magnitude (dB re 1V acc /1V act) -25 -30 -35 -40 -45 -50 -55 Experimental Transfer Function Modeled Transfer Function -60 0 50 100 150 Frequencies (Hz) 200 250 Figure IV.12 Experimental and Modeled Transfer Functions for the Bending Array Control Plant Model Transfer Function for Torsion Actuation -10 Magnitude (dB re 1V acc /1V act) -20 -30 -40 -50 -60 Experimental Transfer Function Modeled Transfer Function 0 50 100 150 Frequencies (Hz) 200 250 Figure IV.13 Experimental and Modeled Transfer Functions for the Torsion Array 191 IV.2.8. Controller Design and Stability Assessment Once the plant model is developed and the actuator authority checked, the controllers are now designed. The type of controller, which is selected for the active tail buffet alleviation experiments, is an acceleration feedback controller (AFC). Further, the type of design for AFC is the design based on the H2 optimization of the closed loop transfer function between modal displacements and disturbance. Two different controllers are designed, one for the bending array and one for the torsion array. For the control of the first bending mode, using the parameters extracted earlier for the bending array, a single degree of freedom acceleration feedback controller was designed. As discussed in Section IV.1.7, only one design parameter is necessary to obtain the controller. In this case this parameter was selected to be the amount of damping in the controller. In order to avoid saturation of the control signal, the damping of the controller was selected to be seven times larger than the damping of the first bending mode of the vertical tail. The parameters of the controller are shown in Table IV-3 and its transfer function is illustrated in Figure IV.14. The second controller was designed for the active damping of the first and second torsion modes using the torsion array. In this case, two separate single degree of freedom controllers are designed, one for each mode, using the parameters extracted earlier for the torsion array. As for the control of the first bending mode, only one parameter is needed for each controller. Following the same argument as before, the damping of each compensator was chosen to be seven times larger than the damping of the associated 192 mode. The parameters of the controller are shown in Table IV-3 and its transfer function is illustrated in Figure IV.14. Table IV-3 Controller Parameters Frequency (rad/s) Damping Ratio (%) Scalar Gain First Bending 157.22 13.464 -14.153 First Torsion 538.05 11.998 -4.8485 Second Torsion 1403.3 12.384 4.7967 Transfer Functions of the Controllers 40 Bending Controller Torsion Controller Magnitude of Controllers (dB re 1Vact/1V acc) 30 20 10 0 -10 -20 -30 0 50 100 150 200 Frequency (Hz) 250 300 350 Figure IV.14 Bending and Torsion Controllers Transfer Functions 193 Figure IV.15 Generalized Root Locus Plot Following the discussion in Section IV.1.7, the stability of the controllers is checked for the use of both controllers at the same time by computing the roots of the global characteristic equation. Then each gain is multiplied by the same number ranging from 0.1 to 3 to obtain the generalized root locus plot illustrated in Figure IV.15. Since the poles at the operating gains show no instability, it considered safe to implement the controllers simultaneously. 194 IV.2.9. Maximum Control Voltage Assessment To avoid any saturation of the control voltage, a maximum control voltage assessment is studied as discussed in Section IV.1.8. The power spectrums of the estimated control voltage are illustrated in Figure IV.16. Estimated Power Spectrum of Control Signals 7000 Bending Controller Torsion Controller 6000 Magnitude (V2/Hz) 5000 4000 3000 2000 1000 0 0 50 100 150 200 Frequency (Hz) 250 300 350 Figure IV.16 Estimated Power Spectrums of Control Voltages Using Equation (IV.22), the maximum RMS of the control voltages are estimated to be 7 Volt for the bending controller and 11.8 Volt for the torsion controller. Even though these values do not all lie below the maximum voltage allowable of 7 Volt, it is to be noted that this estimate is very conservative since once the controlled system settles, the magnitude of the sensor voltage will be much lower than its open loop value. Since the magnitudes of the estimation of the maximum RMS voltages are just slightly higher 195 than the maximum allowable voltage, the control signal should not be clipped when the steady state of the closed loop system is reached. IV.2.10. Controller Implementation using a Digital Signal Processor The equipment that was used during the wind tunnel tests included: • 2 PCB 303-A02 accelerometers with their PCB 480-D06 Signal Conditioners, • 4 Offset Piezoceramic Stack Actuators were based on PI-810.10 (180N, 100V) piezoceramic stacks, • 2 PI E-663 LVPZT-Amplifier to provide DC offset and amplify the control signals, • 1 Alpha-Combo dSPACE digital signal processor unit consisting of: • DS-1003 DSP Board (based on a TMS320C40, 60 Mflops) for I/O handling, • DS-1004 DSP Board (based on a DEC’s Alpha AXP 21164, 600 Mflops) for controller computation, • DS-2001 Analog to Digital Converter Board with 5, 16 bit conversion, independent channels, • DS-2101 Digital to Analog Converter Board with 5, 12 bit conversion, independent channels • 2 Krohn-Hite Model 3343 Analog Filter Units used as anti-aliasing filters for data acquisition, 196 • 1 Pentium based PC for controller design and data post processing, • 1 Wavetek Model 112 Signal Generator to generate white noise, • 1 Tektronic Model 2236 Oscilloscope to check the magnitude of the control signals in real time. The first step of the controller implementation was the wiring of the experiment. As illustrated in Figure IV.17, the accelerometer sensor on the inboard trailing edge tip of the vertical tail is connected to a PCB signal conditioner. In turn, the output of the signal conditioner is connected to both an analog to digital (A/D) channel of the dSPACE system for control implementation and to another A/D channel through an anti-aliasing filter for data acquisition. Four different digital to analog (D/A) channels are used to carry the control signals. Each channel is connected to an input channel of a piezo driver. The two piezodriver units are used to set DC biases and amplify the control signals. Finally, each output of the piezo drivers is connected to a stack actuator. The experimental setup for the active damping of the buffet induced vibrations is illustrated in Figure IV.18. The controllers are implemented on the dSPACE system. The system is made of five A/D converters, five D/A converters and two digital signal processors (DSP). The first DSP is dedicated to the process of the input and output channels, and then sends the data to the second DSP. The second DSP, which runs faster, is used for the implementation of the controllers. 197 PCB Signal Conditioner In 1 In 2 Anti-Aliasing Filters Out 1 Out 2 Out1 Out2 Out3 Piezo Driver 1 In1 In2 In3 Out1 Out2 Out3 Piezo Driver 2 In1 In2 In3 O1 O2 O3 O4 O5 I1 I2 I3 I4 I5 D/A Channels A/D Channels Digital Signal Processor PC Computer Figure IV.17 Wiring for the Active Buffet Alleviation Experiment The programming of the overall control experiment is done using block programming with the Matlab extension called Simulink. As an example, the controllers are entered in the Simulink file as transfer function blocks. As discussed in Section IV.1.9, the algorithm chosen for the time integration is a Runge-Kutta algorithm of order five. Furthermore, since the Alpha DSP board is fast enough, the controller is implemented at a rate of 10kHz. The file is then converted to the DSP machine language and downloaded to the dSPACE system. Once the system starts, the controllers are effective. 198 Figure IV.18 Experimental Setup for the Active Buffet Alleviation Experiment IV.2.11. Worse Buffet Conditions Control Experiment The conditions at which this experiment is performed are dictated by the arguments discussed in Sections IV.2.2. The operating free stream dynamic pressure of 9 psf is set to reflect a full-scale aircraft flying at Mach 0.6 at an altitude of 20,000 ft. The angle of attack of 20 degrees is chosen to validate the controller at the worse buffet conditions for the given free stream velocity. 199 Comparison between Open and Closed Loop at α =20° and q =9psf ∞ 1200 Experimental Open Loop Experimental Closed Loop Auto-Power Spectrum of Acceleration (g 2/Hz) 1000 800 600 400 200 0 0 20 40 60 80 Frequency reduced to full scale (Hz/2.25) 100 120 Figure IV.19 Comparison Between Open and Closed Loop Auto-Power Spectrum of Trailing Edge Tip Acceleration at α=20° and q∞=9psf. The auto-power spectrums of the uncontrolled and controlled trailing edge tip acceleration are illustrated in Figure IV.19. This figure shows that each of the controlled frequency has its auto-power spectrum reduced by a factor of at least 5. Furthermore, in the case of the first bending mode and second torsion mode, the responses are suppressed to a level equivalent to the one that would be obtained in the absence of the modes. IV.2.12. Angle of Attack Sweep Control Experiment (Robustness Issues) Once the controller has been validated at its operating point, its effectiveness is checked for different conditions. For this second experiment, the operating free stream 200 dynamic pressure of 9 psf is kept. However, the angle of attack is varied from 0 to 23 degrees. For small angles of attack, the angle is changed by increments of two degrees between every run. However, above 12 degrees, a sweep is made for every degree. The root mean squares of the trailing edge tip acceleration are illustrated in Figure IV.20. RMS Tip Acceleration versus Angle of Attack 7 Open Loop Experimental Data Open Loop Polynomial Fitting Closed Loop Experimental Data Closed Loop Polynomial Fitting RMS Acceleration (g/ √Hz) 6 5 4 3 2 1 0 0 5 10 15 Angle of Attack (deg) 20 25 Figure IV.20 Uncontrolled and Controlled Root Mean Square of the Trailing Edge Tip Acceleration versus Angle of Attack at q∞=9psf. This control experiment shows that the root mean square of the trailing edge tip acceleration is reduced by 30% for angles of attack below 15 degrees and by about 20% at 20 degrees. These differences are due to the much larger buffet disturbance pressure at the worse angle of attack. However, as is discussed previously, even with 20% reduction in RMS, the peaks associated with each mode are reduced by a factor of at least 5. This 201 experiment confirms that the controllers are effective on the range of angles of attack from 0 to 23 degrees. IV.2.13. Free Stream Dynamic Pressure Sweep Control Experiment (Robustness Issues) Since the controller has been validated for the range of angles of attack from 0 to 23 degrees at the operating free stream dynamic pressure of 9 psf, its effectiveness must be checked for different free stream velocities. For that purpose, different angles of attack are selected. 14, 17, 20 and 23 degrees angles of attack cover the different buffet regimes that the scaled model encounters. For each angle of attack, the free stream dynamic pressure is varied from 5 to 13 psf. Uncontrolled and controlled data are recorded for every 2 psf increment of the free stream dynamic pressure. The root mean squares of the trailing edge tip acceleration are illustrated in Figure IV.21 through Figure IV.25. As before, the results illustrated in Figure IV.21 through Figure IV.25 show that as the disturbance increases the effectiveness of the controllers decrease. However, even at a free stream velocity 25% higher than the operating free stream velocity, the minimum RMS reduction is still 17%. These results prove that the controllers are stable and effective over the full buffet domain which means angles of attack ranging from 14 to 23 degrees and free stream velocity ranging from –25% to +25% of the full-scale equivalent of Mach 0.6 at 20,000 ft. 202 RMS Tip Acceleration versus Free Stream Dynamic Pressure 12 Open Loop Fit Closed Loop Fit α=14° α=17° α=20° α=23° RMS Acceleration (g/ √Hz) 10 8 6 4 2 0 5 6 7 8 9 10 Free Stream Dynamic Pressure (psi) 11 12 13 Figure IV.21 Uncontrolled and Controlled Root Mean Square of the Trailing Edge Tip Acceleration versus Free Steam Dynamic Pressure for Four Angles of Attack RMS Reduction in 0-300 Hz Band 30.00 25.00 20.00 RMS 15.00 Reduction (%) 10.00 5.00 0.00 7 14 17 20 21 23 Angle of Attack (degree) 13 Free Stream Dynamic Pressure (psi) Figure IV.22 Percent RMS Reduction of the Trailing Edge Tip Acceleration of the Left Vertical Tail in the 0-300 Hz Frequency Band 203 RMS Reduction in 17.5-29.5 Hz Band 50.00 40.00 30.00 RMS Reduction (%) 20.00 10.00 7 0.00 14 17 20 21 23 Angle of Attack (degree) 13 Free Stream Dynamic Pressure (psi) Figure IV.23 Percent RMS Reduction of the Trailing Edge Tip Acceleration of the Left Vertical Tail in the 17.5-29.5 Hz Frequency Band around First Bending Mode RMS Reduction in 65-99 Hz Band 50.00 40.00 30.00 RMS Reduction (%) 20.00 10.00 7 0.00 14 17 20 21 23 Angle of Attack (degree) 13 Free Stream Dynamic Pressure (psi) Figure IV.24 Percent RMS Reduction of the Trailing Edge Tip Acceleration of the Left Vertical Tail in the 65-99 Hz Frequency Band around First Torsion Mode 204 RMS Reduction in 205-239 Hz Band 70.00 60.00 50.00 40.00 RMS Reduction (%) 30.00 20.00 10.00 0.00 7 14 17 20 21 23 Angle of Attack (Degree) 13 Free Stream Dynamic Pressure (psi) Figure IV.25 Percent RMS Reduction of the Trailing Edge Tip Acceleration of the Left Vertical Tail in the 205-239 Hz Frequency Band around Second Torsion Mode IV.3. Tests on a Full-Scale Vertical Tail Once the active buffet alleviation system is validated for the wind tunnel model, the combination of acceleration feedback control and offset piezoceramic stack actuators is tested on a full-scale specimen. The school of Aerospace Engineering at the Georgia Institute of Technology possesses a half empennage from an F-15 aircraft that is mounted on a frame designed to simulate the connections to a complete operational aircraft. This structure serves as the full-scale specimen for the active buffet alleviation system study. This half empennage includes the left vertical tail, the left horizontal tail and the rear left section of the fuselage that makes the tail boom and the rear engine mounts. This specimen is illustrated in Figure IV.26. 205 Figure IV.26 Laboratory Sub-Assembly IV.3.1. Full-Scale Control System Objective To assess the feasibility of using acceleration feedback control in combination with offset piezoceramic stack actuators to control vibrations in the full-scale vertical tail, the first task is to identify the modes that should be controlled. Due to a limited amount of piezoceramic stacks, for these tests, a single actuator array is selected. Hence for demonstration purposes, the first bending mode of the vertical tail is to be controlled with the addition of the structurally coupled mode associated with the first bending mode of the horizontal tail. For the experiments, a 50-pound MB Dynamics electromagnetic shaker mounted on the outboard trailing edge tip is to provide the disturbance loads. In 206 order to measure the force applied to the sub-assembly, a PCB load cell, 208 A02, is mounted between the shaker’s sting and the vertical tail. PCB 308 B14 accelerometers mounted on the inboard tip of the vertical tail both at the leading and trailing edges are to provide performance measurements to assess the effectiveness of the control system. IV.3.2. Sensor and Actuator Placements In a manner similar to the choice of sensor location for the wind tunnel scaled model, as discussed in Section IV.2.3, for this full-scale experiment, the control sensor location is chosen to be the inboard trailing edge tip of the vertical tail. This location is also useful for an actual implementation since at this location an extruding fairing exists in which the tail-light is mounted. Hence, for an actual implementation of the system, the accelerometer could be mounted within this extruding fairing. Once the sensor location is decided, the actuator placements are next. For this experiment, six offset piezoceramic stack actuators based on Physik Instrumente PI830.40 (100V, 800N) piezoceramic stack are used. As discussed in Section IV.1.4, the actuator placement is based on reciprocity theory, the objective of the placement is to maximize the transverse acceleration at the trailing edge tip of the vertical tail due to induced bending moments at the actuator locations. Hence, using reciprocity, the tail is transversally actuated by the shaker mounted on the outboard trailing edge tip of the vertical tail and curvature is measured at candidate actuator locations with LDT2-028K PVDF film sensors. 207 To obtain the best actuator locations, two different sets of measurements are performed. First, a chordwise study is performed by measuring curvature in three directions at five different locations along a waterline of the center toque box. Then, a spanwise study is performed by measuring curvature in three directions at four different locations, close to the root, along the elastic axis of the vertical tail. For both sets of measurements, curvature is measured along the waterline and then at angles of 60 and 120 degrees with respect to the waterline. During these experiments, the PVDF film sensor output voltage is conditioned with a Charge Amplifier designed in such a way that the sensor signal is the averaged dynamic curvature under the covered area. Further, the PVDF film is chosen such that its length matched the length of the offset piezoceramic stack actuators that are considered. Transfer functions between the curvature measurements and the input load to the vertical tail as measured by the load cell are taken in each direction at each location. Then all transfer functions are analyzed in a manner similar to the one described in Section IV.1.6 simultaneously for each set of experimental data. As a result, a global value for the poles associated with the first bending mode of the vertical tail and the structurally coupled mode associated with the first bending mode of the horizontal tail are obtained. The resulting modal amplitude for each location and each direction are given in Table IV-4 and Table IV-5 for the chordwise and the spanwise studies respectively. Then, these coefficients are used with Equation (IV.8) to determine the three generalized curvatures, ϕxx, ϕzz and ϕxz. Then, using the three generalized curvatures, angular distributions of the curvature are computed for each location for both the first 208 bending mode of the vertical tail and the first bending mode of the horizontal tail. Finally, these distributions are weighted such that the control authority on the first bending mode of the vertical tail had a coefficient of 1 and the first bending mode of the horizontal tail a coefficient of 20%. The weighted angular distributions are illustrated in Figure IV.27 and Figure IV.28 for the chordwise and spanwise study respectively. Table IV-4 Modal Amplitudes associated with the Chordwise Study of the Actuator Placement First Bending of Vertical Tail (3) First Bending of Horizontal Tail (4) Location 0° 60° 120° 0° 60° 120° 1 -9.390 119.60 -2.97 -9.07 110.83 -0.69 2 -19.63 117.48 4.97 -11.68 139.28 10.65 3 -23.45 122.54 10.21 -25.37 146.67 24.31 4 -22.41 116.54 23.08 -31.97 144.18 65.92 5 -34.56 117.04 19.13 -70.45 159.77 45.91 209 Table IV-5 Modal Amplitudes associated with the Spanwise Study of the Actuator Placement First Bending of Vertical Tail First Bending of Horizontal Tail Location 0° 60° 120° 0° 60° 120° a -17.80 124.98 12.56 -73.96 170.39 23.23 b -25.93 127.04 12.82 -45.38 198.77 28.16 c -24.23 125.97 20.21 -55.05 201.47 17.50 d -33.86 117.95 10.25 -58.46 193.46 27.73 Actuator Chordwise Efficiency Study (for Modes 3, 4, with Weights 1, 0.2) 1 loc loc loc loc loc TE 0.8 LE 1 2 3 4 5 Normalized Efficiency 0.6 0.4 0.2 0 -0.2 <-- Elastic Axis -0.4 0 20 40 60 80 100 120 Orientation (degree) 140 160 180 Figure IV.27 Actuator Placement Optimization Plot for Chordwise Study 210 Actuator Spanwise Efficiency Study (for Modes 3, 4, with Weights 1, 0.2) 1 loc loc loc loc 0.8 a b c d Normalized Efficiency 0.6 0.4 0.2 0 -0.2 <-- Elastic Axis -0.4 0 20 40 60 80 100 120 Orientation (degree) 140 160 180 Figure IV.28 Actuator Placement Optimization Plot for Spanwise Study Based on Figure IV.27, the optimal chordwise location is aft of the elastic axis, close to the rudder location, at an angle with respect to the waterline slightly larger than the one of the elastic axis. It is however to be noted that the distribution of curvature for the first bending mode of the tail has a distribution symmetric with respect to the elastic axis and an optimal placement on the elastic axis itself. The deviation from this symmetry is due to the fact that the authority over the coupled mode due to the first bending mode of the horizontal tail is larger tower the trailing of the tail. This phenomenon is due to the fact that the attachment point of the horizontal stabilator is below the rudder and hence aft of the elastic axis. 211 Figure IV.28 shows that the optimal spanwise location is at about 25% of the span from the root of the vertical tail. This result is not unexpected since the tail has a tapered thickness, which decreases almost uniformly from the root to the tip of the vertical tail. Hence, the maximum curvature was expected not at the very root of the tail but at a somewhat higher spanwise location. Again, the optimal direction along the elastic axis for the control of the first bending modes of both the vertical and horizontal tails is along the direction of the axis itself. As a result of this analysis, the six offset piezoceramic stack actuators are bonded on both inboard and outboard surfaces of the vertical tail in a symmetric manner. The actuators are bonded on the waterline associated with location b and at an angle corresponding to the elastic axis orientation of 67.5° with respect to the waterline. On each surface, the sets of three actuators are bonded symmetrically with respect to the elastic axis even though Figure IV.27 indicated that aft location had more overall authority. IV.3.3. Plant Characterization Once the actuators were bonded on the tail, a plant model of the system is identified. As discussed in Section IV.1.6, for acceleration feedback control, the plant parameters can be derived from the transfer function between the actuators and the control sensor. In order to obtain this plant transfer function, a random signal is fed to the actuators and the response of the control accelerometer is measured. The actuators on the 212 outboard side of the tail, or left side, are driven by the random signal itself while the actuators on the inboard side of the tail, or right side, are driven by the opposite of the random signal. As a result, while on side the actuators are expanding, on the other side of the tail the actuators are contracting. In order to model the dynamics of the piezo-drivers, the offset piezoceramic stack actuators, the sensor and its signal conditioner in addition to the dynamics of the structure, the experimental transfer function is taken between the input voltage to the piezo-drivers and the output voltage of the sensor signal conditioner. Then, using the system identification technique presented in Section IV.1.6, sixteen natural frequencies, modal damping ratios and modal residues are identified. These plant parameters are given in Table IV-6 and a comparison of the experimental and modeled transfer function is shown in Figure IV.29. 213 Magnitude (dB re[1V inp/1Vresp]) Plant Transfer Function Experimental Modeled -20 -30 -40 -50 Phase (degrees) -60 0 10 20 30 40 50 60 70 0 10 20 30 40 50 Frequency (Hz) 60 70 180 90 0 -90 -180 Figure IV.29 Comparison between Modeled and Experimental Plant Transfer Functions 214 Table IV-6 Full-Scale Vertical Tail Plant Parameters Mode Numbers Natural Frequencies (Hz) Damping Ratios (%) Modal Residues (x 10-4) 1 5.12 2.66 1.790 2 7.20 3.57 2.260 3 10.66 1.22 9.140 4 19.00 1.12 6.740 5 26.05 1.34 3.470 6 35.77 0.63 -2.650 7 37.75 0.43 -13.900 8 39.04 0.43 8.830 9 41.73 0.48 -0.840 10 45.47 0.40 -4.850 11 48.19 0.34 -0.880 12 56.20 0.38 -0.611 13 57.87 0.38 -0.805 14 61.98 0.41 4.760 15 70.11 2.22 -3.880 16 75.68 0.70 -8.350 215 IV.3.4. Controller Designs In order to illustrate the multi degree of freedom system controller designs presented in Section II.2.3 and II.2.4, two different controllers are designed for the control of the first bending mode (mode 3) of the vertical tail and the coupled mode (mode 4) resulting from the first bending mode of the horizontal tail. IV.3.4.a Crossover Controller First, an acceleration feedback controller based on the crossover design concept is designed. As discussed in Section IV.1.7, the first step of the design consists in the choice of the controller design parameters. For this particular application, the gains of the controller are chosen as the controller parameters. These gains are chosen in such a way that the controller mostly alleviates the first bending mode of the vertical tail and, as a secondary task, alleviates the coupled mode. In the second step of the design, the closed loop coupling terms are assumed to be negligible and the controller parameters are computed independently for each mode as an initial guess for the following task. Finally, the algorithm developed in Section II.2.3 is used to compute the optimal controller parameters for multi degree of freedom systems. The different controller parameters are given in Table IV-7. 216 Table IV-7 AFC Crossover Design Controller Parameters SDOF Approximation MDOF Optimization Mode Gain Frequency (Hz) Damping (%) Frequency (Hz) Damping (%) 3 48.86 10.66 22.346 10.37 22.203 4 24.21 19.00 13.896 18.69 14.701 A comparison between the SDOF assumption design controller transfer function and MDOF optimization design controller transfer function is shown in Figure IV.30. Controller Transfer Function 45 40 Magnitude (dB) 35 30 25 20 15 SDOF design MDOF design 10 Phase (degrees) 5 0 10 20 30 40 50 60 70 0 10 20 30 40 50 Frequency (Hz) 60 70 180 90 0 -90 -180 Figure IV.30 Comparison of SDOF Assumption and MDOF Optimization Designs AFC Crossover Controller Transfer Functions 217 The poles of both the SDOF assumption and MDOF optimized designs are shown in Figure IV.31. A comparison of the open loop transfer function with the closed loop transfer functions of both the SDOF assumption and MDOF optimized designs is shown in Figure IV.32. Figure IV.31 clearly shows that for the multi degree of freedom optimization, the crossover condition is reached since the two closed-loop poles of the structure and controller coincide exactly. However, from Table IV-7, Figure IV.30 and Figure IV.32, the difference between the SDOF assumption design and the MDOF optimization are very small which means that the closed loop coupling terms can almost be neglected. Comparison Between SDOF Design and MDOF Optimal Poles 500 450 400 struct ol SDOFd cont ol SDOFd strc cl SDOFd cont cl MDOFd cont ol MDOFd strc cl MDOFd cont cl Imaginary Part 350 300 250 200 150 100 50 0 -20 -15 -10 Real Part -5 0 Figure IV.31 Comparisons of Open and Closed Loop Poles for SDOF Assumption and MDOF Optimization Designs of AFC Crossover Controller 218 It is important to notice at this point that, in effect, the multi degree of freedom optimization, when the controller design parameters are the controller gains, consists of tuning the poles of the controller to operate at crossover exactly. Figure IV.31 shows that this tuning is small for this particular application. Magnitude of Open and Closed Loop Transfer Functions OL Modeled SDOFd CL Modeled MDOFd CL Modeled Magnitude (dB) -20 -30 -40 -50 Phase (degree) -60 0 10 20 30 40 50 60 70 0 10 20 30 40 50 Frequency (Hz) 60 70 180 90 0 -90 -180 Figure IV.32 Comparisons of Open and Closed Loop Transfer Functions for SDOF Assumption and MDOF Optimization Designs of AFC Crossover Controller Further, Figure IV.31 shows that the uncontrolled closed loop modes are stable even though the amount of damping of seventh mode, corresponding to the first torsion mode of the structure, has been slightly decrease. However, since this first torsion mode is the most predominant in the tail buffet response of the F-15 aircraft, an actual buffet 219 alleviation system would heavily control this torsional mode. Since the stability of the closed loop system has been checked, the controller can be implemented. Figure IV.32 shows that the first bending mode of the vertical tail would be reduced by 14dB correspond to 80% of suppression while the coupled mode would reduced by 10dB (over 65% of reduction). This figure also shows that all the modes above 40 Hz, which are uncontrolled, are left completely unchanged by the controller. This is due to the roll off and high frequency controller phase characteristics of acceleration feedback control which reduces spillover effects. IV.3.4.b H2 Optimal Controller The second controller that was designed is based on the optimization of an H2 criterion. For this design, the matrices R1 and R2, as defined in Section II.2.4, are chosen in such a way that the controller minimizes a weighted potential energy associated with the two controlled modes. The additional weights are chosen to be the natural frequency of each controlled mode. Hence the weight matrix, R1, is given by: é0 0 ωs ,3 R1 = ê ë0 0 0 0 ωs , 4 0 ... 0ù 0 ... 0 (IV.25) Further, the second weight matrix, R2, is a zero matrix of the same dimension as R1. Hence, the resulting functional, which is minimized, is given by: æ ì1 J s = çç lim E í è t →∞ î t t 0 üö ω ξ 3 (τ) + ω ξ 4 (τ) dτý 4 3 2 4 4 2 220 1/ 2 (IV.26) Then, for the purpose of comparison with the previously designed controller, the gains are chosen to be the same. Again, assuming that the closed loop coupling terms are negligible, the controller parameters are computed using the equations from Section II.1.3. These computed controller parameters are then used as the initial guess for the multi degree of freedom optimization discussed in Section II.2.4. The resulting controller parameters are given in Table IV-8. Table IV-8 AFC H2 Optimal Design Controller Parameters SDOF Approximation MDOF Optimization Mode Gain Frequency (Hz) Damping (%) Frequency (Hz) Damping (%) 3 48.86 10.66 10.564 10.51 13.521 4 24.21 19.00 6.388 18.69 7.295 For further comparison, a second functional, which is associated with the root mean square of the control signal, u(t), is defined. This functional is given by: æ ì1 J c = çç lim E í è t →∞ î t t 0 1/ 2 üö u (τ) dτý 2 (IV.27) Table IV-9 shows the values of the two functional for the single degree of freedom assumption design and for the multi degree of freedom design. A comparison 221 between the controller transfer functions of the single degree of freedom assumption design and the multi degree of freedom optimization design is shown in Figure IV.33. Table IV-9 Comparison of Functionals between SDOF Assumption and MDOF Optimization Designs Structural Functional Js Control Functional Jc Open Loop SDOF Assumption MDOF Optimization SDOF Assumption MDOF Optimization 63.565 33.076 32.693 6716.7 6335.1 % Reduction 47.97 48.57 % Reduction 5.68 Controller Transfer Function 50 Magnitude (dB) 40 30 20 SDOF design MDOF design Phase (degrees) 10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 Frequency (Hz) 60 70 180 90 0 -90 -180 Figure IV.33 Comparison of SDOF Assumption and MDOF Optimization Designs AFC H2 Optimal Controller Transfer Functions 222 Both Table IV-9 and Figure IV.33 show that the multi degree of freedom optimization design for the controller has larger damping coefficients than the SDOF assumption design. This fact explains why the controller functional is reduced much more than the structural functional by performing the MDOF optimization. A comparison of the poles between the two designs, illustrated in Figure IV.34, shows that the MDOF closed loop structural poles appears to the left of their SDOF assumption design counterparts resulting in a higher damping value. Furthermore, the graph shows that when the MDOF optimization is performed, the controller and structural poles are closer than before the optimization. Comparison Between SDOF Design and MDOF Optimal Poles 500 450 400 struct ol SDOFd cont ol SDOFd strc cl SDOFd cont cl MDOFd cont ol MDOFd strc cl MDOFd cont cl Imaginary Part 350 300 250 200 150 100 50 0 -10 -8 -6 -4 Real Part -2 0 Figure IV.34 Comparisons of Open and Closed Loop Poles for SDOF Assumption and MDOF Optimization Designs of AFC H2 Optimal Controller 223 A comparison of the closed loop transfer functions and the open loop transfer functions, as illustrated in Figure IV.35, shows that by performing the MDOF optimization, the transfer function in the area of the controlled modes is smother. Furthermore, the first peak of each controlled pair, which is associated with the structure, is reduced further compared to the SDOF assumption design. Magnitude of Open and Closed Loop Transfer Functions OL Modeled SDOFd CL Modeled MDOFd CL Modeled Magnitude (dB) -20 -30 -40 -50 Phase (degree) -60 0 10 20 30 40 50 60 70 0 10 20 30 40 50 Frequency (Hz) 60 70 180 90 0 -90 -180 Figure IV.35 Comparisons of Open and Closed Loop Transfer Functions for SDOF Assumption and MDOF Optimization Designs of AFC H2 Optimal Controller Figure IV.35 also shows that the magnitude of first bending mode of the vertical tail is reduced by about 16.5 dB, which corresponds to a 85% of reduction, and that the magnitude of the coupled mode due to the first bending mode of the horizontal tail is 224 reduced by 9.5 dB, which corresponds to over 65% of reduction. Furthermore, it is interesting to notice that even though the damping of the seventh mode is slightly decreased, this effect is somewhat smaller than the damping reduction due to the crossover controller. This result can be attributed to the fact that the controller damping ratios are about one half of the ones of the crossover design. Finally, as for the crossover design, the root locus plot, Figure IV.34, shows that all the closed loop modes are stable and hence the controller can be implemented. IV.3.5. Setup and Implementation of the Vibration Control System on the Full-Scale SubAssembly The equipment for the full-scale vibration control experiment of the first bending mode of the vertical tail and the coupled mode resulting from the first bending mode of the horizontal tail consists of the following items: • 2 PCB 308-B14 accelerometers with their PCB Signal Conditioners, • 1 PCB 208-A02 Load Cell with its Signal Conditioner, • 6 Offset Piezoceramic Stack Actuators were based on PI-830.40 (800N, 100V) piezoceramic stacks, • 2 PI E-663 LVPZT-Amplifier to provide DC offset and amplify the control signals, • 1 MB Dynamics 50-lb Shaker and its Amplifier, • 1 Alpha-Combo dSPACE digital signal processor unit consisting of: 225 • DS-1003 DSP Board (based on a TMS320C40, 60 Mflops) for I/O handling and controller computation, • DS-1004 DSP Board (based on a DEC’s Alpha AXP 21164, 600 Mflops), unused for this experiment, • DS-2001 Analog to Digital Converter Board with 5, 16 bit conversion, independent channels, • DS-2101 Digital to Analog Converter Board with 5, 12 bit conversion, independent channels • 2 Krohn-Hite Model 3343 Analog Filter Units used as anti-aliasing filters for data acquisition, • 1 Pentium based PC for controller design and data post processing, • 1 Wavetek Model 112 Signal Generator to generate white noise, • 1 Tektronic Model 2236 Oscilloscope to check the magnitude of the control signals in real time. As illustrated in Figure IV.36, the closed loop wiring consists of an accelerometer mounted on the trailing edge tip of the vertical tail, its signal is fed to a signal conditioner and then directly input to an Analog to Digital converter. The control signal is generated by a Digital to Analog filter and then input to a piezo-driver, which amplifies the control signal before driving the piezoceramic stacks. Simultaneously, external disturbance is 226 provided by a shaker, which is driven by a random signal generated by the signal generator. Performance measurements are obtained in the form of transfer functions between the input force to the structure as measured by the load cell and acceleration response from the control sensor and the secondary performance accelerometer. Load Cell Shaker Control Sensor Signal Conditioner Signal Conditioner Performance Accelerometer Anti-Aliasing Filter Signal Conditioner Anti-Aliasing Filter Anti-Aliasing Filter I5 I4 I3 I2 I1 DSP Board (Controller) O5 O4 O3 O2 O1 OPSAs Shaker Amplifier Signal Generator Ch1 Ch2 Ch3 Piezo Driver PC Computer Ch1 Ch2 Ch3 Piezo Driver Figure IV.36 Wiring for the Vibration Control Experiment of the Full-Scale Vertical Tail As described in Section IV.1.9, the controller is implemented using the Simulink toolbox of the Matlab software. The Simulink block representation of the controller is 227 illustrated in Figure IV.37. In addition of implementing the controller itself, two additional features are added to the file. First, in order to compensate for DC drifts in the sensor side of the closed loop, a constant, which is obtained experimentally, is removed from the sensor signal. Second, in order to insure the safety of the actuators, a saturation block is implemented that saturates the control signal just below the maximum allowable voltage to the piezo-driver. Figure IV.37 Simulink Representation of the Controller Finally, as discussed in Section IV.1.9, the algorithm chosen for the time integration of the controller is a Runge-Kutta algorithm of order five. Furthermore, since the DSP board is fast enough, the controller is implemented at a rate of 10kHz. 228 IV.3.6. Full-Scale Vibration Control Experimental Results For the purpose of validating the control system experimentally on the full-scale vertical tail, the controller designed in Section IV.3.4.b is tested. As discussed in the previous section, the performance of the controller is measured by two transfer functions. The first frequency response function is taken between the control sensor, located on the trailing tip of the vertical tail, and the input force to the tail as measured by the load cell. A comparison between open and closed loop behavior of this transfer function is illustrated in Figure IV.38. The second performance sensor is located on the leading edge of the tail and the comparison between its open and closed loop behavior is given in Figure IV.39. The accelerance at the control sensor location, Figure IV.38 and Figure IV.40, shows a reduction in the peak associated with the first bending mode of the vertical tail of about 12 dB, which correspond to a reduction by about 75%. Simultaneously, the second controlled mode is reduced by 8 dB, correspond to a reduction by 60%. Furthermore, almost every uncontrolled mode remained unchanged but for the seventh and eighth modes which damping coefficients are slightly increased and decreased respectively. The response from the secondary performance sensor is illustrated in Figure IV.39 and Figure IV.41 by its frequency response function. A reduction in the magnitude of the first bending mode of 11.5 dB, which correspond to a reduction of about 72.5%, is obtained. A reduction in the coupled mode resulting from the first bending mode of the horizontal tail of 5 dB is obtained that correspond to about 45% reduction. As for the 229 control sensor location, the accelerance at the leading edge tip of the vertical tail shows that all uncontrolled modes are left almost unchanged. Magnitude of Open and Closed Loop Transfer Functions Magnitude (dB re[1g/N]) 0 -10 -20 -30 -40 -50 Open Loop Closed Loop -60 Phase (degree) -70 0 10 20 30 40 50 60 70 0 10 20 30 40 50 Frequency (Hz) 60 70 180 90 0 -90 -180 Figure IV.38 Experimental Comparison between Open and Closed Loop Accelerance at the Control Sensor Location It is to be noted however, that the root means square of the input disturbance force is about 2 N Hz , which is well below the actual buffet loads, and that the control signal where just below the saturation limits. As a result, the six offset piezoceramic stack actuators based on PI-830.40 piezotranslators would not provide enough control authority for actual tail buffet alleviation. Yet, it is also to be noted that these stacks are too flexible 230 for use on the actual vertical tail with a reasonable offset distance. In fact, using a larger diameter size actuator would not only increase the maximum force that can be generated by the actuator but would also increase the transmission of energy from the stack to the structure with an acceptable lever arm distance. Hence, actual tail buffet loads could be alleviated with arrays of offset piezoceramic stack actuators based on larger stacks than the ones used in this experiment. Magnitude of Open and Closed Loop Transfer Functions Magnitude (dB re[1g/N]) 0 Open Loop Closed Loop -10 -20 -30 -40 -50 -60 Phase (degree) -70 0 10 20 30 40 50 60 70 0 10 20 30 40 50 Frequency (Hz) 60 70 180 90 0 -90 -180 Figure IV.39 Experimental Comparison between Open and Closed Loop Accelerance at the Secondary Performance Sensor Location 231 Magnitude of Open and Closed Loop Transfer Functions 0.045 Open Loop Closed Loop 0.04 0.035 Magnitude (g/N) 0.03 0.025 0.02 0.015 0.01 0.005 0 0 5 10 15 Frequency (Hz) 20 25 30 Figure IV.40 Zoom View of Experimental Comparison between Open and Closed Loop Accelerance at the Control Sensor Location Magnitude of Open and Closed Loop Transfer Functions 0.04 Open Loop Closed Loop 0.035 Magnitude (g/N) 0.03 0.025 0.02 0.015 0.01 0.005 0 0 5 10 15 Frequency (Hz) 20 25 30 Figure IV.41 Zoom View of Experimental Comparison between Open and Closed Loop Accelerance at the Secondary Performance Sensor Location 232 CHAPTER V. CONCLUSION AND RECOMMENDATIONS The main goal of this research work was to develop an active buffet alleviation system for high performance twin-tail aircraft using piezoceramic stack based actuators in combination with acceleration feedback control (AFC) theory. In order to complete this task, the work was divided into three main research areas. These areas were the formulation and the design of acceleration feedback controllers; development and modeling of piezoceramic stack based actuator subassemblies, methods of implementing these controllers in structural systems control, and validation of the actuators and controllers for tail buffet alleviation. For the acceleration feedback control work, new methods for the design of the controller parameters were presented for generalized single degree of freedom systems. These new designs were based on pure active damping and quadratic performance criteria, which are based on structural generalized coordinates, minimization. Then, noncollocated acceleration feedback multi-mode controller design methods were developed for a single sensor and a small number of actuator arrays (for Multi-Input Single-Output systems). Then, a new type of moment inducing actuator, the offset piezoceramic stack actuator, which is based on the use of piezoceramic stacks, was developed to provide the 233 needed control authority for buffet alleviation. This actuator was also designed to satisfy high reliability and maintainability requirements. In addition, a technique was developed to analytically model the actuator on the basis of the modal expansion of the offset piezoceramic stack actuator driving a benchmark structure. The results of this analysis were used to create a low frequency approximation of the offset piezoceramic stack actuator as well as to optimize its offset distance and its placement. Because of the non-availability of reliable mathematical or numerical models for both the controlled structure and the buffet induced loads, a control system design method, which is based solely on the use of experimental data, was developed. Then, two sets of experiments were conducted to show the feasibility of controlling the buffet induced vibrations during high angle of attack operations of a selected HPTTA. The first experiment validated both the effectiveness and the robustness of the developed active buffet alleviation system on an aeroelastically scaled model in wind tunnel studies. The second experiment showed that the combination of offset piezoceramic stack actuators and acceleration feedback control could suppress vibrations in a full-scale vertical tail sub-assembly. Since, during this research work, the controller stability was checked a posteriori, further research in the area of acceleration feedback control should first consists in defining some stability domain for the control multi-degree of freedom systems. This stability domain could be found by using a Lyapunov approach with a Lyapunov function representing the total energy associated with both the structure and the controller. It was also remarked during the experiments that even though the control signal was saturated 234 during the transient regime, the controller would control vibrations and reach a stable steady state regime. Hence, the saturation properties of acceleration feedback control should be studied as well. Further, a closer relation with classical H2 optimal control theory could be studied by solving for the controller gain matrix with the introduction a control signal cost in the quadratic optimization criteria. The control of multi-input, multi-output systems should also be considered and the method of obtaining the control sensors influence matrix studied on the basis of either sensor averaging theory or performance criterion minimization. Another research area for acceleration feedback control is the study of the transient effects, such as settling time and overshoot, of the controller on the closed loop system. Furthermore, the wind tunnel tests demonstrated the robustness of the developed designs for acceleration feedback control. As a result, a study of the robustness of acceleration feedback control for multi-degree of freedom systems should be performed in order to determine the bounds on system uncertainties that will maintain closed loop stability and performance. Such bounds would also enable the use of a single time invariant acceleration feedback controller for a slowly time varying system with small parameter variations. If the robustness bounds are wide enough, a single time invariant controller could control vibrations during different flight regimes and even during dynamic maneuvers. Another approach to alleviating vibrations in systems with time varying parameters, due to different flight conditions of dynamic maneuvers, that should be studied, is the extension of acceleration feedback control concepts to adaptive, or time varying, controllers. Finally, to better implement acceleration feedback control digitally 235 on DSP processors, a sampled-data formulation of the equations governing the controller should be developed to identify the lowest order of time integration and smallest frequency at which the controller can be implemented to insure both stability and efficiency. For the design of optimal offset piezoceramic stack actuators, further studies should be performed to size the dimensions of the contact area between the actuator and the controlled structure. This optimization should maximize the transfer of energy from the piezoceramic stack to the structure while minimizing the local stresses in the area of the actuator. Further, weight optimization could be performed by studying the shape and material of the actuator assembly. Finally, a study of the stress redistribution in the controlled structure due to the presence of the actuators under closed loop conditions should be performed to insure that stresses are not locally increased in a manner sufficient to generate new fatigue critical areas. For future applications, the actuators would not have to be bonded on the surface of the structure, attachment points should be included in the design stage of the controller structure in such a way that either the actuator could be bolted to the structure. Another method could be to include the actuator mounts as lugs in such a way that they would be an integral part of the controlled structure. The applications of acceleration feedback control in combination with offset piezoceramic stack actuator are not limited to tail buffet alleviation. In fact, the large authority of offset piezoceramic stack actuators makes them ideal for vibration suppression in large structures such as large space structures or full-scale aircraft. The 236 high roll-off properties of acceleration feedback control at higher uncontrolled frequencies are compelling for vibration control applications in which only lower frequencies should be suppressed. Finally, before conducting any flight tests of this research buffet alleviation system, the laboratory fin sub-assembly should be tested with actuators powerful enough to alleviate actual buffet loads. Such actuators could be based on Physik Instrumente P243 or P-247 piezoceramic stacks, which can produce loads up to 30kN, and have much larger stiffness than the ones tested during this research work, which would result in reasonable optimal offset distances. Furthermore, the buffet alleviation system should be tested on other twin-tail aircraft, such as F/A-18, to generalize its use to all high performance twin-tail aircraft. Finally, a study of the drag that would result from having the actuators protruding from the surface of the vertical tail even protected by aerodynamic shielding should be performed. 237 APPENDICES 238 REFERENCES [1] Ferman, M.A., Liguore, S.L., Smith, C.M., and Colvin, B.J., "Composite Exoskin Doubler Extends F-15 Vertical Tail Fatigue Life", 34th Structures, Structural Dynamics, and Materials Conf., v. 1, 1993, pp. 398-407. 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Wright, "Convergence Properties of the Nelder-Mead Simplex Algorithm in Low Dimensions", SIAM Journal of Optimization, V. 9, N. 1, 1998, pp. 112-47 [51] Courant, R. and Hilbert, D., “Methods of Mathematical Physics”, V. 1, Interscience, New York, 1961, p.277. [52] Triplett, W.E., “Pressure Measurements on Twin Vertical Tails in Buffeting Flow”, Journal of Aircraft, V. 20, N. 11, 1983, pp. 920-925 [53] Lee, C.K., and Moon, C.K., “Modal Sensor/Actuator”, Journal of Applied Mechanics, V. 57, 1990, pp. 434-441. [54] Vold, H., Crowley, J., and Rocklin, G.T., “New Ways of Estimating Frequency Response Functions”, Sound and Vibration, 1984, pp. 34-38 352 VITA Maxime Bayon de Noyer was born on May the 12th 1971 in Neuilly Sur Seine, near Paris, France. He did his High School studies in Bordeaux and Paris, France, and obtained a Baccalauréat C (Scientific Series) in June 1989. He went on to Mathématique Supérieure and Spéciale at the Collège Stanislas, in Paris. In September 1991, he joined the program at the Euro-American Institute where he received an Associate Degree in May 1992. In May 1994, he obtained a Bachelor of Science in Aerospace Engineering from the Florida Institute of Technology. In September 1995, he obtained a Master of Science in Aerospace Engineering from the Georgia Institute of Technology. Then, he enrolled in the Ph.D. program of the School of Aerospace Engineering at the Georgia Institute of Technology. During this period, he worked as a research assistant. 353