dynamic analysis model of spherical roller bearings with
Transcription
dynamic analysis model of spherical roller bearings with
Behnam Ghalamchi DYNAMIC ANALYSIS MODEL OF SPHERICAL ROLLER BEARINGS WITH DEFECTS Thesis for the degree of Doctor of Science (Technology) to be presented with due permission for public examination and criticism in the Auditorium 1382 at Lappeenranta University of Technology, Lappeenranta, Finland on the 14th of November, 2014 at noon. Acta Universitatis Lappeenrantaensis 596 Supervisors Professor Aki Mikkola Department of Mechanical Engineering Lappeenranta University of Technology Finland Professor Jussi Sopanen Department of Mechanical Engineering Lappeenranta University of Technology Finland Reviewers Professor Viktor Berbyuk Department of Mechanical Engineering Chalmers University of Technology Sweden Associate Professor Morten Kjeld Ebbesen University of Agder Norway Opponents Professor Viktor Berbyuk Department of Mechanical Engineering Chalmers University of Technology Sweden Associate Professor Morten Kjeld Ebbesen Department of Mechanical Engineering University of Agder Norway ISBN 978-952-265-669-8 ISBN 978-952-265-670-4 (PDF) ISSN 1456-4491 Lappeenranta University of Technology Yliopistopaino 2014 Preface The research work of this thesis was carried out during the years 2011 - 2014 in laboratories of Machine Design and Machine Dynamics at Lappeenranta University of Technology. Acknowledgements First of all, I would like to express my sincere gratitude to my supervisors Professor Aki Mikkola and Professor Jussi Sopanen for all the supports they provided during this entire research. Your encouragement and enthusiasm have been a source of inspiration which kept me going forward. I am grateful for the opportunity I was given to work in this interesting research project. I am grateful to Professor, Viktor Berbyuk and Associate Professor, Morten Kjeld Ebbesen for reviewing this thesis and giving valuable comments and suggestions. It is a pleasure to Thanks to all the members of Laboratories of Machine Design and Machine Dynamic over years. Janne Heikkinen, Antti Valkeapää, Emil Kurvinen, Oskari Halminen, Adam Klodowski, Marko Matikainen, Scott Semken, Ezral Bin Baharudin, John Bruzzo, and Elias Altarriba, I feel honored to be part of this particular research groups. Special thanks to my friend and my former college Tuomas Rantalainen for his helps to make me more familiar with these groups atmosphere. The financial support of Tekniikan edistämissäätiö is highly acknowledged. I will be eternally grateful for my parents for their encouragements and supports. Zahra, without you I wouldn’t be where I am. You made it all possible and I will be always grateful for your unconditional love and support. Lappeenranta, November 2014 Behnam Ghalamchi Abstract Behnam Ghalamchi Dynamic Analysis Model of Spherical Roller Bearings with Defects Lappeenranta, 2014 90 pages Acta Universitatis Lappeenrantaensis 596 Dissertation. Lappeenranta University of Technology ISBN 978-952-265-669-8 ISBN 978-952-265-670-4 (PDF) ISSN 1456-4491 Rolling element bearings are essential components of rotating machinery. The spherical roller bearing (SRB) is one variant seeing increasing use, because it is self-aligning and can support high loads. It is becoming increasingly important to understand how the SRB responds dynamically under a variety of conditions. This doctoral dissertation introduces a computationally efficient, three-degree-of-freedom, SRB model that was developed to predict the transient dynamic behaviors of a rotor-SRB system. In the model, bearing forces and deflections were calculated as a function of contact deformation and bearing geometry parameters according to nonlinear Hertzian contact theory. The results reveal how some of the more important parameters; such as diametral clearance, the number of rollers, and osculation number; influence ultimate bearing performance. Distributed defects, such as the waviness of the inner and outer ring, and localized defects, such as inner and outer ring defects, are taken into consideration in the proposed model. Simulation results were verified with results obtained by applying the formula for the spherical roller bearing radial deflection and the commercial bearing analysis software. Following model verification, a numerical simulation was carried out successfully for a full rotor-bearing system to demonstrate the application of this newly developed SRB model in a typical real world analysis. Accuracy of the model was verified by comparing measured to predicted behaviors for equivalent systems. Keywords: Spherical roller bearing, dynamic analysis, defect UDC 624.078.54:621.822:621.822.6:004.94 C ONTENTS 1 2 3 4 Introduction 1.1 Rolling Element Bearings . . . . . . . . . . . . 1.2 Spherical Roller Bearings (SRB) . . . . . . . . 1.3 Objective and Scope of the Dissertation . . . . 1.4 Outline of the Dissertation . . . . . . . . . . . 1.5 Scientific Contributions and Published Articles . . . . . . . . . . . . . . . 15 16 17 19 20 20 Rotor-Bearing System Analysis 2.1 Rotor Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Rigid Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Flexible Rotor . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Solving the Governing Equations of Motion . . . . . . . . . 2.2 Modeling the Spherical Roller Bearing . . . . . . . . . . . . . . . . 2.2.1 Dynamic Model of the Spherical Roller Bearing . . . . . . 2.2.2 Geometry of Contacting Elastic Solids . . . . . . . . . . . . 2.2.3 Geometry of Spherical Roller Bearing . . . . . . . . . . . . 2.2.4 Contact Deformation in Spherical Roller Bearing . . . . . . 2.2.5 Elastic Deformation in Spherical Roller Bearing . . . . . . 2.2.6 Calculating the Stiffness Matrix of Spherical Roller Bearing . . . . . . . . . . . . . . . . . . . . . . 23 23 23 26 29 30 32 34 35 37 39 40 . . . . . . . . . . . . . . . . . . . . . . . . . Modeling of Non-idealities in the Spherical Roller Bearing 3.1 Localized Defects . . . . . . . . . . . . . . . . . . . . . 3.1.1 Outer Race Defects . . . . . . . . . . . . . . . . 3.1.2 Inner Race Defects . . . . . . . . . . . . . . . . 3.2 Distributed Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 43 43 45 46 Numerical Examples and Experimental Verification 4.1 Case 1: Single Bearing Numerical Simulations . . . . . . . . . . 4.1.1 Single Bearing Load Analysis - Contact Forces . . . . . . 4.1.2 Elastic Deformation, Displacement, and Load . . . . . . . 4.1.3 Clearance, Displacement, and Load . . . . . . . . . . . . 4.1.4 Osculation Number, Displacement, and Load . . . . . . . 4.1.5 Number of Rollers, Displacement, and Load . . . . . . . . 4.1.6 Angular Alignment of Side-by-Side Roller Arrays . . . . 4.2 Case 2: Rigid Rotor Supported by Ideal Spherical Roller Bearings 4.2.1 Studied Structure . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . 4.3 Case 3: Single Spherical Roller Bearings with Localized Defects . 4.3.1 Studied Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 47 48 48 50 50 52 52 54 54 56 58 58 . . . . . . . . . . . . . . . . 4.4 4.5 5 4.3.2 Single Bearing Load Analysis - Outer Ring Defects . . . . . . . 4.3.3 Single Bearing Load Analysis - Inner Ring Defects . . . . . . . Case 4: Rotor Supported with Spherical Roller Bearings with Localized Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Studied Structure . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Simulation Results - Rotor with Outer Ring Defects . . . . . . 4.4.4 Simulation Results - Rotor with Inner Ring Defects . . . . . . . Case 5: Rotor Supported with Spherical Roller Bearings with Distributed Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Studied Structure . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 60 62 64 64 65 67 68 69 69 70 Conclusions 75 Bibliography 79 Appendices 87 S YMBOLS AND ABBREVIATIONS S YMBOLS a Aelem A A b B cd Cos C C d D eub e E F F G G I I J kelem ke ks K K L m me melem Mt M ne N N P major axis area amplitude steady state matrix minor axis bearing widht clearance osculation damping coefficent damping matrix diameter depth unbalance mass eccentricity displacement of a bearing elastic modulus force force vector shear modulus gyroscopic matrix moment of inertia (axis in plane) identity matrix moment of inertia (axis perpendicular to plane) single element stiffness matrix elliptic parameter shear correction factor stiffness coefficent stiffness matrix lengt of an element mass of an object semi major axis of an ellipsoid consistent mass matrix total mass mass matrix of the system semi minor axis of an ellipsoid number of rolling elements shape function matrix order of waviness q q̇ q̈ r R t T vector of generalized coordinates first time derivative of generalized coordinates second time derivative of generalized coordinates radius effective radius time transformation matrix G REEK LETTERS ψ β δ ζ θ Θ ν < ξ ρ τ φ0 φ ω Ω I , II tilting motion of a rotor roller angular position displacement / compression elliptic integral parameter angular position momentum Poisson’s ratio ratio between inner and outer radius of rotor element elliptic integral parameter density parameter given by φ from the ellipse center auxiliary angle phase angle rotational frequency rotational velocity 1, 2 S UBSCRIPTS 0 1, 2, 3, 4 A B b c cg con d D e elem ex g gy i in j p P out r rpof rpif R S t T ub W x, y, z initial state 1st , 2nd , 3rd and 4th variable solid A solid B bearing center cage contact difference defect ellipse element external gravitational gyration index of roller in a row inner index of a row pitch order of waviness outer roller roller pass outer ring frequency roller pass outer ring frequency rotor support total tangential unbalance waviness about / in direction of x, y or z -axis S UPERSCRIPTS b i in g n bearing index of body i inner global coordinates number of step in the Newton-Raphson method out tot T outer total transpose of vector or matrix A BBREVIATIONS DOF FEM FFT ICP ODE RIF SF RMS SRB VC Degrees Of Freedom Finite Element Method Fast Fourier Transform Integrated Circuit Piezoelectric Sensor Ordinary Differential Equation Roller Pass Inner Ring Frequency Shaft Rotating Frequency Root Mean Square Spherical Roller Bearing Varying Compliance C HAPTER 1 Introduction Rotating machines are important in industrial machinery that contains several components. Figure 1.1 shows some of those parts that have the most effect on the behavior of the systems, such as the main rotor and bearings. Since there is a wide range of industrial applications for rotor-bearing systems, it is becoming increasingly important to study the dynamic behavior of rotating systems. The initial approach in rotor dynamic analysis was to study different components individually. However, in order to have accurate enough results to compare with real rotor-bearing systems, it is important to consider the interaction between various elements using a computer simulation approach. Figure 1.1: Rotating system with bearings The topic of rotor dynamic analysis has been interesting and useful for researchers for many decades. A large number of books have been published, and research in this field is ongoing. Calculating the critical speed of the rotor as a major concern in designing process of rotating machineries, finding a stable and safe operation region for the system, and reducing the unexpected vibrations due to geometric imbalance in the rotor were 15 16 1 Introduction studied at the early stages [46, 49, 64] of interest. One of the simplest rotor-bearing systems is called the Jeffcott rotor, which contains a bulk mass disk carried by a massless flexible shaft and supported by two linear bearings, one at each end. However, in complex rotating systems, these simplifications are no longer valid. For this purpose, a transfer matrix method and a finite element method were used to predict the natural frequencies, mode shapes, and unbalance response of systems. The transfer matrix method is an analytical approach that is used by Prohl [62]. Ruhl and Booker [67] applied a finite element method to a rotor dynamic analysis for the first time. In this process, the rotor was defined by a divided rigid mass segment. Childs [13] developed the finite element method for complicated models. In this approach, the flexibility of the rotor and also bearing force effects could be described directly. Recently, researchers have tried to develop the finite element methods to improve the computational efficiency of simulation [76, 11]. They used a new modal analysis solution which is effective in determining the dynamic characteristics of a rotor dynamic system while in rotating condition. The recent studies on rotor dynamic systems are mostly focused on considering the effect of nonlinearity effects in the dynamic analysis of rotating equipment. Nonlinearities in rotor dynamic analysis can be caused by different elements, for instance, geometric nonlinearities [52, 56], rotor-base excitations [16, 17, 65], oil film in sliding bearings [2, 79, 50], and magnetic bearings [39]. Another type of non-linearity in rotor systems comes from the non-linear terms of bearing stiffness and damping. This doctoral thesis introduces nonlinear dynamic model of spherical roller bearing to predict the nonlinear dynamic behaviors of a rotor-bearing system. In general, rotor dynamic analysis could be done in both the frequency and the time domain. Working in the frequency domain is computationally efficient compared with a time domain analysis. It can predict the stability and natural frequencies of the rotor system. However, for analyzing the nonlinearity of the system which is come from the bearings, the steady state responses are not adequately capable. It usually needs linearization of the bearing data. Thus, the time transient analysis becomes necessary. For this purpose, by applying modal synthesis method [42, 49, 82], we can decrease the total number of system’s degrees of freedom, and by applying numerical integrators, such as different orders of the Runge-Kutta method, the governing equations of motion for a rotor system can be solved in a straightforward manner. 1.1 Rolling Element Bearings Bearings are one of the more important components in mechanical systems, and their reliable operation is necessary to ensure a safe and efficient operation of rotating machinery [28]. For this reason, a multipurpose dynamic rolling element bearing model capable of predicting the dynamic vibration responses of rotor-bearing systems is important. 17 1.2 Spherical Roller Bearings (SRB) However, bearings introduce nonlinearities, often leading to unexpected behaviors, and these behaviors are sensitive to initial conditions. For rolling element bearings, the significant sources of nonlinearity are the radial clearance between the rolling elements and raceways and the nonlinear restoring forces between the various curved surfaces in contact. A special type of non-linearity is introduced to the system if the contact surfaces have distributed defects, such as waviness, or localized defects, such as inner or outer ring defects. In order to study the rotating machinery vibration caused by the nonlinearity of a bearing, transient analysis on detailed simulation should be used. As an example, El Saidy [18] studied the shaft supported by a nonlinear rolling element bearing. A finite element equation was applied and solved in this model. An analytical model of a high speed rotor with a nonlinear rolling element bearing is introduced by Harsha [29]. Implicit type numerical integration was applied to solve the differential equation of motion. 1.2 Spherical Roller Bearings (SRB) As Figure 1.2 shows, a spherical roller bearing by itself can contain two rows of rollers, an outer ring with a spherical raceway, and an inner ring with two spherical raceways perpendicular to bearing axes, and a cage. The locking feature makes the inner ring captive within the outer ring in the axial direction only. With their two rows of large rollers, they have a high load-carrying capacity, axial forces in both directions, and high radial forces, which make them irreplaceable in many heavy industrial applications. Rollers Cage Exploded view Outer ring Figure 1.2: Components of SRB Inner ring 18 1 Introduction Because of symmetrical rollers and spherical raceways, a spherical roller bearing compensates shaft tilting and misalignments without increasing friction or reducing bearing service life (Figure 1.3). Force Figure 1.3: Shaft tilting and misalignments in SRB Spherical roller bearings are used in many applications wherever a rotation shaft might need to change the alignment of its axis or to carry high loads. Some examples of these applications include different types of stone crushers (jaw or impact crushers), vibrating screens, bulk mass handling, marine equipment, geared transmissions and modern high power wind turbines. By investigating the available literature on the dynamic analysis of rolling element bearings, we could find that extensive research has been conducted on the dynamics of ball bearings while studies related to spherical roller bearings have received short shrift. For example, Goenka and Booker [24] extended the general applicability of the finite element method to include spherical roller bearings (SRBs). In their research, triangular finite elements with linear interpolation functions were used to model the lubricant film. The loading conditions for spherical roller bearings with elastohydrodynamic and hydrodynamic lubrication effects were analyzed by Kleckner and Pirvics [45]. They simulated the mechanical behavior of spherical roller bearings in isothermal conditions. 1.3 Objective and Scope of the Dissertation 1.3 19 Objective and Scope of the Dissertation This doctoral thesis introduces a new general-purpose spherical roller bearing model developed to act as an interface element between a spinning rotor and its supporting structure. Spherical roller bearings experience point contact between the inner race, rolling element and outer race in the no-load condition and elliptical contact when loaded. The modeling approach presented in this dissertation accounts for the loaded condition and has three degrees of freedom. Its simplifying assumptions make the model computationally efficient. It is accurate enough for engineering analysis, since it can capture the most important dynamic properties of the bearing. It is also studied in [21, 22]. Model performance is demonstrated by comparing the results of two basic numerical simulations to the results obtained using both commercial bearing analysis software and the bearing radial deflection formula proposed by Gargiulo [19]. The simulations focused on the more important design parameters: diametral clearance, number of rollers, and osculation. A third numerical simulation of a full bearing system was performed to demonstrate the application of this new SRB model in a typical real world analysis. In the model, bearing forces and deflections were calculated as a function of contact deformation and bearing geometry parameters according to nonlinear Hertzian contact theory. The bearing force calculation routine can be used as a stand-alone program or as part of a bearing stiffness matrix calculation routine in a multibody or rotor dynamic analysis code. For design and analysis purposes, a versatile dynamic model capable of modeling bearing defects is important. Bearing defects are classified as distributed defects, such as surface roughness, waviness, misaligned races, and off-size rolling elements, and local defects such as cracks, pits, and spalls on the rolling surfaces [71, 73]. The defects are usually the result of manufacturing error, improper installation, or abrasive wear. These two categories of bearing imperfections are studied separately in the international journals and conference proceedings, and make up the main core of this doctoral dissertation. Effect of defect shape in dynamic behavior of the bearing is studied in [23]. Two kinds of defects were studied: elliptical and simple defects in both inner and outer rings. For the elliptical shape, the roller path the defect area without losing connection with rings. This leads to varying contact stiffness in the defect area due to variation in the contact geometry between the defect surfaces and the bearing components. In simple defect, it is assumed that the roller immediately lose its connection when it is in the defect area which does not impact contact stiffness. Finally, the proposed model is compared to and verified by measured data taken from a rotor bearing system with a predefined local defect. Bearing waviness as geometric inaccuracies might excite the rotating system to vibrate. This distributed defect is studied in [34]. 20 1.4 1 Introduction Outline of the Dissertation The thesis is divided into seven chapters with the following outline: Chapter 1: Introduction It provides the motivation, background, and objectives of the thesis. The operation principle, characteristics, and applications of spherical roller bearings are introduced. Scientific contributions and publications according to this research are also presented. Chapter 2: Rotor-bearing systems analysis Two general types of rotor modeling are presented in this chapter. Governing equation of motion for both rigid and flexible rotors is introduced. In the following, nonlinear model of spherical roller bearing model is proposed in this chapter. It contains the whole process of calculation of bearing for in general coordinate system as well as introducing the bearing geometry, nonlinear contact Hertzian contact theory and elastic deformation in spherical roller bearing under external loads. Chapter 3: Modeling of non-idealities in spherical roller bearing Different kinds of non-idealities in spherical roller bearing are introduced in this section. It provides information about applying defect parameters in bearing modeling for both local and distributed defects. Chapter 4: Numerical examples and experimental verification This chapter is introduced five different case studies. To verify the new bearing model, a series of verifying numerical calculations were carried out for a single SRB subjected to a simple radial load. Some of the simulation results were compared to results obtained by applying the formula for the spherical roller bearing radial deflection and the commercial bearing analysis software. Also, numerical simulation was carried out of a full rotorbearing system comprising a rigid rotor supported by SRBs on either end of the rotor axle. Finally, rotor-bearing system includes bearing non-idealities are presented. Chapter 5: Conclusions Conclusions and discussion of this doctoral dissertation is provides in this chapter 1.5 Scientific Contributions and Published Articles The doctoral thesis provides the following scientific contributions: • a computationally efficient three-degree-of-freedom model of a spherical roller bearing is proposed. In the model, bearing forces and deflections were calculated as a function of contact deformation and bearing geometry parameters according to nonlinear Hertzian contact theory. Transient dynamic behaviors of a rotor-SRB system and vibration cause by nonlinearity of bearing are studied; 1.5 Scientific Contributions and Published Articles 21 • distributed defects, such as the waviness of the inner and outer ring, and localized defects, such as inner and outer ring defects are taken into consideration in the proposed model. Using this model to derive a catalog of behaviors as a function of defect type and location, it is possible to identify an actual defect by examining its measured behavior and comparing it to the catalog of derived behaviors, and • the effects of important SRB parameters, such as diametral clearance, the number of rollers, and osculation number on ultimate bearing performance are presented. Some of the results presented in this thesis have also been published in the following papers: • Ghalamchi Behnam, Sopanen Jussi, Mikkola Aki, ‘Simple and Versatile Dynamic Model of Spherical Roller Bearing.’ International Journal of Rotating Machinery. 2013. • Heikkinen Janne, Ghalamchi Behnam, Sopanen Jussi, Mikkola Aki, 2014, ‘TwiceRunning-Speed Resonances of a Paper Machine Tube Roll Supported by Spherical Roller Bearings – Analysis and Comparison with Experiments’, Proceedings of the ASME International Design Engineering Technical Conferences IDETC2014, August 17–20, 2014, Buffalo, NY, USA • Ghalamchi Behnam, Sopanen Jussi, Mikkola Aki, ‘Nonlinear Model of Spherical Roller Bearing Including Localized Defects’ , 9th IFToMM International Conference on Rotor Dynamics, 2014, Milan, Italy • Ghalamchi Behnam, Sopanen Jussi, Mikkola Aki, 2014, ‘Dynamic model of spherical roller bearing’, Proceedings of the ASME International Design Engineering Technical Conferences IDETC2013, 2013, Portland,OR, USA • Ghalamchi Behnam, Sopanen Jussi, Mikkola Aki, ‘Nonlinear Model of Spherical Roller Bearing Including Localized Defects’. (To be submitted). 22 1 Introduction C HAPTER 2 Rotor-Bearing System Analysis 2.1 Rotor Dynamic Analysis In scientific research, rotors are described as two general types: rigid and flexible. In the flexible rotor, all flexible eigenfrequencies of a system at low frequencies that are crossed during the run-up and run-down are considered. In the rigid rotor, the effects of these flexible eigenfrequencies are neglected [35]. In fact, a pure rigid rotor is just a simplified model of a real rotor, which could decrease the total number of degrees of freedom in systems and prevent complex mathematical modeling. The governing equation of motion in both rigid and flexible rotors, based on Newton’s II law of motion, can be expressed as follows: Mq̈(t) + (C + ΩG)q̇(t) + Kq(t) = F (t) (2.1) where M is the mass matrix, q is the displacement vector, q̇ and q̈ are the first and second time derivatives of generalized coordinates C is the damping matrix, Ω is rotation speed, G is the gyroscopic matrix, K is the stiffness matrix, F is a force vector, and t is a time variable which will not be shown in the following equations. 2.1.1 Rigid Rotor For a rigid rotor, the effect of internal damping can be neglected [20], so it should be equal to zero ( i.e., C = 0). According to rigid rotor assumptions, the stiffness matrix also should be zero (K = 0). Therefore, the equation of motion of a rigid rotor in the center of mass coordinates can be written as: MRc q̈ Rc + ΩGRc q̇ Rc = F Rc 23 (2.2) 24 2 Rotor-Bearing System Analysis T In this equation, q Rc = x y βx βy , showing the transversal and tilting motions of the rotor in the x and y directions (Figure 2.1), and subscripts R and c respectively, refer to the rotor and center of mass of the rotor. Equation (2.3), which follows, presents the mass matrix MRc , the gyroscopic matrix GRc , and the force vector F Rc in center of gravity coordinates. MRc GRc F Rc mR 0 0 mR = 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 −1 Fx Fy = Θx Θy 0 0 0 0 Ix 0 0 Iy 0 0 I 1 z 0 (2.3a) (2.3b) (2.3c) In Equation (2.3), mR is the rotor mass, and Ix and Iy are the transversal moments of inertia about the x- and y axes, respectively, and Iz is the polar moment of inertia about the z axis. F and Θ denote force and moments on their axes. The rotor is assumed to be axisymmetric, so Ix = Iy . y Bearing II ψy Bearing I z ψx CS KS dI dII CS KS Figure 2.1: Rigid rotor modeling using four degrees of freedom According to Equation (2.2) The equation of motion of rigid rotor in bearing coordinates is as follows: 25 2.1 Rotor Dynamic Analysis (2.4) T Subscript b refers to the bearings. In bearing coordinates, q Rb = xI yI xII yII shows the rotor displacements at the SRBs in positions I and II in the x and y directions. MRb and GRb can be calculated as below: MRb q̈ Rb + ΩGRb q̇ Rb = F Rb MRb = TT 2 MRc T2 (2.5a) GRb = TT 2 GRc T2 (2.5b) T where T2 = T− 1 . Transformation matrix T1 can be defined as follows: 1 0 T1 = 0 −dI 0 1 −dI 0 1 0 0 dII 0 1 dII 0 (2.6) In Equation (2.4), F Rb includes the external forces (F ex ), the bearing forces (F b ), is calculated in Equation (2.67), the gravity forces for the rigid rotor (F g ), and the unbalance forces as shown by Equation (2.7). F Rb = F ex + F b + F g + F ub (2.7) For the bearing housing, the equation of motion also can be written as: MS q̈ S + CS q̇ S + KS q S = 0 (2.8) T where subscript S refers to the supports and q S = xSA ySA xSB ySB , shows the displacements of the bearings housing in the x and y directions. The mass matrix MS , the damping matrix CS , and the stiffness matrix KS are presented as follows: MS = mS I4 (2.9a) CS = CS I4 (2.9b) KS = KS I4 (2.9c) where I4 is a 4×4 identity matrix. Finally, for the whole rotor-SRB system, the assembly matrix according to Equations, (2.4) and (2.8) can be written as follows: MRb 0 q̈ Rb ΩGRb 0 q̇ Rb 0 0 q Rb F Rb + + = 0 MS q̈ S 0 CS q̇ S 0 KS qS 0 (2.10) 26 2 Rotor-Bearing System Analysis yi ψy,i xi ψz,i zi ψx,i z yi+1 ψy,i+1 xi+1 ψz,i+1 zi+1 ψx,i+1 Figure 2.2: Three dimensional beam element 2.1.2 Flexible Rotor In rotor dynamics the finite element method (FEM) is the most convenient method to describe complex real-world machinery. In this study, the basic element is a 12-degreeof-freedom three-dimensional two-node beam element (Figure 2.2), whose stiffness, damping, and gyroscopic matrices can be found in [1]. Each node of the element has six-degrees-of-freedom. The stiffness matrix kelem for the used 3D beam element can be written as: Aelem E L kelem 0 0 0 0 0 = Aelem E − L 0 0 0 0 0 ay 0 az 0 0 GJ L 0 cz −cy 0 0 0 ey 0 ez 0 0 0 0 0 Aelem E L 0 0 0 cz −cy 0 0 0 ay 0 az − GJ L 0 0 0 0 0 GL L fy 0 0 fz 0 0 0 −cy cz 0 0 0 −ay 0 0 −az 0 0 0 cz −cy 0 Symmetric 0 0 ey 0 ez (2.11) 27 2.1 Rotor Dynamic Analysis where Aelem is element area, E is elastic modulus, L is element length, J is perpendicular moment of inertia, and G is shear modulus. Note that the cross-section of the rotor is assumed to be symmetrical. This means that coefficients ay = az , ey = ez , cy = cz , fy = fz . These coefficients are identical as follows: 12EI L3 (1 + φ) (2.12) 6EI + φ) (2.13) ey = ez = (4 + φ)EI L(1 + φ) (2.14) fy = fz = (2 − φ)EI L(1 + φ) (2.15) ay = az = cy = cz = L2 (1 where φ= 12EI ks GAelem L2 (2.16) where I is moment of inertia of the beam cross-section and ks is the shear correction factor that takes into account the non-uniform shear stress distribution over the beam crosssection. The shear correction factor can be defined for a hollow circular cross-section as follows [12]: ks = 6(1 + ν)(1 + <2 )2 (7 + 6ν)(1 + <2 )2 + (20 + 12ν)<2 (2.17) where ν is posson’s ration and < is the ratio between inner and outer radius of rotor element.. The consistent mass matrix melem for the used 3D beam element can be written as: 28 2 Rotor-Bearing System Analysis 1 3 melem 0 Ay 0 0 Az 0 0 0 0 −Cy 0 0 0 Cz = Mt 1 0 0 6 0 B 0 y 0 0 Bz 0 0 0 0 0 Dy 0 −Dz 0 Symmetric Jx 3A 0 0 Ey 0 Ez 0 0 0 0 0 Jx 6A 0 Dy −Dz 0 0 0 1 3 0 0 Ay 0 Az 0 0 0 0 0 Fy 0 0 Fz 0 0 Cy 0 −Cz 0 Jx 3A 0 0 Ey 0 Ez (2.18) where Mt = ρAelem L Ay = Az = By = Bz = Cy = Cz = 11 ( 210 + Dy = Dz = Ey = Ez = Fy = Fz = 13 ( 420 + 1 + ( 105 1 −( 140 + 1 60 φ 1 60 φ + 13 35 + 7 10 φ + 13 φ2 + 65 ( Lr )2 (1 + φ)2 9 70 + 3 10 φ + 16 φ2 − 65 ( Lr )2 (1 + φ)2 11 120 φ + 1 2 24 φ 3 40 φ + 1 2 24 φ + 1 + ( 10 − 21 φ)( Lr )2 )L (1 + φ)2 1 2 120 φ 1 2 120 φ (2.19) (2.20) (2.21) (2.22) 1 − ( 10 − 21 φ)( Lr )2 )L (1 + φ)2 (2.23) 2 + ( 15 + 16 φ + 31 φ2 )( Lr )2 )L2 (1 + φ)2 (2.24) 1 + ( 30 + 61 φ − 16 φ2 )( Lr )2 )L2 (1 + φ)2 (2.25) 29 2.1 Rotor Dynamic Analysis and where the radius of gyration is defined as: r rgy = I Aelem (2.26) Matrices are formed using a standard assembly procedure of FEM. A generalized displacement, that is, the nodal displacement is obtained using a shape function matrix N. q g = Nq (2.27) The superscript g refers to global coordinates. The shape function matrix is constructed using beam elasticity theory ([12, 47]). The complete system consists of the generalized displacements of all the nodes in global coordinates. The equation of motion of the complete system is Mg q̈ g + (Cg + ΩGg )q̇ g + Kg q g = F g (2.28) where q g is the global displacement vector. 2.1.3 Solving the Governing Equations of Motion Rotor dynamic analysis could be done in both frequency and time domain. In general, working on frequency domain is computationally efficient, compare with time domain analysis. It can provide the stability and natural frequencies of the rotor system. By applying the state variable form of a dynamic system, second order differential equations can be expressed as a set of simultaneous first order differential equations. To reduce the differential equation which is shown in Equation (2.28) into a system of first order differential equations, let q = x1 (2.29a) q̇ = x2 (2.29b) q̈ = x3 (2.29c) From Equation (2.29) we have dq = x2 dt dq̇ ẋ2 = = x3 dt ẋ1 = (2.30a) (2.30b) 30 2 Rotor-Bearing System Analysis From Equations (2.29 ) and (2.30), Equation (2.28) can be re-written as: ẋ2 = M−1 F − M−1 (C + ΩG)x2 − M−1 Kx1 (2.31) Finally, the state space form of equation of motion is calculated as follows ẋ1 x1 0 =A + ẋ2 x2 M−1 F (2.32) where A is a state matrix which defined as 0 I A= −M−1 K −M−1 (C + ΩG) (2.33) In this case, according to the eigenvalues solution from Equation ( 2.33 ), a Campbell diagram, which relates the rotational speed and the natural frequencies of the system, and also system mode shapes could be calculated. Frequency Domain Solution: The term frequency response, refers to the steady state response of a system. It is the response of the system to harmonic excitation. In this case, steady state solution can be obtained by defining the displacement vector , by sinusoidal function, as follows: q(t) = asinωf t + bcosωf t (2.34) where the vectors a and b are coefficient vectors and t is time. By deriving the first and second order of Equation ( 2.34 ) and applying those to Equation ( 2.28 ), steady state solution can be obtained. Transient Analysis: In the real world, in rotor bearing systems, it sometimes happens that the frequency domain analysis does not provide adequate capability. Therefore time transient analysis becomes necessary. In this case, the governing equation of motions is integrated directly by applying numerical integrators such as different orders of RungeKutta method. 2.2 Modeling the Spherical Roller Bearing A spherical roller bearing consists of a number of parts, including a series of rollers, a cage, and the inner and outer raceways. Describing each component in detail can result in a simulation model with a large number of degrees-of-freedom. Additionally, as with all radial rolling bearings, spherical roller bearings are designed with clearance. This clearance also increases the computational complexity of the system. However, bearing 2.2 Modeling the Spherical Roller Bearing 31 analysis computation should be efficient so it can be used to simulate the dynamics of complete machine systems. Creju et al. [14, 15] improved the dynamic analysis of tapered roller bearings by improving integration of the differential equations describing the dynamics of the rollers and bearing cage. Their study considered the effects of centrifugal forces and the gyroscopic moments of the rollers. The effects of the correction parameters for roller generators in spherical roller bearings were discussed by Krzemiński-Freda and Bogdan [48]. In their study, they focused on determining a proper ratio of osculation coefficients for both races to obtain the self-stabilization of the barrel shaped roller and to minimize friction losses. Olofsson and Björklund [58] performed 3D surface measurements and an analysis on spherical roller thrust bearings that revealed the different wear mechanisms. A theoretical model for estimating the stiffness coefficients of spherical roller bearings was developed by Royston and Basdogan [66] showing that coefficient values are complicated functions, dependent on radial and axial preloads. While this work is useful for qualitative analysis, it cannot deliver the dynamic insights needed for understanding the high performance machine systems. Olofsson et al. [57] simulated the wear of boundary lubricated spherical roller thrust bearings. A wear model was developed in which the normal load distribution, tangential tractions, and sliding distances can be calculated to simulate the changes in surface profile due to wear. Taking into account internal geometry and preload impacts, Bercea et al. [6] applied a vector-and-matrix method to describe total elastic deflection between double-row bearing races. However this study focused only on static analysis. Therefor, it is not capable of delivering a detailed analysis of the complex dynamic behaviors of spherical roller bearing systems involving nonlinear interactions between rollers and inner/outer races. Cao et al. [9, 10] established and applied a comprehensive spherical roller bearing model to provide quantitative performance analyses of SRBs. In addition to the vertical and horizontal displacements considered in previous investigations, the impacts of axial displacement and load were addressed by introducing degrees-of-freedom in the axial shaft direction. The point contacts between rollers and inner/outer races were considered. These bearing models have a large number of degrees-of-freedom since there is one degree-of-freedom (DOF) for each roller and an additional 3 to 5 DOFs for the inner race. Its high complexity makes this bearing model unattractive for the analysis of complete rotor-bearing systems. For example, a single gear-box can contain up to ten roller bearings. The effect of centrifugal forces on lubricant supply layer thickness in the roller bearings was considered by Zoelen et al. [75]. In particular, this model is used to predict lubricant layer thickness on the surface of the inner and outer raceways and each of the rollers. In 32 2 Rotor-Bearing System Analysis this extended model it is assumed that the lubricant layers for each of the roller raceway contacts are divided equally between the diverging surfaces. Although a large number of ball bearing models exist, there has been little study of spherical roller bearing dynamics. For example, Harsha et al. [30, 32] studied the rolling element dynamics for certain imperfect configurations of single row deep-grooved ball bearings. The study revealed dynamic behaviors that are extremely sensitive to small variations in system parameters, such as the number of balls and the number of waves. A dynamic model of deep-groove ball bearings was proposed by Sopanen and Mikkola [69, 70]. They considered the effects of distributed defects such as surface waviness and inner and outer imperfections. In this study, to improve the computational efficiency of the proposed spherical roller bearing model the following simplifications have been introduced. • cage movement is based on the geometric dimensions of the bearing; therefore, it is assumed that no slipping or sliding occurs between the components of the bearing and all rollers move around the raceways with equal velocity. • the inner raceway is assumed to be fixed rigidly to the shaft. • there is no bending deformation of the raceways. Only nonlinear Hertzian con-tact deformations are considered in the area of contact between the rollers and raceways. • the bearings are assumed to operate under isothermal conditions. • rollers are equally distributed around the inner race, and there is no interaction between them; and • the centrifugal forces acting on the rollers are neglected. 2.2.1 Dynamic Model of the Spherical Roller Bearing The bearing stiffness matrix and bearing force calculation routines are implemented according to the block diagrams shown in Figure 2.3. The bearing geometries, material properties and the displacements between the bearing rings are defined as inputs. For the stiffness matrix calculation routine, the external force on the bearing is given as an input. The bearing force calculation routine can be used as a stand-alone program or as part of a bearing stiffness matrix calculation routine in a multi-body or rotor dynamic analysis code. 33 2.2 Modeling the Spherical Roller Bearing Start Read input Read input • SRB Geometry • Displacements External force • SRB material • Step size Initial Displacements Contact stiffness coefficents Rows Rollers Updated Displacements Contact elastic deformation Bearing force calculation Bearing force No |External force − Bearing force| < tolerance? Elastic compression > 0? No Yes Roller contact force Summation of rollers contact forces Yes Saving KT as bearing stiffness matrix Roller number > total number of the rollers? No Yes Row number > total number of the rows? No Yes Bearing force End Figure 2.3: Block diagram for the bearing stiffness matrix and bearing force calculation In the following sections, the theory behind the bearing force and bearing stiffness matrix 34 2 Rotor-Bearing System Analysis calculation is explained in detail. 2.2.2 Geometry of Contacting Elastic Solids Two solids that have different radii of curvature in two directions (x and y) are in point contact when no load is applied to them. When the two solids are pressed together by a force F , the contact area is elliptical. For moderately loaded spherical roller bearings, the contact conjunction can be considered elliptical [45], as shown in Figure 2.4. The following analysis will assume the curvature is positive for convex surfaces and negative for concave surfaces [25]. F rAx rBy rBx y rAy x Figure 2.4: Elliptical contact conjunctions where rAx , rAy , rBx , and rBy are the radii of curvatures for two solids (A and B). The geometry between two solids in contact can be expressed in terms of the curvature sum R, and curvature difference Rd as follows [8, 26] 1 R Rd 1 1 + and Rx Ry 1 1 = R − Rx Ry = (2.35) (2.36) The curvature sums in x and y are defined as per Equations (2.37) and (2.38). 1 Rx 1 Ry = = 1 + 1 rAx rBx 1 1 + rAy rBy and (2.37) (2.38) Variables Rx and Ry represent the effective radii of curvature in the principal x- and y-planes. When the two solids have a normal load applied to them, the point expands to 35 2.2 Modeling the Spherical Roller Bearing an ellipse with a being a major axis and b being the minor axis. The elliptic parameter is defined as [25]: ke = a . b (2.39) The elliptic parameter can be re-defined as a function of the curvature difference Rd and the elliptic integrals of the first ξ and second ζ kinds as follows [26]: 2ξ − ζ (1 + Rd ) ke = ζ (1 − Rd ) 1/2 , (2.40) Equations (2.41) and (2.42) define the first and second kinds ξ and ζ . −1/2 1 2 1 − 1 − 2 sin ϕ ξ = dϕ and ke 0 1/2 Z π/2 1 1 − 1 − 2 sin2 ϕ ζ = dϕ ke 0 Z π/2 (2.41) (2.42) Brewe and Hamrock [8] used numerical iteration and curve fitting techniques to find the following approximation formulas for the ellipticity parameter ke and the elliptical integrals of the first ξ and second ζ kinds as shown below. ke = 1.0339 Ry Rx 0.6360 , Rx , Ry Ry ζ = 1.5277 + 0.6023 ln . Rx ξ = 1.0003 + 0.5968 2.2.3 (2.43) (2.44) (2.45) Geometry of Spherical Roller Bearing The most important geometric dimensions of the spherical roller bearing are shown in Figure 2.5. Diametral clearance is the maximum diametral distance that one race can move freely. Osculation is defined as the ratio between the roller contour radius and the race contour radius as written in Equation (2.46). Cos = rr rin , rout (2.46) 36 2 Rotor-Bearing System Analysis Subscripts r, in, and out refer to roller, inner race, and outer race, respectively. Perfect osculation is when Cos is equal to 1. In general, maximum contact pressure between the race and the roller decreases as osculation increases. Decreasing contact pressure reduces fatigue damage to the rolling surfaces; however, there is more frictional heating with increasing conformity. A reasonable value for osculation and one that can be used in the roller contour radius definition is 0.98 [28]. rin φ0 cd 4 dr rr rout dp 2 B Figure 2.5: Dimension of spherical roller bearing Figure 2.6 illustrates the radii of curvature between roller, outer race, and inner race of an SRB. out O–1 in ,r out rAx Ax out rAy in rBy in out rBx in rAy in rBx out rBy O–2 Figure 2.6: Radii of curvature between roller, outer race, and inner race 37 2.2 Modeling the Spherical Roller Bearing Figure 2.6 suggests the radii of curvature for the roller-to-inner race contact area can be written as follows: dr , 2 = rr , (2.47) in rAx = in rAy (2.48) dp − dr cos φ0 − 2 cos φ0 = −rin . in rBx = in rBy cd 2 cos φ0 , (2.49) (2.50) Similarly, the equations for the radii of curvature for roller-to-outer race contact can be written: dr , 2 = rr , (2.51) out rAx = out rAy (2.52) dp + dr cos φ0 + 2 cos φ0 = −rout . out rBx = − out rBy 2.2.4 cd 2 cos φ0 , (2.53) (2.54) Contact Deformation in Spherical Roller Bearing From the relative displacements between the inner and outer race the resultant elastic deformation of the ith rolling element of the j th row located at angle βji from the xaxis can be determined. The initial distance, A0 , between the inner and outer raceway curvature centers (O–1, O–2) can be written, again based on Figure 2.6, as given by Equation (2.55). out in + rBy − dr − cd A0 = rBy 2 (2.55) The corresponding loaded distance for roller i in row j can be written as follows: A βji = r i δzj 2 2 i + δrj (2.56) i and δ i are the displacements for roller i in row j in axial and radial directions, where δzj rj respectively, which can be determined using these equations. 38 2 Rotor-Bearing System Analysis (a) ith (b) y A-A plane y A roller ωout βji ωin ez ey x z ey A ex Figure 2.7: (a) Axial and (b) transverse cross-section in the A-A plane of spherical roller bearing i δzj i δrj = A0 sin(φ0 ) + ez , = A0 cos(φ0 ) + (2.57) ex cos(βji ) + ey sin(βji ), (2.58) The variables ex , ey and ez represent displacements in the global xyz-coordinate system, and βji is the attitude angle of roller i in row j (See Figure 2.7). The initial contact angle is negative for the 1st row and positive for the 2nd row of the bearing. The distance between race surfaces along the common normal is given by Equation (2.59). out in + rBy − A βji . d βji = rBy (2.59) Elastic compression becomes: δβ i = dr − d βji . (2.60) j And, the loaded contact angle in each roller element can be defined as follows: φij −1 = tan i δzj i δrj ! . (2.61) 39 2.2 Modeling the Spherical Roller Bearing 2.2.5 Elastic Deformation in Spherical Roller Bearing In a single rolling element, total deflection is the sum of the contact deflections between the roller and the inner and outer races. The deflection between the roller and the race can be approximated as given by Equation (2.62) [45]. δ0 = Fcon Kcon 2/3 , (2.62) the Fcon denotes normal load, and Kcon is the contact stiffness coefficient, which can be calculated using the elliptic integrals and ellipticity parameter in this manner. s Kcon = πke E 0 Rξ , 4.5ζ 3 (2.63) The effective modulus of elasticity E 0 is defined as follows: 1 1 = 2 E0 2 2 1 − νA 1 − νB + EA EB , (2.64) E and ν are the modulus of elasticity and Poisson’s ratio of solids A and B. The total stiffness coefficient for both inner and outer race contact areas can be expressed with the next equation. tot Kcon = 1 in )−2/3 + (K out )−2/3 (Kcon con 3/2 . (2.65) According to Equations (2.60) and (2.65), the contact force for roller i in row j can be calculated in this manner 3/2 tot Fji = Kcon δβ i . j (2.66) Finally, the total bearing force components acting upon the shaft in the x, y, and z directions can be written according to following equations. 40 2 Rotor-Bearing System Analysis Fxb = − Fyb = − Fzb = − 2 X N X Fji cos φij cos βji , (2.67) Fji cos φij sin βji , and (2.68) Fji sin φij , (2.69) j=1 i=1 2 X N X j=1 i=1 2 X N X j=1 i=1 The variable N is the number of rolling elements in each row. In Equations (2.67-2.69), only positive values of the contact force Fji are taken into account and Fji = 0 for the negative values. 2.2.6 Calculating the Stiffness Matrix of Spherical Roller Bearing In general, an accurate estimation of the SRB stiffness matrix is needed which, since it has a significant effect on the static and dynamic analyses of rotating mechanical systems. According to Equation (2.67), for a known given load, the displacements of bearing are calculated using the Newton-Raphson iteration procedure as follows: (n) q (n+1) = q (n) − (KT )−1 F (n) (2.70) (n) The displacement values can be calculated at step n + 1. In Equation (2.70), KT is the tangent stiffness matrix, and vector F (n) includes the bearing forces and external forces at iteration step nth as follows: (n) F (n) = F b − F (n) ex (2.71) The tangent stiffness matrix can be written as: (n) KT = ∂F (n) ∂q (n) (2.72) It can be seen that the tangent stiffness matrix is essentially a matrix of sensitivities. In particular, it is the sensitivities of the internal member forces to perturbations in the nodal displacement DOFs of the system. In this calculation process, the convergence criterion for the iteration is defined as follows: |F | < 0.001 · |F ex | (2.73) C HAPTER 3 Modeling of Non-idealities in the Spherical Roller Bearing Thus far, a large number of bearing dynamic behavior studies have been published that target distributed defects such as waviness. Wardle [78] derived a vibration force analysis approach to determine the wavelength of surface imperfections in thrust loaded ball bearings. In ball bearing defect analysis, most studies have targeted non-linear dynamic bearing behaviors resulting from surface waviness [59, 36, 37, 38, 30, 33, 5, 31]. Typically, a sinusoidal function is defined as the model of waviness. In some cases, waviness studies include the effects of centrifugal force and gyroscopic moment as well as the internal clearances of the ball bearings. Also, there are some vibration studies for angular contact ball bearings [3, 4] and for roller bearings [32, 77] with inner and outer races and with rolling surface waviness. For local defects in ball and roller bearings, several works offer mathematical models for the detection of bearing defects. McFadden and Smith [53, 54] carried out one of the first mechanical modeling studies of localized bearing defects. Initially, they presented a vibration model with single point defects in the inner raceway, also considering the effects of bearing geometry, shaft speed, and bearing load distribution. Finally, they extended their model to include multiple point defects under radial load. Vibration was modeled as the product of a series of impulses that occur at the frequency of the interaction between rolling elements and defects, olso known as the rolling element passing frequency. Tandon and Choudhury [73] presented a model for predicting the vibration frequencies of rolling bearings resulting from a localized defect on the outer race, inner race, or on one of the rolling elements under radial and axial loads. They showed, in the characteristic defect frequencies, that a discrete spectrum has a peak. The “healthy and faulty” model for a rolling element bearing based on a multi-body dynamic approach was proposed by Nakhaeinejad and Bryant [55]. They studied the effects of a localized fault type on vibration response in rolling element bearings. 41 42 3 Modeling of Non-idealities in the Spherical Roller Bearing Sopanen and Mikkola [69, 70] proposed a model for deep-groove ball bearings include localized and distributed defects. They defined the distributed defects as a Fourier cosine series, and the shapes of local defects were described using their length and height. This model could be used in general multi-body or rotor dynamics code as an interference element between the rotor and the housing. Rolling element bearings analysis with various defects on different components of the bearing structure was proposed by Kiral and Karagulle [44]. They employed a finite element vibration analysis to detect defects in rolling element bearings, and also studied the effects on the time and frequency domain parameters of defect location and quantity. Rafsanjani et al. [63] studied the topological structure and stability of rotor roller bearing systems, including surface defects and internal clearance, by applying the classical Floquet theory. They linearized the nonlinear equation of motion and solved the equations numerically by using the modified Newmark time integration technique. For deep groove ball bearings with single and multiple race defects, vibration analysis was presented by Patel et al. [60], wherein the vibration results were presented in the frequency domain; ignored in earlier studies. Kankar et al. [40] studied fault diagnosis for high speed rolling element bearings with local defects in inner and outer races and also rollers by using a response surface technique. This model predicts peaks in a discrete spectrum at the characteristic frequencies. The model was validated with measurement data. Tadina and Boltezar [72] improved the vibration simulation of ball bearing with local defects during run-up. In the proposed model, local defects are defined using an ellipsoidal depression in the inner and outer races and a flattened sphere on the roller, which leads to different contact stiffness in the defect area. Liu et al. [51] studied vibration analysis methods for ball bearings with a local defect by applying a piecewise response function. The amplitude of the impulse generated by a ball bearing passing over a local defect was determined with respect to the shape and size of the local defect. The effects of localized defects, such as micron-sized spalls, on the vibration response of ball bearings were studied by Kankar et al. [41]. They proposed a mathematical model based on Hertzian elastic contact deformation theory to predict the characteristic defect frequencies and amplitudes under radial load. There are also many studies on measurement techniques for the detection of bearing defects and health monitoring. For one of the first studies, Tandon and Choudhury [74] reviewed the vibration and acoustic measurement methods for the detection of localized and distributed defects in rolling element bearings. They compared several time domain parameters, such as overall Root Mean Square (RMS) level, crest factor, probability density, and kurtosis, and found kurtosis to be the most effective. These kinds of studies have continued, and especially in recent years, new methods have been presented [7, 43, 50, 61, 68, 80, 81]. According to literature reviews, extensive research has been conducted to study the 3.1 Localized Defects 43 analysis of ball bearing defects. There are fewer studies related to spherical roller bearings. Cao and Xiao [10] presented a dynamic model for a spherical roller bearing that included the effect of surface defects, preloads, and radial clearance in the dynamic behavior of the bearing. They assumed point and line contacts between rollers and the inner and outer races according to different bearing forces. They also defined race and roller surface waviness as sinusoidal functions. As a regular Hertz point contact problem for the defect point, they were able to consider large point defects in their numerical calculations, assuming two degrees of freedom for each roller, which led to a large number of degrees of freedom for the system and a correspondingly computationally inefficient model. The objective of this study is to propose a simple and efficient dynamic model of the spherical roller bearing with local defects in the inner and outer races. The contact forces between rollers and the inner and outer races are described according to nonlinear Hertzian contact deformation. The effect of radial clearance has been taken into account. The defect geometry for the inner and outer race is an impressed ellipsoid. This leads to varying contact stiffness in the defect area due to variation in the contact geometry between the defect surfaces and the bearing components. In addition, a simple defect shape, which does not impact contact stiffness, is modeled. The results are compared to those of elliptical defects. 3.1 Localized Defects In most cases, the non-idealities, such as the waviness of the rings, misalignments and bearing defects, are the main reason for the vibration in the bearing. In this research, the vibration response according to the local defects on the inner and outer race, and also on rollers, is studied. In most of the previously published studies of rolling element bearings with local defects, it was usually assumed that the rolling elements in the region of the defect lose their connection to the inner or outer race, thus causing a large impulsive force on rotor bearing systems [73]. Some recent studies [72] have proposed modeling the bearing defect as an ellipsoidal shape while taking into account the contact stiffness variation in the defect location. In this study, both these modeling approaches are studied and compared in the case of a spherical roller bearing. 3.1.1 Outer Race Defects It is assumed that the center of defect on the outer race is located at angle φ from the horizontal axis. Correspondingly, the angle occupied by the defect is equal to φD and the maximum depth of the defect is Dd as shown in Figure 3.1. When a roller with an attitude angle of βji passes the defect, the contact elastic deformation can be calculated as follows: 44 3 Modeling of Non-idealities in the Spherical Roller Bearing δ(βji )D = dr − d(βji ) − δ + (βji ) (3.1) In comparing Equations (3.1) and (2.60), δ + (βji ) is the additional displacement in the dent area, which is subtracted from total displacement (see Appendix A). Note that the additional displacement is defined only when φ − φ2D < βji < φ + φ2D . φ βji δ + (βji ) θcon con φD Dd Figure 3.1: Outer race defect As mentioned above, the elliptical defect leads to different contact stiffness in the dent area, since the radii of curvature in the dent area is different compared to that of the race. According to Hertzian contact theory, a new curvature sum and difference as well as new contact stiffness can be evaluated (Equations (2.35) to (2.44)). However, the evaluation of new contact stiffness in the dent area, in the case of the elliptical defect, is straightforward; out (Equation (2.53)) has be re-evaluated as follows [72]: only the radius of curvature rBx out rBx = m e ne 3 2 (m2e sin τ + n2e cos2 τ ) 2 (3.2) where me is the semi major axis, ne is the semi minor axis of the ellipsoid and τ is a parameter given by the polar angle φ from the ellipse center as: −1 τ = tan me tan φ ne (3.3) From Equations (2.66) and (3.1) and by re-calculating contact stiffness, the contact force for roller i in row j can be calculated as follows: 45 3.1 Localized Defects 32 i tot Fj,D = kcon,D δ βji ,D cos βji − θcon (3.4) tot where subscript D refers to the defect area, kcon,D is the contact stiffness, and the θcon is the contact angle of the roller (see Figure 3.1) to get the exact force direction to the center of bearing. 3.1.2 Inner Race Defects Assuming that the elliptical defect on an inner race with a length equal to φD and with a maximum depth of Dd (Figure 3.2), the contact elastic deformation when the roller with attitude angle of βji passes the defect, will be calculated as follows: δ βji ,D (3.5) = δ + βji − dr + d βji and the contact force for roller i in row j can be calculated as follows: 23 i tot Fj,D = kcon,D δ βji ,D cos θcon − βji (3.6) tot where kcon,D is the contact stiffness which follows the same procedure of Equations (3.2) and (3.3). βji Dd θcon δ + βji φD Figure 3.2: Inner race defect 46 3.2 3 Modeling of Non-idealities in the Spherical Roller Bearing Distributed Defects The waviness in the bearing ring can be presented as a Fourier cosine series as follows R(θ) = n X AP cos (P θ + φP ) (3.7) P =1 where AP is the amplitude and φP is the phase angle of P th order waviness, and θ is the angular coordinate of the bearing ring which is shown in Figure 3.3. The amplitudes and phase angles of the harmonic components can be obtained from measurements by analyzing the results with the fast Fourier transform (FFT). y ith roller x Am Figure 3.3: Race waviness model The effect of the waviness of the bearing ring can be included in the elastic deformation of roller i in row j, (Eq. 2.24) as follows: δβ i j ,W = δβ i + n X j P =1 i Ain P cos P βj − θin + φP where the summation describes the waviness of the inner ring. (3.8) C HAPTER 4 Numerical Examples and Experimental Verification 4.1 Case 1: Single Bearing Numerical Simulations This study introduces a computationally efficient, three-degree-of-freedom, SRB model that was developed to predict the transient dynamic behaviors of a rotor-SRB system. To verify the new bearing model, a series of verifying numerical calculations were carried out for a single SRB subjected to a simple radial load. The SRB modeled was a double-row spherical roller bearing (FAG 21322-E1-TVPB) with 16 roller elements in each row. Table 4.1 gives the relevant dimensions and parameters of the roller bearing, which were used to define the model. All numerical calculations were performed using Matlab-2011b. Some of the Matlab results were compared to results obtained by applying the formula for the spherical roller bearing radial deflection (Gargiulo [19]) and the commercial bearing analysis software BearinX provided by the Schaeffler Group. The model verification analysis series comprised six sets of Matlab numerical calculations, each focused on a specific area of behavior. In the first set, roller contact forces were calculated for four levels of radial load. The second set explored the relationship between bearing displacement and load. A third set of calculations established how SRB elastic deformation changes with load as a function of diametral clearance. The effect of osculation number on bearing displacement for different levels of radial loading was the area of focus of the fourth set of calculations. The fifth set looked at changes in displacement as a function of radial load and the number of bearing rollers. Finally, the last pair of verification calculations looked at bearing displacement with respect to time for two separate arrangements of the caged side-by-side roller arrays; when they are aligned and when they are staggered. 47 48 4 Numerical Examples and Experimental Verification Table 4.1: Dimensions and parameters of the spherical roller bearing free contact angle roller diameter inner raceway contour radius outer raceway contour radius roller contour radius bearing width clearance pithc diameter number of rows number of rolling elements in one row modulus of elasticity Poisson’s ratio 4.1.1 φ0 dr rin rout rr B cd dp nz N E ν 7.92 29 106.61 106.61 103.95 50 41 175 2 16 206 0.3 degree mm mm mm mm mm µm mm GPa - Single Bearing Load Analysis - Contact Forces The first set of calculations was performed to verify that the newly developed SRB model would correctly simulate how roller contact forces change with increasing load. Figure 4.1 shows the calculated contact force distribution for radial loads in the y direction of 4, 6, 8, and 10 kN. The figure demonstrates, as theory predicts, that at the input diametral clearance of cd = 41 µm, fewer rollers support the lowest applied radial load, and as the load increases, the number of supporting rollers increases. 4.1.2 Elastic Deformation, Displacement, and Load The second set of calculations explored the relationship between bearing displace-ment and load. Figure 4.2 illustrates the predicted relationship between the applied radial force and bearing displacement, and shows that displacement increases with the increasing load. This eccentric displacement of the rotating bearing centers is a result of bearing radial clearance (cd = 41 µm) and the elastic deformations occurring at the regions of roller-to-race contact. In Figure 4.2, the red curve shows the behavior predicted by the BearinX commercial bearing software and the black one shows the behavior predicted by the Gargiulo [19] formula for the spherical roller bearing radial deflection. 49 4.1 Case 1: Single Bearing Numerical Simulations F = 975N F = 975N F = 565N F = 565N F = 565N F = 565N F = 1380N F = 1380N F = 883N F = 883N F = 883N F = 883N F = 1.5N F = 1.5N F = 1.5N F = 1.5N Fex = 4kN Fex = 6kN F = 2097N F = 1752NF = 1752N F = 2097N F = 1464N F = 1182N F = 1464N F = 1182N F = 1464N F = 1182N F = 1464N F = 160N F = 1182N F = 66N F = 160N F = 160N F = 66N F = 66N F = 160N F = 66N Fex = 10kN Fex = 8kN Figure 4.1: Contact forces of the rolling elements in case of different radial loads, where the shaded surface represents the spherical surface of the outer racel 12 10 Force [kN] 8 SRB, Table 1 parameters BearinX Gargiulo 6 4 2 0 20 22 24 26 28 30 32 34 Displacement [µm] Figure 4.2: Radial force with respect to displacement 50 4.1.3 4 Numerical Examples and Experimental Verification Clearance, Displacement, and Load The third set of calculations established how SRB elastic deformation changes with load as a function of diametral clearance cd . Figure 4.3 shows elastic deformation and radial loading force plotted for four different values of clearance. The simulations demonstrate that elastic deformation is not affected significantly by changes in cd . This conclusion is supported by the BearinX prediction and the Gargiulo bearing radial deflection estimation. Although elastic deformation seems to be insensitive to changes in cd , diametral clearance does affect displacement between the bearing races as expected. 14 cd = 39 µm cd = 41 µm 12 cd = 44 µm Force [kN] 10 cd = 46 µm BearinX Gargiulo 8 6 4 2 0 0 2 4 6 8 10 12 14 Elastic Deformation [µm] Figure 4.3: Effect of clearance on elastic deformation of the bearing. In this case the cd elastic deformation is calculated by ey − 2 Figure 4.4 shows bearing stiffness behavior respect to radial load for four different values of clearance. 4.1.4 Osculation Number, Displacement, and Load The effect of osculation number on displacement for different levels of radial loading was investigated with the fourth set of calculations. Figure 4.5 shows a family of displacementto-load curves representing four different values for osculation number. The prediction reveals that osculation significantly affects bearing stiffness. An osculation number value of Cos = 0.96 seems to correspond to the reference solution obtained using the BearinX software. 51 4.1 Case 1: Single Bearing Numerical Simulations 8 4 x 10 3.5 Stiffness [N/m] 3 2.5 2 cd = 39 µm 1.5 cd = 41 µm 1 cd = 44 µm cd = 46 µm 0.5 0 0 2 4 6 8 10 12 14 Force [kN] Figure 4.4: Stiffness with respect to radial force 15 Cos = 0.98 Cos = 0.99 Cos = 0.97 10 Cos = 0.96 Force [kN] BearinX 5 0 20 22 24 26 28 30 32 34 Displacement [µm] Figure 4.5: Effect of osculation number on radial force-displacement relationship 52 4.1.5 4 Numerical Examples and Experimental Verification Number of Rollers, Displacement, and Load The fifth set of verification calculations investigated how displacement changes with radial load and the number of bearing rollers. The results are presented in Figure 4.6. They indicate that SRB load carrying capacity increases with its number of rolling elements. In this case, the BearinX software predicts slightly higher displacement values. 12 N = 16 N = 18 N = 17 N = 14 BearinX 10 Force [kN] 8 6 4 2 0 20 22 24 26 28 30 32 34 Displacement [µm] Figure 4.6: Effect of number of rollers on the radial force-displacement relationship 4.1.6 Angular Alignment of Side-by-Side Roller Arrays The final set of verification calculations looked at bearing displacement with respect to time for two separate arrangements of the caged side-by-side roller arrays; when they are aligned and when they are staggered, as Figure 4.7-a illustrates. On the left in the figure, the Type A arrangement shows the twin roller arrays in alignment. On the right, Type B shows an 11.25° angular offset between them. For this pair of calculations, an external force of Fy = -2000 N and an angular shaft velocity of ωin = 100 rad s were applied. The outer bearing races were fixed, that is to say, ωout = 0 rad .Figure 4.7-b plots the calculated s y displacements as a function of time assuming pure rolling motion between the bearing rollers and the inner and outer raceways. 53 4.1 Case 1: Single Bearing Numerical Simulations y y F ω in F ωin x x One side shifted equal to 11.25o Both side at the same level Type A Type B (a) 25.6 Type A Type B Displacement in y direction [ µm] 25.58 25.56 25.54 25.52 25.5 25.48 25.46 25.44 25.42 25.4 0 5 10 15 20 25 30 35 40 45 50 Time [msec] (b) Figure 4.7: Effect of roller position on bearing force The Type A bearing shows a varying compliance (VC) vibration at a frequency of 123.6 Hz. This is equal to the roller-pass-outer-ring frequency of the bearing. As expected, bearing Type B vibrates at twice the roller-pass-outer-ring frequency. In the Type A bearing, the displacement variation due to the VC effect is 0.34%. In contrast, the 54 4 Numerical Examples and Experimental Verification displacement variation is 0.05% for the Type B bearing. The 11.25° angular shift between the roller arrays seems to reduce the VC effect significantly. 4.2 Case 2: Rigid Rotor Supported by Ideal Spherical Roller Bearings To demonstrate the application of the newly developed SRB model in a typical real world analysis, a numerical simulation was carried out of a full rotor-bearing system comprising a rigid rotor supported by SRBs on either end of the rotor axle. 4.2.1 Studied Structure The rotor bearing system, shown in Figure 4.8, can be described with eight degrees of freedom where the four-degree-of-freedom are defined for both bearings housing dis-placements in the x and y directions, and the next four-degree-of-freedom for rotor displacements that may be selected in many ways (see part 2.1.1, Rigid Rotor). One possibility is to use the center of mass translations in the x and y directions and the two rotations about those axes. Another possibility, which was selected for this work, is to use the translational coordinates of two bearing locations as the system’s degree-of-freedom y yI yII mub ψy B d1 ψx eub aub d2 z dI CS KS dII CS KS Figure 4.8: Rigid rotor supported with two spherical roller bearings The bearing housings were connected to the ground using linear spring-dampers, whose stiffness and damping coefficients are KS and CS , respectively. The angular velocity of the rotor about the z-axis was assumed constant. Table 4.2 lists the dimensions and mass properties for the modeled rotor-SRB system. According to Equation ( 2.7 ), in this particular case, the individual force components can be written with these equations. 4.2 Case 2: Rigid Rotor Supported by Ideal Spherical Roller Bearings 55 Table 4.2: Dimensions of the rigid rotor length left bearing lacation right bearing lacation end part diameter middle part diameter density mass of rotor transverse moments of inertia polar moment of inertia L dI dII d2 d1 ρ mR Ix = Iy 500 225 225 110 130 7850 49.138 0.9846 kg kg m2 Iz 0.0993 kg m2 kg m3 F ex Fb Fg F ub 0 −250 = 0 −250 I Fx I Fy = F II x mm mm mm mm mm FyII 0 − mr 2 = 0 − m2r (4.1) (4.2) (4.3) cos (Ωt + θ0 )Lub sin (Ωt + θ0 )Lub = mub eub Ω2 cos (Ωt + θ0 )Rub sin (Ωt + θ0 )Rub (4.4) where mub , eub are the mass and eccentricity, and θ0 is the phase angle of the unbalanced mass, which is assumed zero. Lub and Rub are calculated as follows: Lub = Rub = dII − aub dII + dI dII + aub dII + dI (4.5) (4.6) 56 4 Numerical Examples and Experimental Verification The unbalance mass parameters used in the rigid rotor model and the properties of the supporting structure are listed in Table 4.3. Table 4.3: Unbalance mass parameters of the rotor and support properties unbalance mass eccentricity of the unbalance mass unbalance distance to center of mass of rotor phase angle of the unbalance mass mub eub aub θ0 0.005 0.1 0.12 0 kg m m rad The properties of the supporting structure are listed in Table 4.4. Table 4.4: the properties of the supporting structure 4.2.2 bearing mass support stiffness mS KS support damper CS 11 1 · 107 5 · 103 kg N m Ns m Simulation Results Using Matlab-2011b, the dynamics of the rigid rotor and two SRBs were solved for rotation speed of 3000 rpm. The differential equations of motion (Equation (2.10)) were solved using an ode45 time integrator scheme. Figure 4.9 shows the predicted horizontal and vertical translational rotor displacements at the A and B spherical roller bearing locations as a function of time. After a brief initial transient vibration, the rotor settles into steady-state harmonic vibration as a result of unbalanced forces. The rotor axis deflects in an elliptical orbit. Because of the positioning of the unbalanced load, the deflections for SRB A are greater than those for SRB B. Figure 4.10 illustrates the orbital motion of the rotor axis. 4.2 Case 2: Rigid Rotor Supported by Ideal Spherical Roller Bearings 15 SRB−A, x−disp SRB−B, x−disp Displacement (µm) 10 5 0 −5 −10 −15 0 0.1 0.2 0.3 0.4 0.5 time (s) 0 SRB−A, y−disp SRB−B, y−disp Displacement (µm) −20 −40 −60 −80 −100 −120 −140 0 0.1 0.2 time (s) 0.3 0.4 0.5 Figure 4.9: Simulated horizontal and vertical displacements of the rotor 57 58 4 Numerical Examples and Experimental Verification Orbit at SRB−A Orbit at SRB−B −78 −80 y (µm) −82 −84 −86 −88 −90 −5 0 x (µm) 5 Figure 4.10: Simulated horizontal and vertical displacements of the rotor Figure 4.9 and Figure 4.10 show that total rotor displacement in the y direction is about 84 µm on average. Much of this displacement is due to elastic compression of the support springs. Theory predicts a value for support compression of about 60 µm. The additional 20 µm displacement is bearing clearance. Therefore, the elastic compression of the SRB structure must be only a few micrometers in this case. However, the elliptical orbit of rotor axis displacement (Figure 4.10) shows greater displacement in the x-axis direction than in the y-axis direction, which implies differences in bearing stiffness. This difference is the result of bearing clearance, which is taken up in the y-axis as the y-direction radial load acts on the bearing, and not taken up in the x-axis direction. 4.3 4.3.1 Case 3: Single Spherical Roller Bearings with Localized Defects Studied Structure In this section, a series of numerical calculations are carried out for a single SRB subjected to a simple radial load to verify the local defect analysis of the bearing. The investigated spherical roller bearing type is the SKF 22216-EK roller bearing. The relevant dimensions and material properties of the studied spherical roller bearing are shown in Table 4.5 while the dimensions are depicted graphically in Figure 2.5. It should be noted that some of the parameter values are based on manufacturer data and some are based on experimental measurements on a real bearing. The measurements were carried out using a three-dimensional coordinate measuring machine as shown in Figure 4.11. In the first set 4.3 Case 3: Single Spherical Roller Bearings with Localized Defects 59 of numerical calculations, the effect of outer ring defect on bearing force was calculated and the second example explores the same effect for a bearing with inner ring defect. Table 4.5: Dimensions and parameters of the spherical roller bearing, 22216-EK free contact angle roller diameter1 inner raceway contour radius1 outer raceway contour radius1 roller contour radius bearing width clearance bearing pitch diameter number of rows number of rolling elements in one row modulus of elasticity Poisson’s ratio φ0 dr rin rout rr B cd dp nz N E ν 8.25 15.544 63.2 63.2 61.49 33 41 132 2 21 206 0.3 degree mm mm mm mm mm µm mm GPa - [1] Based on experimental measurements on real bearing (Figure 4.11) Figure 4.11: Bearing geometry parameters measurements 60 4.3.2 4 Numerical Examples and Experimental Verification Single Bearing Load Analysis - Outer Ring Defects The first set of calculations was performed to consider the vibration behavior of the bearing with outer race defect. Figure 4.12 shows two types of defects (simple and elliptical defect) on one side of the outer ring and hence only one row of rollers is in contact with the defects. The dimension of the defects on the outer ring is presented in Table 4.6. In both type of defects, the maximum depth of the defect is defined as equal to 0.05 mm and it is big enough to lose connection between the roller and the rings. The only difference is that, for the simple defect, it happens whenever the roller enters a defect area, but for the elliptical defect, it does not happen immediately. This model runs with a constant displacement in the vertical direction which is equal to ey = -0.0324 mm and an angular shaft velocity, ωin = 78.54 rad s (750 rpm) for both type of defects that are shown in Figure 4.12. y a Dd Elliptical defect b ey ωin x Dd Wd Simple defect Figure 4.12: Dimension of defect on outer ring In the following, Table 4.6 lists the dimensions of the outer ring defect. Table 4.6: Dimensions of the outer ring defect Elliptical defect Simple defect Ellipse major axis Ellipse minor axis Defect position in the outer ring Defect macimum depth Defect position in the outer ring Defect width Defect depth a b φ Dd φ Wd Dd 20 12 280 0.05 280 9.26 0.05 mm mm degree mm degree mm mm Figure 4.13 shows the system response for both types of defects. The results presented in the frequency domain clearly show peaks at the roller pass outer ring frequency, its double frequency and also a minor peak at the cage frequency (See appendix B). 4.3 Case 3: Single Spherical Roller Bearings with Localized Defects 61 Figure 4.13: Force amplitude in time and frequency domain for outer race defect (a) Elliptical defect-time domain; (b) Elliptical defect-spectrum; (c) Simple defect-time domain; (d) simple defect-spectrum The spectra shown in Figure 4.13 for both elliptical and simple defect contain peaks at the outer ring defect frequencies. In this case, no side band frequencies are generated because 62 4 Numerical Examples and Experimental Verification the outer ring is fixed to the bearing housing [70]. The second harmonic spectra for roller pass outer ring frequency are also shown. Furthermore, it seems that in a spherical roller bearing, the local defect shape (simple or elliptical) does not have significant effect on dynamic behavior of system because of the large number of rollers in both side of the bearing. 4.3.3 Single Bearing Load Analysis - Inner Ring Defects In this section, the effect of inner ring defect on bearing vibration behavior was investigated. Similarly to the previous example, it is assumed that the bearing rotated with a constant rotation speed equals to ωin = 78.54 rad s (750 rpm) and the pre-defined displacement in the vertical direction as ey = -0.0324 mm. Both elliptical and simple defects are modeled on one rolling surface of the inner ring as shown in Figure 4.14. y Dd a ωin b Elliptical defect ey Dd x Simple defect Figure 4.14: Dimension of defect on inner ring Dimensions of the inner ring defect are shown in Table 4.7 Dimensions of the inner defect. Contrary to the outer ring defect, the inner race defect rotates with the same frequency of the rotor. Thus, it is assumed in this example that the initial angular position of the defect is φ0 = 0. Figure 4.15 shows the effect of inner race defect on bearing force in y direction. 63 4.3 Case 3: Single Spherical Roller Bearings with Localized Defects Table 4.7: Dimensions of the inner defect Elliptical defect Simple defect Ellipse major axis Ellipse minor axis Defect position in the outer ring Defect macimum depth Defect position in the outer ring Defect width Defect depth a b φ0 Dd φ0 Wd Dd 20 12 0 0.05 0 9.26 0.05 mm mm degree mm degree mm mm Figure 4.15: Force amplitude in time and frequency domain for inner race defect (a) Elliptical defect-time domain; (b) Elliptical defect-spectrum (c) Simple defect-time domain (d) simple defect-spectrum 64 4 Numerical Examples and Experimental Verification The roller pass inner ring frequency and its multiple, as well as the shaft rotating frequency and its multiples for both kind defects, are clearly visible in Figure 4.15. The roller pass inner ring frequency and its harmonic have sidebands, which are made up of the difference between RIF and shaft rotation frequencies. 4.4 4.4.1 Case 4: Rotor Supported with Spherical Roller Bearings with Localized Defects Studied Structure Figure 4.16 depicts the rigid rotor-bearing system used in the simulation and experimental verification of the bearing model. The rotor-bearing system is supported by two SRBs at both ends of the shaft. In general, the dynamic behavior of a rotor bearing system containing bearing defects is a summation of steady-state harmonic vibration due to harmonic unbalance forces and transient vibration caused by bearing defects. Therefore, a time transient analysis using numerical integration is selected to predict the nonlinear dynamic behavior of the rotor-bearing system. l l3 l2 l4 l5 l6 l1 y d1 d2 CS z KS d3 d4 CS d5 KS Figure 4.16: Schematics of the rotor supported with two spherical roller bearings In simulations, the bearing housings are connected to the ground using linear springdampers, whose stiffness and damping coefficients are KS and CS , respectively. The angular velocity of the rotor about the Z-axis was assumed constant. Table 4.8 lists the dimensions for the modeled rotor-SRB system. The rotor bearing system, shown in Figure 4.16, is modeled using the beam finite elements that have four degrees of freedom per node. The system has 72 degrees of freedom where four degrees of freedom are used for bearing housing displacements (i.e. translation in the x and y directions), and the remaining 68 degrees of freedom are used for rotor 4.4 Case 4: Rotor Supported with Spherical Roller Bearings with Localized Defects65 Table 4.8: Dimensions of the rigid rotor d1 = 70 mm d2 = d4 = 80 mm d3 = 88 mm d5 = 70 mm l = 600 mm l1 = l6 = 100 mm l2 = l5 = 130 mm l3 = l4 = 160 mm displacements (i.e. 17 nodes). It is also assumed that there are no displacements in the rotor axial direction. In this example, the bearing houses are connected to the ground using a linear spring-damper, whose stiffness and damping coefficients are KS and CS , respectively. The properties of the rotor bearing system, including its supporting structure, are listed in Table 4.9. Table 4.9: Properties of the rotor and supporting structure 4.4.2 Rotor mass Rotor’s material density Rotor Poisson ratio Support mass Support stiffness mR ρ ν mS KS Support damping CS 24.299 7850 0.3 11 1 · 107 5 · 103 kg kg m3 kg N m Ns m Measurement Setup Two measurements setups for spherical roller bearings with elliptical inner and outer ring defects, for which the dimensions are shown in Table 4.6 and Table 4.7, was built and simulated with Matlab. Figure 4.17 shows defect machining on outer ring By using numerical integrator (ode15s), the differential equation of the rotor bearing system’s motion (Eq. ( 2.28 )) is solved. The measurement setup is shown in Figure 4.18. The faulty bearing is located in right side of the rotor. The rotor was rotated by an external rotational belt drive. The measurement data was recorded when system ran at a steady rotation speed. In the measurement run, the steady rotation speed was 768 rpm measured by contact tachometer. In this particular case, average external force equal to 500 ± 10 N is applied in the vertical direction in the middle of the rotor. Applied external force was measured by washer load cell (HC 2001 DS Europe) as shown in Figure 4.18. Moreover, A ICP accelerometer with a sensitivity of 98 mV g is used to measure the vibrations, which is mounted on top of the housing of the test bearing. The accelerometer is connected to the vibration analyzer (CSI 2130), the output of which is connected to a computer. The signals are sampled at 1024 Hz for 4 seconds (i.e.4096 samples).High pass filter of 1 kHz is used to remove the unwanted high frequency noise. 66 4 Numerical Examples and Experimental Verification Figure 4.17: Defect machining on outer ring Figure 4.18: Test setup for bearing defect experiments 4.4 Case 4: Rotor Supported with Spherical Roller Bearings with Localized Defects67 4.4.3 Simulation Results - Rotor with Outer Ring Defects Figure 4.19 presents and compares the simulation result with measurements for the outer ring defect in the frequency domain. In this case, by applying the Fast Fourier Transform (FFT) analysis with the same FFT properties have used in measurement (signals are sampled at 1024 Hz for 4 seconds and Hanning window applied to have better frequency resolution), the magnitude of displacements in the y direction is calculated and presented in the frequency domain. The spectrum for simulation results shows peaks exactly in the roller pass outer ring frequency (118.1 Hz) that is listed in Table 4.10 and its second harmonic spectra. Measurement results also have peaks around roller pass outer ring frequency (115.1 Hz). Internal slipping between bearing components might be the reason of this difference. Figure 4.19: Spectrum of the vertical displacement for a bearing with outer ring defect a) Simulation result; b) measurement result 68 4.4.4 4 Numerical Examples and Experimental Verification Simulation Results - Rotor with Inner Ring Defects Figure 4.20 depicts the comparison of the measured and simulated spectrums for the bearing with inner ring defect. Spectrums are taken on vibration signals in the vertical direction. To separate fault signals from extra noises, the bearing defect frequencies are present in Table 4.10 (See appendix B). Figure 4.20: Spectrum of the vertical displacement for bearing with inner ring defect a) Simulation result; b) measurement result Figure 4.20 shows peaks exactly equal to shaft rotation frequency and its multipliers for both simulation and measurement results. Simulation results demonstrate the roller pass inner ring frequency (RIF), and its side bands (RIF±SF) at 149.5 Hz. The same behavior 4.5 Case 5: Rotor Supported with Spherical Roller Bearings with Distributed Defect69 as explained in Figure 4.19, RIF value in measurement results has some difference with the calculated value where the slipping between bearing components would be the reason for this difference. Table 4.10: Faulty bearing frequencies Rotation Speed (rpm) Shaft Frequency (Hz) Cage Frequency (Hz) Roller Pass Outer Ring Frequency (Hz) Roller Pass Inner Ring Frequency (Hz) 4.5 763 12.72 5.61 117.96 149.08 Case 5: Rotor Supported with Spherical Roller Bearings with Distributed Defect This study introduces resonances in the subcritical rotational speed range arising from the bearing waviness. The waviness components cause excitations that are multiples of the rotational speed. This might lead into a situation where the rotor starts to vibrate in the subcritical rotational speed meaning the rotational speed below the operational speed. In some applications, such as in paper machines, subcritical resonances are of practical significance since they may be within the operating speed range and can influence the quality of the final product. This case study utilizes a finite element method to model a paper machine tube roll system containing a symmetric tube and wavy spherical roller bearings. The waviness of the bearing rings fall into the commonly used manufacturing tolerances. A transient analysis is performed for the system and the results are converted into the frequency domain in which the results are compared with the experimental ones. Results are shown waterfall plots. Results indicate that the bearing waviness is dominating the superharmonic frequencies and the uneven mass and stiffness distribution of the rotor plays a minor role. However, in order to study the subcritical responses more accurately the non-idealities of the rotor should be included in the finite element model. 4.5.1 Studied Structure The rotor is modeled using a finite element method. The interactions between tube roll and spherical roller bearings are described using contact force which is defined using nonlinear Hertzian contact theory (Chapter 2). The finite element analysis is performed in MATLAB software. The type of spherical roller bearings is 23026CCK/WW3. It contains rolling elements in two rows and 25 rolls in a row. Table 4.11 gives the relevant dimensions and parameters of the roller bearing, 70 4 Numerical Examples and Experimental Verification Table 4.11: Dimensions and parameters of the spherical roller bearing, 23026CCK/WW3 free contact angle roller diameter1 inner raceway contour radius outer raceway contour radius roller contour radius bearing width clearance bearing pithc diameter number of rows number of rolling elements in one row modulus of elasticity poisson’s ratio φ0 dr rin rout rr B cd dp nz N E ν 8.07 17.5 93.5 93.5 91.5 35 40 164 2 25 206 0.3 degree mm mm mm mm mm µm mm GPa - The element type is used in rotor model is Timoshenko beam element. The used matrices are taken from the ANSYS Theory, referring namely to the element type BEAM4 (2001). The finite element model comprised 21 nodes on the rotor having 4 DOFs per node and additional mass elements to describe the support of the bearing housing. Axial displacements and rotation around x-axis was constrained in all rotor nodes and only radial translation of bearing housing was free resulting in a total of 88 DOFs. In transient analysis, large matrices are computationally expensive. The number of DOFs can be reduced using a reduction method such as Guyan reduction or the modal reduction used in this analysis. The number of DOFs was decreased in the modal reduction using the eigenmodes of the system [42]. The eigenvalue analysis was performed for the full finite element model. The rotor system under investigation consists of the tube roll supported by spherical roller bearings in both ends. The parameters of the real bearings are emulated in the simulation model of the system. This is achieved by measuring the non-idealities of the bearings and incorporating them into the simulation model. The studied paper machine tube roll is shown in Figure 4.21. The measurement equipment consists of a PC-based data collection system, laser sensors fitted with amplifiers, a connection panel, and a guide trail where the sensors are installed. The measurement results for the middle of the roll are considered in this study. The locations of the measurement points are shown in Figure 4.21. The angular velocity of the rotor is obtained using pulse sensor. 4.5.2 Simulation Results The roller test rig is modeled using MATLAB software. The simulation model includes a flexible symmetrical tube roll, the support of the spherical roller bearings, and nonlinear 4.5 Case 5: Rotor Supported with Spherical Roller Bearings with Distributed Defect71 Figure 4.21: The locations of the measurement points in millimeters. Cut view of the tube roll is presented below the centerline contact forces between the roll and the bearings. The parameters of the rotor model are introduced in the Table 4.12. The stiffness of the bearing pedestals are measured from the actual machine by applying static load and measuring the displacements. Damping ratios are determined by simulating the force impulse and using the logarithmic decrement. A more accurate description of the pedestal can be found in [27]. Even though the rotor non-idealities are not modeled the subcritical vibrations can be studied since the waviness of the bearings is contributing mainly for these vibrations. Table 4.12: Properties of the finite element model Material properties Young’s modulus Mass density Poisson’s ratio Dimensions of the rotor Total length of the rotor Length of the shell Diameter of the shell Thickness of the shell Pedestal Vertical stiffness Horizontal stiffness Damping ratios Pedestal mass 2.07 · 1011 N m kg m2 5 4 0.3225 18.72 m m m mm 262 · 106 N m N m 7801 0.3 63 · 106 2 127 % kg The waviness components of the spherical roller bearings inner rings in both rolling rows in both ends are presented in Table 4.13. In the simulation model, attention is paid to the waviness components of 2nd to 4th orders only because the amplitudes of 72 4 Numerical Examples and Experimental Verification higher components are insignificantly small. The second order waviness has the highest amplitudes. The phases of different orders of waviness are close in both rolling sides in both tending and serving sides which increases the overall effect of waviness. Table 4.13: Bearing inner ring waviness components of orders 2nd to 4th Tending side k 2 3 4 k 2 3 4 Inner rolling race Amplitude ck [µm] Phase φk [°] 5.11 351.50 1.04 234.86 1.00 158.21 Inner rolling race Amplitude ck [µm] Phase φk [°] 2.41 356.61 0.91 25.58 0.21 91.33 Outer rolling race Amplitude ck [µm] Phase φk [°] 4.67 348.90 1.66 210.13 0.47 148.87 Outer rolling race Amplitude ck [µm] Phase φk [°] 2.70 346.09 0.90 36.11 0.39 191.25 The simulation results are compared to measurements using spectrum maps of the displacements. The spectrum maps include the frequency range of the twice-running speed resonance between 14 and 18 hertz. The measured and simulated spectrum maps of the horizontal displacements are shown in Figure 4.22. The corresponding spectrum maps in vertical directions are shown in Figure 4.23. All the modeled excitations can be seen in the simulated spectrum maps. The measured tube roll is balanced to meet the standards but there is some unbalance left as can be seen from the first harmonic component of the displacement. However, the amount of unbalance does not affect the higher harmonic components, and thus, unbalance is not considered in this study. The amplitudes of the horizontal second harmonics are greater than those of the vertical ones in all the simulation cases as well as in the measurements. This is caused by the support of the roll which stiffness in horizontal direction is less than in vertical direction. When simulated and measured results are compared, it can be seen that the twice-running-speed resonance can be found at the expected rotational speed in both horizontal and vertical direction. The response amplitude in the horizontal direction is greater in simulated case. In vertical direction, the measured amplitude is higher than the measured one in the resonance. These relatively small differences can be caused by the rotational speed increment of 0.2 hertz which means that most likely the exact resonance frequency of the system is not hit and the amplitude is diminished from the actual resonance peak in both simulation and experiments. It can be also seen that the resonance of the FFT in simulation is less than in measurements due to computational reasons. The running speed resonance is noticeable in the measured spectrum maps. The running speed response arises from the unbalance but also from the eccentricity of the bearings. Eccentricity component of the bearing is not measured why it is not included in the waviness description of the bearing model. 4.5 Case 5: Rotor Supported with Spherical Roller Bearings with Distributed Defect73 The unbalance was not considered in the simulation why the rotating speed frequency response cannot be seen. Three-times-running-speed resonance (3X-resonance) is visible in the measured spectrum. It can be also weakly seen from the simulated spectrum maps but the amplitudes of the responses are much smaller than the twice-running speed resonance which is used in scaling the figure axes a) Amplitude [µm] 250 200 150 100 50 0 0 20 40 Frequency [Hz] 60 14 16 18 Speed [Hz] 16 18 Speed [Hz] b) 300 Amplitude [µm] 250 200 150 100 50 0 0 20 40 Frequency [Hz] 60 14 Figure 4.22: Measured and simulated vibrations in horizontal direction. 74 4 Numerical Examples and Experimental Verification a) Amplitude [µm] 100 80 60 40 20 0 0 20 40 Frequency [Hz] 16 18 60 14 60 18 16 14 Speed [Hz] Speed [Hz] b) Amplitude [µm] 250 200 150 100 50 0 0 20 40 Frequency [Hz] Figure 4.23: Measured and simulated vibrations in vertical direction C HAPTER 5 Conclusions This doctoral dissertation introduces a comprehensive and computationally efficient, three-degree-of-freedom, SRB model that was developed to predict the transient dynamic behaviors of a rotor-SRB system. The simulation results demonstrate that the SRB model is sufficiently accurate for typical rotor bearing systems. The new SRB model can be used as an interface element between a rotor and its supporting structure in an analysis of rotor dynamics. The model is simple and useable for either steady state or transient analyses. Furthermore, local defect in inner and outer ring and also waviness as a distributed defect are added as non-idealities to the SRB model, and rotor dynamic analysis has also done with bearings non-idealities. The dynamic model describes contact forces as nonlinear Hertzian contact deformations. It takes into account the influences of roller angular position on bearing contact forces. The equations of motion were solved numerically. To verify the new bearing model, a series of verifying numerical calculations were carried out for a single ideal SRB subjected to a simple radial load. Physical parameters such as contact force, bearing displacement, elastic deformation, diametral clearance, osculation number, and the number and arrangement of bearing rollers were examined to verify the model. The verification calculations supported or revealed the following. • As theory predicts, roller contact forces change with increasing load. Fewer rollers support lower applied radial loads, and more rollers come into play as load increases. • Bearing displacement increases with increasing load. • Elastic deformation is not affected significantly by changes in cd . Although elastic deformation seems to be insensitive to changes in cd , diametral clearance does affect displacement between the bearing races. 75 76 5 Conclusions • Osculation significantly affects bearing stiffness, and the force and displacement responses are heavily dependent on bearing clearance and osculation number. These parameters must be considered for an accurate assessment of system performance. • SRB load carrying capacity increases with its number of rolling elements. • Even an ideal spherical roller bearing experiences varying compliance (VC) vibration with a frequency equal to the roller-pass-outer-ring frequency of the bearing. However, the VC effect can be reduced with an angular offset of the side-by-side roller arrays. An 11.25° angular shift seems to reduce the VC effect significantly. To demonstrate the application of the newly developed SRB model in a typical real world analysis, a numerical simulation was carried out of a full rotor-bearing system comprising a rigid rotor supported by SRBs on either end of the rotor axle. The governing differential equations of motion for this specific rigid rotor-SRB system were solved numerically. The predicted bearing displacements are consistent with the general theory of rotor dynamics. After a brief initial transient vibration, the rotor settles into steady-state harmonic vibration as a result of unbalanced forces. The rotor axis deflects in an elliptical orbit. However, the elliptical orbit of rotor axis displacement shows greater displacement in the x-axis direction than in the y-axis direction. This difference is the result of bearing clearance, which is taken up in the y-axis as the y-direction radial load acts on the bearing, and not taken up in the x-axis direction. Proposed SRB dynamic simulation model, including an inner or outer race surface defect, was verified by comparing measured-to-predicted behaviors. Four defect cases were run: a simple surface cavity on the inner race, a simple surface cavity on the outer race, an elliptical surface concavity on the inner race, and an elliptical surface concavity on the outer race. The SRB model assumes that roller-to-race-surface contact over the elliptical surface concavities is not lost immediately compared to simple defect. It means that in this type of local defect, whenever the depth of defect is less than elastic compression between rollers and races and also bearing internal clearance, the roller keeps its connection to races and this roller contact force should also be taken into account to calculate the bearing force. Simulation results provided a catalog of behaviors as a function of defect type and location, making it possible to identify an SRB defect by comparing the observed bearing behavior with the various behaviors described by the catalog. Numerical results also revealed that each local defect would result in vibration at a frequency equal to the bearing defect cycling frequency. The predicted amplitude of vibration for the elliptical defect is slightly greater than the vibration amplitude for an equivalent simple defect. The measurements showed spectra peaking at different frequencies than predicted for the bearing with the outer and inner ring defects. The percentage variation between the theoretical and experimental defect frequency is found to be 1.7–2.7%. This may 77 be because of the internal slipping between rollers and inner and outer races which effect on cage frequency and also roller pass inner and outer ring frequencies. However, the real rotor bearing system has different unknown parameters which appeared in the measurement results. As a last numerical example, tube roll that is supported by spherical roller bearing in both ends was studied. In this example bearings have waviness in the inner ring which are measured and introduced in simulation model using Fourier cosine series. In the model only 2nd to 4th order of waviness were considered. The dynamics of a paper machine tube roll is analyzed in terms of twice-running-speed subcritical vibration. Final results show that simulation and measurement results are comparable in both horizontal and vertical direction. Future Studies: Study about spherical roller bearing and rotor-bearing systems can be improved by the following suggestions: • Performance of spherical roller bearing can be considers in non-axial rotation due to shaft tilting and misalignment. • Improve the contact theory for internal components connections • Considering the effect of lubricant film damping in rolling contacts and centrifugal force of rollers in dynamic behavior of bearing, might make the model more realistic. 78 5 Conclusions B IBLIOGRAPHY [1] ANSYS Theory Manual, 2001. [2] A DILETTA , G., G UIDO , A. R., AND ROSSI , C. Nonlinear dynamics of a rigid unbalanced rotor in journal bearings. part 1: Theoretical analysis. Nonlinear Dynamics 14, 1 (1997), 57–87. [3] A KTURK , N. The effect of waviness on vibrations associated with ball bearings. 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Computers & Structures 80, 32 (2002), 2537–2549. 86 BIBLIOGRAPHY 87 APPENDICES Appendix A: calculating the defect depth according to roller angular position O ψ1 βji F1 a ψ2 F2 P δ + βji Figure 1: Defect depth calculation According to Fig.1 and the geometry properties of the ellipse, the following equations can be written: P F 1 + P F 2 = 2a 2 PF1 2 PF2 2 = OP + 2 = OP + 2 OF 1 2 OF 2 − 2OP · OF 1 · cos ψ1 − 2OP · OF 2 · cos ψ2 Finally, OP can be calculated to get defect depth depend on roller angular position. (1) (2) (3) 88 APPENDICES Appendix B: characteristic frequencies of bearing The rotational frequency of the cage is calculated as follows fcg fR = 2 dr 1+ cos φ0 dp (4) and roller pass outer ring frequency frpof fR =N 2 dr 1− cos φ0 dp (5) and also roller pass inner ring frequency frpif fR =N 2 dr 1+ cos φ0 , dp (6) where fR is the rotor rotation frequency and dp is a bearing pitch diameter which is defined as dp = Rout + Rin (7) ACTA UNIVERSITATIS LAPPEENRANTAENSIS 554. 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