measurement techniques in structural acoustics

MEASUREMENT TECHNIQUES
IN STRUCTURAL ACOUSTICS
SESSIONS
Vibration Damping, Energy and Power Flow
G. Pavic
LVA, INSA de Lyon, 69621 Villeurbanne, France, [email protected]
The quadratic quantities of interest in vibration analysis are kinetic and potential energy and power flow density (intensity).
All of these quantities can be measured. The classical procedures for the measurement of energy in vibrating linear structures are
based on static representation of the Hooke’s law. As a rule, structural damping is not taken into account in a correct manner
which can lead to large underestimation or overestimation of true energy levels.
The influence of structural damping is demonstrated for some characteristic cases of plates and beams.
P ∝ A+2 − A−2
ENERGY, POWER AND DAMPING
The input power P, overall energy E, frequency ω
and damping η are related by an approximate formula:
P ≈ 2 E kinη ω ≈ Eη ω
(1)
Expressed in terms of modal parameters – natural
frequencies υ, modal shape φ, modal force f and mode
amplification factor Ω – the overall kinetic energy and
total input power are given by:
2
Ekin
2
fq
ω2
fq
ω
2
,
=
P=
∑
∑η q υ q 2
4m q Ω 2
2m q
Ωq
q
f q = ∑ Fn (ω )φ q (z n ) Ω q = υ (1 + jηq ) − ω
2
q
(2)
(3)
where A denotes the wave velocity amplitude.
If damping is admitted, (1) becomes more complex;
the difference of amplitude squares is changed into:
A+2 e − kxη 2 − A−2 e kxη 2 + ηA+ A− sin( ∆ϕ − 2kx) (4)
where η - loss factor, k - wavenumber, ∆ϕ - wave
phase difference and x - distance along the rod’s axis.
The power variation is both exponential and sinusoidal.
The power flow in flexural vibration is of much
more complex nature as not only the travelling waves
but also the decaying ones carry energy. Fig. (2) shows
the distribution of power flow in a clamped-clamped
1m long steel beam, excited at 0.2m, loss factor 1%.
2
n
The simple formula (1) holds for kinetic energy
only at resonances. Its extension to the total energy is
valid in cases where the kinetic and potential energies
are similar, i.e. again close to resonances, Fig. 1.
Fig. 1. Energy in a multi-DOF system. Dotted – resonances.
Eq. (1) is valid for either the whole structure or its
subassemblies exhibiting resonant behaviour. It cannot
be applied at the local level, i.e. for analysis of energy
distribution within structure / subassemblies. The role
of damping on energy distribution has to be examined.
Fig. 2. Power flow in a clamped-clamped beam normalized
to unit input power. Loss factor 1%. Negative and positive
values indicate divergence in power flow at the excitation.
Fig. 3 gives the energies along the beam at 1kHz.
SPATIAL DISTRIBUTION OF ENERGY
AND POWER
The power flow in lossless rods expressed in terms of
amplitudes of propagating vibration waves reads:
Fig. 3. Spatial distribution of energy in a clamped-clamped
beam. Frequency 1kHz.
SESSIONS
Away from the excitation region and boundaries
the two energies are almost identical, while near the
singularities the difference between the two increases.
Fig. 4 shows the frequency distribution of power
flow, normalised to the input power, at 53 equidistant
positions covering the entire beam span.
Fig. 7. Intensity in a plate shown by the intensity divergence.
Plate loss factor: left 0,1% (enhanced 10 times), right 10%.
The presence of structural damping couples in
power travelling and evanescent waves. This applies
not only to plates and beams but to elastic structures in
general.
DAMPING vs MEASUREMENTS
Fig. 4. Normalized power flow in a clamped-clamped beam
at equidistant positions along the beam span.
Fig. 5 shows the total kinetic and potential energies
in a 1.5m×1.25m×20mm steel plate excited at 4 points.
Energy, energy density and vibration intensity can
all be measured in certain types of structures1-6. When
measuring these quantities structural damping must not
be disregarded.
Fig. 8 shows the measurement error for the beam as
defined in the previous section, caused by neglecting
structural damping. The curves concern two classical
techniques, the 4-point and the 2-point one.
Fig. 5. Ratio of kinetic and potential energies in a plate.
The damping is seen to little affect the ratio of the two
energies which stays close to unity except in the region
of very low relative resonance density. The damping
however does affect to a great deal the distribution of
energy density, Fig. 6 and of power flow, Fig. 7. These
two figures refer to the frequency of 500 Hz where the
overall kinetic and potential energies are of close level.
Fig. 6. Lagrangian energy density in a plate at 500 Hz.
Loss factor: left 0,1%, right 10%. n excitation points.
Fig. 8. Error of power measurement in a clamped-clamped
beam caused by neglecting damping. Spacing: 4 cm.
REFERENCES
1. D.U. Noiseux, J. Acoust Soc. Am. 47, 238-247 (1970).
2. G. Pavic, Proc. 3 Intl. Congress on intensity techniques,
Senlis, France, 1990, pp 21-28.
3. E.G. Willimas, H.D. Dardy and R.G. Fink, J. Ac. Soc.
Am., 78, 2061-2068 (1985).
4. J.C. Pascal, T. Loyau and X. Carniel, J. Sound Vib.,161,
527-531 (1993).
5. J.R.F. Arruda, J.P. Campos and J.I. Piva, Proc. Intl. Conf.
Noise Vib. Eng., Leuven, Belgium, 1996 Vol. 1, 641652.
6. Yu.I. Bobrovnitskii, Acoustical Physics, 45, 260-271
(1999).
SESSIONS
Energy source localization using the reactive structural
intensity
P. S. L. Alves and J. R. F. Arruda
Department of Computational Mechanics, State University of Campinas,
C.P. 6122, 13083-970 Campinas, SP, Brazil [psl,arruda]@fem.unicamp
Structural Intensity analysis can be a powerful tool in noise and vibration control problems. The active part of the structural intensity
is usually measured. It corresponds to the period-averaged value of the force times the velocity, and is related to energy propagation
in a waveguide. The reactive part of the structural intensity, related with the energy of standing waves within the waveguide, is usually
not analyzed. The authors have recently shown that if properly separated into its wave components, reactive intensity maps agree
well with operational modes, clearly indicating the nodal lines in thin flat plates. It is well established that active intensity plots show
the energy flow paths and the divergence of the active intensity can indicate the location of energy sources and sinks. For highly
reverberant structures, however, measuring active intensity becomes awkward, and intensity plots fail to indicate the region where the
energy is injected or dissipated in the structure. In this paper, the possibility of localizing energy sources based on the divergence of
the reactive structural intensity and on the distribution of the potential and kinetic energy densities within the structure is investigated.
Numerical and experimental results are presented for beams with point excitation. The structural intensity formulation is based on
the Euler-Bernoulli theory. A finite element model is used in the numerical simulation. The conditions under which it is possible to
effectively detect the location of point energy sources using the proposed method are addressed.
POWER FLOW DEFINITIONS
In the frequency domain, power flow is a complex vector defined as the product of a force (or moment) and the
complex conjugate of the corresponding velocity. The expression for its active component is given by:
1
I = ℜfF (ω)V (ω) g
2
(1)
This component corresponds to the period-averaged
value. It provides information about the main energy flow
paths within the structure, thus enabling the identification
of the energy sources and sinks. Active power flow per
unit area is also referred to as ‘structural intensity’. Structural intensity is defined as the product of the stress tensor and the velocity. For lightly damped structures, analogously to acoustic intensity, it is awkward to measure
the structure intensity [1]. In this case, the component of
the energy injected in the structure is mainly related with
the reverberation of energy and is usually called reactive
power, which is given by the following expression:
1
PR = ℑfF (ω)V (ω) g
2
(2)
This component originates the standing waves or modes
shapes [2,3].
Since the active and reactive structural intensity components are vectorial quantities, the corresponding energy
flow across a closed contour around a point may be obtained. This quantity corresponds to the energy flow balance and may be computed by the divergence of the corresponding map. The active energy continuity equation
for the active component is given by following expression [4]:
(3)
∇ a = I hΠi
where I represents the active power due to the body forces
and hΠi the time average of the power dissipated per unit
volume. Analogously, the energy continuity equation for
the reactive compenent is given by [4]:
∇ r = PR
2ω(hU i
hT i)
(4)
where PR is the reactive power and hU i and hT i are the
time averages of the strain energy density and the kinetic
energy density, respectively.
Active and reactive components of the power per unit
width for Bernoulli-Euler beams are obtained by substituting the expressions of moment and shear force:
Ix (x; ω) =
EI
∂2 w ∂2 w
ℜf
2
∂x2 ∂x∂t
+
∂3 w ∂w
g
∂x3 ∂t
(5)
EI
∂2 w ∂2 w ∂3 w ∂w
ℑf
+
g (6)
2
∂x2 ∂x∂t
∂x3 ∂t
Period-averaged values for the strain and kinetic energy are given by:
PRx (x; ω) =
2 hU i = 14 EI ∂∂xw2
∂2 w
∂x2
hT i= 14 ρAω2w w
(7)
NUMERICAL RESULTS
A numerical simulation was performed using a
0.0319x0.00315x1.5 m aluminum beam (E=70 10 9
SESSIONS
N/m2 , ρ=2780 kg/m 3 ) modeled with 60 two-node
Bernoulli-Euler elements (Fig. 1). In order to simulate a
semi-anechoic termination, localized stiffness and damping coefficients were used: KT =1000N/m; CT =100Ns/m
(translational); KR =100Nm/rad, CR =10Nms/rad (rotational).
The divergence of the reactive power and the term related with the distribution of the strain and kinetic energies were obtained at 105Hz (Fig. 2a). The difference
between the two curves shows the location where the energy is injected in the structure (Fig. 2b).
0.6
0.7
0.5
0.6
0.4
0.5
0.3
0.4
0.2
0.3
0.1
0.2
0
0.1
−0.1
0
−0.2
0
0.5
1
1.5
−0.1
0
0.5
1
[m]
1.5
[m]
(a)
(b)
FIGURE 2. (a) Divergence of the reactive power ‘:’ and the
distribution of strain and kinetic energies ‘-’; (b) difference between the two curves.
EXPERIMENTAL RESULTS
An experiment was conducted using an aluminum
beam plunged into a sand box to simulate a semianechoic termination (Fig. 3). The point where the force
is applied and the beam dimensions are the same considered in the numerical simulation. FRF’s were measured
with white noise excitation in the frequency range 0-800
Hz and responses were measured with a laser Doppler vibrometer.
The experimental results shown in Figs. 4a and b indicate that it was possible to localize the energy source
using the reactive power divergence. However, this was
only possible in higher frequencies and with very accurate measurements. Using the reactive power implies
computing third order spatial derivatives from measured
data. It is known that differentiation amplifies noise and,
thus, computing the third order derivative is critical. Very
accurate results and careful curve fitting of the spatial data
are absolutely essential, as well as high density spatial
data. The authors are currently investigating the practical
limitations of the method and extending it to plate structures.
FIGURE 3. Experimental setup.
0.04
0.08
0.02
0.06
0
0.04
−0.02
0.02
−0.04
−0.06
0
0
0.5
1
−0.02
1.5
0
(a)
0.5
1
1.5
(b)
FIGURE 4. (a) Divergence of the reactive power ‘:’ and the
distribution of strain and kinetic energies ‘-’; (b) difference between the two curves at 317Hz.
REFERENCES
ACKNOWLEDGMENTS
The authors are grateful to the Fundação de Pesquisa
do Estado de São Paulo (FAPESP) for the financial support.
1. F. J. Fahy, Sound Intensity, Elsevier Applied Science, London, GB.
2. P. S. L. Alves, J. R. F. Arruda, L. Gaul and S. Hurlebaus,
Power Flow Estimation Using Pulse ESPI, In: Proc. of the
4th Int. Conf. on Vibration Measurements by Laser Techniques: Advances and Applications, pp.362-373, (2000),
Ancona, Italy.
3. P. S. L. Alves and J. R. F. Arruda, Active and Reactive
Power Flow Estimation Using Mindlin Plate Theory, In:
Proc. of the 9th Int. Symposium on Dynamic Problems of
Mechanics, pp.459-464, (2000), Florianópolis, Brazil.
4. K. S. Alfredsson, Active and Reactive Structural Energy
Flow, Trans. ASME, Volume 119, pp. 77-79.
FIGURE 1. FEM model.
SESSIONS
Measuring the Potential Energy of Structure Vibrations
Yu.I.Bobrovnitskii
Laboratory of Structural Acoustics, Mechanical Engineering Research Institute of
Russian Academy of Sciences, 101990 Moscow, Russia, E-mail: [email protected]
While the kinetic energy of a forced vibrating structure can be estimated by measuring the vibration velocity at a number of
points, direct measurement of the potential energy of vibrations is a problem due to difficulties in mounting sensors and
measuring the stresses all over the structure. In this paper, a method is reported that allows one to estimate the potential energy of
vibrations, as well as other energy characteristics, in a rather economic way - through the measurement of the input impedance of
the structure. It is shown in computer simulation and experimentally that the method gives reliable and accurate estimates of the
potential energy in the low and middle frequency range.
INTRODUCTION
of the driving point velocity response be known (or
The potential energy of a forced vibrating structure
characterizes the stresses averaged over the structure
and therefore can be useful in many practical
applications. At present, the potential energy can not
be measured directly by the existing methods because
of the necessity to measure six strain components over
the entire structure. Recently, a very efficient method
has been proposed [1,2] that allows one to estimate the
energy characteristics of linear vibrating structures
using minimum data measured with usual facilities. To
obtain the potential energy, loss factor or other energy
characteristics, one does not need, in this method, to
measure or compute the vibration response over the
entire structure; neither does one need to possess a
vibrational model of the structure. The only quantities
needed are the complex amplitudes of the external
forces (or the input impedances of the structure with
respect to these forces) and the complex amplitudes of
the velocity response at the driving points. In the
simplest case, when the structure is driven by a single
external force it is sufficient to measure only two
quantities – the amplitude of the force and the
amplitude of the driving point velocity. The method is
mathematically well based and verified in computer
simulations and experimentally. Here, the method is
further developed, the accent being made on estimation
of the potential energy.
measured). The parameters of the system as well as the
system response at other points are supposed
unavailable. According to the proposed method, one
can, using the given quantities, f and v, to estimate the
potential energy U as
MAIN RELATIONS
Consider a linear elastic system with continuous or
lumped parameters and with any type of damping,
performing harmonic vibrations under the action of the
external point force fexp(-iωt). Let the complex
amplitude f of the force and the complex amplitude v
1
 ∂z (ω ) z (ω ) 
−
| v | 2 Im α
∂ω
ω 
8

(1)
1
 ∂y (ω ) y (ω ) 
−
U ≅ − | f | 2 Im β
.
∂ω
ω 
8

(2)
U ≅−
or
Here, the input impedance z(ω) and input mobility y(ω)
are the functions of the measured quantities:
z(ω) = f / v, y(ω) = v / f .
α and β are closed to unity correction coefficients
introduced in paper [2].
It has been proved [1] that when the elastic system
under study has no damping, relations (1) and (2) are
mathematically correct, i.e. they give the exact values
of the potential energy at all frequencies, the correction
coefficients α and β being equal to unity . When the
elastic system has damping, these relations give good
results at the frequencies where the system is mass or
rigidity controlled. In the vicinity of the natural
frequencies where the system is damping controlled the
true value of the potential energy are restored with the
help of the correction coefficients. The frequency
range of validity of the estimates and of the whole
method at its present stage comprises the frequencies
where the resonance peaks of the system vibration
SESSIONS
response are distinctly separated from each other, and
the difference between two adjacent resonance
frequencies is at least three times greater than the width
of the resonance peaks. For not heavily damped
structures (with the material loss factor less than 0.1)
these are low and middle frequencies.
COMPUTER SIMULATION AND
EXPERIMENT
To illustrate the accuracy of the proposed estimates
for the potential energy consider a straight uniform rod
of length l free of stresses at one end which executes
longitudinal vibrations under the action of an external
harmonic load applied to the other end. The vibrations
are assumed to be governed by the classical equation of
Bernoulli with complex Young’s modulus, Ec=E(1-L0)
0 being the material loss factor. Figure 1 shows the
numerically in a computer. The main problem
encountered during the experiment was differentiation
of the imaginary part of the input impedance and
mobility with respect to frequency. Differentiation is a
poor conditioned operation extremely sensitive to
errors in the data: small deviations in the data
(measured with inevitable noise) may lead to large
errors in the derivatives. This problem has been
overcome with the help of the Pade approximation as a
smoothing procedure for the input impedance and
mobility. The Pade approximation, i.e. the
representation by a ratio of two polinomials, is a
“natural” descriptor for impedances and mobilities
much more efficient than usually used polinomials.
When the total order of the two polinomials was twice
the number of resonance (antiresonance) frequencies in
the frequency range under study, their ratio provided
very accurate approximation to the input impedance
and mobility (<1.5%) as well as to their derivatives
with respect to frequency (<5%). Such obtained
estimates of the potential energy of the experimental
beam is shown in Figure 2. There is a good
FIGURE 1. Potential energy of the rod vs frequency: exact –
solid line, estimated – crosses. k is the wavenumber; material
loss factor is equal to 0.05.
potential energy of the kinematically excited rod: the
exact values computed analytically are presented by
the solid line and the estimates via the input impedance
are shown by crosses. The estimated potential energy is
practically equal to the exact one at low frequencies
(kl<8) and does not differ from it more than 10% at
middle frequencies (kl<20). At higher frequencies,
where the elastic wavelength is less than one third of
the rod length, the difference increases.
The estimates (1) and (2) have been verified in a
laboratory experiment with a flexurally vibrating beam
(4x5x150 cm) driven by a shaker. Measured were the
amplitudes and phases of the driving force and
acceleration at the driven point. Based on them, the
velocity, the input impedance, its reciprocal – the input
mobility, their derivatives with respect to frequency
and the estimates of the potential energy were obtained
FIGURE 2. Potential energy of the flexurally vibrating
beam: measured by the proposed method (solid line) and by
two independent methods (dashed line and stars).
agreement between the estimated values and the values
obtained by other methods. The experiment confirmed
that the proposed method gives reliable estimates of
the potential energy at least at low and middle
frequencies.
REFERENCES
1.Yu.I.Bobrovnitskii, J.Sound Vibr. 217, 351-386
(1998).
2.Yu.I.Bobrovnitskii, M.P.Korotkov, Acoust. Physics,
46, 655-662 (2000).
SESSIONS
Techniques for Characterising Vibration Inputs to
Structures in Multiple Source Situations
P.R. Wagstaff, R. Dib and J-C. Henrio
Department G.S.M., University of Compiègne, BP 60319, 60206 Compiègne Cedex, France. ([email protected])
Practical techniques of characterising unknown dynamic inputs to structures rely on accurate measurements of the response of the
system and the application of inverse techniques to identify the different forces and moments applied at the interfaces between the
sources and the receiving structure. The sources may be the result of several generating mechanisms contained within a single
mechanical unit linked to the structure or the combination of several different components, that are connected independently.
This paper discusses some aspects of methods that may be used to separate and evaluate the characteristic force inputs generated
by a specific source at the structure interface in a multiple source situation. Inverse methods of force identification are linked with
the signals from reference transducers to reduce the effects of interference from secondary sources. It is shown that different types
of estimator produce different types of result if signal conditioning is applied to the inverse method. The application of these
techniques is illustrated by experiments conducted on a test bench intended for electric motors. The aim of the experiments was to
test different ways of reducing errors in identifying the force inputs generated by the motor due to the vibrations of the generator
used as a brake on the test bench.
INTRODUCTION
APPLICATION
The characterisation of the structure borne excitation
resulting from primary or secondary vibration sources
of industrial machinery and other problems linked to
noise and vibration has been a popular topic of research
in recent years. The resolution of an inverse problem is
often necessary to obtain a satisfactory estimate of the
excitation under realistic operating conditions. With
larger systems and structures it may be possible to
place force transducers and accelerometers between the
primary source and the receiving structure without
markedly changing the characteristics of the
interactions, but in most cases it is preferable to retain
the same conditions of liaison between the source and
the structure as those of normal operation.
Some of the physical limitations and the methods of
obtaining the types of measurement required using the
classical direct inverse problem approach are described
and discussed as well as methods of improving the
quality of the results in multiple source situations.
The particular application that has been the object of
our investigations is the characterisation of the
structural excitation induced by an asynchronous
electric motor fixed to a test bench. The problems of
qualifying the structural excitation of the motor in this
situation are similar to those faced by many
applications dealing with characterising noise and
vibration in industrial machinery. In this case the motor
is the primary source which interests us and the brake
(II in figure 1). used to provide the load on the motor
and thus simulating normal operation is a secondary
source of vibration, which for certain frequencies is
completely independent of the motor excitation and for
other frequencies is completely correlated with it. The
coupling between the brake and the motor and the
structure is relatively strong because both are mounted
on flanges bolted rigidly to the structure, so any
reference signals designed to differentiate between the
sources are liable to be affected by both..
VI
V
II
2
III
3
1
5
IV
4
P o s itio n s o f
acc e lero m e te rs
I
C h arg e
a m p lifie rs
Pow er
A m p lifiers
i
D ata a cq u is itio n
r1
1
r2 2
P o s itio n s o f
ex c itatio n s an d
refe ren c e p o in ts
FIGURE 1. Set-up used for characterising the transfer between each point of excitation and the structure
SESSIONS
BASIC PRINCIPLES
The measurement of the frequency response functions
between each point of excitation on the flange and the
accelerometers characterising the response is carried
out using a shaker and a force transducer in the absence
of the primary source. The motor excitation is
assimilated to 4 force inputs normal to the flange, each
force centred on the positions of the four bolts fixing
the motor to the flange. If the motor is the only source
of excitation of the structure, these forces may be
identified by measuring the cross spectral response
matrix of the accelerometer responses [G xx ] during
normal operation and solving the pseudo inverse
problem to find the unknown force input matrix
G ff in expression (1) with the aid of the measured
processing this data are available, either the principal
component or partial coherence techniques and the
pseudo-inversion technique is then applied to the
conditioned accelerometer response matrix data instead
of the directly measured response using the same FRF
matrix as before.
EXPERIMENTAL RESULTS
0
-1 0
-2 0
-3 0
dB
-4 0
-5 0
-6 0
[ ]
-7 0
-8 0
FRF (frequency response function) matrix [H].
[Gxx] = [H ]H [Gff ][H ]
M ×M
M ×N
-9 0
-1 0 0
(1)
N × N N ×M
The pseudo inversion of the FRF matrix is achieved
with the aid of SVD (singular value decomposition).
The principles of this kind of identification are well
known, but some points require a reminder. Firstly the
inertia elements upstream of the points of force
application during FRF measurements should be
removed. In the case of rotating machinery this means
removing the shaft if one is trying to identify the
excitation applied by the shaft to the bearings. The
drawback is that the structure is no longer under the
same static gravity loading which may result in
measured FRF values being different from those
associated with true running conditions. This static load
may be simulated during the FRF measurement with
the aid of elastic tension elements.
The problems associated with the effects of multiple
sources may be illustrated by the MIMO (Multiple
Input Multiple Output) model presented below.
0
1000
2000
3000
4000
F re q u e n cy (Hz )
5000
6000
7000
FIGURE 3. Measured force (black), direct inverse (blue)
conditioned inverse (green).
The above results were obtained with one reference and
one shaker replacing the motor excitation in order to
investigate the different errors possible and compare
directly with the known input measured with a force
transducer. The conditioned inverse estimation is
completely coincident with the measured force
spectrum whereas the direct inverse underestimates the
force in certain zones due to the effects of noise. For
two independent sources and shakers the results for one
of the identified force spectra is presented below using
different estimates for the maximum and minimum
force contributions coherent with the references on the
force transducers.
0
-20
L12
Y1 ( f )
Σ
H 21
Amplitude dB
-40
H 11
-60
-80
X 1( f )
H 12
L12
X 2( f )
Y2 ( f )
Σ
H 22
H 1n
L12
-120
.
.
.
.
.
.
.
Σ
-100
0
1000
2000
3000
4000
Fréquence (Hz)
5000
6000
7000
FIGURE 4. Measured force (black), maximax (green), max
(blue), mini (red).
Yn ( f )
H 2n
FIGURE 2. MIMO model for two inputs and N outputs.
The input signals representing the characteristics of the
two different sources are used to condition the
measured output response spectra and to distinguish
between the forces due to each source. Two methods of
In this case the maximum estimator is collinear with
the measured force and seems the better choice, but the
reference signals are normally accelerometers which
are more contaminated by the secondary sources at the
modal frequencies. In the fuller version of this paper,
extensions of these methods are presented to try and
identify the real force contribution of each source, even
in the presence of inter-source reference contamination.
SESSIONS
Vibrational Power Transmission in Structures Built by
Dynamically Mismatched Substructures
J. Lianga and B.A.T. Peterssonb
a
Department of Aeronautical and Automotive Engineering, Loughborough University, U.K.
b
Institute of Technical Acoustics, Technical University of Berlin, Germany
The vibrational power transmission in structures made of mismatched substructures is discussed. It is revealed that power
supplied by a force into such structures is localised in the directly driven substructure.
INTRODUCTION
In the field of solid-state physics, a phenomenon called
Anderson localisation [1] was discovered that, in the
presence of local defects, renders the normal modes of
a nominally periodic lattice to be confined in the
region close to driving point and therefore the
magnitude of response will decay exponentially away
from this region. In a series of studies, the authors of
references [2,3] attempted to bring this case to
structural acoustics, and proved that for periodic
structures with extended disorder such a behaviour
appears, under the condition that the structure is large.
Because of the imposed condition, the results obtained
are less interesting to vibration engineers wishing to
see that vibration can be confined in a small region.
This leads to the present work through which a more
effective way is sought to localise structural vibration
in the vicinity of the driving point. The configuration
of the structure to be considered is such that the
substructures are cascaded and the dynamics of each
substructure are mismatched to those of its neighbours
and strictly, therefore, no periodicity of the structure is
required.
ANALYSIS
YJJ(1) and YJJ( 2 ) , the coupling between the two
substructures can be described by the ratio [4]
cp21 = YJJ(1) / YJJ( 2 )
(1)
The interface force at the joint is found to be
ìcp - cp 2 + L
cp 21
(2)
FJblock = FJblock í 21 -1 21 - 2
FJ =
1 + cp 21
î1 - cp 21 + cp 21 - L
where FJblock is the reaction from the joint J when it is
blocked. If the dynamics of the two substructures are
mismatched, i.e.
-1
cp21
<< 1 , FJ
cp21 << 1 or
is
approximated to zero order as FJ » 0 for cp21 << 1 ,
-1
or FJ » FJblock for cp21
<< 1 . Having determined the
point
mobility
of
the
structure
from
(1)
(1)
Ypp = Ypp - YpJ FJ / Fp , one obtains the zero order
approximation of input power as
ìï Pin(1), free for cp21 << 1
2
1
Pin = Re Y pp Fp » í (1),block
(3)
-1
<< 1
for cp21
2
ïî Pin
where the superscript free denotes the situation where
the joint J is dynamically free. Equation (3) states
that the power input into such a structure is
approximately equal to that for the case where the
second substructure is removed, leaving the joint either
free or blocked.
( )
The power transmitted into the second substructure is
Fp
p
(1)
(2)
J
Figure 1 A two-element structure coupled at J
Commence with a simple case where the structure is
made of two substructures coupled at a joint J , as
depicted in Figure 1. By means of the uncoupled point
mobilities of the two substructures at the contact point
( )
2
determined by Pt = 12 Re YJJ( 2) FJ . It is noted that the
zero order approximation of
FJ
vanishes for
cp21 << 1 and hence for this case it is necessary to
approximate
FJ
to the first order such that
block
21 J
FJ » cp F
. Thus, the transmitted power is found
to be of the order
SESSIONS
±1
Pt ~ cp21
Pin
(4)
where the positive sign is used for cp21 << 1 whereas
the negative sign is associated with
-1
cp21
<< 1 .
Equation (4) indicates that the power transmitted into
the second substructure is smaller than the input power
±1
by the factor cp21
.
By carrying out such analysis recurrently for the
structures made of n mismatched substructures
arranged on a chain, it can be demonstrated that the
transmitted power obeys
Pt
J ( i +1 )
~ cp(±i1+1)i Pt J i ,
i = 1,2, L n
(5)
This means that a significant reduction of vibrational
power can be achieved in the area a few spans of
substructure away from the vicinity of the excitation
more than one point or even a continuous line or
surface. If the multiple points or the continuous line or
surface is scattered or distributed over an area with
typical dimensions substantially smaller than the
wavelength of the governing wave, the multiple-point
or continuous connection will present negligible effect
and can be treated as a single point connection [5]. If,
on the other hand, the typical dimensions of the
connecting area are comparable with or larger than the
wavelength, the effects of interactions between
coupling points and components of motion and
excitation must be taken into account. Upon focussing
on the overall vibration transmission, extensions to
encompass the multi-point and component cases have
been proposed [6] Further investigations, however, are
required to gain more insight into the influence of the
interaction and to be able to circumvent restrictive
assumptions imposed.
CONCLUDING REMARKS
DISCUSSIONS
The present analysis is subject to certain conditions.
First, it is apparent that the results given in equations
(3-5) may fail in cases where the dissipation in one or
more substructures vanishes. Such a failure is caused
by the truncation of the series of the interface force
because the power transmissions associated with the
higher order terms of this force are neglected. With a
decrease of the losses, the neglected components of the
power become increasingly important. Therefore, the
application of the present analysis should be restricted
to situations where the damping in each substructure is
sufficiently large such that the power dissipated in a
substructure is higher than the power transmitted into
its downwards neighbour c.f. the relaxed concept of
weak coupling [5]
Second, for the situation of broadband excitation,
judging the dynamic mismatch by means of
cp(±i1+1)i << 1 , i = 1,2, L
is not adequate because
cp(±i1+1)i is an oscillatory function of frequency. This
difficulty may be overcome by assessing the overall
Despite some unresolved practical problems, the
results obtained have some potentially useful
implications. First, in companion with damping
treatment for each substructure, mismatching can
localise power in the vicinity of the excitation and
hence significantly reduce power transmission in
structures. Second, to estimate the power fed to a
structure constituted by a set of mismatched
substructures, it is sufficient to consider the driving
substructure only. Third, an efficient way to apply
damping treatment for such built-up structures is to
add damping to the directly driven substructure, or in
the vicinity of excitation.
REFERENCES
1.
2.
3.
4.
value of cp (±i1+1)i , i = 1,2, L in the frequency range of
5.
interest. As a consequence, the analysis can only be
regarded applicable for the overall vibration
transmission.
6.
P.W. Anderson, Physical Review 109, 1492-1505(1958)
C. H. Hodges, JSV 82(3), 411-424(1982)
C. H. Hodges and J. Woodhouse, Report on Progress in
Physics 49, 107-170(1986).
J. M. Mondot and B.A.T. Petersson, JSV 114(3), 507518(1987).
L. Cremer and M. Heckl and E. Ungar, Structure-borne
sound. Berlin: Springer-Verlag, 2nd Edition, 1988
J. Liang and B.A.T. Petersson. Dominant dynamic
characteristics of built-up structures. (to appear in JSV)
Third, herein point connections are assumed between
substructures, which differs in some aspects from
practical situations where the connections involve
SESSIONS
The Method of Local-Global Homogenization (LGH)
for Structural Acoustics
D. B. Bliss and L. P. Franzoni
Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, USA
A homogenization method for complex structures, valid at all frequencies, is being developed with emphasis on structural
acoustics. Applications include naval and aerospace structures. The approach differs from classical homogenization and utilizes
a local-global decomposition facilitated by adding and subtracting canceling smooth forces. The smooth global problem, which
can be solved independently, has an infinite-order structural operator; and periodic discontinuities are replaced by equivalent
distributed suspension terms. The global problem can be solved efficiently since all rapidly varying scales have been removed.
The local problems provide transfer function information, and are only solved afterward if needed. Several problems have been
homogenized: a beam with periodic impedance discontinuities, a fluid-loaded membrane with discontinuities; and a cylindrical
shell with periodic ribs. An interesting feature is that the effects of fluid radiation can be transferred entirely to the smooth global
problem, while evanescent modes are absorbed into the global operator. [Sponsored by the U.S. Office of Naval Research].
INTRODUCTION
Many important structural systems having
discontinuities, braces, and/or attachments at regular
intervals are spatially periodic, or very nearly so.
Prime examples are aircraft fuselages, truss
structures, and ribbed hulls. Even when forced at a
single frequency, the response occurs in a broad
spectrum of spatial wavenumbers. Structures such as
fuselages and hulls are fluid loaded, and their
response is altered accordingly.
Not only the
structural motion, but also the acoustic radiation,
scattering, or interior sound field, may be of interest.
Calculating the motion of such structures is a
complex and computationally expensive task. The
disparity of scales requires high numerical resolution.
The forcing may be a series of locally applied forces
or continuously distributed. If the structure is
spatially periodic, the response will exhibit stop- and
pass-bands and have a discrete wavenumber
spectrum. If the structure is not strictly periodic,
there will be a distributed wavenumber spectrum and
the structural response may exhibit localization.
Often the low wavenumber (long wavelength)
portion of the response is of primary interest, since it
models the gross vibratory response. For fluid loaded
structures, the low wavenumber part of the response
is most efficiently coupled to the acoustic field, since
low wavenumbers give supersonic phase speeds.
The goal is to isolate the low wavenumber
problem from the overall problem. Because the low
wavenumber problem is smooth and contains transfer
function information from the high wavenumber part
of the problem, the approach is a homogenization
method. It differs from classical homogenization,
and it is valid for the full frequency range. The low
wavenumber problem is smooth and global; namely
it spans the structure. The high wavenumber problem
can be thought of as a series of contiguous local
solutions between the discontinuities. Therefore the
method is called Local-Global Homogenization.
Since the global problem has a known degree of
smoothness, there are potential advantages in
accuracy and efficiency, if the approach can be
extended to numerical methods. Philosophically, the
approach is an analytical reformulation, prior to
solution, to allow the direct and efficient calculation
of the most important aspects of complex problems.
FORMULATION
For illustration, Fig. 1 shows a 1-D primary
structure, such as a beam, string, or membrane, with
impedance discontinuities and distributed forcing.
L
L
L
L
L
L
discrete impedances
FIGURE 1. Forced structure with discontinuities.
The equation of motion for the structure, assuming
harmonic time dependence, is of the form:
L[hG + åh L ] = å d (x - xm )iwhZ m + Fe -iax + f s e -igx - f s e -igx
Here L is the structural operator and the displacement
h has been written as a sum of the global
displacement and sets of local displacements between
the discontinuities. On the right-hand-side a smooth
force has been both added and subtracted, so the
SESSIONS
original equation is unchanged. The wavenumber of
the applied forcing is a ; for the smooth (slowly
varying) force (fs) it is g = a - 2np/L, where n is
chosen to minimize g.
The wavenumber g
corresponds roughly to the smallest wavenumber
(longest wavelength) that passes through the
discontinuities. By superposition, the problem can be
separated into a global equation and local equations
that apply on open intervals between discontinuities,
L[hG ] = f s e -igx and L[h L ] = Fe -iax - f s e -igx .
To solve the local problem, a system of equations
is constructed. The number of unknowns is equal to
the number of constants in the homogeneous solution
plus the number of smooth forces introduced. The
number of conditions that must be satisfied due to
geometric and natural boundary conditions equals the
number of homogeneous constants; therefore one
additional constraint can be added per smooth force.
For example, it may be desirable to remove the
lowest wavenumber from the contiguous local
solutions. This constraint adds another equation to
the linear system and allows for solution of the
unknown smooth force in terms of the applied
forcing and the global motion. In addition to this
condition, the system of equations is comprised of
matching conditions between sub-intervals at the
discontinuities and a phase-shifting condition.
Returning to the global equation, the smooth force
on the right-hand-side can then be replaced by the
expression for fs found from the local system of
equations. Re-arranging so that all of the terms
containing the global variable are on the left side and
terms containing the forcing are on the right side, the
resulting global equation is then inverse transformed.
In spatial variables, the equation governing the global
motion consists of a new structural operator that
contains an infinite number of even spatial
derivatives plus a suspension term. For example, the
global equation for a membrane is given by:
FLUID-LOADING EFFECTS
The local and global solutions can be structured so
that the lowest wavenumbers are contained only in
the global solution and the higher wavenumbers are
contained entirely in the contiguous local solutions.
Because the lowest wavenumbers have the highest
phase speed, these are the modes that radiate into the
fluid. Thus fluid radiation can be confined entirely to
the global problem.
Furthermore, through an
accurate approximation, the evanescent fluid modes
can have their loading effect absorbed into the local
solutions and thereby used to modify the transfer
function effect appearing in the global operator.
Thus the effect of evanescent fluid modes can be
moved to the global operator, and only radiating
modes need to be considered in the global problem.
RESULTS AND CONCLUSIONS
Example problems based on membranes, beams,
and cylindrical shells with periodic discontinuities
have been homogenized. For the membrane, unequal
mass discontinuities were homogenized; in all other
cases thus far, the discontinuities have been identical.
For these problems, the dispersion relation for the
global problem will exhibit the expected stop- and
pass-band behavior of Bloch waves; see Fig. 3.
Re[gL]
3
2
1
5
10
15
20
-1
b bL
-2
-3
Im[gL]
0.4
0.2
5
10
15
20
bL
-0.2
æ
iwz
L2 n ¶ 2 n ö
iwz (cos gL - cos kL )
÷h =
ç1 - cos kL sin kL + å
Fe - igx
2n ÷
ç
2
k
(
L k2 - g 2 k2 - a 2
n 2n )! ¶x
ø
è
(
)(
)
On the right-hand-side, the original forcing is
operated on by a filtering function so that the forcing
is now slowly varying.
Schematically, the
homogenized original problem is shown in Fig. 2.
L
L
L
L
L
L
New operator includes suspension
FIGURE 2. Global problem after homogenization.
-0.4
FIGURE 3. Dispersion relation for the beam with
periodically spaced mass discontinuities.
Local-Global Homogenization is a promising
new approach for separating an original problem that
contains fast and slow spatial variations into
components with different wavenumber content. The
smoothly varying global problem often contains all of
the information that is desired and it is easily
converged, thus providing computational savings.
SESSIONS
Diagnostics of structure vibrations in acoustic frequency
range with the aid of self-organizing feature maps
S. N. Baranov, L. S. Kuravsky
Problem Laboratory of Mathematical Modeling
attached to the Computer Center of Russian Academy of Sciences,
c/o “Rusavia”, 6 Leningradskoye Shosse, 125299 Moscow, Russia. E-mail: [email protected]
Failure diagnostics for the structures suffered vibrations in acoustic frequency range is presented. Normalized spectral
characteristics of structure response measured in checkpoints are used as indicators to be analyzed. Self-organizing feature maps
(Kohonen networks), for which output variables are not required, detect faults. Simultaneous application of different networks
duplicating each other makes it possible to improve the quality of recognition. Principal component analysis is employed to
reduce the number of variables under study. An aircraft panel with different combinations of attached defective dynamic
suppressors is considered to demonstrate features of the approach. Tests have demonstrated high effectiveness of the presented
way of recognition and showed the advantages of neural networks over cluster analysis in recognition problems.
Technical diagnostics is one of the most typical
spheres where neural networks are used. Under
consideration here is failure diagnostics of the
structures suffered vibrations in acoustic frequency
range. This diagnostics is carried out on the base of
spectral characteristics measured in structure
checkpoints. It is supposed that neither all possible
types of damages nor corresponding changes induced
in the spectral characteristics may be predicted
beforehand.
Because of multiplicity of structure types and
their applications, it is impossible to generalize
considerably the problem solutions. Therefore
employed for method demonstration is a specific
system including a simply supported steel sandwich
rectangular panel and two attached 1-degree-offreedom elastic vibration suppressors with fluid
friction. Its dynamic behavior was simulated on the
basis of models and methods described in paper [1].
Positions of suppressors were optimized. Wide-band
random processes represented test acoustic loads.
System conditions were estimated via
standardized power spectral densities of accelerations
in a checkpoint. (In general case, some checkpoints
may be used.)
Initial data to estimate such
characteristics may be obtained with the aid of
accelerometers. Standardizing spectral densities makes
it possible to analyze only qualitative shape of
structure response spectra and not to take into account
the response level.
The following system conditions were
simulated: OK – both suppressors work properly,
Only1 – suppressor 2 is defective, Only2 – suppressor
1 is defective, Panel – both suppressors are defective,
Nonlin – non-linear suppressor response (shock
interaction of the moving element and stopper). The
first variant corresponds to normal operating mode,
and the following four ones represents system
damages.
Since all the damages are not assumed to be
known before diagnostics, it is impossible to apply
ordinary neural networks with supervised learning for
their detection. Self-organizing feature maps (Kohonen
networks) [2-3], for which output variables are not
required, may be useful in this case.
Self-organizing feature maps have an output
layer of radial units [4]. This layer is also called a
Topological Map and, as a rule, is laid out in a 2- or 1dimension space.
Starting from an initially random set of centers,
the Kohonen algorithm successively tests each training
case and selects the nearest (winning) radial unit
center. This center and the centers of neighboring units
are then updated to be more like the training case. As a
result of a consequence of such corrections, some
network parts are attracted to the training cases, and
similar input situations activate the groups of units
lying closely on the Topological Map.
A self-organizing feature map is taught to
“understand” input data structure in such a way and to
solve the classification problem. The idea, on which
this network is based, was originated by analogy with
some known features of the human brain.
If clustering of input data is completely or
partially ascertained, semantic labels might be attached
to certain units of the Topological Map.
When a classification problem is solved, so
called accept threshold is set. It determines the greatest
distance on which recognition occurs. If the distance
from the winning element to an input case is greater
than this threshold, it is supposed that the network has
not made any resolve. When units are labeled and
SESSIONS
thresholds are determined properly, the self-organizing
feature map may be used as a detector of new events: it
informs about input case rejection only if this case
differs from all labeled radial units significantly.
The given approach supports diagnostics of both
known in advance and unknown damages.
Simultaneous application of different networks
duplicating each other makes it possible to improve the
recognition quality. Frequency ranges are used as
variables, and the values of normalized power spectral
densities at the centers of these ranges – as cases. Thus,
each complete case represents a separate power
spectral density.
In the test example, initial variants of neural
networks with 3×3, 4×4 and 7×7 output layer
dimensions were trained to recognize the states OK and
Only1. Later on, the conditions Only2, Panel and
Nonlin arose successively. After detection of new
damage types, network training was carried out again,
with corresponding labels being assigned to units of
the Topological Maps1.
It is convenient to estimate the recognition
quality via the percentage of correctly identified
situations. Two sorts of errors may occur: errors of the
1st type, when some unknown system state is identified
as known one, and errors of the 2nd type, when some
system state that has been known before is identified as
unknown one or incorrectly. Application of networks
duplicating each other2 made it possible to avoid errors
of the 1st type in 99-100% of analyzed cases and errors
of the 2nd type – in 98-99% of such cases.
When all variants of system damages are known
before, the problem is essentially simpler. One can
employ traditional neural networks with supervised
learning to solve it. Perceptrons were the best for the
test problem: networks of 100%-recognition were
revealed. Radial basis function networks turned out to
be less accurate.
Neural networks are, of course, not the only way
to solve recognition problems. The same purposes may
be achieved by means of other procedures – for
example, cluster analysis that is intended for partition
of an initial object set into classes following a given
criterion. Comparison of both techniques makes it
possible to draw the following conclusions:
♦ cluster analysis does not yield distinct criteria for
classification: one cannot always distinguish
qualitatively new and old-type damages – the
result depends on critical distance selection;
♦
cluster analysis is less reliable than neural
networks;
♦ cluster analysis needs more computer resources
than neural networks.
Principal components analysis and factor analysis
are employed to reduce the number of input variables
under study if the number of frequency ranges to be
taken into account is too great and worsens network
characteristics. These methods extract few latent
hypothetical variables that explain approximately all
the set of observed ones. As for the test problem, input
data capacity might be reduced up to 2 latent variables,
with the errors being avoided in 93-100% of cases.
Nonlinear transforms on the base of autoassociative
neural networks [5] are used in more complicated
situations.
As a rule, reduction of problem dimension
prunes the number of neurons and, therefore, improves
characteristics of network training.
CONCLUSIONS
1.
2.
3.
4.
REFERENCES
1.
2.
3.
4.
1
Working with a real structure, examination to reveal the failure
nature must be fulfilled before new training and label assigning.
Otherwise, the network will only be able to inform of an appearance
of some new, unknown earlier, damage type.
2
Recognition results were selected “by a majority”.
Self-organizing feature maps, whose training data
do not contain output variables, make it possible to
diagnose conditions of vibroacoustic systems in
situations where neither all possible damage types
nor corresponding changes induced in observed
characteristics are not predictable beforehand.
If all types of system damages are known
beforehand, ordinary neural networks with
supervised learning (perceptrons, radial basis
function networks) may be used for diagnostics.
Test results showed that neural networks were
more efficient recognition tools than cluster
analysis.
Reduction of problem dimension (with the aid of
principal components analysis, etc.) improves
characteristics of network training.
5.
Kuravsky, L. S., and Baranov S. N., “Selection of
optimal parameters for acoustic vibration suppressors”,
in Proceedings of the 7th International Conference on
Recent Advances in Structural Dynamics, Southampton,
United Kingdom, 2000.
Kohonen, T., Biological Cybernetics 43, 59-69 (1982).
Kohonen, T., “Improved versions of learning vector
quantization”, in Proceedings of the International Joint
Conference on Neural Networks, San Diego, USA,
1990.
Haykin, S., Neural networks: a comprehensive
foundation, Macmillan Publishing, New York 1994.
Kramer, M. A., AIChe Journal, 37, 233-243
(1991).
SESSIONS
On the Sound Transmission through a Truncated Conical
Shell and its Coupling to a Cylindrical Shell
P. Neplea, C. Lesueurb
a
EADS AIRBUS SA, 316 Route de Bayonne, 31060 Toulouse Cedex 03, France
LRMA ISAT, 49 rue Mademoiselle Bourgeois BP 31, 58027 Nevers Cedex, France
b
The present paper deals with the vibroacoustic behaviour of a truncated isotropic conical shell either rigidly backed or coupled
to a stiffened cylindrical shell for the purpose of understanding airborne sound transmission through aircraft cockpit. To
understand sound transmission mechanisms through a truncated cone and quantify the influence of its coupling to a cylinder on
them, experimentation has been carried out on three cases : (a) a rigidly backed truncated cone, (b) an assembly of a truncated
cone and a stiffened cylinder, (c) a stiffened cylinder. The lower and upper ring frequencies of the cone are fR2=1800Hz and
fR1=4500Hz and its critical frequency is fc=12000Hz. fR2 and fR1 correspond to the ring frequencies of cylindrical shells with
radius equal to the larger and the smaller radii of the cone. The Noise Reduction (NR) of the truncated cone for cases a and b,
and the NR of the cylinder for case c, have been measured under diffuse sound field conditions in the 100Hz-10kHz range. The
two major results are the superposition of the NR curves of the truncated cone for cases a and b above 0.56*fR2 and the
similitude of the transmission behaviour of the truncated cone to that of a cylinder.
INTRODUCTION
In the context of understanding airborne sound
transmission through aircraft cockpit, the present work
deals with the vibroacoustic behaviour of a truncated
isotropic conical shell, either rigidly backed or coupled
to a stiffened cylindrical shell. The aircraft cockpit is
modeled by a truncated isotropic conical shell of
constant thickness and the aircraft structure (cockpit +
cabin) by an assembly of a truncated cone and a
stiffened cylinder. In the literature, no information
could be found concerning sound transmission through
such structures. So we set up a data bank in order to
understand transmission mechanisms through a
truncated cone and quantify the influence of its
coupling to a cylinder on them.
Experimentation (Noise Reduction (NR) + cavity
modes) has been carried out on three cases : (a) a
rigidly backed isotropic truncated cone, (b) an
assembly of an isotropic truncated cone and a stiffened
cylindrical shell, (c) a stiffened cylindrical shell.
DESCRIPTION OF THE
EXPERIMENTS
Figure 1 shows the geometry and the dimensions of
the shells.
The NR of the truncated cone for cases a and b, and
the NR of the cylinder for case c, have been measured
under diffuse sound field conditions in the 100Hz10kHz range. For each case, the excitation was a white
continuous noise generated by a loudspeaker, and four
microphones (two external, two internal) were used.
For each microphone, the measurement has been made
for 12 circumferential different positions, with the
same radial and longitudinal positions (cf. figure 2).
The Noise Reduction (NR) has been obtained by :
NR (f ) = 10 log10
2
pext
2
pint
S
(1)
V
<pext2>S and <pint2>V are respectively mean external
and internal quadratic sound pressures with S and V
being the cone surface and volume for cases a and b,
and being the cylinder surface and volume for case c.
.
INTERPRETATION OF THE
EXPERIMENTAL NR CURVES
To refer to the transmission mechanisms of a
cylinder, let’s consider the lower and upper ring
frequencies of the truncated cone, fR2=1800Hz and
fR1=4500Hz, and its critical frequency fc=12000Hz.
fR2 and fR1 correspond to the ring frequencies of
cylindrical shells with radius equal to the larger and
the smaller radii of the cone.
Figure 3 shows the Noise Reduction (NR) in the
SESSIONS
100Hz-10kHz range for the three cases. The two major
results are the superposition of the NR curves of the
truncated cone for cases a and b above 0.56*fR2 (fig.3,
curves 1, 1’) and the similitude of the transmission
behaviour of the truncated cone for cases a and b to
that of a cylinder (fig. 3, curves 1, 1’, 2).
Note however that the plateau ends with the ring
frequency for the cylinder (fig. 3, curve 2), while it
ends at 1.56*fR2 (or 0.62*fR1) for the cone (fig. 3,
curves 1, 1’, zone 2), that is between the lower and
upper ring frequencies. Moreover, the NR increases
h
D1
hc
Scone
h1 L
Hemispheric
nose
PERSPECTIVES
An
analytical
approach
and
full
scale
experimentation on an airplane cockpit at EADS
AIRBUS SA France will soon complete our study.
24 stringers
Vcone
D2
hn
Scylinder
with a 12 dB/octave (fig.3, curves 1, 1’, zone 3), which
is not the classical mass law tendency behaviour
(6dB/octave) observed for a cylinder (fig.3, curve 2).
Vcylinder
End plate
h2
End
plates
h3
Cone :
h=1mm
D1(external)=382mm
D2(external)=942mm
L=950mm
h1=25mm
h2=30mm
hn³10mm
Cylinder :
hc=1mm
D2(external)=942mm
Lc=1800mm
h3=30mm
The 24 stringers are
equally spaced.
Both shells and stiffeners are made of aluminium
Lc
FIGURE 1. Thin isotropic conical shell and stiffened cylindrical shell – Geometry and dimensions
Case a
Case b
Fixation bar for microphones
l
l=150mm
l
l l
l
l
Case c
l
l=150mm
l
External microphones (distance from the skin»10mm)
l
l=150mm
Internal microphones (distance from the skin »50mm)
FIGURE 2. Microphones positions for Noise Reduction measurements
(1)
Zone 1
Zone 2
Me mbra ne
be ha viour
with minima
due to c a vity
mode s
P la te a u
10 dB
Zone 3
Zone 4
Inc re a se De c re a se
with a
be low fc
12 dB/ oc t
slope
(1)
10 dB
(3 )
(3 )
(4 )
(4 )
(1')
fR2
100
1000
F re que nc y ( H z)
(2 )
fR1
10000
100
fR2
1000
fR1
10000
F re que nc y ( H z)
FIGURE 3. Measured Noise Reduction (NR) under diffuse sound field conditions. (1) NR of the truncated cone case a, (1’) NR
of the truncated cone case b, (2) NR of the stiffened cylindrical shell, case c, (3) Mass law under diffuse sound field conditions
(6dB/octave slope), (4) 12 dB/octave slope. (1), (1’), (2) are analysed in narrow band Df=4Hz.
SESSIONS
Sound Radiation from Visco-elastically Damped Plates
Excited by a Random Point Force
K Akamatsua and T Yamaguchib
a
Machinery Acoustics, 1-1-2-314 Obanoyama Shinohara, Nada-ku, Kobe 657-0015, Japan
b
Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma 376-8515, Japan
The acoustic radiation from baffled finite plates with free viscoelastic damping layers excited by a random point force is studied
in order to evaluate damping treatment performances under the condition of neglecting fluid loading. The vibration response of
the plates with damping layer is obtained by the finite element method. Modal damping ratios are estimated from undamped
normal mode results by means of the modal strain energy method. Expressions for the surface acoustic intensity and the radiated
sound power are derived in the transform formulation and evaluated numerically. An experimental study is carried out to
measure the surface intensity distributions and to compare them with the analytical results.
ANALYTICAL METHOD
The surface acoustic intensity distribution on a
baffled finite plate of arbitrary configurations excited
by a stationary random force is derived. The surface
acoustic pressure radiated from the plate is described
in the frequency domain as [1]
ωρ
P(x,y,ω ) =
2
(2π )
=ωρF
−1
∞
V˜ (ς x ,ς y ,ω )
∫∫
k −ς −ς
2
−∞
2
x
2
y
e
(
j ς x x +ς y y
) dς dς
x
y
[ V˜ (ς ,ς ,ω )G˜ (ς ,ς )]
x
y
x
y
(1)
where F −1[ ] denotes the inverse Fourier transform,
G˜(ς x ,ς y ) is the Green function and V˜(ς x ,ς y ,ω ) is the
Fourier transform of the plate vibration velocity
V(x,y,ω ) in the spatial co-ordinate. V(x,y,ω ) is given
by
V(x,y,ω ) = jωF(ω )∑ H r (ω )φ r (x0 ,y 0 )φ r (x,y)
(2)
r
H r(ω ) =
1
m rω (1 −ω ω 2r + jηω ω r )
2
r
(3)
2
where ω r , m r , φ r (x,y) , η r are the radian natural
frequency, modal mass, mode shape function and
modal loss factor for r th mode, and F(ω ) is the
applied force onto point (x 0 ,y0 ) .
The acoustic intensity at the position (x,y) on the
plate is given by
1
∗
(4)
I(x,y,ω ) = P(x,y,ω )V (x,y,ω )
2
Numerical Results
In the numerical analysis, an aluminum plate with
free layer damping treatment with dimensions of
480 × 360 mm and thickness of 2 mm clamped in a
frame with thickness of 10 mm was used. A square
baffle of length 2.5 m was placed around the plate.
The undamped mode shapes and modal parameters
were computed for the composite plate with the viscoelastic material treated as it were purely elastic, then
the modal loss factors were obtained by the modal
strain energy method [2].
The FFT algorithm and the averaged Green function
developed by Williams [3] are used to compute the
acoustic pressure on the source plane. The baffle plate
area is divided into a lattice of 128 ×128 points with
the lattice spacing of 20 mm.
Figure 1 shows the acoustic intensity distributions
with F(ω ) =1 N for the octave band centered at 250 Hz.
The total acoustic power radiated from the plate is
obtained by a summation of the acoustic intensity
distribution over the plate surface. Two normalized
values of the power, the radiation efficiency and the
power conversion efficiency, are shown in Figures 2
and 3. The power conversion efficiency is defined as
the ratio of the acoustic power Wa to the vibratory
power Wk supplied to the plate,
ς = Wa Wk
Wk = Wa + Wk0
(5)
Wk0 is the input power to the plate neglecting the back
reaction of the radiated acoustic pressure. It is shown
that the radiation efficiency is independent of damping,
while the power conversion efficiency depends on
damping.
EXPERIMENT
In order to evaluate the surface intensity patterns
and the acoustic power for consistency of predicted
trends, an experiment measuring the acoustic intensity
field near a vibrating plate driven by a point force with
white spectrum was carried out.
SESSIONS
without damping layer (250 Hz)
20
10
0
19
-10
10
1.5
with damping layer (250 Hz)
y
1
0.5
the driving point accelerance. The nearfield estimate of
the surface acoustic intensity of the baffled plate was
measured using the two-microphone technique. The
vibratory input power is given by
2
1
F A
∗
(6)
Wk = Re{FV } =
sinφ
2ω F
2
where A F is the magnitude of and φ is the phase of
the driving point accelerance.
Figure 4 shows the experimentally measured
surface intensity patterns. The overall agreement
between the predicted and measured intensity
distributions is good. Table 1 compares the predicted
and measured power conversion efficiency. The trend
of effects of damping is consistent except 125Hz band.
without damping layer (250 Hz)
1.5E-04
0
1.0E-04
-0.5
5.0E-05
Power conversion efficiency [-]
Sound radiation efficiency [-]
-1
17
FIGURE 1. Surface intensity patterns, octave band at 250
Hz, upper; undamped, lower; damped.
0.0E+00
-5.0E-05
0
10
6
5.0E-05
-1
with damping layer (250 Hz)
10
2.5E-05
-2
10
0.0E+00
-3
10
without damping layer
with
damping layer
-4
10
60
80 100
300
Frequency [Hz]
FIGURE 4. Measured surface intensity patterns, octave band
at 250 Hz, upper; undamped, lower; damped.
FIGURE 2. Sound radiation efficiency.
0
10
Table 1. Power conversion efficiency.
125 Hz band
250 Hz band
500 Hz band
undamp damp undamp damp undamp damp
Predicted 0.596 0.022 0.089 0.0059 0.124 0.0126
Measured 0.050 0.009 0.130 0.0068 0.200 0.0169
-1
10
-2
10
-3
10
without damping layer
with
damping layer
-4
10
-2.5E-05
500
60
80 100
300
Frequency [Hz]
500
FIGURE 3. Power conversion efficiency.
The plate assembly was hung vertically at the center
of a baffle with dimensions of 1.5 ×1.8 m . A minishaker was attached to the plate via an impedance head
and driven by a white noise source with frequency
range up to 1000 Hz . The output signals from the
impedance head are fed to the FFT analyzer to obtain
CONCLUSIONS
The acoustic radiation from plates with free viscoelastic damping has been studied analytically and
experimentally. The overall agreement between the
predicted and measured is good.
REFERENCES
1. H. Peng and R. F. Keltie, J. Acoust.Soc.Am. 85 (1), 57–67
(1989).
2. J. Kanazawa, T. Yamaguchi and K. Akamatsu, J. Acoust. Soc. Am.
100(4 Pt. 2) p 2754 (1996).
3. E G. Williams and J. D. Maynard, J. Acoust.Soc.Am. 72 (6),
2020-2030 (1982).
SESSIONS
Nearfield effects on acoustic radiation modes of a structure
O. Schevina , P. Herzogb and M. Rossia
a Laboratory of Electromagnetics and Acoustics, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland
b Laboratoire de Mécanique et d’Acoustique, CNRS, 31 chemin Joseph Aiguier, 13402 Marseille Cedex 20, France
The use of acoustic radiation modes to characterize the behaviour of a structure has received increasing attention since the beginning
of the 90’s, especially for active control applications (ANC). The main advantage of this expansion is that it involves independant
radiation contributions, so that a truncated series leads to a well-defined accuracy for the sum of its terms. However, these modes
are defined from the active power radiated by the vibrating surface; they are thus related to the far-field pressure, or to the real
part of the radiation impedance on the surface itself. As many ANC systems minimize a weigthed sum of squared inputs, their
performances can at first glance be guaranteed only if they involve specific structural sensors, or microphones located in the far
field. However, none of these solution is as cost-effective as could be microphones located close to the structure. Conversely,
near-field microphones pick up the reactive components of the pressure field, and the transfer matrix estimated is in that case is no
more characteristic of the radiated power. This paper proposes to discuss how radiation modes are related to the near field, and the
consequences on the radiation control of a structure from bringing the microphones close to it.
INTRODUCTION
ANC systems are usually aimed to control the acoustic pressure in the far field, and thus they should use
microphones located far away from the noise source,
in such a way that they pick up a signal representative
of the noise to be cancelled. If microphones are located in the vicinity of the source, they pick up local
acoustic phenomena which are not propagated further.
These phenomena, frequently grouped under the term
"nearfield", tend to modify the behaviour of the control system, owing to the fact that the controller seeks
to reduce an acoustic pressure which is not representative of the acoustic power radiated to the far field. Because cheap microphones may be used reliably at low
frequencies, and that reducing the necessary amount of
wiring has a significant effect on the cost of the system,
it was found interesting to study the possibility of bringing such sensors closer to the source, and to analyze the
consequences of this design.
Equation (1) is a good estimate only if the microphones
are placed in the far field and spaced homogeneously
around the noise source.
By expressing transfer functions Z between the pressures p at each microphone and velocities vs at each
point of a discrete model of the source surface, equation (1) can be written in the following matrix form [1] :
Ŵa
vH
s H vs
(2)
1 H
where H ρc
Z ∆S Z is the radiation operator, and ∆S
is a diagonal matrix containing areas associated with
each microphone. The notation H denotes the hermitian
operator. By introducing the eigenvalue/eigenvector decomposition of the operator :
H
QH ΛQ
(3)
in which Q is an orthogonal matrix of eigenvectors and
Λ a diagonal matrix of eigenvalues λi , equation (2) can
be written as
Ŵa
cH Λ c
N
∑ λ2i ci 2
(4)
i 1
RADIATED POWER
Controllers available on the market are usually designed to minimize a sum of squared pressures picked
up by a certain number of microphones. Weighted by
the medium characteristics and the area implicitely associated with each measurement position, this sum is an
estimate of the active power radiated by the source :
N
Ŵa
p2
∑ ρc ∆Si
i 1
(1)
where c Q vs is a vector of coupling coefficients between the surface velocity vs and each eigenvector. The
radiated power estimate is therefore formulated as a sum
of radiation modes which have the property of radiating
power independently of each other.
In order to study the influence of these modes in the
nearfield, we consider the trace of the radiation modes
on the source surface, and compute their effects, on a
surface conformal [2] to the source at a given distance,
as the quadratic estimate defined by equation (1). This
estimate is then compared to the true active power radiated, computed in the far field.
SESSIONS
Source
0
0.1
1
10
distance ( m)
FIGURE 1. Evolution of the acoustic field with distance. Left : vibration pattern of the source surface. right : acoustic field
radiated by the source on 3 different conformal surfaces.
NEARFIELD EFFECTS
We consider here a planar rectangular surface
1.7 x 3 m vibrating and radiating an acoustic field at
100 Hz. Figure 1 illustrates the evolution of the pattern
of the acoustic field with increasing distance, showing
the complexity of the local components in the nearfield.
When bringing the microphones closer, this reactive
component of the acoustic field becomes non-negligible
in comparison with the active one, and this causes the
controller to get a bad estimate of the radiated power if
the microphones are positionned closer than about 1 m.
The pressure picked up in the nearfield, for each radiation mode pattern, is the sum of the pressure related
to its contribution to radiated power, and the reactive
pressure corresponding to local disturbances. These two
components are out of phase, and thus the pressure magnitude in the nearfield is systematically overestimated and so is the related power contribution. Table 1 gives
in the case of that planar surface, the power attenuation
that would be achieved by the controller if microphones
were in the nearfield or in the far field, for a given number of modes included in the ANC design.
Table 1. ANC performances with microphones in the nearfield or far field.
Number
of modes
1
2
5
10
Attenuation (dB)
d 0 5 m d 10 m
1.55
3
7
13
3
4
12
20
Conversely, for a given mode, distance and frequency, the ratio between the active and reactive components of the pressure is fixed by the trace on the surface
of the radiation mode considered. Assuming that the reactive components generated by different modes do not
interact significantly, the overestimation can be fully determined and taken into account in the controller as an
additional weighting factor.
In practice, such a weighting factor has been found
necessary only if microphones have to be placed in
the very nearfield. For intermediate distances and reasonable microphone locations, the weighting factors
mentionned in equation (1) seem sufficient. This has
been confirmed in the case of more complex non-planar
sources, which behaviour revealed no major differences
with planar ones.
CONCLUSION
We have shown that active power radiated by a radiation mode tends to be overestimated by an ANC controller using close microphones. Conversely, a method
is proposed to take this into account by a suitable correction of the controller input weights.
ACKNOWLEDGMENTS
Authors wish to thank the swiss institutions CTI and
PSEL for their financial support, as well as the ABB
Sécheron company for their active participation.
REFERENCES
1. S. J. Elliott and M. E. Nelson, J. Acoust. Soc. Am. 94,
2194-2204 (1993).
2. C. I. Holmer, J. Acoust. Soc. Am. 61, 465-475 (1977).
SESSIONS
Sound reflection characteristics of suspended panel array
T. Yokotaa, S. Sakamotoa, and H. Tachibanaa
a
Institute of Industrial Science, Univ. of Tokyo, Komaba 4-6-1, Meguro-ku, Tokyo 153-0041, JAPAN
The frequency characteristics of sound reflection by suspended panel arrays often equipped in halls are investigated by numerical
analysis based on the Fresnel-Kirchhoff diffraction theory. Three types of panel arrays with different shapes and arrangements are
set and their reflection characteristics are compared.
INTRODUCTION
In order to reinforce the sound reflection, suspended
panel arrays are often equipped in halls. However, it is
known that the sound reflection frequency characteristics of suspended panel arrays are apt to be uneven due
to the interference between discrete reflections from each
panel [1,2]. In this study, this problem is investigated by
numerical analysis based on the Fresnel-Kirchhoff diffraction theory by setting three kinds of suspended panel
arrays with regular and irregular spatial distributions.
From the results, the way of making the reflection characteristic flat is considered.
Type-1
Type-2
Type-3
CALCULATION CONDITION
As the typical suspended panel arrays, three kinds of
variations (Type-1: circle, Type-2: triangle and Type-3:
random) shown in Fig. 1 were set in this study. The reflection frequency characteristic for each panel array was
calculated by the diffraction theory based on the FresnelKirchhoff approximation according to Babinet’s principle.
Figure 2 shows the position of the sound source, calculation area and the projection area of the panel arrays. In
the calculation area, 2809 (53x53) observation points were
assumed at an interval of 0.5 m. At each observation point,
the velocity potential of the reflected sound was calculated for 84 frequencies from 90 Hz to 2240 Hz which were
chosen at an equal interval on logarithmic scale. After
the calculation, the results were averaged in every 1/3
octave band and they were normalized by the value calculated for the direct sound from the mirror image of the
source (diffraction factor: DF).
(open area ratio : 50%)
FIGURE 1. Arragements of the reflectors
Section
Panel Array
R1
Source
Plan(half area of the sound field)
Calculation Area
(observation points : 0.5m intervals)
C.L.
RESULTS AND DISCUSSIONS
Figure 3 shows the spatial distribution of DF in the
calculation area in the case of Type-1, in which different
R1
Source
01 2
5
10 [m]
FIGURE 2. Calculation set-up
SESSIONS
500 Hz (1/3 oct. band)
0
2k Hz (1/3 oct. band)
DF[dB]
125 Hz (1/3 oct. band)
Source
Source
-30
FIGURE 3. Distributions of reflected sound level
10
0
DF[dB]
-10
Type-1
Type-2
Type-3
-20
-30
125
250
500
Frequency [Hz]
1k
FIGURE 4. Reflection frequency characteristics at R1
Type-1
10 Small
Type-2
CONCLUSIONS
From the results of this numerical study, it has been
confirmed that the suspended panel arrays regularly distributed in space are apt to produce uneven reflection
frequency characteristic due to the interference between
discrete reflections. This problem can be mitigated to some
extent by designing the panels in irregular shape and
distributing them randomly in space.
2k
SD [dB]
interference patterns are clearly seen in each frequency
band. These patterns are caused by the interference between the reflections from the regularly distributed discrete panels.
Figure 4 shows the reflection frequency characteristics at the specular reflection point of the center reflector
(R1 in Fig.2) for the three types of panel arrays. In the
cases of Type-1 and Type-2, remarkable dips are seen at
around 250 Hz, whereas the frequency characteristic is
relatively even in the case of Type-3.
In order to evaluate the evenness of the reflection frequency characteristic at each observation point, standard deviation (SD) of the value of DF in each 1/3 octave
band in the frequency region was calculated. As a result,
Fig.5 shows the spatial distributions of SD for each type
of panel array. In the results for Type-1 and Type-2, it is
seen that a lot of areas colored dark (SD is large) are
regularly distributed. It means that there are many areas
where the reflection frequency characteristic is much
uneven. On the other hand, in the case of Type-3, the
distribution pattern seems to be vague and areas colored
light (SD is small) are large. It means that the area where
the reflection frequency characteristic is relatively flat is
expanded.
0 Large
Type-3
Evenness of frequency characteristics
Source
REFERENCES
[1] R.W.Leonard, L.P.Delsasso, and V.O. Knudsen, J.Acoust.
Soc. Am. 36 (12), 2328 - 2333 (1964)
[2] B. G. Watters, L. L. Beranek, F. R. Johnson, and I. Dyer,
Sound 2, 26 - 30 (1963)
FIGURE 5. Distributions of SD
SESSIONS
Absorption and transmissibility of coupled
microperforated plates
T. DUPONT, G. PAVIC and B. LAULAGNET
Laboratoire Vibrations et Acoustique, INSA de Lyon, France
The microperforated plate, backed up by a cavity and a rigid wall, is usually used for sound absorption. Based on this
configuration, the well-known mathematical model by Maa [1, 2] has shown a good agreement with measurements.
However, this model fails if a microperforated plate is coupled to another flexible plate. The latter case is the subject of the
present paper. At first a one-dimensional case is studied, that of a microperforated plate coupled with a rigid but movable
spring-supported piston. A comparison done between modelling and Kundt tube experiment has shown a good agreement.
The main effect which enables absorption of a microperforated plate is the viscosity of air flowing through perforations. Two
methods have been suggested to increase this effect. A two-dimensional model in oblique incidence has been then developed
considering infinite plates. A wave approach using plate impedance has been applied to express the principal acoustic
indicators: transmission, absorption and reflection. A parallel impedance scheme has been used to take in account the
vibration of the microperforated plate. Different cases have been analysed: the isolated microperforated plate, coupled with a
thin plate, a thick plate and a rigid wall.
1. ONE DIMENSIONAL SYSTEM
A mathematical model of absorption and transparency
of a micro-perforated plate (MPP) system was
produced. The model uses the MPP’s impedance as
given by Maa [1,2].
In order to understand the acoustics of the system and
to validate the developed model, an industrial MP
made by a Swedish manufacturer Sontech was tested in
a Kundt’s tube. The MPP was backed by a cavity and a
rigid cap. A very good match was found between the
experiment and the model, Fig. 1.
Fig 1. Simulation (---) and measurement (-o-) of absorption
in the Kundt tube. Sontech MPP: slit perforations and louvre
structure coupled to a rigid wall via 10cm deep cavity. Dural,
thickness 1,25 mm, estimated equivalent diameter of circular
perforation ∅ 0,25mm, estimated perforation rate 3,5 %.
implies that it is difficult increasing absorption without
modifying the MPP’s parameters.
Secondly, the absorption model had been extended by
replacing the rigid wall by a piston (mass with spring
and damping). The piston is used here to represent the
action of a second plate in the 1D model. The mass, the
spring and the piston damping represent a resonant
system equivalent to one of the modes of the second
plate, i.e. one additional degree of freedom. The piston
was found to affect absorption only near its resonance
frequency. In this region, the absorption is completely
modified w/r the previous case; the piston’s resonance
and anti resonance are clearly identifiable (Fig. 2).
Fig 2. Computed absorption coefficient of a cavity-backed
MPP: (dural, circular holes, thickness 1,25 mm, perforation
rate 3 %. Cavity termination: --- rigid wall; -•- 1cm thick
steel piston loaded by 2.42e6 N/m spring. Structural damping
0,01. Cavity depth: 10 cm.
The perforation’s parameters (perforation rate and hole
size) were found to be the key factors governing the
2. TWO DIMENSIONAL SYSTEM:
system’s absorption. As the viscosity of the fluid inside
INFINITE PLATES
the perforations is the principal absorption effect here,
two methods were attempted to increase it: a) heating
the plate and b) replacing the air in the perforations To account for possible variations in the angle of sound
with a more viscous fluid (oil). No major improvement incidence, a further study was made of an infinite MP
was found either by experiment or by simulation. This plate in tandem with an ordinary infinite plate. The
transmission loss, the reflection and the absorption
SESSIONS
factors were computed. The classical formula for the
MPP impedance does not take into account the plate
vibration. The motion of a MPP can modify the part of
velocity responsible for the viscous effect (global
viscosity). If this motion is of comparable level and
phase with the acoustic velocity, the viscous effect will
be reduced, while the transparency will be affected too.
In order to include this effect in the model and to keep
the classical MPP impedance model valid, a parallel
impedance branch was added to the existing one:
frequencies a screen effect takes place producing high
reflection of the composed system. However, the
higher the frequency the narrower the absorption band,
thus at high frequencies the reflection dominate the
absorption implying an increase of the transmission
loss with frequency.
The coincidence occurs around 1200 Hz, Fig. 3 – top.
At this frequency and at certain angles the transmission
loss becomes very small. It can be seen that the MPP
used reduces the coincidence effect without requiring
1 = 1 + 1 , where Zvib is the plate vibration any absorption material: in the example shown the TL
difference MPP system – simple plate is around 15 dB.
Ztotal Z vib Z MPP
A
reversed system has been also tested: source – air –
impedance under sound wave excitation and ZMPP is
simple
plate – air - MPP. Although reflection and MPP
the MPP impedance of a single MPP based on the
absorption reciprocity do not apply, one can show that
Maa’s model.
the transmission loss is here perfectly reciprocal. For
the reversed system the reflection is very high: it tends
very quickly towards unity with frequency, exactly like
the single plate reflection, which in turn induces very
low pressure in the air gap and consequently a very
low MPP absorption. Because of very high reflection,
the inverse configuration has lost all of the MPP
advantages. Thus when using a MPP, it has to be
mounted facing the side of the acoustic source, or
inside a room which needs sound isolation.
3. CONCLUSION
A MP used in a lightweight composite panel is a good
solution of noise control. Its presence leads to non
negligible system absorption without the need to add
any classical absorption material. In addition, it gives
rise to transmission loss, reduces the reflection effect
and weakens the coincidence effect. A MPP
application could be a viable solution to the low
frequency sound transmission in thin plates. Moreover,
contrary to porous materials commonly used for
absorption, a MPP is inflammable and shows very
good hygienic features. This system could be used in
transport industry, civil engineering, and in acoustic
panel design.
REFERENCES
[1]
Fig 3. Transmission loss (top) and reflection coefficient
(bottom) at 45° incidence. -•- simple plate (thickness: 2mm, [2]
damping: 1%); --- the same simple plate preceded by a 15 cm
deep cavity and a MPP (steel, circular perforation, thickness
[3]
1.5mm, ∅ 0.5mm, perforation rate 0.75).
One can see on Fig.3 up to 15 dB rise of transmission [4]
loss in the presence of the MP. In contrast to the simple
plate case, the MPP system does not produce a [5]
reflection coefficient which sharply jumps to unity
because of the absorption effect, Fig. 3 - bottom.
[6]
As the frequency rises further, the difference in the
transmission loss globally increases (at 4000 Hz the
difference exceeds 20 dB). The reflection increases
with frequency owing to the MPP alone. At very high
MAA, Da You. Theory and design of microperforated panel
sound-absorbing constructions. Beijing, China : Scientia
Sinica, 1975, vol. 18, n°1, p 55-71.
MAA, Da You. Wide-band sound absorber based on
microperforated panels. Beijing, China : Chinese journal of
acoustics, 1985, vol. 4, n°3, p 197-108.
MAA, Da You. Potential of microperforated panel absorber.
JASA, 1998, vol. 104, n°5, p 2861.
NILSSON, A., NILSSON, E. Sound transmission through
honeycomb panels. Proceedings of Congress: Modern practice
in stress and vibration analysis Dublin, 1997.
KANG, J., FUCHS, H. V., Predicting the absorption of open
weave textiles and microperforated membranes backed by a air
space. Stuttgart: JSV, 1999, vol. 220, n°5, p 905-920.
FUCHS, H.V., ZHA, X. Acrylic-glass sound absorbers in the
plenum of Deutscher Bundestag. Stuttgart : Applied Acoustics,
1997,vol 51, n°2, p 211-217.
SESSIONS
Unsteady dynamics using local and global energy models
M. N. Ichchoua , F. S. Suia and L. Jezequela
a Department
of Mechanical Engineering, Ecole Centrale de Lyon, 69130 Ecully, France
A new energy method, called as transient local energy approach (TLEA) is proposed to predict the structural transient dynamic
response in the time domain. For the purpose of comparative studies, the energy expressions got from three methods (TLEA, TSEA
and exact results) are given, respectively. These studies indicate that the difference between TLEA and TSEA depends on the different
description of time-varying energy flow transferred between two coupled subsystems. The comparisons show that TLEA is much
more accurate than TSEA.
INTRODUCTION - LITTERATURE
POSITION OF THIS WORK
SECOND ORDER TRANSIENT LOCAL
AND GLOBAL EQUATIONS
The transient behavior, such as shock and impact, can
cause a broad frequency structural vibration and noise,
especially in high and mid frequency band. In the early
development of the well known statistical energy analysis (SEA), Manning and Lee (1968) proposed a method
based on steady-state power balance equation to deal with
the mechanical shock transmission [1]. However, as was
pointed out by Manning and Lee [1], TSEA was not developed formally, because the definition of coupling loss
factor in transient condition was just "transplanted" from
steady-state SEA, it seems not to be always appropriate
and reliable. More recently, Pinnington and Lednik [2, 3],
published additional TSEA study by comparing with exact results for two-degree-freedom model, the conclusion
they got were not always satisfactory to predict the transmitted energy precisely. This paper will present a new
Transient Local Energy Approach (TLEA) and its discretized format. The two oscillators system and the corresponding two coupling subsystems are studied to illustrate the reliability of TLEA by comparing the solution
with the exact results and TSEA.
Second order local energy equation
FIRST ORDER TRANSIENT SEA
EQUATIONS
dE1
+ η1 ωE1 + η12 ωE1
dt
η21 ωE2
I s t) =
1 ∂~I (~s; t )
ηω ∂t
c2
∇W (~s; t )
ηω
(3)
The energy equation is obtained from the energy balance
as
∂2W (~s; t )
∂t 2
∂W (~s; t )
2
+ (ηω) W (~s; t ) = 0
∂t
(4)
then, formula (4) is called TLEA equations. This equation is different when comparing it to Nefske’s equation
[7]. Precisely, the later and the classic TSEA do not consider the time-varying part of the energy flow term, and
this ignorance results some inevitable errors. The detailed
discussions can be seen in the reference [4, 5].
c2 ∇2W (~s; t ) + 2ηω
The TLEA equation will be discretized so that it can
be used in the interconnected subsystems or multi-DOF
(1)
dE2
+ η2 ωE2 + η21 ωE2 η12 ωE1
(2)
dt
The TSEA energy expressions and the parameters definition are almost the same as that got by Pinnington and
Lednik [2], therefore, the comparison can be made easily
between TLEA, TSEA and exact solution.
Πin2 =
~(~;
Second order global energy equation position with TSEA
The model in Figure 1 is used for TSEA study. The
energy balance associated to this model is:
Πin1 =
Let us define W (~s; t ) as the total energy density associated and ~I (~s; t ) as the active energy flow. c is the energy
velocity, the same as the group velocity of waves in the
slight damping media. After some mathematics it can be
shown [4]:
1
0
0
1
0
1
0
1
0
1
0
1
F2(t)
F1(t)
k1
k2
k
m1
m2
c1
c2
x1(t)
x2(t)
1
0
0
1
0
1
0
1
0
1
0
1
0
1
FIGURE 1. Two-degree-of-freedom model.
SESSIONS
Table 1. Parameters of two different oscillators
Test
Oscillators
Mass
(kg)
Inherent
loss
A
1
2
2.5
2
0.08
0.093
Coupling stiff.
(N/m)
Block freq.
(rad/s)
CLF
ηi j ; η ji
Coupling rate
† η =η
ij i
1000
1072.4
0.26
0.243
3.25
2.60
5105
† : i; j = 1; 2; i 6= j
50
0.25
(a)
40
Transmitted energy power(W)
Energy(J)
0.2
0.15
0.1
0.05
0
0
0.02
30
20
10
0.04
0
0.08
−10
(b)
0
0.005
0.01
0.015
0.02
0.025
0.03
Time(s)
Energy(J)
0.06
FIGURE 3. Comparison of transmitted energy flow (from element 1 to 2) with the coupling ratio r = 2. —, TLEA solutions;
, TSEA solutions;
, exact results.
0.04
0.02
0
−0.02
0
0.02
0.04
Time(s)
REFERENCES
FIGURE 2. Comparison of energy results of two different oscillators. Test A. (a): Input energy, (b): Transmitted energy. —,
, TLEA solutions.
exact results; , TSEA solutions;
oscillator. Precisely, the distributed structure can be divided into some discreted element. Supposed the nodal
value is zero order in the finite elements and the concept
of total energy rather than energy density turns the TLEA
equation into the form:
Πin1 =
1 d 2 E1
dE1
+2
+ η1 ωE1 + η12 ωE1
η1 ω1 dt 2
dt
η21 ωE2
Πin2 =
1 d 2 E2
dE2
+2
+ η2 ωE2 + η21 ωE2
2
η2 ω1 dt
dt
η12 ωE1
(5)
(6)
The coupling loss factor used in TLEA and TSEA is the
same definition in steady state condition.
NUMERICAL SIMULATIONS - FIRST
VERSUS SECOND ORDER MODELS
1. J. E. Manning and K. Lee, Shock and Vibration bulletin.
37 (4):65–70, 1968. Predicting mechanical shock transmission.
2. R. J. Pinnington and D. Lednik. Journal of Sound and Vibration, 189(2):249–264, 1996. Transient statistical energy
analysis of an impulsively excited two oscillator system.
3. R. J. Pinnington and D. Lednik. Journal of Sound and Vibration, 189(2):265–287, 1996. Transient energy flow between two coupled beams.
4. M. N. Ichchou, F. Sh. Sui, and L. Jezequel. CAA’2000
Congress, Sherbrooke, September, 2000. Transient local
energy: theory and application.
5. M. N. Ichchou. Oral presentation at SEANET meeting,
KTH, June, 2001. Unsteady SEA: Formulations and numerical examples.
6. F. Sh. Sui, M. N. Ichchou and L. Jezequel. Journal of Sound
and Vibration, In press, 2001. Prediction of vibroacoustics
energy using a discretized transient local energy approach
and comparison with TSEA.
7. D. J. Nefske and S. H. Sung. NCA Publication, 3, 1987.
Power flow finite element analysis of dynamic systems: Basic theory and application to beams.
The time-varying energy results of three methods are
compared numerically (see Table 1). The comparisons
show that TLEA is much more accurate than TSEA (see
Figure 2 and Figure 3).
SESSIONS
Optimal Design of Stockbridge Dynamic Vibration
Neutralizer: Comparation Between BFGS and Genetical
Algorithms
S.E. Floodya and J.J. de Espíndolab
a
Acoustics Department, Universidad Tecnológica V. Pérez Rosales, Brown Norte 290, Ñuñoa. Santiago, Chile
b
Vibration and Acoustics Laboratory, Universidade Federal de Santa Catarina, Campus Trinidade.
Florianópolis, SC Brasil
The present paper deals with an extension in the study of vibration dynamic vibration neutralizers applied to complex structural
systems, that are independent of geometry, mass, stiffness and damping distribution, introducing the concept of Equivalent
Generalized Quantities. The basic idea is to transform the mechanical impedance that the dynamic vibration neutralizer
transfers in the primary system’s point of connection, in generalized quantities of mass and damping, depending of the
frequency. This methodology was applied to design a viscoelastically modified Stockbridge dynamic vibration neutralizer,
applying the Finite Element Method, extending the proposed theory of generalized equivalent quantities for several degrees of
freedom that can be presented due to the characteristics of this dynamic vibration neutralizer. The cost function was made using
the maximum absolute values of the principal coordinates of the joint system, over a frequency range, this cost function has a
lot of locals minimums in the viable region. Comparations between a minimization algorithm using derivates, the BFGS and
Genetic
Algorithm were
made, to obtain the optimal dimensions of this
mechanical
device.
INTRODUCTION
The dynamic vibration neutralizers, also called
vibration absorbers, are devices or structures
(secondary systems), that are fixed to another
structure (primary system), to reduce vibration
levels. They act over the primary system applying
reaction forces and dissipating vibration energy. The
classic theory of vibration neutralizers introduced by
Den Hartog [1] for viscous neutralizers, called MCK
is difficult to apply; therefore is inadequate for
complex mechanical systems, where many modes
can contribute in the response of the primary
system. Espíndola and Silva [3] introduced the
concept of Generalized Equivalent Quantities, in the
study of vibration neutralizers applied to complex
structural systems, that are independent of geometry,
mass and stiffness distribution. The basic idea is to
transform the mechanical impedance, of the
neutralizer’s coupling point to the primary system,
in generalized quantities of mass and damping that
depend of the frequency. With the generalized
quantities, is possible to formulate the compound
equations of motion in terms of the generalized
coordinates of the primary system only. After the
equations are written in the principal coordinates,
retaining them that correspond to the frequency
band of interest, where the problem of high response
resides. Then, the computations are made in a modal
subspace, with a minimum number of equations.
This method will be used to project a
viscoelastically modified Stockbridge vibration
neutralizer, the viscoelastic material will be added to
increase de dissipation of the vibration energy. The
effects of the frequency dependence in this type of
materials
will
be
studied
FEM MODEL OF A
STOCKBRIDGE VIBRATION
NEUTRALIZER
Basically the neutralizer is composed by a central
mass, two sandwich (metal – elastomer) beams and
two tuning masses, as been showed in Fig. 1. The
finite element method has been used to model in the
most general way possible the neutralizer’s
behavior. The complex and frequency dependent
stiffness matrix is a result of the presence of the
viscoelastic material in the structure. This lead to an
associated eigenvalue problem, which has frequency
dependence and will be solved using the technique
presented in Espíndola and Floody [4] paper. The
motion equation of the secondary system is:
Mq
DD + K (ω )q = f (t )
(1)
SESSIONS
Central Mass
Elastomer
Tuning
Mass
Steel
FIGURE 1 Stockbridge vibration neutralizer Finite Element model
FIGURE 2 FRF Compound System Genetic
Algorithm.
The dynamical stiffness at the root of the
viscoelastically modified Stockbridge neutralizer isb
given by Espíndola, Floody [4].
PROJECT VARIABLES AND
COST FUNCTION
The project variables to optimize are the physical
dimensions of the neutralizer; this can be
represented as a vector:
x = [l1 , l 2 , l 3 , h1 , h2 , h3 , h4 , t ]
T
(2)
Where l ´s represent the lengths, h ´s correspond to
the heights of the metal and elastomer layers and t
is the width of the neutralizer. The cost function to
minimize, proposed by Espíndola and Bavastri [3],
is the modulus of a vector formed by maximum
values of the principal coordinates of the compound
system. This can be expressed in the equation (3).
The minimization method used was the genetic
algorithm to avoid the great number of local
minimum of the cost function. Comparations
between a minimization algorithm using derivates,
like the BFGS were made. The results are shown in
the figure 2 and 3.
min
β (x , Ω )
2
Ω1 ≤ Ω ≤ Ω 2
0 < l i ≤ Li
0 < hj ≤ H j
0<t ≤T
i = 1, l ,3
j = 1, l ,4
β i (x, Ω ) = max Pi , s (x, Ω )
(3)
FIGURE 3
Algorithm
FRF
Compound
System
BFGS
REFERENCES
1 Den Hartog, J.P., 1956, “Mechanical Vibrations,
McGraw-Hill, New York.
2 Espíndola, J.J, Silva, H.P., 1992, “Modal
Reduction of Vibrations by Dynamic
Neutralizers: A General Approach”, 10th
International Modal Analysis Conference, San
Diego, California, pp. 1367-1373.
3 Espíndola, J.J., and Bavastri, C.A., 1997,
“Reduction of Vibration in Complex Structures
with Viscoelastic Neutralizers - A Generalized
Approach and a Physical Realization”,
Proceeding of DETC’97, 1997 ASME Design
Engineering Technical Conferences, September
14 - 17, 1997, Sacramento, California.
4 Espíndola, J.J, Floody, S.E., 2001, “On the
Modeling of Metal – Elastomer Composites
Strutures : A Finite Element Method Approach”,
PACAM IV – DINAME VI, Pan American
Congress in Applied Mechanics, Rio de Janeiro,
Brasil.
SESSIONS
Some Aspects concerning a Method for SEA Loss Factor
Estimation
K-O. Lundberg
Department of Engineering Acoustics, LTH, Lund University
P.O.B. 118 SE-22100 Lund Sweden, Karl-Ola [email protected]
A method for experimental estimation of SEA loss factors based on CMTF:s (complex modulation transfer functions), has
been earlier reported [1,2]. In the method the low frequency part of the CMTF curve is fitted to a SEA model.
The used SEA model has minimum-phase transfer functions. The CMTF curve, however, is not minimum-phase due to a
propagation delay. To yield better SEA parameter estimations, the delay should be estimated and removed by shifting the
origin of the squared impulse response. Alternatively, using the Hilbert transform, the minimum-phase version of the CMTF
can be computed from its magnitude function.
The SEA loss factors are estimated for the non-modified, the time-shifted and the Hilbert manipulated response.
In the fit procedure the CMTF curve up to the "knee" has been used, rather arbitrary. The damping can be found in the very
low frequency part of the CMTF curve, which is obvious from the moment theorem. It means that the centre of gravity time
of the squared impulse response is equal to minus the slope of the phase function of CMTF at zero frequency.
The centre of gravity time, the slope of the phase function and the decay constant from decay rates of the reverberation curve
are computed.
INTRODUCTION
A wavefront emanating from a source of sound
travels at the velocity of sound and reaches a
receiver after some time. Thus due to the finite wave
velocity there is a propagation delay between the
source and a receiver. If the system under test is
finite, the response to a pulse of short duration is the
direct sound followed by a series of reflections.
From this response, or rather the squared impulse
response, the damping in the system can be found.
-9
7
x 10
first time
of arrival
1
0
0
0.005
0.01
0.015
time (s)
0.02
0.025
0.03
FIGURE 1. Initial part of a squared sound pressurevolume velocity impulse response. Measured in a
reverberation chamber at a source-receiver distance of
3.15 m in a 1/3-octave band centred at 1250 Hz. The first
time of arrival is at 0.0102 s.
The used SEA model consists of lumped boxes,
while the system under test is a wave propagation
system. When estimating SEA parameters,
therefore, the propagation delay should be removed.
SHIFTING THE ORIGIN OF TIME
One possibility is to visually inspect the squared
impulse response and then shift the origin of its
time record to the moment at which the response
starts to build up. This is usually done when
damping is derived from decay rates of the
reverberation curve; the origin is then shifted to the
first time of arrival.
An impulse response of a physically realizable
system is a causal function. The squared impulse
response is causal as well. Then some relations in
signal theory can be used. Let g (t ) be a causal
if (w )
function and G (w ) = A(w )e
its Fourier
transform. If G is minimum-phase-shift, then the
phase function can be uniquely determined from the
natural logarithm of the magnitude function. They
are related by the Hilbert transform, i.e.
f = Hilbert{ln A} . Thus the minimum-phase
version of the phase function can be computed from
a given magnitude function.
The CMTF is defined as the Fourier transform of the
squared impulse response, low-pass filtered.
SESSIONS
-9
14
x 10
THE MOMENT THEOREM
The centre time t s is the centre of gravity along
the time axis of the squared impulse response. For a
single-degree-of-freedom system with a decay
constant d, the inverse of the centre time is 2d.
The moment theorem [3] relates the first moment of
a time function to the frequency derivative of its
Fourier transform at zero frequency: The centre
time of the squared impulse response is equal to the
slope of the phase function of CMTF at zero
frequency.
2
0
-2
0
0.005
0.01
0.015
time (s)
0.02
0.025
0.03
FIGURE 2. The Hilbert-manipulated squared impulse
response.
½CMTF½
1
1_
Ö2
Table 1. Estimates of 2d (=wh
factor)
from nonmodified
squared
impulseresponse
5.80
0.170
from leftshifted
0.0102 s
squared
impulseresponse
6.22
0.160
1 / t s (1/s)
5.89
6.26
Slope of
ÐCMTF(0) (s)
-1/slope (1/s)
-0.170
-0.160
-0.158
6.26
6.09
6.38
6.31
a0 (1/s)
t s (s)
0.99 Hz=W0mag/2p
0
0
0
1
5.89
W 0 ph (1/s). See 5.47
2
3
modulation frequency (Hz)
4
5
=
-p/4
-p/2
0
0.87 Hz = W 0ph/2p
1
2
3
modulation frequency (Hz)
4
5
FIGURE 3. The CMTF. The solid line corresponds to
the non-shifted, the dashed to the Hilbert-manipulated
and the dotted to the left-shifted 0.0102 s, squared
impulse response. Upper: Magnitude function (All three
curves coincide) Lower: Phase function. The dasheddotted line is the used SEA model
6.24
CMTF (0)
2
See Fig.3,upper
Reverberation time
T [-5,-25] dB (s)
13.82/T (1/s)
T [-5,-20] dB (s)
13.82/T (1/s)
T [-5,-15] dB (s)
13.82/T (1/s)
EDT [0,-10] dB (s)
13.82/EDT (1/s)
slope= –0.160
b0
jW + a 0
where the
polynomial coef. are derived from a least square curve fit
of the Hilbert-manipulated CMTF using the interval
from zero to the frequency where ÐCMTF is about -40°.
From
Hilbertmanipulated
squared
impulseresponse
6.35
Fig. 3, lower
CMTF (W0 mag ) =
Ð CMTF
where h is the loss
2.21
6.25
2.12
6.52
2.16
6.39
2.18
6.34
ACKNOWLEDGEMENTS
The author is grateful to Prof. S. Lindblad for
fruitful discussions.
REFERENCES
1. Lundberg K-O, Acustica united with acta acustica, 83, 1-9
(1997).
2. Lundberg K-O, Building Acoustics 8 57-74 (2001).
3.Skudrzyk E. The Foundations of Acoustics.
SESSIONS
Identification and quantification of damping mechanisms of
active constrained layer damping
Hélène Illaire, Nicolas Poulain, Wolfgang Kropp
Department of Applied Acoustics, Chalmers University of Technology, S-41296, Göteborg, Sweden
To optimise ACLD (Active Constrained Layer Damping) treatments, it is essential to identify and quantify the different
mechanisms leading to the reduction of vibrational energy in the structure. Shen [1] showed that the power input by an external
excitation in a beam treated with ACLD is dissipated by the shearing of the viscoelastic layer and by the absorption of
mechanical energy by the actuator.
In this work, it is shown that the actuator also change the structure impedance, and thus the power input by external disturbances.
The motion of a fully treated beam is calculated with a wave approach model. Then, the power balance of the structure is derived,
and ACLD treatments’ mechanisms of action are identified and quantified.
INTRODUCTION
The principle of ACLD is to replace the constraining
layer of conventional constrained layer treatment by an
actuator (Fig. 1). The actuator will increase the shear in
the viscoelastic layer; it will also apply a force on the
beam, so that energy is also dissipated through active
damping.
- The viscoelastic layer is only shearing, with a
constant shear angle across its depth.
The equation of motion for the transversal deflection
is derived by writing the equilibrium of forces and
moments on each layer. It is a differential equation of
order 6. Solving the characteristic equation gives 6
wave numbers associated with 6 waves propagating on
the structure. Six boundary conditions are necessary to
determine the amplitudes of these waves. In this work,
each end of the beam is resting on a spring. These
boundary conditions are chosen because they are easy
to implement experimentally.
FIGURE 1. Principle of ACLD.
Experimental validation of the model
Understanding and quantification of ACLD
treatments’ mechanisms of action is essential for their
optimisation. These mechanisms consist in dissipation
of energy by the viscoelastic layer, in absorption of
energy by the actuator, and as shown here, in reduction
of external input power due to the change of structure
impedance induced by the actuator.
The motion of a fully treated beam is calculated with
a wave approach model. Then, the power balance of
the structure is derived, and ACLD treatments’
mechanisms of action are studied.
DESCRIPTION OF THE MODEL
The model is based on the work of Baz [2]. It uses a
wave approach to calculate the motion of a beam fully
treated with ACLD. The model makes use of several
assumptions:
- The transversal deflection is the same for all layers,
i.e. there is no compression with respect to the
thickness.
- Only bending waves propagate in the base and cover
beam. Classical Euler-Bernouilli assumptions apply for
these layers.
FIGURE 2. Acceleration at the driving point of the test
beam: model (plain) and measurements (dotted).
The model is validated for a passive configuration.
An aluminium beam of dimensions 0.4 x 0.03 x 0.003
m was treated on its whole length with a 3M damping
tape. The beam was resting on springs made of foam
and was excited at one end with a shaker. The
displacement at the excitation point was measured with
an accelerometer. The mass-loading effect of the
SESSIONS
measurement equipment was included in the model. A
first measurement with the untreated beam was
performed in the range 0-2 kHz, to determine the
stiffness of the springs, and to tune the Young’s
modulus of the aluminium. Then the measurement was
repeated with the treated beam. Fig. 2 shows a good
agreement between the results of the measurement and
of the model.
Experimental validation of the model for an active
configuration is part of ongoing work.
input power is strongly reduced by the active control in
this case.
IDENTIFICATION AND
QUANTIFICATION OF DAMPING
MECHANISMS
In this work, a quantification of the different
damping mechanisms based on an energy approach is
proposed. The power balance in the structure has to be
fullfiled:
Wext + Wa = Wshear
,
FIGURE 3. Displacement at the driving point of the test
beam: without control (dotted) and with control (plain).
(1)
where Wext is the power input of the external force
applied by the shaker, Wa is the power input of the
actuator, Wshear is the power dissipated in the
viscoelastic layer, and <⋅> denotes the time averaging.
Wa can be positive or negative, depending on the
amplitude and phase applied to the actuator.
The force applied by the actuator is changing the
impedance of the structure and thus the power input of
the external force. This change of the power input can
be quantified as
∆Wext = Wext p − Wext ,
(2)
where Wext p is the power input of the external force
FIGURE 4. Power balance of the beam at the 1st mode
when the active control is turned off.
Simulations
Simulations are calculated on a test beam to illustrate
the damping mechanisms. The test beam is the same as
the beam used to validate the model, except that freefree boundary conditions are now assumed for the sake
of simplicity. The actuator is driven with a voltage of
amplitude 50 Volts. The phase shift between the
control voltage and the external force is π.
The displacement of the driving point with and
without control is shown in Fig. 3. The amplitude of
the 1st mode is decreased of 9 dB, and this of the 2nd
mode of 6 dB. The even modes are not affected by the
control, since the actuator covers the whole length of
the beam.
The power balance is then calculated at the 1st mode.
The quantities Wext , Wext p , Wshear and Wa
are plotted in Fig. 4. The results show that the external
CONCLUSION
This study shows that the reduction of input power
induced by the actuator can substantially contribute to
the reduction of vibrational energy in the structure.
Therefore, taking this mechanism into account when
optimising ACLD treatments could lead to a
substantial improvement of the efficiency of such
treatments.
ACKNOWLEDGEMENTS
This research is sponsored by Volvo Research
Foundation, Volvo Educational Foundation and Dr Pehr G
Gyllenhammar Research Foundation.
REFERENCES
1. Shen, I.Y., Journal of Vibration and Acoustics 119, 192199 (1997)
2. Baz, A., Active constrained layer damping, in Proceedings
of Damping’93, San Francisco, 1993, pp. IBB 1-23
SESSIONS
A Wavenumber Approach for the Response of Aircraft
Sidewalls to Random Pressure Fluctuations
C. Maury, P. Gardonio and S.J. Elliott
Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, U. K.
Under cruise conditions, aircraft fuselages are exposed to high-level fluctuating pressures mainly due to aerodynamic
turbulence and jet noise. In order to analyse the resulting sound transmission through aircraft sidewalls, simulations
have been carried out to predict the response of a fuselage panel for two types of random pressure field, namely a
diffuse acoustic field and a turbulent boundary layer excitation.
An analytical model is first presented for determining the vibro-acoustic response of an aircraft panel to a large class of
random excitations in the wavenumber domain. From the simulations, several trends have been pointed out: (i) both
resonant and non resonant modes contribute to the panel response over a broad frequency range, below the acoustic
coincidence frequency for a diffuse field excitation, and over the hydrodynamic coincidence area for a turbulent
excitation; (ii) the panel structural modes are more efficiently excited with a diffuse acoustic field than with turbulent
wall-pressure fluctuations.
INTRODUCTION
This paper is concerned with the use of a
wavenumber-frequency approach to determine
the vibro-acoustic response of a fuselage panel to
wall-pressure fluctuations either due to a fully
developed turbulent boundary layer (TBL) or to
an acoustic diffuse field. The first configuration
is encountered either during flight conditions or
in a wind-tunnel facility. The second case is
usually achieved in a transmission suite facility.
The objective of this paper is to show that, in
order to simulate the response of a TBL-excited
panel in a laboratory, it is not sufficient to
generate a reverberant field of similar level than
the wall-pressure fluctuations due to a TBL.
THEORY
Measurements performed by Wilby et al. on
the forward part of an aircraft fuselage under
cruise conditions have shown that at high
subsonic Mach number ( U ∞ = 225 m s ) and for
frequencies above 400 Hz, the vibrations induced
by the TBL wall-pressure fluctuations imparted
on the fuselage shell are only correlated over a
fuselage panel, i.e. the area between two adjacent
frames and stringers [1]. Therefore, a simplified,
but still representative model is to consider an
array of uncorrelated fuselage panels [2, 3].
Each fuselage panel is modelled as a flat
aluminium panel set in an infinite rigid baffle,
simply-supported along its boundaries and
tensioned because of the cabin pressurisation
effect. The results presented here have been
calculated for a panel with Young’s modulus
E = 71GPa , mass density ρ s = 2700 kg / m 3 ,
Poisson ratio ν = 0.33 and a structural damping
of 1%. The panel dimensions are l x = 200 mm ,
l y = 170 mm . The panel thickness is h = 2 mm .
The panel is excited either by a TBL or an
acoustic diffuse field. The Efimtsov model has
been used for the spectrum of the TBL excitation [4]. The excitation spectrum for an acoustic
diffuse field is given by spatially Fourier
transforming a sinc correlation function and so,
is represented in the wavenumber domain by a
function which is constant within the disk
k x2 + k y2 ≤ ω co and zero-valued elsewhere.
Using a modal formulation, the spectrum for
the sound power radiated can be written as:
Φ Wr (ω ) =
ρ 0 c0 ω 2
Tr Y H JYR ,
2
[
]
(1)
where Y stands for the modal resonance matrix,
R is the radiation matrix and J is the matrix of
the modal joint-acceptances an element of which
is given by:
2
j mn (ω ) = ∫∫ Φ p (k;ω ) Smn (k) d 2 k l x2 l y2 Φ 0 (ω ) (2)
∞
where Φ p (k; ω ) is the excitation spectrum and
th
Smn (k) is the spatial Fourier transform of the mn
panel mode. jmn (ω ) represents a measure of the
SESSIONS
coupling between the main wall-pressure
fluctuations and the major sensitivity region of
the mnth structural mode.
0
−5
Modal Jointfflacceptances
−10
RESULTS
Figure 1 represents the sound power inwardly
radiated by the panel and normalised by the
power spectrum of the excitation Φ 0 (ω ) for both
types of excitations. From the differences
observed, it is clear that, in order to simulate the
sound transmission through a TBL-excited panel,
it is not sufficient to use a reverberant acoustic
field of similar mean-square pressure level. The
sound power radiated by the fuselage panel
(SPR) is 10 to 20 dB higher for an acoustic
diffuse field excitation than for a TBL excitation,
and the difference is accentuated when the
frequency increases.
−30
10Log10(ΦW /Φ0(ω)) (dB rel. 1 W/Pa2)
−40
−50
−15
−20
−25
−30
−35
−40
−45
−50
0
1000
2000
3000
4000
5000
6000
Frequency (Hz)
FIGURE 2. The modal joint-acceptances for the
modes (1,1), (1,2), (4,3) and (7,2) of the panel; Bold
curves: acoustic diffuse excitation; thin curves: TBL
excitation.
All these properties are summarised in Figure 3
which represents the eigenfrequencies of the
panel modes against streamwise mode number,
along with the acoustic and hydrodynamic
coincidence lines. The proximity of the eigenfrequencies to the coincidence lines confirms the
frequency ranges over which highly excited
modes mainly contribute to the panel response
for both kind of excitations.
Diffuse field
−60
r
6000
−70
5000
TBL
−80
Acoustic matching lines
4000
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
Frequency (Hz)
−90
6000
Frequency (Hz)
3000
2000
Hydrodynamic matching lines
FIGURE 1. The sound power inwardly radiated by a
fuselage panel normalised by the excitation spectrum.
Bold curve: acoustic diffuse excitation; thin curve:
TBL excitation.
An interpretation can be given in terms of the
wavenumber sensitivity of the panel structural
modes to a given form of excitation, also
represented by the modal joint-acceptances. They
have been represented in Figure 2 for some panel
modes for which the resonant frequencies have
been located by a tip. For each mode, the
maximum occurs at a frequency for which the
spacewise variations of the mode are mostly
coincident with those of the forcing field. For a
TBL excitation, it can be seen that, below 2 kHz,
both resonant and non-resonant highly excited
modes mainly contribute to the panel response
whereas, above 2 kHz, the panel modes are
inefficiently excited. For a diffuse field excitation and up to 10 kHz, both resonant and nonresonant modes are efficiently excited by the
reverberant field and so the SPR levels do not
decrease with frequency as in the TBL case.
1000
0
0
1
2
3
4
5
6
7
8
Streamwise mode number
FIGURE 3. Location of the panel eigenfrequencies in
the frequency-streamwise wavenumber domain: -o-,
m=1; -|-, m=2; -◊-, m=3; -∇-, m=4.
REFERENCES
1.
J. F. Wilby and F. L. Gloyna, J. Sound and Vib.
23(4), 443-466 (1972).
2.
C. Maury, P. Gardonio and S. J. Elliott, Modelling of the flow-induced noise transmitted
through a panel ISVR Technical Report no 287,
2000.
3.
C. Maury, P. Gardonio and S. J. Elliott, Active
control of the flow-induced noise transmitted
through a panel AIAA Journal, 2001 (In press).
4.
B. M. Efimtsov, Soviet Physics Acoustics 28(4),
289-292 (1982).
SESSIONS
Experimental characterization of the coupling between
extensional and bending waves in a beam with discontinuity
M.-H. Moulet, F. Gautier and J.-C. Pascal
Laboratoire d’Acoustique de l’Université du Maine, UMR CNRS 6613,
Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France
In order to predict vibratory behavior of complex assembled structures, accurate models of the junctions between sub-structures
are required. Because of the junctions, out-of-plane and in-plane vibrations are coupled. With a wave approach, this coupling
can be described with a scattering matrix, whose terms are interpreted as reflection, transmission and coupling coefficients. The
objective of this paper is to present a specific measurement technique allowing us to determine the scattering matrix terms. This
technique is developed and validated in the simple case of a beam with a discontinuity, which induces a coupling between
bending and extensional waves.
INTRODUCTION
In aeronautical or automotive sectors, complex
assembled structures are widely used. Junctions
between sub-structures couple out-of-plane and inplane vibratory displacements. In the case of beams,
junctions couple bending and extensional waves. A
simple analytical model of such coupling is developed
and applied to periodic structures in [1]. In [2], power
flow is measured in T-beams, where bending and
extensional waves are coupled, without the junction
being characterized. Characteristics of a joint, which
constitutes a junction between two beams, are
measured for bending waves only, in [3].
The aim of this paper is to characterize experimentally
the coupling between bending and extensional waves
in a beam with a discontinuity. First, the scattering
matrix of a junction is presented in the case of a beam
with a mass, which constitutes the studied
discontinuity. Then, an experimental technique
provides us the scattering matrix terms. Finally,
experimental terms are compared with theoretical
predictions.
SCATTERING MATRIX OF A
JUNCTION
We study the case of a prismatic beam with an added
mass (see figure 1), whose centre of gravity, G, of the
mass is not on the neutral axis of the beam. Such a
discontinuity, which is a simple junction separating
two parts of the beam, couples extensional and
bending waves.
-
F
[A- C- E-]
w (x)
y
.
O.
G
u-(x)
[B+ D+ F+]
w+(x)
yG
u+(x)
x
xG = 0
[B- D- F-]
[A+ C+ E+]
FIGURE 1. Beam with an added mass
.
On both sides of the discontinuity, the longitudinal
displacement u and the transverse flexural
displacement w are expressed as:
u ± ( x) = A ± e jke x + B ± e − jke x ,
(1)
w± ( x) = C ±e jkb x + D±e− jkb x + E ±e−kb x + F ±ekb x , (2)
where ke and kb are the wave numbers associated to
longitudinal and flexural wave motion. The
superscripts + and – refer to front (x>0) and back
(x<0) of the discontinuity respectively.
The discontinuity can be intrinsically characterized by
a scattering matrix defined by:
V out = SV in .
(3)
The amplitude vectors Vin = [A-;B+; C-; D+; E-; F+] t and
t
Vout = [A+;B-; C+; D-; E+; F-] represent waves coming
into and coming out of the junction.
In the case of a widened junction, when far field
approximation is valid, the discontinuity can be
~
characterized by a reduced scattering matrix S defined
~
by [ A+ ; B − ; C + ; D − ]t = S [ A− ; B + ; C − ; D + ]t . Making
use of the motion equations of the added mass, it can
~
be shown that the scattering matrix S has the
theoretical following form:
SESSIONS
 Te
 R
~
S = e
 Ceb

− Ceb
Re
Cbe
Te
− Cbe
− Ceb
Tb
Ceb
Rb
− Cbe 
Cbe 
Rb 

Tb 
|Te|
|Re|
(4)
kb l
|Tb|
phase (rad)
phase (rad)
|Ceb|
kf l
kb l
u
out-of-plane
vibrometer
beam
w(ω)
FIGURE 2. Experimental set-up
The beam is excited by a shaker at one end and is free
at the other end. The induced vibratory field is scanned
by two laser vibrometers: the first one measures
flexural velocity w; the second one measures
longitudinal velocity u. For each vibrometer and for
each part of the beam (x<0 or x>0), two measurement
points are needed in order to determine the amplitude
vectors Vin and Vout.
Because such a measurement provides us only 4
relations between the 6 scattering matrix terms defined
in (4), two configurations of the excitation source have
to be used. For the first configuration, a harmonic
force is applied in the longitudinal direction (Ox). For
the second configuration, the force is applied in the
transverse direction (Oy). The amplitude vectors Vout
and Vin being determined in the two configurations, the
six scattering matrix terms can be calculated with an
inversion procedure.
EXPERIMENTAL RESULTS
kf l
kb l
FIGURE 3. Magnitude of the scattering matrix terms
associated to the discontinuity
(continuous line: experimental, dotted line: theoretical)
phase (rad)
in-plane
vibrometer
kf l
kb l
|Cbe|
phase (rad)
u(ω)
(1)
w
shaker
e(ω)
kf l
kb l
phase (rad)
mirror
added mass
analyzer
N
1= 1
MEASUREMENT TECHNIQUE
In order to measure the scattering matrix terms, a
specific experimental set-up is defined on figure 2.
kb l
|Rb|
f ( x ) = ∑ α n sin( nπl x )
phase (rad)
where Ri are reflection coefficients, Ti are transmission
coefficients and Cij are coupling coefficients.
Subscripts b and e are related to bending and
extensional waves respectively.
As shown in figure 3, the magnitudes of the
experimental kscattering
matrix terms have
kf l the same
fl
frequency evolution and the same order of magnitude
as the magnitudes of the theoretical terms. However,
for some frequencies, discrepancies remain which can
be explained by ill conditioning of the matrix to be
inverted and by position error of in-plane vibrometer;
because of this error, longitudinal motion induced by
extensional waves can not perfectly be separated from
longitudinal motion induced by bending waves.
CONCLUSION AND PROSPECTS
A specific experimental set-up is developed and tested
in order to measure the scattering matrix terms of an
added mass on a beam. The qualitative agreement
between measures and theoretical prediction could be
improved with a spatial filtering to separate the
different waves and regularization methods to avoid
the ill conditioning of the matrix to be inverted.
REFERENCES
-4
The studied beam is 2m long (l), has a 1.6 10 m
cross-section, and is made with aluminum whose
Young’s modulus is E=67.5 GPa and whose density is
ρ=2700kg.m-3. The position of the added mass
(M=74.10-3 kg) is defined by the coordinates
yG=3.45.10-2 m and xG=0 (see figure 1).
1. D. J. Mead and Š. Markuš, J. Sound Vib, 90(1), 1-24
(1983).
2. R. P. Szwerc, C. B. Burroughs, S. A. Hambric and T. E.
McDevitt, J. Acoust. Soc. Am., 107(6), 3186-3195 (2000)
3. G. Pavic, Proceedings of International Conference
NOVEM 2000, 31Aug.-2Sept.2000, Lyon, France
SESSIONS
Diagnostics of structure vibrations in acoustic frequency
range with the aid of self-organizing feature maps
S. N. Baranov, L. S. Kuravsky
Problem Laboratory of Mathematical Modeling
attached to the Computer Center of Russian Academy of Sciences,
c/o “Rusavia”, 6 Leningradskoye Shosse, 125299 Moscow, Russia. E-mail: [email protected]
Failure diagnostics for the structures suffered vibrations in acoustic frequency range is presented. Normalized spectral
characteristics of structure response measured in checkpoints are used as indicators to be analyzed. Self-organizing feature maps
(Kohonen networks), for which output variables are not required, detect faults. Simultaneous application of different networks
duplicating each other makes it possible to improve the quality of recognition. Principal component analysis is employed to
reduce the number of variables under study. An aircraft panel with different combinations of attached defective dynamic
suppressors is considered to demonstrate features of the approach. Tests have demonstrated high effectiveness of the presented
way of recognition and showed the advantages of neural networks over cluster analysis in recognition problems.
Technical diagnostics is one of the most typical
spheres where neural networks are used. Under
consideration here is failure diagnostics of the
structures suffered vibrations in acoustic frequency
range. This diagnostics is carried out on the base of
spectral characteristics measured in structure
checkpoints. It is supposed that neither all possible
types of damages nor corresponding changes induced
in the spectral characteristics may be predicted
beforehand.
Because of multiplicity of structure types and
their applications, it is impossible to generalize
considerably the problem solutions. Therefore
employed for method demonstration is a specific
system including a simply supported steel sandwich
rectangular panel and two attached 1-degree-offreedom elastic vibration suppressors with fluid
friction. Its dynamic behavior was simulated on the
basis of models and methods described in paper [1].
Positions of suppressors were optimized. Wide-band
random processes represented test acoustic loads.
System conditions were estimated via
standardized power spectral densities of accelerations
in a checkpoint. (In general case, some checkpoints
may be used.)
Initial data to estimate such
characteristics may be obtained with the aid of
accelerometers. Standardizing spectral densities makes
it possible to analyze only qualitative shape of
structure response spectra and not to take into account
the response level.
The following system conditions were
simulated: OK – both suppressors work properly,
Only1 – suppressor 2 is defective, Only2 – suppressor
1 is defective, Panel – both suppressors are defective,
Nonlin – non-linear suppressor response (shock
interaction of the moving element and stopper). The
first variant corresponds to normal operating mode,
and the following four ones represents system
damages.
Since all the damages are not assumed to be
known before diagnostics, it is impossible to apply
ordinary neural networks with supervised learning for
their detection. Self-organizing feature maps (Kohonen
networks) [2-3], for which output variables are not
required, may be useful in this case.
Self-organizing feature maps have an output
layer of radial units [4]. This layer is also called a
Topological Map and, as a rule, is laid out in a 2- or 1dimension space.
Starting from an initially random set of centers,
the Kohonen algorithm successively tests each training
case and selects the nearest (winning) radial unit
center. This center and the centers of neighboring units
are then updated to be more like the training case. As a
result of a consequence of such corrections, some
network parts are attracted to the training cases, and
similar input situations activate the groups of units
lying closely on the Topological Map.
A self-organizing feature map is taught to
“understand” input data structure in such a way and to
solve the classification problem. The idea, on which
this network is based, was originated by analogy with
some known features of the human brain.
If clustering of input data is completely or
partially ascertained, semantic labels might be attached
to certain units of the Topological Map.
When a classification problem is solved, so
called accept threshold is set. It determines the greatest
distance on which recognition occurs. If the distance
from the winning element to an input case is greater
than this threshold, it is supposed that the network has
not made any resolve. When units are labeled and
SESSIONS
thresholds are determined properly, the self-organizing
feature map may be used as a detector of new events: it
informs about input case rejection only if this case
differs from all labeled radial units significantly.
The given approach supports diagnostics of both
known in advance and unknown damages.
Simultaneous application of different networks
duplicating each other makes it possible to improve the
recognition quality. Frequency ranges are used as
variables, and the values of normalized power spectral
densities at the centers of these ranges – as cases. Thus,
each complete case represents a separate power
spectral density.
In the test example, initial variants of neural
networks with 3×3, 4×4 and 7×7 output layer
dimensions were trained to recognize the states OK and
Only1. Later on, the conditions Only2, Panel and
Nonlin arose successively. After detection of new
damage types, network training was carried out again,
with corresponding labels being assigned to units of
the Topological Maps1.
It is convenient to estimate the recognition
quality via the percentage of correctly identified
situations. Two sorts of errors may occur: errors of the
1st type, when some unknown system state is identified
as known one, and errors of the 2nd type, when some
system state that has been known before is identified as
unknown one or incorrectly. Application of networks
duplicating each other2 made it possible to avoid errors
of the 1st type in 99-100% of analyzed cases and errors
of the 2nd type – in 98-99% of such cases.
When all variants of system damages are known
before, the problem is essentially simpler. One can
employ traditional neural networks with supervised
learning to solve it. Perceptrons were the best for the
test problem: networks of 100%-recognition were
revealed. Radial basis function networks turned out to
be less accurate.
Neural networks are, of course, not the only way
to solve recognition problems. The same purposes may
be achieved by means of other procedures – for
example, cluster analysis that is intended for partition
of an initial object set into classes following a given
criterion. Comparison of both techniques makes it
possible to draw the following conclusions:
♦ cluster analysis does not yield distinct criteria for
classification: one cannot always distinguish
qualitatively new and old-type damages – the
result depends on critical distance selection;
♦
cluster analysis is less reliable than neural
networks;
♦ cluster analysis needs more computer resources
than neural networks.
Principal components analysis and factor analysis
are employed to reduce the number of input variables
under study if the number of frequency ranges to be
taken into account is too great and worsens network
characteristics. These methods extract few latent
hypothetical variables that explain approximately all
the set of observed ones. As for the test problem, input
data capacity might be reduced up to 2 latent variables,
with the errors being avoided in 93-100% of cases.
Nonlinear transforms on the base of autoassociative
neural networks [5] are used in more complicated
situations.
As a rule, reduction of problem dimension
prunes the number of neurons and, therefore, improves
characteristics of network training.
CONCLUSIONS
1.
2.
3.
4.
REFERENCES
1.
2.
3.
4.
1
Working with a real structure, examination to reveal the failure
nature must be fulfilled before new training and label assigning.
Otherwise, the network will only be able to inform of an appearance
of some new, unknown earlier, damage type.
2
Recognition results were selected “by a majority”.
Self-organizing feature maps, whose training data
do not contain output variables, make it possible to
diagnose conditions of vibroacoustic systems in
situations where neither all possible damage types
nor corresponding changes induced in observed
characteristics are not predictable beforehand.
If all types of system damages are known
beforehand, ordinary neural networks with
supervised learning (perceptrons, radial basis
function networks) may be used for diagnostics.
Test results showed that neural networks were
more efficient recognition tools than cluster
analysis.
Reduction of problem dimension (with the aid of
principal components analysis, etc.) improves
characteristics of network training.
5.
Kuravsky, L. S., and Baranov S. N., “Selection of
optimal parameters for acoustic vibration suppressors”,
in Proceedings of the 7th International Conference on
Recent Advances in Structural Dynamics, Southampton,
United Kingdom, 2000.
Kohonen, T., Biological Cybernetics 43, 59-69 (1982).
Kohonen, T., “Improved versions of learning vector
quantization”, in Proceedings of the International Joint
Conference on Neural Networks, San Diego, USA,
1990.
Haykin, S., Neural networks: a comprehensive
foundation, Macmillan Publishing, New York 1994.
Kramer, M. A., AIChe Journal, 37, 233-243
(1991).
SESSIONS
Cluster Control of Total Acoustic Power Radiated
from a Planar Structure Using Smart Sensors
Nobuo Tanaka
Department of Mechanical Engineering, Tokyo Metropolitan Institute of Technology, Tokyo, Japan
This paper deals with the minimization of total acoustic power radiated from a vibrating distributed-parameter planar structure. This
paper presents a novel control method utilizing both the smart sensors and cluster actuation, thereby enabling one to suppress the total
acoustic power without causing observation/control spillover problems. First, the acoustic power matrix of a planar structure is shown
to be expressed in a form of a block diagonal matrix by reordering the columns and rows of the matrix, and hence the suppression of the
power mode defined in each block matrix leads to the suppression of the total acoustic power. Then, cluster control consisting of cluster
filtering and cluster actuation is introduced, which permits one to control each cluster independently. Finally, the experiment is conducted, demonstrating the validity of the proposed method for suppressing the total acoustic power.
INTRODUCTION
Minimization of the structural kinetic energy of a vibrating structure does not necessarily mean the minimization
of the total acoustic power radiated from the structure.
Sometimes the sound level increases while the vibration
level decreases. Furthermore, the use of point sensors and
actuators for controlling the structural vibration causes both
the observation and control spillover problems leading to
the instability of a control system.
To overcome the problems, this paper presents a novel
control method utilizing both smart sensors [1] and cluster actuation [2], thereby enabling the suppression of the
total acoustic power without causing observation/control
spillover problems. The smart sensor implemented with
signal processing functions is realized by shaping PVDF
film sensors with the aim to extract the acoustic power
mode that is the common thread connecting directly the
vibration field and acoustic field. Therefore, the suppression of the power mode leads to that of total acoustic power.
The cluster actuation is a control strategy to activate the
specific targeted cluster without inducing control spillover
in the sense of cluster, its conceptual background being
based upon cluster filtering; that is, all the structural vibration modes of a rectangular panel, for instance, may be
filtered into four clusters - odd/odd modal cluster, odd/even
modal cluster, even/odd modal cluster and even/even modal
cluster. Among these, the odd/odd modal cluster is the
greatest contributor to the total acoustic power. By introducing smart sensing and cluster control, the extraction
and suppression of each cluster without causing spillover
can be performed. This paper begins by discussing the
acoustic power modes of a vibrating panel, presenting a
method for suppressing the total acoustic power by suppressing the acoustic power mode. By employing the cluster control comprising both smart sensors and cluster actuation, the smart cluster feedback control system is constructed. Finally, the experiment is carried out, demonstrating the validity of the proposed method for suppressing the total acoustic power of a vibrating panel.
ACOUSTIC POWER MODE
Consider some generic structure, subject to harmonic excitation by an unspecified primary forcing function. Then
the total acoustic power radiated from the structure is writ-
ten as
Pw = v HAv = v HQΛ
ΛQ – 1 v = u HΛu
(1)
where v is the vector of complex modal velocity amplitudes, H is the matrix Hermitian, A is some real, symmetric, positive-definite acoustic matrix, u is the power modal
amplitude vector, Q is the modal matrix and Λ is the diagonal matrix obtained by the orthonormal transformation
A = QΛ
ΛQ –
1
(2)
As can be seen from Eq. (1), u is given by
u = Q – 1v = Q Tv
(3)
Accordingly, the velocity at x of a structure is expressed
by
v(x) = Ψ T(x)v = Φ T(x)u
(4)
where Ψ and Φ are the vectors of the eigen function and
power mode function, respectively. The relevance between
these vectors are given by
Φ(x) = Q TΨ(x)
(5)
The total acoustic power in Eq. (1) is, then, expressed as
Nm
Pw =
Σλ
i=1
i
ui
2
(6)
where λ is the eigenvalue of A, which is always positive
and real due to the property of A, and hence Pw is always
suppressed when ui, the power modal amplitude, is reduced.
Furthermore, the acoustic power modes are combinations
of like-index structural modes (odd/odd modes, odd/even
modes, even/odd modes and even/even modes) which contribute independently to the acoustic radiation.
CLUSTER CONTROL
The difficulty in controlling the vibration of a distributed-parameter structure lies in the infinite number of
eigenfunctions (structural modes) present in a “real” system. The approach presented in this paper for tackling this
problem is “cluster control” that consists of both “cluster
filtering” and “cluster actuation”. Cluster filtering places
structural modes into a finite number of clusters, each clus-
SESSIONS
ter possessing some common property, while employing
cluster actuation excites targeted clusters independently.
Thus, by using both cluster filtering and actuation, cluster
control avoids observation/control spillover in the sense
of cluster. The end result means that groups of modes can
be treated independently, and it becomes possible to preferentially direct control effort to the most bothersome clusters. This is an important result for structural acoustics, as
the clusters containing volumetric modes can be preferentially dampened using a DVFB-like approach with guaranteed stability.
Unlike a conventional modal control approach using
point sensors and point actuators for suppression of structural modes, cluster control aims to suppress the cluster of
interest, leading to suppression of all structural modes belonging to the cluster.
With a view to giving further insight into the significance of cluster control, it is worthwhile using the specific
example of controlling the odd/odd modal cluster. First,
in order to construct the cluster control system, 4 sensors
for cluster filtering and 4 actuators for cluster actuation
are needed. The output signal eo/o of a cluster filtering on
the odd/odd modes is described by
4
4
REFERENCES
[1] N.Tanaka, S. D. Snyder and C. H. Hansen, “Distributed Parameter Modal Filtering Using Smart Sensors”
Transactions of the ASME, Vol.118 pp.630-640, 1996
[2] S. D. Snyder, N. Tanaka and Y. Kikushima, “The Use
of Optimally Shaped Piezo-electric Film Sensors in the
Active Control of Free Field Structural Radiation, Part 1:
Feedforward Control” ASME J. Vib. Acoust. Vol.117,
pp.311-322, 1995
∞
Σ w(r i,t) =iΣ= 1 kΣ= 1 ϕk(r i)η k(t)
i=1
∞
(7)
Σ ϕ ok / o(r 1)ηok / o(t)
k=1
Phase deg
=4
Mobility
Then, using Eq. (7) as a feedback control signal weighted
with the feedback gain of go/o, the control force is given
by
∞
4
f(r,t) = – 4 g o / o Σ ϕ ok / o(r 1)η k (t) Σ Fiδ(r – r i)
o/o
(8)
Introducing the force polarities, multiplying Eq. (8) by
ϕi(r), integrating it over the domain D and substituting the
resulting control force into the equation of motion, it is
expressed in a form of a modal coordinate system as
i=1
η i(t) +
ωi2η i(t)
∞
=
– 16g o / o Σ ϕ ok / o(r 1)η k (t) ϕ i(r 1)
o/o
Gain dB
Gain dB
Observe from Eq. (9) that the output signal from the odd/
odd modal cluster is fed back only to the odd/odd modal
cluster, thereby avoiding the control/observation spillover
in the sense of cluster.
In exactly the same way, control of the other clusters is
performed by introducing the polarities of the sensors and
actuators .
Figure 1 shows the experimental result for suppressing
smart sensor outputs extracting odd/odd modal cluster, odd/
even modal cluster, even/odd modal cluster and even/even
modal cluster by using the direct feedback control. As is
seen from the figure, all the sensor outputs corresponding
to the power modes are suppressed significantly without
causing instability of the feedback control system.
Spectrum
0
0
– 20
(b ) O dd/ o dd mo da l c l us te r
– 50
– 70
(c) O d d/ev en m od al c l u ste r
– 50
– 70
– 30
(d ) E ve n/od d mo da l cl us te r
– 50
– 70
– 30
Gain dB
if η i(t) corresponds to the odd/odd modes
.
–18 0
20
– 30
k=1
if η i(t) does not correspond to the odd/odd modes
(9)
0
– 40
– 30
Gain dB
k=1
(a) D riv i ng p o int mo b il i ty
1 80
Gain dB
e o / o(t) =
CONCLUSIONS
A new control approach for suppressing the total acoustic
power radiated from a planar structure using both distributed-parameter sensors and cluster actuation has been presented. It was found that the smart sensors based upon the
shaped PVDF film may extract the targeted power mode.
It was also found that the cluster actuation enables the independent excitation on the targeted cluster without causing control spillover in the sense of cluster. Experimental
results demonstrate the validity of the proposed method
for suppressing the total acoustic power radiated from a
vibrating panel.
(e) E ve n/ev en mod al cl u ste r
– 50
– 70
10
1 00
Frequency
5 00
Hz
FIGURE 1 Smart sensor outputs after cluster control
SESSIONS
Zonal and global control of vibrational structural intensity
in an infinite fluid-loaded elastic plate
Jungyun Won and Sabih I. Hayek
Active Vibration Control laboratory
Department of Engineering Science and Mechanics
Penn State University, University Park, PA 16802, U.S.A.
In this paper, the active control of active vibrational structural intensity (VSI) in an infinite elastic plate in contact with a heavy
fluid is modeled by the Mindlin plate theory of bending. The plate is excited by a point force, which generates a vector active
VSI field in the plate. Point force actuator(s) at arbitrary location(s) on the plate are utilized to minimize the VSI in a desired
zone or in a large global area of the plate. The influence of the number of controller actuators and the location of these
controllers relative to the source region for the minimization of the total VSI in a region is explored.
INTRODUCTION
Vibrational energy flow, in the form of structural
intensity, is one of the most useful ways to understand
the paths of vibration transmission and propagation.
When a structure is coupled to a heavy fluid, the energy
flows from a structure, through a coupling medium to
the farfield and sometimes flows back to the structure.
In this paper, the structural intensity of fluid-loaded
Timoshenko-Mindlin plate with harmonic point forces
are calculated for an infinite plate. The cost function to
be minimized is the magnitude of structural intensity
vector at chosen reference point(s).
z
Acoustic Media
y
Fc
Fo
the acoustic pressure on the plate, Fo is the applied
force, and F is ∑y/∑r+y/r.
The acoustic pressure, p, which acts on the plate is
described by the scalar wave equation:
where k is the acoustic wavenumber, and p is the
acoustic pressure. A boundary condition is needed to
couple the acoustic pressure on the surface of the plate
to its vibration:
∂p ∂z
ψb =
FORMULATION
ρ sh3 2
ω ) Φ − G ′h ∇ 2 w = 0
(1)
12
δ (r )
− Pa = − ρ s h ω 2 w
G ′h ( ∇ 2 w + Φ ) + Fo
2π r
where D is the bending stiffness, G´ is the adjusted
shear modulus, rs is the density of the plate, h is the
thickness of the plate, w is the forcing frequency, Pa is
(3)
= ρ oω 2 w
ρ ⋅ f1 ( ρ ) H 0(1) (rkρ )
w Fb
dρ
=
ε
h
2 −∫∞ f ( ρ ) −
ρ
f
1
(
)
2
ρ 2 −1
(4)
ψ Fb ∞
ρ 2 H1(1) (rkρ )
dρ
=
ε
kh 2 −∫∞ f ( ρ ) −
ρ
f
1
(
)
2
ρ 2 −1
(5)
∞
wb =
FIGURE 1. Physical geometry of plate
( D ∇ 2 − G ′h +
z =0
To solve these equations, it is necessary to apply
Hankel transform. After solving the transformed
equations simultaneously with the boundary condition
and applying inverse Hankel transform, the
displacement and shear deformation can be obtained.
The normalized solutions are given by:
x
Two simultaneous time-harmonic TimoshenkoMindlin plate equations of motion describing the
motion of the plate in terms of the displacement, w and
the shear deformation angle, y are [1]:
(2)
(∇ 2 + k 2 ) p = 0
where f1 ( x) = 1 + K s 2 Ω 2 x 2 − K s 2 K d 2 Ω 2
2
2
f 2 ( x) = ( x 2 − K s )( x 2 − K d ) − 1 Ω 2
and Ω = ω ω c . wc is a classical coincidence
frequency.
Time-averaged structural intensity is expressed as:
SIb =
SI
1
6(1−υ) dwb
*
= − Re[κ 2
+ψb ) ⋅ iΩwb ]
(
3 2
Dk h ωc
2
(kh)2 dkr
(6)
ψ
dψ
1
*
− Re[( b +ν b ) ⋅ iΩψb ]
dkr
kr
2
SESSIONS
6pi
-5pi
-4pi
2
0
4
2
pi
2pi
3pi
4pi
5pi
6pi
-pi
0
X*kf
pi
2pi
3pi
-2
-2
2
-2
0
3pi
4pi
5pi
6pi
FIGURE 2. VSI and VSI reduction in dB
: Controller at (p, 0), Ref. Point at (4p, 0)
If one uses two controllers and one reference point,
the resultant minimization of VSI is comparable to one
controller/reference point. In this case, the value of the
optimal control force is not unique, and the control
algorithm gives the closest value from the starting point
-2pi
-2pi
-3pi
2
pi
2pi
3pi 4pi 5pi 6pi
-4
-10 -8
-12
8
-4
-12
-10
-8 -6
-4 -2
02
10
4
4 126 12412268024 8
8 10
-2
-4
68
4
02
0
2
-2
-8
-6
-5pi
0
X*kf
-12
-3 -10
-1-800 -4
-2 -8
-6
-4pi
-6pi -5pi -4pi -3pi -2pi -pi
10
-2 0 2
-6
22420
-2
-4
-2
0
-8
10
-pi
-6
-4
0
-2
-4
0
Y*kf
0
2
4
0
-pi
-4
-8
-2
-4
-2
40
1280
-22
-24
-26
-28
-30
-2
-4
-2
0
-4
2pi
6pi
-6pi
-6pi -5pi -4pi -3pi -2pi
-pi
-4
0
X*kf
pi
-6
pi
5pi
-6
0
X*kf
4pi
30
2286
Y*kf
pi
0
-6
-4
-2 -4
2pi
2
4
2pi
186
0
-2
3pi
-2
0
3pi
10
-6
-4
-6pi
-6pi -5pi -4pi -3pi -2pi -pi
3pi
2
Y*kf
4pi
-6
Y*kf
5pi
4pi
-6
2pi 3pi 4pi 5pi 6pi
6pi
5pi
-6
2
2
2pi
-4
4
pi
-28
-12
-14
-16
-18
-20
-22
-24
-26
-6
6
0
X*kf
-8
pi
6pi
-6pi
30
2826 24 22
20
18
16
14
12
10
8
4
6
-pi
-6
0
X*kf
6pi
-5pi
0
-6pi -5pi -4pi -3pi -2pi -pi
5pi
30
281428 12
26
64
12
10
8
-5pi
-6pi
4pi
pi
2
-4
-4pi
-5pi
3pi
2
0
-4pi
2pi
64
-3pi
26
2224
86
2
-2
-2pi
-3pi
10
8 64 2
0
0
-2pi
2
4
-pi
0
-60
-8
-4
8
6
4
2
-2
-1
0
-pi
pi
10
8
2
0
-4
6
Fig. 4 shows the results for two controllers at (≤p, 0)
and two reference points at (≤4p, 0). It shows the
minimized VSI field and the significant reductions are
attained in the region near the entire x-axis.
68
1012
1416
18
24
60
114
121
0
2
64
8
-4
-2
pi
pi
0
X*kf
-4
10
12
24
4
FIGURE 4. VSI and VSI reduction in dB
: Controllers at (≤p, 0), Ref. Points at (≤4p, 0)
4
2pi
2pi
26
24 22
20
18
16
14
12
-5pi
-4pi
0
26
-4
0
2
4
16
14
12
24
0
28
10
-2
30
28
Y*kf
4
10
8
6
-6pi
-6pi -5pi -4pi -3pi -2pi
28
30
-4pi
-6pi
24
320 26
8
86
-5pi
-4
-2
-3pi
108
2
0
-4pi
0
16
14
12
4
-2pi
0
8 10
2
114
16 18
20
22
0
68 1
24
28
26
22 24
20
18
16
14
12
-pi
-3pi
2830
26 24
-2
-pi
-2pi
10 8
6
26
0
18
20
22
0
0
pi
pi
4
18
20
22
2pi
2pi
6
3pi
1412
18 16
20
24
22
6
-4
4
6
3pi
4pi
2
10 8
4pi
-6pi -5pi -4pi -3pi -2pi -pi
0
3pi
6pi
0
2
0
5pi
5pi
-2
4pi 6
8
3pi
4pi
4
5pi
6pi
6pi
-4
5pi
2
4pi
When two controllers are used to control the VSI at
two reference points, the resulting reduction in VSI is
more global than two previous cases.
-3pi
5pi
10
4
-6pi
-6pi -5pi -4pi -3pi -2pi
2
0
X*kf
FIGURE 3. VSI and VSI reduction in dB
: Controllers at (≤p, 0), Ref. Point at (4p, 0)
6pi
6pi
2
4
30
12
16
18
20
26
30
6
-4
-6pi -5pi -4pi -3pi -2pi -pi
RESULTS
The control of VSI depends on the relative location
of the source and the controller. By varying the
location of the controller relative to the source, it was
found that efficient reductions were attained at the
locations which are multiples of half fluid-loaded
structural wavelengths. The closer the controller is to
the source, the more global is the reduction in VSI.
Fig. 2 shows the VSI and VSI reduction in dB with
the controller at (p, 0) and the reference point at (4p, 0),
and it shows VSI reduction in the region near the xaxis.
8
6
1210
8
-2
-5pi
30 26
24
22
20
141618
12
0
-6pi
28
6 81
0
4
-4
4
2
-3pi
142224
-22
-24
-26
-28
-30
-8
-402468
-2-6
-1
-2
-2
121086
0
-4pi
30
28
26
24
22
2018
1614
2
-2pi
1214
16
18
2022
24
26
6 8 10
4
0
-3pi
0
24
26 28
-2
-2
Y*kf
0
-pi
-4
164
28
0
-pi
-2pi
12108 6
122802
30
pi
24
2pi
0
-2
-2
3pi
8
262
1
164
3pi
18
20
2pi 22
24
pi
4pi
6 8
0
4pi
5pi
Y*kf
4
6
5pi 108
12
6pi
-4
2
The integration paths and branch cuts for the
integrations above have been used by several authors
such as in [1].
After using the paths, several
integrations occur during the calculation of wb and yb,
and can be calculated using numerical techniques. For
the horizontal axis of plots, normalized distance, kf r,
which is normalized to fluid-loaded structural
wavelength, is used. Hence, 2π in the horizontal axis
means a full fluid-loaded structural wavelength.
For numerical calculation, parameters of a steel plate
in water are used. The excitation frequency is W=0.2.
One/two point-force controllers are used to control the
structural intensity at one/two reference points
The structural intensity at the reference point is a
second order polynomial in terms of a real and an
imaginary part of the control force(s) when one
controller is used. To assure that the VSI is minimized,
the square of the magnitude of the VSI vector is
minimized. The steepest gradient search method is
used to find the minimum of the function, which is a
numerical optimization technique.
which is zero in this paper. Fig. 3 is the result of two
controllers at (≤p, 0) and one reference point at (4p, 0).
Y*kf
NUMERICAL CALCULATION
2pi
3pi
4pi
5pi
6pi
FIGURE 5. VSI and VSI reduction in dB
: Controllers at (p, 0), (0, p), Ref. Point at (4p, 0)
In fig, 5, two controllers of the same forces are
located at (p, 0) and (0, p) with two reference points at
(4p, 0) and (0, 4p). The results are symmetric with
respect to x-y diagonal. The VSI was reduced near the
positive x and y-axes.
REFERENCES
1.
C. Seren and S. I. Hayek, J. Acoustic Soc. Am.
86(1), 195-209 (1989)
SESSIONS
Multiple-Aspect Acoustic Scattering from
Fluid-Filled Cylinders Measured at Sea
A. Tesei
SACLANT Undersea Research Centre, V.le S. Bartolomeo 400, 19138 La Spezia, Italy. [email protected]
Multiple-aspect backscattering is studied at low-intermediate frequency from fluid-loaded, fluid-filled, thin-walled cylinders. The
dynamics of predicted elastic waves are described in the frequency-aspect domain by extending to liquid-filled shells the thinshell theory originally developed for air-filled shells. The models are validated by at-sea measurements of a water-filled cylinder.
INTRODUCTION
Multi-aspect elastic backscattering from fluid-filled,
thin-walled, cylindrical shells is studied in the ka range
(1,20). A canonical water-loaded, flat-ended, thinwalled, steel cylinder is considered. In previous work
[1][2] the elastic contribution from a water-filled shell
insonified at broadside was predicted and in [3]
theoretical considerations were validated on at-sea
data. This work extends the analysis to the shell
response at oblique incidence. The aspect-dependent
dynamics of the expected elastic waves are formalized
by extending thin-shell theory to liquid-filled shells.
The proposed models have been validated by the
analysis of at-sea measurements.
THEORETICAL CONSIDERATIONS
A number of elastic wave families were proven [1] to
be generated by infinite cylindrical shells insonified at
broadside. For ka∈(1,20), in the case of a thin-shell
steel cylinder with relative thickness h=d/a=0.024 (d
being the wall thickness and a the radius), the wave
families common to both air-filled and liquid-filled
shells are the outer-fluid-borne A Scholte-Stoneley and
shell-borne A0 and S0 Lamb-type waves [1]. If the shell
is liquid-filled, the filling causes the generation of a
number of additional elastic waves (Fig. 1): the inner
and outer Scholte-Stoneley S wave, the inner ScholteStoneley A wave and multiple periodical internal
bounces rl’ of the first, second, and higher orders [2].
The frequency modes f nl of each wave l (n is the
modal order) can be expressed in terms of the
properties of target and inner and outer media [3].
FIGURE 1. Wave travel paths at normal incidence.
FIGURE 2. (a) Wave number decomposition at oblique
incidence. (b) Path of a generic helical wave.
According to thin-shell theory [1][4][5] developed
for air-filled shells, at oblique incidence the elastic
response of the cylinder is mainly characterized by
shell-borne waves (Lamb-type S0 and shear S waves).
Outer-fluid borne Scholte-Stoneley A wave modes are
predicted at low frequency. At oblique incidence (Fig.
2) the wave travel path around the shell from circular is
expected to be helical, according to the decomposition
of the wave number k along radial (θ) and axial (z)
directions [1][4][6]:
k 2 = k z2 + kθ2 , k =
2πf nl
2πf nl
, kz =
sin α
c ext
c ph
(1)
where cext is the sound speed of the outer medium and
α the incident angle. As phase matching of the helical
wave occurs when kθ=2πn/P, where P is the perimeter
of the projection of wave travel path on the radial plane
[6] (Fig. 1), from Eq. (1) we obtain:
(
sin 2 α = c ext / c ph
)2 1 − (n c ph )2 / ( f nl P )2 
(2)
If the wave is (quasi-)nondispersive (e.g., S0, shear
and S waves) its phase speed cph (approximately) tends
to a constant sound speed that depends on the wave
nature and the physical properties of the medium (e.g.,
for shear waves cph→cs). Hence the mode loci of
(quasi-)nondispersive waves are obtained by
substituting cph in Eq. (2) with the appropriate speed.
Each locus is a quasi-parabolic curve centred at
broadside with an asymptote at its critical angle [4].
SESSIONS
Thin-shell theory is extended here to liquid-filled
shells off broadside. The dynamics of inner-fluid-borne
S wave and periodical internal bounces rl’ are expected
to be similar to those described in Eq. (2) for air-filled
shell waves. However these waves have a travel speed
tending to the sound speed of the inner medium only
when they travel along the shell walls. Then, under the
assumption of thin shell, they propagate through the
shell walls and are expected to re-radiate outside the
shell with a phase speed tending to either the shear or
the compressional speed of the shell itself, i.e., with
cph→cp|s. Under this hypothesis the model of mode loci
of inner Scholte-Stoneley S and periodical internal
bounces is obtained by extending the procedure
described above for empty thin-shells:
(
)
( )
2
2
sin 2 α = cext / c p|s 1 − (n cin )2 / f nl P 


FIGURE 4. Identification of inner-fluid-borne waves.
(3)
where P = 2π (a − d ) for the inner-fluid-borne surface
wave S and P = 4l ' sin (π /(2l ) )(a − d ) for a periodical
internal reflection rl’ (l’=1,2,…).
EXPERIMENTAL RESULTS
Free-field measurements were performed in a basin
from a water-filled, 0.25m radius x 2m, 6mm thinwalled steel cylinder [3]. The object, suspended
underwater from a floating frame, could rotate around
its minor axis while insonified by a parametric sonar
with a Ricker 8kHz pulse. Backscattered response was
received by a hydrophone in monostatic configuration.
The multiple-aspect spectral representation of the data
(Fig. 3) shows a regular, patterned texture generated by
a number of elastic wave mode loci, the presence of
which is a first indication that the insonified object is a
shell with circular cross-section. A more detailed
analysis allows the identification of mode loci
belonging to S0 Lamb-type, shear and outer-fluid A
waves in accordance with thin-shell theory (Fig. 3).
FIGURE 5. Model-based interpretation of the response of a
water-filled cylinder insonified at broadside.
Additional shear and compressional wave mode loci
are identified as belonging to inner-fluid-borne surface
waves and periodical bounces (Fig. 4) from the
resonance mode identification of the response at
broadside (Fig. 5). The good agreement between theory
and experiment encourages the extension of the
analysis to cylinders lying proud on or buried in the sea
bottom and to more complex objects.
ACKNOLEDGMENTS
The author is grateful to A. Maguer, B. Zerr and W.
Fox, who was the scientist in charge of the trial, for the
fruitful scientific discussions and the continuous
support. Many thanks go to J. Fawcett for his
fundamental contribution to target scattering modeling.
REFERENCES
FIGURE 3. Elastic wave identification of water-filled shell
response from thin-shell theory.
1.
N.D. Veksler, Resonance Acoustic Spectroscopy, Springer
Verlag, Berlin, 1993.
2.
J.-P. Sessarego, J. Sageloli, C. Gazanhes, H. Überall, J. Acoust.
Soc. Am. 101 (1), 135-142 (1997).
3.
A. Tesei, W.L.J. Fox, A. Maguer, A. Løvik, J. Acoust. Soc. Am.
103, 2813 (1998).
4.
M. L. Rumerman, J. Acoust. Soc. Am. 93 (1), 55-65 (1993).
5.
J.A. Fawcett, SACLANTCEN Internal Report 273 (1998).
6.
C.N. Corrado, Ph.D. Thesis, MIT, Cambridge (1993).
SESSIONS
Vibro-Acoustic Wave Transmission and Reflection in a
Hose-Pipe System
Yun-Fan Hwang
Applied Research Laboratory, Penn State University,
P.O. Box 30, State College, Pa 16804-0030, U.S.A.
An analysis of sound and vibratory transmission and reflection losses in a fluid filled planar piping system which consists of
straight pipe segments, flexible hose, elbows and/or U-joints is discussed in this paper. Calculation of the transmission losses for
a hose-pipe system has been widely discussed in the literature. The reflection characteristics of a hose-pipe system, however,
have not received proper attention. In this paper, we calculate the reflection and transmission coefficients of a piping system
simultaneously. The numerical example shows that very large pressure and bending wave transmission losses occurred in a hosepipe system are actually caused by reflection rather by attenuation or dissipation by the system.
INTRODUCTION
Analyses of sound and vibratory transmission and
reflection losses in a fluid filled planar piping system
which consists of straight pipe segments, flexible hose,
elbows and/or U-joints are concerned of this paper. In
the approach defined in Ref. [1,2] for a planar (in y-z
plane) piping system, the axial force (Fz), transverse
force (Fy), moment (Mx), displacements (Uy, Uz),
rotations (ψx), fluid pressure (p) and its particle velocity
(v) at both ends of a pipe or hose segment are related by
a transfer matrix. Waves in fluid are treated as plane
waves, which are dynamically coupled with the pipe or
hose wall. The fluid wave speeds are calculated using
the modified bulk modulus resulting from the Poisson
coupling with the pipe wall. Timoshenko beam theory
is use to represent the bending vibration of the pipe. An
elbow or U-joint is handled by representing it as several
straight segments. The axis for any of the segments is
rotated with respect to that of the previous segment
with an incremental angle. The sum of all of the
incremental angles is equal to the turning angle of the
elbow or U-joint (90o or 180o, respectively).
Accordingly, the transfer matrix for each of these
segments must be multiplied by a point matrix (to
correct the rotation of the axis) so that it will be
dynamically comparable with all other segments.
The relationship between the dynamic parameters at
two ends of a piping assembly is represented by the
system transfer matrix, T, which is obtained through a
successive multiplication of the transfer matrices
representing the variouspipe segments, from up to down
stream, of the piping system. That is,
S d = TSu
(1)
where S=[Uz, p, v, Fz,Uy,Ψx, Mx, Fy]T, the superscript T
denotes the transpose of a matrix, and Su and Sd denote
S at the up and down stream terminations of the piping
system, respectively.
From this, the response of the entire system to an
external excitation can be solved by specifying the
boundary conditions and excitation. In order to
determine the transmission and reflection losses, the
system must be connected to a semi-infinite pipe at its
up stream where an incident wave is launched toward
the piping assembly, and there is a semi-infinite pipe at
the down stream where the transmitted waves will
propagate away from the system. For the eiωt time
harmonic function, the incident, reflected and
transmitted waves may be defined as
[ Fz , p, Fy ]inc = [ A1 e
−iλ1u z
[ Fz , p, Fy ]rfl = [ B1 e
B6 e
, A2 e
−iλ1u z
− λu3 z
−iλu2 z
, B2 e
, A3e
−iλu2 z
−iλu4 z
, B5e
],
−iλu4 z
(2)
,
],
(3)
−iλd z
−iλd z
−iλd z
[ Fz , p, Fy ]trn = [ B3 e 1 , B4 e 2 , B7 e 3 ,
B8e− λ3 z ] ,
d
(4)
respectively where λi accompanied with superscripts u
and d are the up and down stream propagating
wavenumbers for various types of waves defined in
Ref. [1]. With some algebraic manipulation, the
reflection and transmission constants,
B = [B1 B2 B3 B4 B5 B6 B7 B8 ]T
(5)
can be determined by solving
B = ( D − TC ) −1 TE
(6)
where
E = [iλ1u ( ρ p Apω 2 ) −1 A1 , A2 ,−iλu2 ( ρ f ω 2 ) −1 A2 ,
− A1 , iλu4 (( ρ p Ap + ρ f A f )ω 2 ) −1 A3 ,
( ρ p I p ω 2 − ( λu4 ) 2 EI p ) −1 A3 ,
iλu4 EI p ( ρ p I pω 2 − (λu4 ) 2 EI p ) −1 A3 ,− A3 ]T . (7)
C and D are square matrices where
SESSIONS
C11 = − E1 , C 22 = 1, C 32 = − E 3 , C 41 = −1,
C 55 = − E 5 , C56 = −λu3 (( ρ p Ap + ρ f A f )ω 2 ) −1 ,
C65 = ( ρ p I pω 2 − ( λu4 ) 2 EI p ) −1 ,
C66 = (( ρ p Ap + ρ f A f )ω 2 + (λ3u ) 2 EI p ) −1 ,
C75 = −iλu4 EI p C65 , C76 = −λ3u EI p C66 ,
C85 = −1, C86 = −1 ;
D13 = iλ1d ( ρ p Apω 2 ) −1 , D24 = 1,
D34 = −iλd2 ( ρ f ω 2 ) −1 , D43 = −1,
D57 = iλd4 (( ρ p Ap + ρ f A f )ω 2 ) −1 ,
calculated transmission and refection coefficients of
the pressure and bending waves. It is shown that both
pressure and bending waves are effectively transmitted
to the other end only at low frequencies. At higher
frequencies, the transmission losses are high because
most of incident wave energies are reflected.
This analysis indicates that the large transmission
losses of the hose-pipe system shown in Fig.1 should
attributed mainly to the system reflection of the
incident waves, rather than to the system dissipation.
In other words, the incident wave energy in the
upstream (left side of the system) remains in the
upstream. In this case a dissipative element installed in
the upstream of the hose-pipe system may be
necessary to reduce the reflected energy.
D58 = λ3d (( ρ p Ap + ρ f A f )ω 2 ) −1 ,
D67 = ( ρ p I pω 2 − ( λ4d ) 2 EI p ) −1 ,
Reference
D68 = ( ρ p I pω 2 + ( λ3d ) 2 EI p ) −1 ,
1. M.W. Lesmez, D.C. Wiggert, and F.J. Hatfield, J.
Fluid Eng. 112, 311-318 (1990).
2. M.L. Munjal and P.T. Thaeani, J. Acoust. So. Am.
101(5), Pt.1, 2524-2535 (1997).
.
D77 = iλd4 EI p D67 , D78 = λ3d EI p D68 ,
D87 = −1, D78 = −1 , and where all of the Cij and
Dij not shown above are zeros and all nomenclatures not
defined above are same as that of Ref. [1]. From Eq. (2)
to (4), it is obvious that lB4/A2l2 and lB2/A2l2 are the fluid
pressure wave transmission and reflection coefficient,
respectively. The transmission and reflection
coefficients for other types of wave can be defined in
the similar manner.
A hose-pipe system shown in Fig. 1 consists of six
segments. Segments 1, 3 and 6 are steel pipes, each of
them is 20 cm long. Segments 2 and 4 are 90o elbows
and the radius of the bend for both is 15 cm. Segment 5
is a rubber hose which is 50 cm long. For simplicity, all
pipe and hose segments are having the same outside
diameter (3 cm) and the same wall thickness (3 mm).
The Young’s modulus, Poisson ratio, and specific
gravity of the hose are assumed to be 2x108 Pa, 0.48
and 1.5, respectively. Each of the elbows is
approximated by four straight segments. Both the left
and right terminations of the assembly were assumed to
fit with semi-infinite steel pipes of the same diameter
and thickness. The fluid in the piping system is water.
A unit incident pressure or bending wave inside the
left semi-infinite pipe is assumed to propagate toward
the assembly. The incident wave will be transmitted to
the right semi-infinite pipe in terms of fluid pressure,
bending and longitudinal waves. Fig. 2 shows the
4
z
5
6
3
2
1
FIGURE 1. A simple pipe/hose assembly where the arrows
indicate the direction of wave propagation.
10
10
10
COEFFICIENTS
NUMERICAL EXAMPLES
y
10
10
10
10
10
2
PRESSURE TRANS
PRESSURE REFLT
BENDING TRANS
BENDING REFLT
1
0
-1
-2
-3
-4
-5
0
200
400
600
FREQUENCY, HZ
800
1000
FIGURE 2.
Predicted transmission and reflection
coefficients of a pipe/hose assembly (Fig. 1).
SESSIONS
Coherence of Seismoacoustic Waves from Explosive Sources
V.S.Averbakha, N.N.Gerdyukovb, I.N.Didenkulova, V.I.Dudinb, V.N.Erunovb,
S.A.Lobastovb, A.P.Marysheva, S.A.Novikovb, A.A.Stromkova, V.I.Talanova
a
Institute of Applied Physics, Nizhny Novgorod, Russia
Russian Federal Nuclear Center-VNIIEF, Sarov, Russia
b
Explosive sources are widely used for excitation of seismic waves. It is usually assumed that higher charge of the source
allows one to ensure deeper seismoprospecting. However the main problem in this case is high energy losses in the
shock wave resulting in weak dependence of seismic waves intensity measured far from explosive charge source.
Coherent methods in seismoprospecting have attracted increasing attention during last years. Coherent signal processing
permits an increase in the signal-to-noise ratio. Special electromagnetic sources were mainly used in coherent
seismoacoustics so far. The present paper describes a new direction in coherent seismoacoustics that is based on the use
of controllable explosive sources. Experimental investigations were performed with the system of synchronized smallcharge explosive sources. The results show high coherence of seismoacoustic waves generated by explosive sources, and
the possibility of using it for seismoacoustic prospecting
INTRODUCTION
EXPERIMENT AND RESULTS
Explosive sources have been used for excitation of
seismic waves since long for seeking resources and
studying subsoil assets [1]. In this case, the use of
high-power sources allows us to increase the seismic
sounding depth. The main obstacle in this way is weak
dependence of the amplitude and generated seismic
waves on the charge power Q [1]. It is known also
that only from 0.01 to 1% of the total explosion energy
is transferred to the seismic wave energy [2].
Moreover, to examine subsoil assets, it is desirable to
use directed seismic waves.
Coherent methods of seismic diagnostics have been
recently developed [3,4]. They are based on the
formation of seismic fields with controllable spacetime structure, which allows us to perform coherent
processing of signals scattered from soil
inhomogeneities. Such a processing allows us to
receive rather weak scattered signals and,
correspondingly, increase the sounding depth. In this
case, it is also possible to substantially increase the
working frequency range up to several hundred Hz,
which improves the spatial resolution. Until recently,
the studies in the field of coherent high-frequency
seismoacoustics were conducted with the help of
coherent electromagnetic-type sources developed at
the Institute of Applied Physics [3,4].
In this paper we consider a new direction in
coherent seismoacoustics related to the possibility of
using controlled low-power explosive sources
developed in VNIIEF.
The scheme of the experiments was as follows.
Explosive sources (with power from 1 to 50 g) were
located at a depth of 30 cm in sandstone-loam soil.
The charges were exploded by an electric detonator
ensuring time synchronization of about 1 µs. Seismic
receivers were located along the radial line with
respect to the charge. The interval between the
receivers was 0.5 m. Therefore, the receiving system
was a linear antenna. The distance from the charge to
the first seismic receiver was 1 m.
Experimental series with "point" charges 1, 2, and
8 g of explosive were carried out. An explosion is
known to generate different-type seismoacoustic
waves such as longitudinal, shear, surface, and deeprunning [1]. These waves propagate at different
velocities and can also be reflected from the layered
soil structure and scattered from various
inhomogeneities. All this results in the delay of
explosion signal. It was observed that for small time
intervals from the explosion time, the spectrum is up to
400 Hz. At later times, the spectrum narrows mainly
due to a decrease in the upper frequency boundary, and
then a region of dominating frequencies is gradually
formed in the interval approximately from 25 to 50 Hz.
To investigate the coherence of seismoacoustic
signals from different explosions we subsequently
exploded three point charges with mass 1 g of
explosive under the same conditions. The results are
shown in Fig. 1. The correlation factor values between
signals are not worse than 0.97 for different receivers.
SESSIONS
500
5.5
400
5
300
4.5
Distance (m)
Amplitude
200
100
0
-100
4
3.5
3
2.5
2
-200
1.5
-300
1
-400
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0
Time delay (s)
FIGURE 1. Seismic signals from 3 consequent explosions.
Such a high correlation is due to the explosion
synchronization system used.
The obtained data allow us to speak of the
possibility of coherent processing of signals from
different explosions. Consider this issue in more detail.
In accordance with the law of explosion identity [1,2]
we have u=k(Qp)n, where u is the particle velocity in
the seismic wave field, k is the proportionality factor,
n is the exponent of power, Qp=Q1/3/r is the reduced
charge mass, Q is the charge value, and r is the
distance from the explosion point to the observation
point. For body seismic waves we have n=1 [1]. From
the identity law it is obvious that the use of a series of
small explosions is more efficient than simultaneous
blasting of a charge with total power of the whole
series. Assuming that we conducted N similar
experiments with charges Q let us determine the
effective value of the charge Qeff required for
generating a signal of the value coinciding with that of
the resulting signal obtained from coherent addition of
signals in a series of N explosions. From the identity
law it follows that after the coherent addition of
signals from N explosions the resulting value of u is
equivalent to one experiment with the explosive mass
Qeff=N3Q. Therefore, coherent addition of signals is by
a factor of Qeff/NQ=N2 more efficient than the blasting
of one charge with total power NQ that is similar to
that in a series of N explosions. For example, adding
the signals from ten subsequent experiments with
Q =10 g, we obtain the same result as that for one
experiment with Q =10 kg of explosive.
The possibility of coherent addition of signals from
explosion sources allows us also to form controlled
seismic fields, in particular, directed seismic beams.
Due to this, we can further increase the seismic
sounding depth. Such an experiment was conducted
with a plane square-shaped distributed explosive
source 0.5 0.5 m2⋅ with total charge value 50 g of
explosive. In Fig. 2 we show halftone image of signals
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Time delay (s)
FIGURE 2. Halftone image of seismic signals received by
ten aligned receivers separated by 0.5 m (the vertical axis) as
a function of time delay (the horizontal axis).
received by ten receivers as a function of delay time
counted off from the charge blasting time. In the
figure, one can see the equiphase lines (hodographs) of
waves. The first hodograph corresponds to the
Rayleigh wave propagating at a velocity of 270 m/sec,
and the second hodograph corresponds to a body wave
refracted at the boundary between dry and moisturesaturated sand at the 2.5 m depth.
CONCLUSION
Along with the use of small-power charges that do
not significantly interfere with the soil structure, the
high degree of the time synchronization of explosion
allow us to ensure high coherence of seismic signals.
The high coherence of signals allows us to perform
accumulation of seismic waves from a subsequent
series of similar explosions and form seismic fields
with the required space-time structure. All this allows
us to substantially increase the efficiency of seismic
sounding, reduce the total power of used charges, and
improve safety during explosive experiments.
ACKNOWLEDGMENTS
This work was supported by ISTC (project 1369) and
in part by RFBR (99-02-16957, 00-15-96741).
REFERENCES
1.
2.
3.
4.
Sadovsky M.A. Selected Papers. Geophysics and
Physics of Explosion. Nauka, Moscow, 1999, pp.20-80.
Kuzmenko A.A., et al. Seismic action of explosions in
rocks. Nedra, Moscow, 1990, pp.325-390.
Averbakh V.S., et al. Acoustical Physics, 44, 725-732
(1998).
Averbakh V.S., et al. Acoustical Physics, 47, 435-441
(2001).
SESSIONS
Experimental Determination of SEA Parameters for
the Prediction of Noise Pressure Level in Cylindrical Cavity
A. Botteona, S. Gergesa
a
Vibrations and Acoustics Lab, Mechanical Engineering Department, UFSC, Florianópolis, Brazil
Statistical Energy Analysis – SEA is a noise and vibration prediction method effective at mid and high frequencies. Some
acoustic and structural systems have its response to random excitation extended to the high frequency region, for instance the
response of aircraft fuselage to the propulsion system and excitation by turbulent boundary layer. The necessary parameters to
the application of the method are modal density, damping loss factor, power input and coupling loss factor, they describe the
dynamic characteristics of structural components and its connections. They can be measured or calculated using
theoretical derivation. This paper presents the experimental determination of modal density and damping loss factor, in a 0.95
diameter, 1.20 height aluminum structure closed in both sides, that simulates one section of a airplane fuselage and its inner
cavity. The results are compared to theoretical prediction. The validation of this simplified model is the first step to obtain a
reliable complex model for the prediction of SPL inside airplanes. The complex model will be used as a prediction tool at
design stage in the development of new aircraft.
INTRODUCTION
In SEA a system is divided in subsystems, each
of these represents a mechanism to store energy. SEA
is based on a power balance. In the steady state, the
power input in a subsystem is either dissipated
internally or transmitted to another subsystem. The
dissipated power is proportional to the energy level of
the subsystem and its damping factor. The power flux
between the subsystems is proportional to the
coupling loss factor and to the difference of the
averaged modal energy level. Through this power
balance its possible to obtain a system of linear
equations, in which the level of energy of each system
is the unknown variable.
In this experiment the model conceived
composes of 3 subsystems, cylindrical shell, inner
cavity and surrounding air. The test set up, suspended
cylinder and detail of impedance head, rod and shaker
is shown in figure 1.
FIGURE 1
MODAL DENSITY
Modal density is defined as n( f ) =
c
N Df
, where fc
Df
is the central frequency and NDf is the number of
modes within Df. When a structure is excited by a
broadband white noise, a very large number of modes
appear in the high frequency region, making difficult
to compute the response of each mode individually,
therefore the approach used is based on methods that
consider averaged values of the mobility.
The concept of point mobility represents the
capacity a component has to absorb power. The modal
density was determined according to equation (1).
n (f ) =
n(f)
MA
1 f2
4 * MA * Re < Y >
Df òf 1
(1)
Modes/Hz
total mass of the cylinder
Re<Y> real part of the drive point mobility
The drive point mobility was measured using
single point excitation by shaker and a impedance
head. Figure 1(right).
The results are presented (Figure 2) in constant
bands of 100 Hz. These are compared to Szechenyi
[1] semi-empiric results, and data given by
commercial software. Damping has been added to the
structure in order to avoid negative components in the
real part of the drive point mobility The added mass
existing between the transducer and the structure has
to be compensated by according to following
SESSIONS
equation Yreal =
Ymeasured
1 - iwM added Ymeasured
(2)
[3]
loss was not considered because in SEA it is related to
the coupling loss factor. Impedance head was used
because the phase difference between force transducer
and accelerometer can cause errors in the results. The
drive point mobility was measured in three different
points and the acceleration was measured in six
different locations for space averaging.
The obtained data, presented in Figure 3, tend to
values between 0.0005 and 0.002 for 1 < f < 4.5 kHz.
Clarkson obtained a constant value of 0.001 for non
stiffened cylinders.
FIGURE 2: modal density ( constant bands 100Hz)
****** experimental data
++++ data given by commercial sofwae AutoSEA
.…….. results for the equivalent flat plate when f® ¥
-------- results calculated by Szechenyi formula
For cylindrical shells, just before the ring frequency,
high values of the modal density are observed due the
grouping of structural ressonances.
DAMPING LOSS FACTOR
Damping is defined as being the ratio
between the energy dissipated per cycle and the
maximum vibratory energy.
h=
Ediss/ cicle
Wdiss
Wdiss
(3)
=
=
2
2
2pM < v > 2pf .M < v > w.M < v2 >
The damping loss factor has specific values
for each mode. However according to Lyon [4] an
average value per band is required. The structural
damping is responsible for the dissipation for the
system vibratory energy. As it varies depending on
the material and geometry it is normally determined
experimentally. There are two methods to determine
structural damping, power injection method and decay
method. In this work we have used the first., the
former is the subject of the next tests.
2
Win = Frms Re < Yreal >
(4)
1 f 2 Win .w.df
h(f ) =
Df òf 1 MA < a 2 >
(5)
FIGURE 3: Damping Loss Factor - Win method
It can be inferred that indirect methods can be
used successfully to determine modal density and
damping loss factors.
These work continues with the determination of
the input power and the measurement of the coupling
loss factor. Having the data we are able to calculate de
energy levels in each subsystem (solving a system of
linear equation), and from that derive the SPL in the
in the inner cavity, which is our final objective.
ACKNOWLEDGMENTS
This work is supported by CNPq – Brazilian Research
and Development Agency.
REFERENCES
1.
2.
h(f)
band averaged damping
Frms
complex force amplitude measured in the impedance head
<a>
spatial averaged acceleration
It was assumed that Win = Wdiss, i.e all the dissipated
energy was through structural damping. Radiation
3.
4.
5.
E. Szechenyi,. 1971, Modal densities and radiation efficiencies
of unstiffened cylinders using statistical methods Journal of
Sound and Vibration., 1971, 19, pp 65-81.
B.L.Clarkson and R.J. Pope, Experimental determination of
modal densities and loss factors of flat plates and cylinder.
Journal of Sound and Vibration, 1981 77(4), pp 535-549..
K.T. Brown and M.P. Norton. Some comments on the
experimental determination of modal densities and loss factors
for Statistical Energy Analysis applications. Journal of Sound
and Vibration, 1985, 102(4), pp. 588-594.
R.H.Lyon, Statistical energy analysis of dynamical systems:
Theory and Application, MIT Press, 1975
P. R Keswick and M. P Norton. A Comparison of Modal
Densities Measurements. Applied acoustics, 1987, 20 137-153.
SESSIONS
Linear and Nonlinear Investigating of Concrete
J.C. Lacouture, P. Johnson and F. Cohen Tenoudji
Laboratoire Environnement et Développement, Université D. Diderot, Tour 33-43, Case courrier 7087,
2 Place Jussieu, 75251, Paris Cedex, France e-mail : [email protected].
In the present work, we are monitoring simultaneously with the use of sonic waves, the linear and nonlinear viscoelastic behavior
of concrete during curing from just after mixing and well into the solid state. The concrete is contained in a cell with a Lucite
base. Using the complex reflection coefficient of short ultrasonic pulse between the Lucite base wall and the concrete, the linear
compressional and shear wave viscoelastic moduli are determined in the linear part of the experiment; the moduli of concrete are
obtained during the whole curing process. For the nonlinear experiment part, a high power, low frequency continuous sine
compressional wave is transmitted through the medium to investigate the evolution of its nonlinear properties during the cure.
Quasi-continuous-wave harmonic generation at different excitation amplitudes are used to extract classical and hysteretic
nonlinear parameters which are put in relation with the moduli calculated after the linear measurements.
Concrete is well known as a complex and multiscale
material. The evolution of this type of material does
not stop with time but its long term properties depend
strongly upon the curing period. This is the reason why
monitoring the properties of concrete during curing is a
major topic of interest [1,4].
The hardening of concrete caused by the reaction of
hydration of the cement is now a very well known
process. Many papers deal with the linear elastic
properties of concrete. Furthermore, it is known that
hardened concrete shows nonlinear properties [2,3]
mainly caused by the presence of microcracking.
However, a crossed analysis of the two type of elastic
properties has not yet been performed and more
generally, to our knowledge, the consolidation
transition of a material has not been studied
simultaneously by the two methods.
This paper begins with a description of the
experimental set-up. The different results are then
discussed : the evolution of the temperature in the core
of the concrete and the linear and nonlinear properties.
EXPERIMENTAL PROCEDURE
A box of square section 16*16 cm with a base of
Lucite contains the concrete poured in it just after
mixing (Fig. 1). The experimental set-up can be
divided into a linear and a nonlinear part. For the linear
measurements, two compressional and shear wide band
piezoelectric transducers are used (500 kHz central
frequency). These transducers are operated in pulse
echo mode. We are looking at the evolution of the
reflection coefficient between the Lucite and the
concrete over time for compressional and shear waves.
From the reflection coefficient and knowing the
density,
PC with A/D converter
concrete
cell
PZT transducer
Amplifier
Wave generator
Nonlinear part
plexiglas
echo
ASA
Linear part
FIGURE 1. Experimental configuration.
we calculate the elastic moduli of the concrete. In the
nonlinear portion of the experiment, we transmit pure
tone compressional waves across the sample at
different amplitudes and perform the analysis to extract
nonlinear effects as the harmonic generation. The
fundamental frequency is 8 kHz. Data collection is
done every minute and stored into a computer. The
PZT emitter and receiver are in direct contact with the
concrete. The temperature is also monitored with a
thermocouple embedded in the sample and the ambient
temperature is measured with another thermocouple.
The concrete is a reactive powder concrete made of
Portland cement, thin sand, fume silica and
superplasticizer.
RESULTS AND ANALYSIS
In Fig. 2 are plotted the temperature evolution and
the shear wave reflection coefficient versus time. The
main chemical reaction starts at approximately 20
hours ; here the concrete begins to structure and
SESSIONS
establishes gradually the connections between solid
particles.
3
2.8
36
35
1
34
0.9
0.8
32
0.7
31
|R| (-) 0.6
T
30
29
0.5
28
0.4
27
Temperature (°C)
33
S - waves
temperature
1.4
1.2
1
0.8
2f
0.6
3f
28
29
30
31
32
33
34
35
36
37
38
Time (h)
22
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
FIGURE 4. Harmonic amplitude dependence between 28
and 38 hours.
Time (h)
FIGURE 2. Shear wave reflection coefficient and the
temperature in the sample between 0 and 95 hours.
0.5
2.0E+10
0.4
1.5E+10
0.3
Poisson's ratio
1.0E+10
Young's modulus
0.2
Young's modulus (Pa)
The shear and compressional wave reflection
coefficients leads to the calculation of the velocities
and the elastic moduli (Fig. 3).
Poisson's ratio (-)
1.6
23
0
5.0E+09
0.1
0.0E+00
25
2
1.8
0
24
0.1
2.2
0.2
25
0.2
2.4
0.4
26
0.3
Harmonic Amplitude dependence (-)
2.6
1.1
26
27
28
29
30
31
32
33
34
35
36
37
38
Time (h)
FIGURE 3. Young's modulus and Poisson's ratio between
25 and 38 hours.
As seen in Fig. 3, the material keeps a fluid character
for 29 hours. Because of the shrinkage of the cement
paste, the concrete debounded from the Lucite after
thirty eight hours and did not allow us to follow the
evolution after this time.
In the nonlinear part of the experiment, transmission
of 8 kHz sinusoidal wave is possible only after 23
hours, the amplitude of the second harmonic is reliable
after 28 hours. In Fig.4 are plotted the variation with
time of the power law coefficient of the second and
third harmonic amplitudes dependence versus the
amplitude of fundamental after 28 hours.
The third harmonic amplitude dependence
coefficient of two or nearly two shows that we are in a
non classical case of nonlinearity instead of the
classical case where the dependence is three. We
believe that the observed dependence is not exactly
two because of the perturbation of the eigen modes
within the cell.
The harmonic dependence after 28 hours
corresponds to a transition period for the material
where the largest particles are already connected and
the hydrates develop in the pore space as deduced from
the linear results [4].
CONCLUSION
With the monitoring of different parameters as
temperature, linear and nonlinear elastic properties, we
collected information about the evolution of the
concrete during curing. We have seen a correlation
between nonlinear response and linear response that
relates to known microstructure.
REFERENCES
1. Boumiz A., Vernet C., Cohen Tenoudji F., Advanced Cement
Based Materials, 3, 94-106 (1996).
2. Johnson, P.A., Materials World, the Journal of the Institute of
Materials, 7, 544-546 (1999).
3. TenCate J., E. Smith and R. A. Guyer, Physical Review Letters.,
85, 1020-1024 (2000).
4. Morin V., Cohen Tenoudji F., Richard P., Feylessoufi A. and
Vernet C., Proceedings of Intern. Symp. On HPC and RPC
edited by P.C. Aïtcin et al. RILEM publisher, Sherbrooke, 1998,
3, pp. 119-126.
SESSIONS
Generalized solvable models in fluid loaded structures
Ivan V. Andronov
Institute in Physics, Univ. of St.Petersburg; Ulianovskaja 1-1, St.Petersburg, 198904, Russia
The generalized point models of inhomogeneities in elastic plates are suggested. These models take into account geometrical size of
the inhomogeneity and correct in some cases classical point models.
INTRODUCTION
The problems of acoustic waves diffraction by elastic
plates or shells with edges, conjunctions, cracks, inclusions and stiffeners attract wide interest caused by the
needs of shipbuilding, noise analysis, acoustics of the
ocean and other sciences. Much attention is paid to problems that can be solved in a closed form of integrals or
series [1, 2]. Then the analysis of physical effects is the
most simple. Such are the problems using point models
formulated by means of contact conditions fixed in a midpoint of the obstacle. However real inhomogeneities have
finite size.
Here we present more precise model of crack and describe a class of generalized point models. These models
are formulated as operator extensions in the form of zerorange potentials [3, 4].
two terms vanish at orthogonal direction to the plate,
while the correction remains nonzero. Moreover for thin
plates asymptotic analysis of D j , A and B involved in
(1) shows the last term in (1) to be the principal one
by the parameter kh (h is thickness of the plate). Figure 1 presents the effective cross-sections Σ computed
for a 0 10 20 10 10 10 5 10 3 0 01 and 0 1m (from
lower to upper curves). Difference is seen already for
very narrow cracks.
0.1Hz
10Hz
100kHz
1kHz
-25
-50
DIFFRACTION BY A CRACK
Consider infinite elastic plate
bounding an acoustic
half-space. Let on a segment a a the plate be absent
and the edges of the two half-plates be free. The field is
generated by an incident at ϕ0 plane wave. Asymptotic
by ka 1 analysis allows the far field amplitude of the
scattered field to be found [5]
-75
FIGURE 1. Frequency dependencies of Σ (dB) for narrow
cracks in 1cm steel plate in water for ϕ0 20
iν k2 ss0
k4 2 2 k6 3 3
c c
c c
π L ϕ L ϕ0 D4 0 D6 0
π M ϕ Ac2 M ϕ0 Ac20 ν
ln ka 4 B
Ψ
Here s sin ϕ, s0
sin ϕ0 , c cos ϕ, c0
(1)
cos ϕ0 ,
L ϕ ik sin ϕ0 M ϕ ν M ϕ k cos4 ϕ k04 4
The following parameters in (1) describe the system
plate–fluid: k and k0 are the wave numbers in acoustic
media, and in isolated plate, ν ρ0 ω2 D, D is bending
stiffness of the plate, ρ0 is the density of fluid, ω is frequency, D4 , D6 , A and B are contact integrals depending
only on k, k0 and ν.
The first two terms in (1) coincide with the exact expression for scattering by point model [2]. The correction is logarithmically by ka smaller, but the first
GENERALIZED POINT MODELS
The following approach is often used in boundaryvalue problems for fluid loaded plates. First the general
solution U that satisfies all the equations and conditions
except the contact conditions is introduced. It contains
parameters that are then found an algebraic system originated from contact conditions. Generally U satisfies the
problem
∆U k2U 0 d 4 4 ∂U
k0 dx4
∂y
y
νU
0
3
dj
∑ c j dx j δ x
y 0
j 0
Here δ x is the Dirac delta-function and c j are arbitrary
parameters. Physically U is the field of point sources on
SESSIONS
the plate. To formulate the point model that takes into
account the geometry of the obstacle, one can add a passive source in the fluid. That is, the first equation in the
problem for U is replaced by
∆U
k2U
!
!
C ln " r & S # o " 1 #'
r%
0
(2)
Here C is arbitrary and S is fixed positive. If S ! 0, the
generalized model is reduced to a classical one.
It should be noted that mathematically the condition
(2) fixes some self-adjoint extension of the operator in
the form of a zero-range potential [3, 4, 6].
The main question that appears in the formulation of a
model of particular obstacle is the value of S in (2). The
most rigorous way to find S for narrow crack is by comparing the far field asymptotics of scattering by a generalized point model with the asymptotics (1). One can find
that logarithmically small corrections in (1) correspond to
S ! a& 2 $
The following generalized models are suggested:
1. Point model of crack (see [2])
C
!
0
ξ( ("*) 0 #+! 0
'
ξ( ( (,"-) 0 #+! 0 $
'
2. Model of a narrow crack (free edges)
U
C ln " 2r & a # o " 1 #'
.
ξ( (
"-)
0 #+! 0
'
ξ( ( (
r%
"-)
4
0'
0 #/! 0 $
0.05
5
5005
4
-20
πCδ " x # δ " y #$
An additional parameter C presented in the new model
should be defined by an additional “boundary” condition. Mathematically correct formulation appears by setting the following condition in terms of the asymptotics
of the solution U when r % 0
U
Σ (dB)
3
4
3
5
2
-30
2
-40
1
5
1
FIGURE 2. Effective cross-sections Σ (dB) as functions of halfwidth a (mm) for the models 1–5
CONCLUSION
The generalized point models were considered for
two-dimensional problems of diffraction by fluid loaded
plates with small obstacles. In these models additionally
to classical boundary-contact conditions an additional
condition was specified for the logarithmic derivative of
the acoustic field in the central point of the obstacle. The
parameter S in this condition was noted independent of
the general parameters of the boundary value problem. It
is defined only by the width of a gap in the plate. This fact
allows the parameter S to be chosen from the analysis of
a simpler problem for absolutely rigid plate.
The method of the zero-range potentials [3, 4] allows three-dimensional defects in fluid loaded plates to
be modelled in a similar manner. See e.g. [7], where the
model of short crack is introduced.
3. Fixed point (see [1])
C! 0
'
ξ "-) 0 #+! 0
'
ξ(0"-) 0 #+! 0 $
4. Model of a narrow slit with fixed edges
U
.
C ln " 2r & a # o " 1 #' r % 0 '
ξ "-) 0 #1! 0 '
ξ(0"-) 0 #+! 0 $
5. Model of a bubble
C
ln " 2r & a # o " 1 #' r % 0 '
U.
ξ 2 C3 " R #$
π
Note that for a ! 0 the models 2 and 4 are close to the
classical models 1 and 3. However, already for very small
ka scattered field are significantly different. The bubble
(model 5) can not be described by classical boundarycontact conditions. Figure 2 presents the dependence of
effective cross-section on half-width a for all the formulated above models. The incidence of 10kHz plane wave
at ϕ0 ! 10 3 on a 5mm steel plate in water is taken as an
example.
REFERENCES
1. I.P.Konovalyuk and V.N.Krasilnikov Problems of diffraction and waves propagation, Leningrad Univ. Press, 4,
149–165 (1965).
2. D.P.Kouzov Appl.Math.Mech. 27, 1037–1043 (1963).
3. B.S.Pavlov Uspekhi matem. nauk 42, 99–131 (1987).
4. S.Albeverio and P.Kurasov Singular perturbations of differential operators and solvable Schroedinger type operators, Cambridge Univ. Press, London Math. Soc. Lecture
Notes 221 (2000).
5. I.V.Andronov, B.P.Belinskiy and J.P.Dauer Wave Motion,
24, 101–115 (1996).
6. I.V.Andronov Appl.Math.Mech. 59, 451–463 (1995).
7. I.V.Andronov J.Math.Sci. 73, 304–307 (1990).
SESSIONS
Vibroacoustical Identification of a Double Wall Panel
S.J. Pietrzko
EMPA-Swiss Federal Laboratory for Materials Testing and Research
Ueberlandstrasse 129, CH 8600 Dübendorf
This paper focuses on the identification of the structural modal behavior of double wall panels, as well as the acoustic mode
shapes in the cavity between the walls. Additionally the model of a whole panel including dynamics of sensors and actuators
used for active control was identified. This general state space model was used to design a control system to improve sound
transmission of a double wall panel around mass-air-mass resonance.
TEST PANELS UNDER STUDY
The first panel under study consisted of two
monolithic glass panels (717x1091 mm) with a
nominal thickness of 6 mm, and a critical frequency of
approximately 2106 Hz. The interpanel spacing of
80 mm was filled with air. The panel was mounted in
wooden sashes with dimensions x = 1480 mm and
y = 1230 mm. For the experiments, the wooden sashes
were installed in a concrete wall of a semianechonic
3
chamber with a volume of 78.1 m . The double wall
resonance frequency of the pane was calculated to be
at 66 Hz, which corresponds well to the observed
transmission loss drop in the range around 71 Hz, see
Fig.1, where three measured frequency response
functions (FRF) are given.
FRF's (sound pressure in the cavity/point force). The
second panel under study was a double glazed window
consisting of a monolithic glass pane with nominal
thickness of 10 mm, critical frequency » 1210 Hz),
and a monolithic glass pane of 4 mm thickness, critical
frequency » 3160 Hz. The interpane spacing of 16 mm
was filled with Argon gas. The double wall resonance
frequency of the window was estimated at 180 Hz,
which corresponds well with the observed drop of the
transmission loss in the 200 Hz one-third octave band.
PLANT DESCRIPTION
The plant used in the experiment has three inputs
i.e. a noise disturbance exciting the system from the
sending room and two control inputs generated by
loudspeakers inside the cavity. The output from the
panel is measured in the cavity by two microphones an
in the receiving room by a microphone-array, Fig.2.
FIGURE 1. Structural (solid) and acoustical frequency response functions of the panel due to a single
point force acting on a pane.
For this measurement, the sending pane was excited in
a corner by a point force, and the driving point
acceleration as well as the microphone signals inside
the cavity were measured. The driving point response
shows clearly the dominating vibratory behavior of the
panel around the mass-air-mass frequency. This was
also observed in simultaneously measured acoustical
FIGURE 2. A double wall plant. Input loudspeakers
inside the cavity placed in the corners. Output
microphones in the cavity and a microphone-array
placed on a hemisphere.
SESSIONS
MODAL PARAMETERS
NOMINAL PLANT IDENTIFICATION
For the second panel, 16 pole values (damped
natural frequencies, damping ratios) and modal
participation factors were calculated using the Least
Squares Complex Exponential Method. In the
frequency range from 180 to 230 Hz, i.e. around the
double wall resonance frequency, there are 6 modes
which contribute significantly to the dynamic
response. Estimates of the damped natural frequencies
stabilized within 3 % and estimates of damping ratios
within 5%. In the second stage of parameter estimation, modal vectors were estimated using the Least
Squares Frequency Domain Technique. They
stabilized within an accuracy of 5%. The modal
parameters of the window close to 200 Hz compared
with the fundamental mode at 38.2 Hz are presented in
the Table 1; the corresponding complex mode shapes
are given in Fig. 3.
The whole plant (double wall structure including
2 loudspeakers and 3 microphones in the cavity) was
identified to build a suitable state space model for
controller design. Identification of the system without
reconstruction filters established a for this structure
typical model of order 50. This model was not suitable
to implement for control, because of computational
time overflow. After reduction of sampling frequency
and using reconstruction filters it was possible to get a
good quality low order balanced state space model.
The subspace identification methodology was
preferred [1]. This model was used to design a robust
controller with good performance.
Table 1: Panel modal parameters around 200 Hz
Freq. Damping
[Hz]
(%)
182.0
197.8
201.0
219.5
1.7
1.3
1.3
2.3
Modal
mass
[kg]
1.0
1.0
1.0
1.0
Modal
damping
[kg/s]
37.80
32.68
33.13
64.04
Modal
stiffness
[N/m] e+06
1.308
1.545
1.595
1.902
FIGURE 4. Example of an identified FRF (solid)
superimposed with a calculated one for a collocated
loudspeaker and microphone in a corner.
CONCLUSIONS
FIGURE 3. Identified modal shapes of the window
around the double wall resonance frequency, in the
200 Hz one-third octave band.
Identified modal vectors (shapes of the panel walls)
were used to verify and update lumped parameter and
FEM models.
For the identification of complex vibroacoustical
structures like double wall panels, the subspace identification method gives reasonable state space models
which can be directly used for robust control design.
Additionally these models can be verified and updated
with experimental modal analysis. Combination of
these two methods is recommended.
REFERENCES
1. P. van Overschee, B.D. Moor, Subspace Identification for Linear Systems: Theory, Implementation, Applications, Kluver Academic Publishers, Norwell, MA,
1996.
SESSIONS
Experimental Investigation on Depth Evaluation of Perforation
Wenxiao QIAO, Xiaodong JU, Guangsheng DU and Jun FANG
Department of Geo-information, University of Petroleum, Dongying, Shandong, 257061 , P.R. China
As an important procedure of oil-well completion, perforation is a key technique in petroleum engineering. To date there is no
in situ technique to evaluate the quality of perforation. In this paper, a method to measure the depth of perforation is
investigated experimentally by using Ultrasonic Pulse Echo Techniques (UPET). The prototype equipment, whose
measurement scope is between 0 and 1000mm, is developed successfully to measure the depth of the perforation in steel and
sandstone targets. The transducers and the related generators needed, as well as parameters of the data acquisition system, are
determined by experiment. Techniques to recognize the perforation and to recognize scattered acoustic wave from bottom of
perforation are bring forward for the first time. Furthermore, two criteria are used to search for the scattered wave
corresponding to bottom of perforation: Firstly the energy of the scattered wave must be strong enough. Secondly the dominant
frequency of the scattered wave must be in the frequency domain of the incident acoustic pulse. All these techniques are
testified successfully by experiments.
INTRODUCTION
As one of the important procedures of oil-well
completion, perforation is a key technique in
petroleum exploration and production. Of extremely
complicated geometrical shapes, perforation tunnels
usually have wedgelike outlines with a diameter of
about 10 mm at the mouth and a depth of up to 700
mm. A comprehensive inspection of perforation
quality, especially the depth evaluation of perforation,
is always an interesting problem. BHTV and the new
generation of imaging logging tools, such as USI and
CBIL, are capable of evaluating the inside wall and
circularity of casing and of evaluating the perforation
positions, but they can not be used to determine the
depth of perforations. There does not exist a device
that can directly measure the downhole perforation
depth[1~3]. Therefore, it is of great economical and
social significance to study methods of downhole
perforations depth evaluation and develop a
corresponding device so as to help objectively
evaluate the perforation effectiveness, optimize
perforations techniques and improve oil-gas recovery
efficiency. Ultrasonic pulse echo techniques could be a
very promising method to provide a downhole
perforations depth evaluation. In this article, an
experimental research method and the corresponding
results are introduced on the evaluation of perforations
depth using ultrasonic pulse echo techniques.
MEASUREMENT SYSTEM
The system shown in figure 1 is mainly composed
of a signal generator, an amplifier, an acquisition
system, a processing system and a positioning system,
the operations of which are all controlled by a
computer. The signal source can generate the signal
with frequency between 100 kHz to 1 MHz and an
amplitude range of 50 to 460V. The resolution and the
highest sampling rate of the data acquisition system
are 12 bits and 10 MHz respectively. The positioning
system is responsible for the translating and rotating
the transducer.
Computer
Controler
Driver
Step
Motor
Signal
Generator
Data
acquisition
Casing
Steel Target
Tank
Rock Target
Perforation
Ultrasonic
Transducer
Perforation
FIGURE 1. The experimental measurement system
The experimental casing has a length of 352 mm,
an external diameter of 160.24 mm and a wall
thickness of 10.20 mm. Two holes (each is 20 mm in
diameter) are drilled in the casing, which could be
connected in alignment with cylindrical steel and rock
perforation targets, for the purpose of simulating two
perforations of different depths. 4 Steel targets and 4
rock targets are used in this experiment. The steel
targets are 150 mm long and their external diameters
are 100 mm. In each target there are 3-4 holes, the
depths range and the opening diameters of which are
10 to 150 mm and about 10 mm respectively. As for
the rock targets, the external diameters are 150 mm
and the lengths are 500 mm. In each target there is
only one hole, which has a diameter of about 10 mm
and a depth of between 130 and 500 mm (penetrated).
SESSIONS
All the operations such as the rising and rotating
of the transducer holder and data acquisition and
display are digitally controlled by means of a
computer-controlled positioning system and a data
acquisition system. When the transducer is rotated for
360 degrees, it is raised 5 mm and 100 times
measurements of the pulse echo are conducted.
Therefore the casing’s inside wall is scanned. Fi gure 2
shows the reflected ultrasonic waveforms in two
different situations: when the transducer’s radiation
direction is in alignment with and is not in alignment
with the perforations. In the figure, the waveform in
the middle is the reflected ultrasonic waveform when
the transducer radiation direction is in alignment with
the perforation. The waveforms on the top and at the
bottom are the reflected ultrasonic waveforms when
the transducer radiation direction is not in alignment
with the perforations. It is shown in the figure that
there are evidently reflected waves at 40 µs if the
transducer is not in alignment with the perforation and
there is no such reflection when the transducer is in
alignment with perforation.
1.5
Amplitude
1.0
0.5
Reflected from Casing
0.0
Scattered from Perforation
-0.5
-1.0
0
10
20
30
40
50
60
70
80
9 0 100 110 120
B in figure 3) and their dominant frequencies are close
to that of the incidence waves. The depth of
perforation could be calculated according to the time
when point B is reached and the acoustic speed in the
liquid inside the perforation. Each ultrasonic echo
waveform is processed so as to determine whether
there exist perforations. If there exits a perforation, the
corresponding perforation depth is calculated out. All
the data such as measurement depth and position,
whether there is a perforation and the perforation
depth are saved.
0.8
Amplitude(V)
EXPERIMENTAL RESULTS
0.4
B
0.0
-0.4
-0.8
0
30
60
90
120
150
Time (µs)
FIGURE 3. The scattered waveforms by a perforation in
steel target
DISCUSSION AND CONCLUSIONS
After the experimental study on the downhole
perforation depth evaluation, a computer-controlled
prototype equipment is developed successfully to
measure the depth of the perforations. The key
parameters of perforations evaluation device are
determined. The digitized signal processing techniques
are developed to determine the existence and
non-existence of perforations and the depth of a
perforation. The technique of tracing the scattered
signals (which come from the perforation bottom)
backwards from the ending point of the received
scattered waveforms is proposed.
Time ( µ s )
FIGURE 2. The reflected waveforms of when the radiation
direction is in alignment and is not in alignment with the
perforations
When the transducer’s radiation direction is in
alignment with the perforations, the scattered
waveforms could be received as shown in figure 3,
where point B is the ending point of the scattered wave.
Because of the irregularity of the inside of the
perforations, there is usually no significant acoustic
wave reflection from the bottom of the perforation, but
the incidence waves could be scattered all over the
inside wall of the perforation. After detailed analysis
and comparison, it could be concluded that the
scattered waves that are received last are from the
bottom of the perforation (corresponding to the point
ACKNOWLEDGMENTS
Other members of the research group include
postgraduate students Cheng Xiangyang, Zhao
Huagang, etc.
REFERENCES
1.
2.
3.
Asheim Harald et al., Determination of Perforation Schemes
To Control Production and Injection Profiles Along
Horizontal Wells, SPE Drilling & Completion, March 1997,
13~17.
Halleck, P. M., Recent Advances in Understanding Perforator
Penetration and Flow Performance, SPE Drilling &
Completion, March 1997, 19~126.
Halleck, P. M., Wesson, D. S., Snider, P. M. and Navarette, M.,
Prediction of In-Situ Shaped-Charge Penetration Using
Acoustic and Density Logs, SPE 22808, 483~490, 1991.
SESSIONS
Interface mobilities for source characterisation; matched
conditions.
B.A.T. Petersson1 and A.T. Moorhouse2
1 Institute of Technical Acoustics, Technical University of Berlin, Einsteinufer 25, D-10587 Berlin, Germany.
2 Acoustics Research Unit, Liverpool University, Liverpool L69 3BX, U.K.
Interface mobilities have been demonstrated useful quantities in conjunction with the analysis of vibration
transmission at multiple point and large area interfaces. The extension of the approach to the related area of
source characterisation is logical. Thereby significant conceptual as well as practical simplifications result. The
interface mobilities which are the ratios of the spatially Fourier decomposed velocity field to excitation field
components, thus retain the formal simplicity and transparency of the single point case. This is at the expense of
a slightly more elaborate post-processing of the constituent, computational or experimental data. Herein, the approach is critically examined for a source-receiver configuration, involving multiple contact points. Particular
interest is directed to the case where both subsystems have matched or close to matched dynamic characteristics.
INTRODUCTION
In a suite of recent publications an approach to
treat large, continuous, closed contour interfaces has
been addressed from various viewpoints, e.g. [1-5]. It
is demonstrated that with the introduction of the concept of interface mobility [4], a concept intimately
related to the class of Fredholm integral equations, a
significant simplification can be obtained for small
and intermediate Helmholtz numbers. The main reason being that the transmission problem is transposed
into that of an equivalent, single point and single
component of motion and excitation, cf. [6]. Its applicability to cases with multiple point connections
between the subsystems therefore becomes interesting. Of foremost interest is the application of interface mobilities in conjunction with source characterisation as recognised in [7].
SOME FUNDAMENTALS
For continuous velocity fields and force distributions over a closed contour, the interface mobility is
defined as,
1
− ik s − ik s
Y pq = 2
Y (s | s0 )e p e q 0 dsds0
C C C
Consider a multi-point interface between a source
and a receiver structure such that a closed contour
can be formed, passing all the points. The interface
forces can be brought on to a form of a continuous
distribution through,
F (s) =
Fmδ (s − sm ) ,
∫∫
∑
Fq =
(∑ F e
− ikq sm
m
) C,
where C is the perimeter of the contour. Similarly,
the vibrations along this contour can be expanded as,
v p = (1 C )
∫
v (s)e
− ik p s
ds .
C
The complex power transmitted from the source to
the receiver is then found to be given by,
Q = C2
(∑ Y
*
pq F p Fq
)
2.
From previous studies it is seen that for small
Helmholtz numbers, kR , where R is a typical radius
or dimension of the interface, the cross-order terms
are generally small compared with those of zero and
first order. For large Helmholtz numbers, moreover,
the cross-order terms asymptotically vanish. With the
cross-order terms small compared with the paired
terms, the transmission problem and the source characterisation is simply subdivided into a sequence of
orders, for instance,
C2
2
2
2
(Y F + Y11 F1 L + Y−1−1 F−1 L)
2 00 0
This means that for many engineering applications,
the formal simplicity of the single-point, singlecomponent case can be retained and terms included
as required. Finally, it opens a viable scheme for
source characterisation since all paired orders can be
treated as individual source contributions.
Q=
such that the coefficients of the associated Fourier
series become,
SESSIONS
EXPERIMENTAL EXAMINATION –
MATCHED SUBSYSTEM DYNAMICS
10
|Y vF|, [m/Ns]
10
10
10
10
-1
10
10
Power, [W]
With particular focus on the implications of
closely matched dynamic characteristics of the
source and receiver, a simple source was constructed.
The mobility of the source system – a finite plate
driven by a miniature shaker at an eccentric position
– was adjusted by means of point masses to align
with that of the receiver – a larger finite plate of the
same thickness as that of the source. The source was
connected to the receiving plate at four contact
points, also positioned eccentrically and the interface
contour chosen was a circle passing through these
points. The high degree of matching is exemplified in
Figure 1 via the ordinary point mobilities of source
and receiver at one contact point. Interface mobilities
were formed from measured point and transfer mobilities of both substructures. To estimate the transmitted power to the receiver, also the free velocity of
the source was registered and Fourier decomposed.
As a reference, the power transmitted to the receiver
was additionally measured directly at the contact
points in the assembled state.
10
10
10
-4
-6
-8
-10
-12
10
1
2
10
Frequency, [Hz]
10
3
Figure 2. Comparison of estimated an measured power
transmission. (—) measured in the assembled state, (- - -)
estimated using zero and first order interface mobilities and
(·······) calculated using the matrix formulation
The interface mobility appears to handle the
matched condition as well as the full matrix calculation, which means that the estimates are sensitive in
regions where the subsystems characteristics are
equal in magnitude but phase conjugated.
-2
CONCLUDING REMARKS
-3
-4
-5
10
10
-2
1
2
10
Frequency, [Hz]
10
3
Figure 1. Ordinary point mobilities of source (——) and
receiver (- - -) at a contact point.
In Figure 2 are compared the power estimated
from zero and first order interface mobilities, and
from a full matrix computation with that measured in
the assembled state. Although the interface mobility
estimate of power, truncated after the first order
terms, requires significantly less data it is mostly as
accurate as the full matrix mobility calculation and is
even more accurate at the higher frequencies. In the
upper range moreover, it is more numerically stable
than the matrix estimate.
By decomposing the source activity, in this case
the free velocity, and the structural dynamic characteristics, a set of source orders is established. In a
first approximation these can be considered as independent source components to be superimposed. The
source components correspond to physically interpretable motion components of the source. It is demonstrated that for common noise and vibration control applications, the few primary orders suffice to
handle the structure-borne sound transmission of
multi-point installations.
REFERENCES
1
2
3
4
5
6
7
Petersson, B.A.T., Proc. Inter-Noise, Honolulu, 1984, pp.
553-558.
Hammer, P. and Petersson, B., JSV 129, 1988, pp 119-132
Petersson, B., JSV 176, 1994, pp 625-639.
Petersson, B., JSV 202, 1997, pp 511-537.
Fulford, R. and Petersson, B.A.T., JSV 232, 1999, pp 877895.
Pinnington, R.J. and Pearce, D.C.R., JSV 142, 1990, pp 461479.
Petersson, B.A.T., Proc. 6th Congress of Vibr. and Sound,
Copenhagen, 1999, pp 5, 2175-218
SESSIONS
Development of Sound Radiation Prediction Program using
PFFEM Analysis Results
Ho-Won Lee, Suk-Yoon Hong and Young-Ho Park
Department of Naval Architecture & Ocean Engineering, Seoul National University, Seoul, Korea
The sound radiation prediction program is developed using the power flow finite element method(PFFEM) analysis results.
PFFEM is a new method used for the prediction of the vibration energy density and intensity of arbitrary shape structures in
medium to high frequency ranges. The boundary element method is used for the sound radiation program developed here, and
the analysis results calculated by PFFEM is used as the boundary condition to analyze the vibration. Then, the sound radiation
analysis can be simultaneously performed. With this program, the vibration and radiation characteristics of complex system
structures such as submarine are predicted.
1. INTRODUCTION
Commercial analysis programs used in the vibroacoustic field are split into two parts of software such
as for the structural vibration analysis and for the
sound radiation analysis. Mostly they have been
individually developed and used. The purpose of this
paper is to structure the system which is able to
analyze the vibrational problem and the sound radiation problem at the same time. The power flow
finite element method(PFFEM) which is used for the
prediction of the vibration energy density and intensity
of a complex structure in the medium to high frequency band is used here as a key method of the
vibration analysis. The sound field around the structure
is studied by solving the surface pressure about the
structure by the acoustic boundary element method
(BEM). At this time, the vibrational energy density
calculated by power flow finite element method is
used as boundary condition(e.g. surface normal velocity) of BEM. Then, computations are performed for
complex structure such as submarine.
where ω is the exciting frequency, η the structural
damping loss factor and m represents the wave type.
e is the time- and locally space-averaged vibrational
energy density and c gm the group velocity. Equation
(1) may be written as matrix form with finite element
techniques as
[K ] {e }= {F }+ {Q }
(e)
(e)
(e)
(e)
(2)
where each term of equation (2) can be written by
K
F
Q
(e)
mij
(e)
mij
2
æ c gm
ç
òD ç ηω ∇ φ i ∇ φ
è
=
=
ò (Π
m
j
ö
+ ηωφ i φ j ÷ dD
÷
ø
(3 )
φ i )dD
(4)
D
(e)
mij
=
2
æ c gm
ò çç ηω
Γ
è
ö
φ i ( − n ) ⋅ ∇ e ÷d Γ .
÷
ø
(5)
2.2 Boundary Element Method
If sound wave propagates through a three dimensional domain with small amplitude, wave equation can
be expressed as
2. THEORY
2.1 Power Flow Finite Element Method
When the vibrational power is input with a unit
plate element of structure and in a steady state,
governing energy equation can be written as
∇2 p + k 2 p = 0
Equation (6) can be modified for the boundary
element method as follows :
N
−
2
c gm
ηω
∇
2
em
+ ηω
em
= Π
m
(1 )
(6)
9
Aα
åå
α
mj
m =1
=1
N
ö
æ
p mα − ç 4π + å C mj ÷ p j =
m =1
ø
è
N
9
Bα vα
åå
α
mj
m =1
m
(7)
=1
SESSIONS
where p j is the sound pressure at j-th node, N the
number of element and vmα is the surface normal
α ,
α ,
velocity at m-th element. Amj
B mj
C mj of equation (7)
are defined as follows ;
α
A mj
=
∂
òS ∂ n r
m
m
α
B mj
= i ωρ
C mj =
− ikR
ö
÷N
÷
ø
Figure 3. Vibration energy density for inner plate
α
(ξ ) J (ξ ) d ξ
(8 )
(ξ )
e mj
òS R mj (ξ ) N α (ξ ) J (ξ ) d ξ
m
∂
ò ∂n
Sm
æ e − ikR mj ( ξ )
ç
ç R (ξ )
mj
è
rm
ö
æ
1
÷ J (ξ ) d ξ
ç
ç R (ξ ) ÷
ø
è mj
(9 )
(10 )
Figure 4. Surface pressure distribution
where J is the jacobian and N α (ξ ) the shape function.
For the energy density of power flow finite element
method to be used, it is transformed into velocity.
3. COMPUTATIONAL EXAMPLES
Computations are performed for the submarine
model which is excited by a harmonic point force
located in the middle of the engine room. It is assumed
that the structure is in the water. The excitation frequency is f = 100 Hz, and the damping loss factor
is η = 0.05 at the inner plate and η = 0.1 at the
outer plate. The submarine model for vibration and
sound radiation analysis is shown in Figure 1. The
vibration energy density obtained from the PFFEM are
shown in Figures 2 and 3. Figure 4 shows the surface
pressure distribution obtained from the boundary
condition of Figure 2. At this time, the sound pressure
and intensity distribution around submarine are shown
in Figures 5 and 6, respectively.
Figure 5. Sound pressure around submarine
Figure 6. Sound intensity around submarine
4. CONCLUSIONS
Figure 1. Submarine model for the vibration and sound
radiation analysis.
Figure 2. Vibration energy density for outer plate
The program for the sound radiation prediction
using PFFEM’s vibration analysis results is developed.
The vibration analysis of arbitrary shape structure
composed of plates is performed by using the power
flow finite element method, and these vibrational
results are used as the boundary conditions and the
sound pressure field around the structure is calculated
by the acoustic boundary element method. This
algorism is used for the program of sound radiation
prediction. With this program, computations are
performed for the case of an underwater submarine
model which is excited by a point force located in the
engine room, and the sound radiation characteristics of
its model are successfully predicted.
SESSIONS
Order Extraction as Order Waveform Based on the Digital
Tracking Filter of Low Orders (II) *)
A.A. Petrovskya, A.V. Stankevichb, S.J. Luznevb
a
Department of Real-time Systems, Bialystok Technical University,
Wiejska St. 45A, 15-351 Bialystok, Poland
b
Department of Computer Engineering, Belarusian State University of Informatics and Raadioelectronics,
6, P. Brovky St., 220027 Minsk, Belarus
The order extraction as order waveform in the rotating machine monitoring systems based on the new digital low order filtering
are presented in this paper. For increasing of an analyzer dynamic range in the given report is offered to execute resampling
amplitude of signal in new designed time moments by the filtering in the order domain, unlike the known approach, where
digital filter was used in the time domain. The given digital tracking filters of low orders allows high-performance tracking of
harmonic responses of periodic in mechanical acoustical systems. The tracking capabilities are independent of the rate of
change of the rotational speed (slew rate). The experiments have shown the possibility to receive a practical dynamic range
more than 100 dB.
INTRODUCTION
The goal of order tracking is to extract selected
orders in terms of amplitude and phase, called Phase
Assigned Orders, or as waveforms. The order
functions are extracted without time delay (no phase
distortion), and may hence be used in synthesis
applications for sound quality, multiplane balancing,
measurements of operational deflection shapes, and
dopplerized engine noise tracking.
PROCESSING IN ORDER DOMAIN
Due to improvements in microprocessor
performance, in particularity, digital signal processor,
it is feasible to design a digital processor of computed
synchronous resampling and order tracking that has
negligible internal phase noise. The digital processor
can be used to:
#1) collect measured data at some fixed rate, digital
filtering of the signals using multirate filters;
#2) measure and store the arrival times of each
synchronizing tachometer pulse simultaneously (it is
important to measure the arrival time of each
tachometer pulse very accurately to reduce the effects
of time jitter);
#3) calculate the new digital resampling time
points on the base of different rotation models and to
store them;
#4) interpolate the stored measurement data in
some optimum manner to obtain new samples at the
desired time points;
#5) compute the spectrum in the order domain.
THE RESAMPLING AMPLITUDE
ALGORITHM
The order tracking front-end algorithms in the
rotating machine monitoring systems based on the
novel digital low order filtering are presented in paper
[1]. For increasing of an analyser dynamic range in the
given article is offered to execute resampling
amplitude of signal in new designed time moments by
the filtering in the order domain, unlike the known
approach, where digital filter was used in the time
domain [2]. As in the case with the frequency domain
for a filter in the order domain we shall enter the term
“Filter of the Low Orders” (FLO). The proposed
approach excludes methodical errors, because in the
order domain such filter will have the constant order
characteristics and for further processing will be
allocated only by part of a spectrum. As for time
domain the given filter realizes automatic tracking for
the change of the object rotation velocity in the
correspondence
with
an
accepted
rotation
mathematical model.
The resampling amplitude algorithm by the
interpolation digital filter in order domain has the
following steps:
#1) to calculate a revolution angle for i resampled
point from beginning of a revolution the base of the
linear model of object rotation;
#2) to calculate a current angle from beginning of
j-th revolution for the nearest points stored with
SESSIONS
constant sampling time period to the resampled point
on the base of the linear model of object rotation;
#3) to check up the angular distance from the
resampled point up to some i-th nearest point stored
with constant sampling time period, whether the
angular distance is no more than half of length of the
impulse response;
#4) to calculate the value of the impulse response
of FLO with angle defined on the step #2 for the signal
points stored with constant sampling frequency;
#5) to calculate resampled sample of a signal in i
point as a sum input signal points stored with constant
sampling frequency weighted with impulse response.
FIGURE 1a. Magnitude response of the FLO in the
frequency domain
TESTING RESULTS
The testing results of an asynchronous electric
engine in the stage of start-up and some time later in
the stage of rated load are shown on the figure 1 (a-d).
The range of generated frequencies was closed to
vibrations of symmetrical engine with 24 slots of
stator. For simulation the classic model of
asynchronous engine was used. The results (figure 1)
also show that the developed method provides stability
of amplitude and rule of spectral lines in order domain
do not depend on operations mode of object.
The magnitude characteristic of FLO in frequency
domain for the given example is shown on the
figure 1a. The curve 1 corresponds to the first
resampled point, the curve 2 - 32-nd, and the curve 3 64-th resampled points respectively. On the figure 1a
the following labels are accepted: w - normalized
frequency, mag - amplitude in dB.
FIGURE 1b. The dependence of rotation speed
CONCLUSION
The dynamic range of analyzer with resamling
amplitude by the FLO does not depend on the number
of experimental points on one complete object
revolution, but correct choice of sampling frequency
for maximally possible rotation speed of object is
important.
FIGURE 1c. The tracking contours of the first, tenth
and twenty fifth orders
REFERENCES
1. Petrovsky A., Stankevich A., Balunowski J. Proc.
of the 6th International congress “On sound and
vibration”, ICSV’99, 5-8 July 1999, Copenhagen,
Denmark. –pp.2985-2992
2. Potter R. Sound and Vibration, September 1990,
pp. 30-34.
8)
This work was supported by KBN under the grant 7T07B 033 18
FIGURE. 1d. The spectral map in order domain
SESSIONS
Response of Rectangular Thin Plate to a Point Force and
Distributed Excitation
P. R. Bonifácio, A. Lenzi
Departamento de Engenharia Mecânica, Universidade Federal de Santa Catarina, 88000 Florianópolis, Brasil,
[email protected], [email protected].
Several techniques are used to study the frequency response of plate, however, they limit with all side simply supports. In this
paper, analytical analyses are presented for various combinations of simply supported (S), clamped (C) and free (F) edges in
thin rectangular plate through parametric analysis and is obtained the effect of boundary conditions in the response of finite
plates. Although there are several combinations, only three of boundary conditions are shown in this work, as SSSS, SSSC and
SSSF, and it can be extended to other cases. Therefore, this parametric analysis concerned with appendage in the form of
frequency response of plate by classical Kirchoff’s theory. Numerical calculations are used to check the obtained analytical
results. The predicted resonance frequencies curves and mode shape are compared against the Finite Element Method (FEM)
and good agreement is found.
INTRODUCTION
At high frequency in structural analysis, where
wavelengths are short, predict the response of
structures by analytical methods is often preferable
once the solutions by FEM are computationally
intensive.
In the present paper the curves of frequencies that
describe the displacement of the plate are determined
for each values of Lx/Ly and for each set of boundary
condition in the plate, calculated in relationship with
eigenvalue parameters λ = Lx 2ω ρ h / D .
The generalized coordinates are expressed through
parametric analysis for each set of boundary condition
in the plate. Generalized forces of the problem are
found to a point force and to a distributed excitation
along the plate.
Mathematical Procedure
ratio for the assumed isotropic material, h, the plate
thickness, and (xo,yo) is the excitation position.
The displacement at any point on the structure can be
represented as [2,3]
w( x, y, t ) =
m
D=
Eh 3
(1 + iη )
12(1 − ν 2 )
(1)
(2)
where w(x,y) is the flexural displacement, D, the
complex bending stiffness, η, the lost factor, ρ, the
material density, E, Young’ modulus, ν, is Poisson’s
mn ( x , y )
e −iω t
(3)
n
where x = x / Lx and y = y / Ly
The case of steady-state harmonic vibration and time
dependence of the form e-iwt is considered in the
present work. This solution (Eq. 3) must satisfy the
boundary conditions by Φ , which represents the base
functions that describe the shapes vibrations in the
plate.
Substituting the strain energy U and kinetic energy T
into Lagrange’s equations (Eq. 4), the dynamic
response of the plate, excited by point or a distributed
force, can be calculated [2].
For the stationary time harmonic response, the
kinetic energy T and the strain energy U, can be clearly
determined by [3].
The flexural displacement of a thin, transversely
vibrating plate, excited by time harmonic point force
F(xo,yo), and to a distributed force F(x,y), oscillating
with a circular frequency ω, is governed by [1]
D∇ 4 w( x, y ) − ρ hω 2 w( x, y ) = Fδ ( x − xo )δ ( x − yo )
∑∑ qmn (t ) Φ
d  ∂T  ∂T
∂U
+
= Qmn

−
dt  ∂q!mn  ∂ qmn ∂ qmn
(4)
where qmn represents the mth-nth-generalized
coordinate and Qmn is the mth-nth-generalized force in
both directions.
The generalized force to a point and distributed
excitation is expressed respectively as [2]
Qmn =
1 1
∫ ∫ δ ( x − x )δ ( y − y ) Φ
o
0 0
Qmn =
1 1
∫∫
0 0
o
mn ( x , y ) dydx
F ( x, y ) Φ mn ( x , y ) dydx
(5)
(6)
SESSIONS
100000
10000
1000
10
w (λ )
Plate SSSF
Lx/Ly=1
Lx/Ly=2
Lx/Ly=3
1E-3
1E-8
2
2
Plate SSSS
Lx/Ly=1
Lx/Ly=2
Lx/Ly=3
1E-4
1
w (λ )
2
w (λ )
1
1E-6
50
100
150
2
200
0
50
100
1/2
λ=Lx ω(ρ h/D)
150
2
Plate SSSC
Lx/Ly=1
Lx/Ly=2
Lx/Ly=3
1E-3
1E-7
200
50
100
1/2
λ=Lx ω(ρ h/D)
150
2
200
1/2
λ =Lx ω(ρ h/D)
FIGURE 1. Parametric analyses
Displacement (m)
Displacement (m)
1E-4
1E-5
1E-6
1E-7
Plate SSSF
Proposed Model
FEM
1E-5
1E-6
Plate SSSC
Proposed Model
FEM
1E-4
Displacement (m)
Plate - SSSS
Proposed Model
FEM
1E-4
1E-5
1E-6
1E-7
1E-7
0
200
400
600
800
1000
0
200
Frequency (Hz)
400
600
800
0
1000
200
400
600
800
1000
Frequency (Hz)
Frequency (Hz)
FIGURE 2. Displacement of plates and comparison with FEM
being F(x, y) the distributed force acting on the plate.
The generalized coordinate on the structure can be
obtained substituting from Eq. 3, Eq 5 and Eq. 6 into
the Lagrange’s equations (Eq. 4), resulting in
q mn (λ ) =
Qmn λ 2
1 1  ∂ 2Φ
 ∂ 2Φ ∂ 2Φ
1 1
∂ 2Φ 2
∂ 2Φ 2  
− 2(1 − ν ) 
−(
)  dx dy − λ 2 ∫ ∫ Φ 2 dy dx
2
2
0 0
∂ x∂ y  
 ∂ x ∂ y

∫0 ∫0 ( ∂ x2 + ∂ y 2 )

results are compared with FEM, which show excellent
agreements in all cases.
Lx
S
S
S
S
Lx
Ly
S
S
F
S
Lx
S
S
C
S
Ly
Ly
FIGURE 3. Boundaries conditions in the plates
where λ = Lx 2ω ρ h / D ;
Conclusions
The parametric analysis consists of finding the
mean-square quadratic spatial response of the plate in
relationship with λ , as seen in Eq. 7, expressed in
(ms2/kg)2. These results can be seen in the Figure 1,
which shows results for ν = 0.3 , and several values of
Lx/Ly.
w2 ( λ ) =
1
1
0
0
∑∑ ∫ ∫ q
m
n
2
2
mn (λ )Φ mn ( x , y ) dx dy
The phenomenon of the vibration in plates is easily
studied through a parametric analysis, thus the main
variables of the problem can be analyzed in a better
way. Therefore, this work can be used to predict a
more complex system as plates coupled in beams.
(7)
This analysis is considered for three different sets of
boundary conditions, as in Figure 3. The results of the
displacements are shown in Figure 2, for a plate with
Lx=1m, Ly=0.5m, h=0.003m, ρ= 2710 kg/m3, ν=0.3,
η=0.01, E=7.2 1010 Pa, xo=0.1m, yo=0.14m. The
REFERENCES
1. Graff, K., Wave Motion in Elastic Solids, New York,
1975, pp. 240-250.
2. Meirovitch, L., Analytical Methods in Vibrations,
London, 1967, pp. 30-54.
3. Leissa, A., Vibrations of Plates, Washington, 1969.
SESSIONS
Numerical Modelling of Sound Transmissions in Buildings
at Low Frequencies
O. Chielloa, F. C. Sgarda, N. Atallab
a
Laboratoire des Sciences de l'Habitat, DGCB URA CNRS 1652, Ecole Nationale des Travaux Publics de l'Etat,
69518 Vaulx-en-Velin CEDEX, FRANCE.
b
Groupe d'Acoustique de l'Université de Sherbrooke, Department of Mechanical Engineering, Univ. de Sherbrooke,
Sherbrooke, QC,J1K2R1,Canada
This paper presents a general method to investigate the vibroacoustic behavior of complex structures coupled to acoustic cavities
at low frequencies. A finite element description is used for structural displacements and fluid pressure fields. The coupled forced
system is expanded on the uncoupled structure and rigid cavity modal basis. The structural modal analysis is performed using a
free-interface component mode synthesis technique. The method proves to be accurate and well suited to parameter studies on
structural complexities. In this paper, the method is applied to investigate the sound transmission paths in buildings at low
frequencies. In addition, a general elastic joint between the separating wall and the lateral walls is accounted for in the model.
The original free-interface technique has been modified both in the definition of the attachment modes and in the assembly
procedure to account for the mass-less joint which is considered as a substructure in the component mode synthesis. An example
is presented to validate the approach. Additional applications will be detailed in the poster session.
INTRODUCTION
THEORY
The estimate of the influence of structural complexities
(joints, stiffener etc) on the vibroacoustic behavior of
elastic structures coupled to acoustic cavities is a
common issue encountered by engineers. In building
acoustics especially, where standards are more and
more strenuous, structural complexities related to the
mounting conditions of partitions may have important
effects on the acoustic insulation at low frequencies.
Deterministic methods such as finite elements coupled
to classical modal analysis are appropriate at low
frequencies but are often limited by the problem size
and not well suited to parametrical studies. Indeed,
when the effect of a structural parameter on the
vibroacoustic behavior of a system needs to be known,
the discretized quantities relative to the whole structure
have to be recalculated before to redo the modal
analysis accounting for the structural changes. In this
paper, component mode synthesis (CMS) is shown to
be an adequate tool for solving this kind of problems
and deserves to be used whenever possible for the
structural analysis. In addition, a free interface method
[1,2] is used that allows for a removal of the junction
degrees of freedom between the sub-structures in the
final system. The coupling of structures with closed
acoustic cavities is accounted for by a modal expansion
over the uncoupled structural modes and rigid cavity
modes. This technique proves to be valid for light fluid
and allows for a significant reduction of the system
size [3]. The combination of these two techniques
enables one to calculate low frequency responses of
complex problems with a good accuracy and a reduced
computation time. In the following, the two methods
are described successively. Then a validation example
is presented. During the poster session, a typical
configuration encountered in building acoustics will be
focused on.
Consider the in vacuo free vibrations of a discretized
structure, undamped and composed of Nc substructures. In order to calculate the eigenmodes of this
structure, displacement vectors u(c) of each substructure c are decomposed in terms of the truncated
modal basis of the isolated F(c) completed by a pseudostatic approximation of the unkept modes [2].
u(c) = F(c) q(c)+Gd(c)fjunc(c)
(1)
where q(c),Gd(c) and fjunc(c) are the modal coordinates
vector, the residual flexibility matrix of sub-structure c
and the force vector applied on sub-structure c by all
the other sub-structures, respectively. Applying
continuity relations of forces and displacements at the
interface between each sub-structure allows for an
elimination of force vectors fjunc(c) in the system of
equations together with an expression of the substructure displacements in terms of a global modal
coordinates vector qr = {q(1) … q(Nc) }T only, such that
u(c)=R(c)qr where xT denotes the transposed of x.
Assuming a temporal dependency of the form ejwt, the
invocation of the stationnarity of the energy functional
of the whole structure expressed in terms of vector qr
leads to an eigenvalue problem of the type:
(Kr - w2 Mr ) qr = 0
(2)
The eigenvectors of the whole structure on the degrees
of freedom of each sub-structure c, F(c), are then given
by Fs(c) = R(c) Fr, where Fr is the modal basis computed
from eigenvalue problem (2). This method is very
advantageous compared to a global modal analysis of
the whole structure when physical parameters of a
single sub-structure are changed since it is only
necessary to perform the modal analysis of the substructure of interest and to use the component mode
synthesis technique to obtain eq(2). This step requires
SESSIONS
the resolution of a reduced size eigenvalue problem
compared to classical global modal analysis. In the case
of high number degrees of freedom problems, the
technique allows for an important decrease of the
memory used for the computation. Finally, the method
is robust and accurate, provided that the number of kept
modes of each sub-structures be adequately selected.
The forced response of a weakly damped system is now
considered. The system comprises a global elastic
structure which is coupled to a discretized acoustic
cavity. The equations of the system are projected over a
dual modal basis composed of the truncated modal
basis of the whole structure obtained by CMS and the
truncated modal basis of the rigid walled cavity, Ff
obtained by a classical modal analysis. They read:
~
2
éW
ù
ì Fm ü
Cm
s -w I
ï
ê
ú ìíq s üý = ïí 1
(3)
1
~
~
ý
ê CTm
W f - I ú îq f þ ï 2 S m ï
2
êë
úû
îw
þ
w
~
In (6), W s denotes the generalized complex stiffness
whole structure, each plate has been considered as an
isolated sub-structure. By retaining the sub-structure
modes up to 1500Hz, (1.5 times the upper frequency
limit of the excitation spectrum), errors on the
eigenfrequencies of the whole structure are less that 1%
up to 1000Hz. The calculation of the vibroacoustic
indicators has been validated by comparing the present
approach to the results of commercial software I-DEAS
(SDRC©) for an identical mesh in the case where one
wall of the cavity is excited by a decentered point force.
Fig.1 exhibits an excellent agreement between both
approaches for the mean square velocity of one of the
non excited plates. Software I-DEAS results have been
obtained from a classical modal analysis of the whole
structure, the fluid coupling being accounted for by a
BEM.
matrix accounting for structural damping such that:
Ns
Ns
~
~
T~
Ws = å W(sc ) = å F(sc ) K ( c )Fs( c )
~
c =1
(4)
c =1
where K ( c ) is the complex stiffness matrix of the
isolated sub-structure c. I denotes the identity matrix,
~
W f is a diagonal matrix filled with the kept eigenvalues
~
of the cavity, I is the generalized complex
compressibility matrix accounting for dissipation within
the cavity under the form of a structural damping and
Cm is a generalized coupling fluid-structure matrix
which can be written as:
Ns
Ns
c =1
c =1
Cm = å C(mc ) = å F(sc ) C( c )Ff
T
(5)
where C(c) is the coupling matrix between sub-structure
c and the cavity. Sm is the generalized source vector in
the cavity and Fm is the generalized force vector applied
on the whole structure. Eq(3) is then solved at each
frequency to get the modal coordinates qf and qs
associated to the cavity and to the structure
respectively. These quantities are used to compute the
vibroacoustic indicators. In the following, all the cavity
and structure modes up to 1.5 times the upper frequency
of the excitation spectrum are kept in the expansions.
RESULTS
A validation example is now presented. Consider a
rectangular cavity with dimensions 0.48´0.40´0.56m
and flexible steel walls. The steel characteristics are
E=2.1011Pa, n=0.3 and r=7800kgm-3. The cavity is
filled with air with density r0=1.2kgm-3 and sound
speed c0=340ms-1. The elastic walls are discretized
using quadratic 8 nodes shell elements (12´10, 12´14
or 10´14 according to the walls). The cavity mesh is
constituted of 6´5´7 quadratic 20 nodes hexaedric
elements. The coupling matrices are calculated over the
fluid mesh (6´5, 6´7 or 5´7) using quadratic 8 nodes
surface elements. The frequency band of interest is
[10-1000 Hz]. For the calculation of the modes of the
FIGURE 1. Mean square velocity of a non excited wall
CONCLUSION
This paper has presented a general approach to study
the vibroacoustic behavior of complex elastic structures
coupled to elastic cavities at low frequency. The main
features of the approach lie in the modal analysis of the
whole structure using a free surface component mode
synthesis together with the use of an uncoupled dual
modal basis for the fluid-structure coupling. These two
techniques can be combined to speed up the resolution
of a complex fluid-structure problem by decreasing the
number of degrees of freedom and prove to be
appropriate for parameter studies. The convergence of
the approach is excellent. Numerical results have been
compared successfully to those obtained with a
commercial software in the case of a flexible walled
cavity filled with air and excited mechanically. During
the poster session, a typical building configuration will
be considered.
REFERENCES
1. Craig, R.R. Jr, and Chang, C.-J. Substructure coupling for dynamic
analysis and testing. CR 2781. NASA. 1977.
2. Tournour, M.A. and Atalla, N., Noise Control Engineering
Journal, 46, 83-90 (1998).
3. Atalla, N., Tournour, M.A., and Paquay, S., A modal approach for
the acoustic and vibration response of an elastic cavity, Proceedings
16th ICA/135th ASA meeting, 1998, pp.189-190.
SESSIONS
On point excited plates
Jonas Brunskog, Per Hammer
Division of Engineering Acoustics, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden
The present paper presents an expression for the point mobility of infinite plates driven by a perfectly rigid indenter. The
problem is of general interest in connection with excitation and transmission of structure-borne sound. The indenter is assumed
to be circular, weightless, stiff compared to the stiffness of the plate, and small compared to the wavelength of the bending and
quasi-longitudinal waves of the plate. A detailed three-dimensional analysis is used. Traditionally the problem is solved
approximately by means of assuming a pressure distribution in the interface between indenter and plate. In the present study the
pressure distribution is also assumed, but an optimal choice of the pressure amplitude is found by means of a variational
formulation.
INTRODUCTION
The object of the present paper is to derive
expressions for the point mobility of infinite plates
when driven by a perfectly rigid indenter. The problem
is of general interest in connection with generation and
transmission of structure-borne sound. This work,
however, have been prompted by a special need; a
accurate description of the imaginary part of the point
mobility is important e.g. for impact situations where a
'spring' will make the impacting body rebound. The
hypothesis is that especially the imaginary part of the
mobility is incorrect in the previous derived
expressions found in the literacy. The indenter is
assumed to be circular, weightless, stiff compared to
the stiffness of the plate, and small compared to the
wavelength of the bending and quasi-longitudinal
waves of the plate. The plate is assumed to be
isotropic.
The mobility is defined as the complex ration of
velocity and force at the intersection between indenter
and plate. By means of the classical Kirchhoff thin
plate equations, it has been shown [1, 2] that the point
mobility of an infinite plate can be written as
v
1
(1)
Y≡ z =
F 8 m′′ B
where vz is the vibration velocity, F the driving force,
m'' the mass per unit area and B is the bending stiffens.
In [3] Ljunggren discusses the accuracy of equation (1)
and compare it with Mindlin theory and a threedimensional theory where only the poles
corresponding to the bending waves are taking into
account.
In all analysis describing the motion of 'thin'
structures it is assumed that at each point, the two
sides of the structure have exactly the same
displacement. This is an approximation and when the
excitation is concentrated in an area which is
comparable with or smaller than the structural
thickness, additional weakness effects can occur.
Moreover, when the radius of the indenter is
comparable or lager than the thickness, additional
inertia effects can occur.
A more detailed three-dimensional analysis has been
used by Paul [4] and Ljunggren [5] for rigid indenter,
and Heckl [6] for soft indenter. Heckl and Petersson
[7] investigated the influence of different choices of
pressure distributions. The boundary value problem for
a soft indenter is simpler than that resulting from a
rigid indenter. For a soft indenter the pressure
distribution under the indenter is constant. Therefore,
both outside and under the indenter the boundary
condition are Neumann conditions, i.e. prescribed
pressures. For a rigid indenter the displacement under
the indenter and the pressure distribution outside the
indenter is prescribed, i.e. mixed Neumann and
Dirichlet conditions.
Both Heckl [6], Ljunggren [5] and Petersson and
Heckl [6] avoided the problem of solving integral
equations by means of assuming a pressure
distribution under the indenter. As there is no
guarantee that this assumption actually results in a
uniform displacement under the indenter, this case will
be denoted quasi-rigid. In [5] a comparison between
the different results can be found (Figure 9 in the
reference). Calculated values of the local reaction for
the same numerical values are presented. There is
hardly any agreement at all between the results. As
Ljunggren points out, this lack of agreement is hardly
astonishing in view of the different presumptions used
in the different cases.
In the present paper the pressure distribution in the
interface between indenter and plate is found by means
of a variational formulation.
SESSIONS
FORMULATION
The system to be solved is a rigid circular indenter
acting on a plate with finite thickness, shown in figure
1. We seek the point mobility Y of the excitation
situation, and thereby the force acting in the interface
between indenter and plate when the indenter is
displaced a distance wieiωt, where the time dependence
eiωt is henceforth suppressed.
R
wi
x
d
z
For a stiff indenter, the boundary conditions are
w0 (x, y ) = wi , for r ≤ R and z = 0

 p(x, y ) = σ z = 0, for r > R and z = 0
(2)
where r=(x2+y2)1/2, and a additional condition is
p(x,y)=σz=0 for z=d. The tangential shear stresses at
the both surfaces are assumed to be zero, i.e. also
under the indenter. It should be noticed that the
displacement of the indenter wi is a real constant (if the
time dependence eiωt is suppressed). The force is found
as the integral of the pressure field under the indenter.
With the aid of Hankel transforms, the displacement
can be written
1
2π
~
p
∫0 iω (AA + AS )J 0 (k r r )k r dk r
∞
(3)
where
AA =
AS =
(k
(k
)
2
r
− iωαk 2 2 µ
+ β 2 tanh (αd 2) − 4αβkr2 tanh (βd 2)
2
2
r
)
R
∫ p( s) s K (r , s)ds = w , r ≤ R .
i
0
This is a Fredholms equation of the first kind.
VARIATIONAL FORMULATION AND
SOLUTION
The varational formulation follows the method
described in Morse and Ingard [8]. With the new
variables q(r)=p(r)r and v=iωw, a variational
formulation of the problem is
R
R
R
R
0
0
0
0
V = ∫ q(r )v * dr + ∫ q * (r )vdr − ∫ q * (r )∫ q(s )K (r , s )dsdr
FIGURE 1. Rigid indenter acting on plate
w0 (r ) =
1
iω
− iωαk 2 2µ
+ β 2 coth (αd 2 ) − 4αβkr2 coth (βd 2)
2
Define a kernel K as
∞
K (r , s) ≡ ∫ J 0 (k r s)( A A (k r ) + AS (k r )) J 0 (k r r ) k r dk r
0
The boundary conditions (2) and equation (3) can now
be used to rewrite the problem as
Assume a pressure distribution that corresponds to the
semi-infinite case, used by Ljunggren [5], p(r)=c/(R2r2)1/2. A stationary point is solved for the constant c,
which is the best fit for the present pressure
distribution. The input mobility can then be
determined as
Y=
1
2πR 2
r K (r , s )s
RR
∫∫
2
R − r 2 R2 − s2
0 0
dsdr
Make use of the relation
R
∫ s J (xs)
0
R 2 − s 2 ds = sin (Rx ) x ,
0
Thus, the result is
Y=
1
sin 2 (kr R )
(AA (kr ) + AS (kr ))dkr .
2π R 2 ∫0
kr
∞
It seems not be possible to calculate this integral
analytically, but it is possible to use numerical
integration.
REFERENCES
1 Boussinesq, J., Applications des Potentiels, GauthierVillars, Paris (1885)
2 Cremer, H. and Cremer, L., Frequenz, 2, 61-84 (1948)
3 Ljunggren, S., Acta Acoustica, 3, 531-538 (1995)
4 Paul, H.S. J. Acoust. Soc. Am., 42 (2), 412-416 (1967)
5 Ljunggren, S. J., Sound Vib., 90 (4) 559-584 (1983)
6 Heckl, M., Acoustica, 49, 183-191 (1981)
7 Petersson, B.A.T. and Heckl, M., J. Sound Vib., 196 (3)
295-321 (1996)
8 Morse, P. and Ingard, U., Theoretical Acoustics, Prinston
University press, Prinston, New Jearcy, 1968
SESSIONS
Uncontrollable Modes in Double Wall Panels
Oliver E. Kaisera , A. Agung Juliusb , Stanislaw J. Pietrzkoc , Manfred Moraria
a ETH
– Swiss Federal Institute of Technology, Automatic Control Laboratory, CH-8092 Zurich, Switzerland
b Department of Applied Mathematics, University of Twente, Enschede, The Netherlands
c EMPA - Swiss Federal Laboratories for Materials Testing and Research, CH-8600 Dubendorf, Switzerland
Double-glazed windows have a poor transmission loss at low frequency. Since the passive means are more or less exhausted one could
think of using an active controller to increase the transmission loss. In the work presented here two speakers in the cavity between the
panes are used as actuators. A modal model is derived and validated with data from a laser scanner and measured transfer functions
on the structure. From an analysis of this model it is shown that for certain configurations of the double wall panel some modes
of the coupled system are uncontrollable and unobservable by speakers and microphones in the cavity, thus limiting the achievable
controller performance. This theoretical result is verified by feedforward control experiments on two types of double-glazed windows.
For the fully controllable window the transmission loss achieved by the active controller is about twice as large as for the window with
uncontrollable modes.
INTRODUCTION
One way to tackle the control of stochastic noise in
three dimensions is to reduce the sound transmission to
the zone of interest. In buildings, windows are often the
weak link in protecting the interior from outside noise.
In particular, double glazed windows have a poor sound
insulation at low frequency around the mass-air-mass resonance (double wall resonance). Since the passive means
for windows are exhausted, an active controller that increases the transmission loss in the low frequency range
is an attractive approach to reduce the noise level in buildings [2].
In the work presented here two speakers in the cavity
between the panes are used as actuators similar as in [4]
and [3]. The full experimental set-up and its dimensions
are given in Fig. 1.
Two double panels were investigated. The symmetric configuration consisted of a 6 mm-panel, a cavity of
84 mm, and a second panel of 6 mm. For the asymmetric
configuration the second panel was replaced by a 3.2 mmpane.
MODELING OF DOUBLE GLAZED
WINDOWS
For the modeling, a double panel structure can be divided into five subsystems, namely the excitation dynamics, the first panel, the cavity, the second panel, and the
radiation. Each subsystem is relatively well understood
[1] and can be modeled with a modal approach. The models of the subsystems can then be assembled to a model
of the double panel structure as suggested in [3]. In addition, we included models of the speakers and transformed
the model into state space form [2].
Validation
FIGURE 1. Experimental set-up. The two panes have a size
of 717 × 1091 mm and are mounted with wooden sashes in the
opening between the sending and the receiving room.
The model was validated with a laser vibrometer and
by measuring transfer functions. The model not only predicted the mode shapes correctly but also the eigenfrequencies (cf. [2] for details). In Fig. 2 the transfer function from a speaker in the corner to a microphone in the
same corner is shown. Apart from a difference in gain
which is due to an unknown speaker parameter, the prediction from the model and the measurement agree very
well.
To make sure that this agreement is not accidental the
validation was repeated for different double panel configurations, i.e. the thickness of the panels and the interpanel spacing was varied. In all cases the agreement between the model and the measurement was similar to the
one in Fig. 2.
SESSIONS
magnitude [dB]
20
2
0
1
−20
0
−40
2
10
−1
phase [deg]
0
−100
−2
0.6
−200
0.4
−300
0.2
−400
0
−500
0.4
0.8
0.6
1
FIGURE 4. Poorly controllable mode (symmetric config.)
frequency [Hz]
FIGURE 2. Transfer function from a speaker in the cavity to a
microphone in the cavity. Thin: model. Thick: measurement.
contr. symmetric config
0.2
2
10
contr. asymmetric config
0
10
Table 1. Performance comparison of the different controllers.
As quality measure the attenuation in dB around the mass-airmass resonance at 80 Hz is used.
0
−10
Symmetric
panel
Asymmetric
panel
Feedforward controller with
error mics in receiving room
8.5 dB
18 dB
Feedforward controller with
error mics in cavity
4 dB
7.5 dB
Controller
−20
−30
50
100
150
200
250
300
350
400
450
500
10
0
−10
−20
−30
50
100
150
200 250 300 350
frequency [Hz]
400
450
500
FIGURE 3. Controllability of the coupled modes for a speaker
in the corner (logarithmic scale with base 10). Some of the
modes of the symmetric configuration (top) are poorly controllable, whereas all modes of the asymmetric configuration are
well controllable.
UNCONTROLLABLE MODES IN
DOUBLE PANEL STRUCTURES
For the optimization of the sensor and actuator locations the controllability and observability grammians calculated from the validated model were used. It was then
noticed that some modes of the symmetric configuration
have poor controllability for all actuator locations.
An analysis of the validated model revealed that the
poorly controllable modes correspond to modes where
the two panels move in-phase as in Fig. 4. Such modes
do not occur in the asymmetric configuration. There, all
the modes are well controllable.
EXPERIMENTAL RESULTS
For both the symmetric and the asymmetric configuration feedforward controllers were implemented with
three different actuator locations. These locations were
the best three locations found in the actuator optimization [2]. While the performance varied only very little
for the different actuator locations a substantial difference between the symmetric and the asymmetric config-
uration was noticed. Due to the uncontrollable modes the
controller is substantially less efficient for the symmetric
panel than for the asymmetric panel (Tab. 1).
CONCLUSIONS
In [4] and [3] it is pointed out that the positioning of
the actuators in the cavity between the panels plays an
important role in order to achieve good performance. We
showed, that in addition the performance of an active controller for a double glazed window can be substantially
improved if the structure is designed for control. For our
experimental set-up the performance at the mass-air-mass
resonance could be doubled if the panel was designed to
have well controllable modes only.
REFERENCES
1. C. R. Fuller, S. J. Elliott, and P. A. Nelson. Active Control of
Vibration. Academic Press Limited, London, 1996.
2. O. E. Kaiser. Active Control of Sound Transmission through
a Double Wall Structure. PhD thesis, Swiss Federal Institute
of Technology Zurich, 2001.
3. J. Pan and C. Bao. Analytical study of different approaches for active control of sound transmission through
double walls. Journal of the Acoustical Society of America,
103(2):1916–1922, Apr. 1998.
4. P. Sas, C. Bao, F. Augusztinovicz, and W. Desmet. Active
control of sound transmission through a double panel partition. Journal of Sound and Vibration, 180(4):609–625, 1995.
SESSIONS
Testing of the Tyre Vibration States by Speckle
Interferometry
J. Slabeyciusa, P. Koštiala, M. Držíkb and M. Rypákc
a
Faculty of Industrial Technologies, Univ. of Trenčín, SK-020 01 Púchov, Slovakia
Institute of Construction and Architecture, Slovak Acad. Sci., SK-842 20 Bratislava, Slovakia
c
Rubber Research Institute, MATADOR, a.s., SK-020 32 Púchov, Slovakia
b
The frequencies of the basic vibration modes of pneumatic tyres were measured by the method of electronic speckle correlation
interferometry. A powerful loudspeaker fed on continually tuned harmonic generator was used for non-contact excitation of the
tyre. The setup of electronic speckle correlation interferometer consisting of green light Nd:YAG laser, optical elements and
CCD camera has been adjusted. The interference pattern of vibrating tyre surface was obtained by PC processing both the
original speckle field image and the image of deformed speckle field caused by vibrations of tyre surface. The obtained results
are discussed.
INTRODUCTION
An automobile tyre should satisfy a number of
functional requirements. It must work reliably in
a large region of dynamic conditions – from static
loadings to high frequency exciting of mechanical
vibrations. The pneumatic tyre today is a highly
sophisticated engineering structure, which viscoelastic,
anisotropic and nonhomogeneous properties have to be
taken into account. Consequently, the reliable
theoretical and computer analysis of mechanical
properties of tyre is extremely difficult [1]. The
performance requirements of the system cannot be
adequately designed without knowledge of its real
dynamic characteristics, where the experimental
measurements are necessary. For the measurement of
the shape changes of tyre caused by its vibrations, the
optical methods are very convenient, because there are
non contact and their sensitivity is very high [2].
average method, double exposure technique and realtime visualisation method.
The
method
analogous
to
holographic
interferometry based on the properties of laser speckles
and digital processing of image is known as electronic
(digital) speckle pattern interferometry [2,5]. In this
case, the electronic images of both wave field
(corresponding to unloaded and loaded object) are
compared in PC by appropriate software.
The electronic speckle pattern interferometry tyre
testing system, adjusted in our laboratory, consists of
Nd:YAG diode pumped laser (50 mW, 532 nm),
optical elements, CCD-camera, PC with framegrabber
and image processing software. The equipment for tyre
clamping and exciting of vibration consists of steady
holder, powerful loudspeaker and continually tuned
harmonic generator with amplifier. A block scheme of
the optical setup is shown in Figure 1.
PC
CCD
Mirror
METHOD
Mirror
Mirror
LASER
Holographic interferometry is an optical method
based on the interference of two optical wave fields –
first one is scattered by object in primary state, the
second one by object in load state [3,4]. Naturally,
these fields cannot really exist in the same time. Hence,
at least one of them is reconstructed by holographic
way. The interference pattern allow to compute the
shape deviation of object in every point. Therefore, the
interferogram stores the full information about the
deviation of the object surface. The holographic
interferometry includes three basic techniques: time-
Diffuser
Lens
Mirror
Test Object
FIGURE 1. The block scheme of the optical setup for
the holographic speckle correlation interferometry.
SESSIONS
RESULTS AND DISCUSSION
The testing object was the tyre MP-12, 175/70
R 13. The tyre fixed on the holder was illuminated by
a broadened laser beam. A part of spreaded light beam
was separated by using of mirror and directed through
another mirror to the small ground screen. Passing
through the ground screen the light creates the speckle
field directed by the semitransparent mirror into the
objective lens of CCD camera. Through this
semitransparent mirror the image of the object is also
passing and is projected onto the CCD matrix where
the interference of both the speckle fields happen.
The next image, figure 3, shows the fifth order
mode with frequency of 216 Hz and, in the end, figure
4 corresponding to mode of sixth order with frequency
282 Hz. The interference pattern of n-th order
vibrational mode consist of 2n separated maxima,
symmetrically arranged on the tyre perimeter.
FIGURE 4. Interference pattern corresponding to
radial vibrational mode of sixth order at frequency
282 Hz.
FIGURE 2. Interference pattern corresponding to radial
vibrational mode of fourth order at frequency 155 Hz.
Any movements of object surface in the line of
sight will create the changes in optical paths and thus
gives rise to the pattern of interferogram. The tyre was
excited by loudspeaker fed on continually tuned
harmonic generator. The images corresponding to the
motionless tyre and the vibrating one are electronically
subtracted and the resulting image with interference
pattern is created.
As the illustrative examples we present interference
patterns
for three vibrational modes of tyre
investigated: Figure 2 shows a fourth-order radial tyre
mode, which frequency was 155 Hz.
The contrast of images, obtained by electronic
interferometry is not as good as that obtained by
classical holographic interferometry. On the other side,
the digital electronic interferometry allow avoid the
recording media. The process of measurement is
convenient and relatively fast.
From the irregularities of interference pattern we
can judge on non-uniformity of tyre tested. The
frequencies of natural vibration modes differ from
those of tyre in contact with road. Nevertheless, the
reducible experiment, in which the tyre is steady in the
massive holder and all the perimeter of tyre is free, can
bring useful information about tyre quality.
REFERENCES
1. Gardner, I., and Theves, M., Tire Sci. & Technol., 17,
86-99 (1989)
2. Kreis, Th.: Holographic Interferometry. Acad. Verlag,
Berlin, 1996
3. Potts, G.R., Tire Sci. & Technol., 1, 255-266 (1973)
4. Keprt, J. and Bartoněk, L., Holographic testing of tyres,
in Proc. 35th Int. Conf. on Experimental Stress Analysis.
Olomouc 1997, p.162-167
5. Urgela, S., Slabeycius, J., Koštial, P., The Application
of Holographic Methods for the Non-destructive
Testing of Tyres, in Proc. Slovak Rubber Conf. ’99,
Púchov, 1999, p.11-19.
FIGURE 3. Interference pattern corresponding to radial
vibrational mode of fifth order at frequency 216 Hz.
SESSIONS