American Institute of Aeronautics and Astronautics 1
Acoustic Characterisation of Perforates using Non-linear System Identification Techniques
Hans Bodén
1MWL, Aeronautical and Vehicle Engineering, KTH, S-100 44 Stockholm, Sweden
This paper discusses the use of nonlinear system identification techniques for determination of linear acoustic impedance and non-linear acoustic properties of perforates and other facing sheets used in aircraft engine liners.
I. Introduction
N Ref. 1 experimental techniques for determining acoustic impedance and flow resistance for perforates under non-linear conditions were discussed. Experiments were made using both pure tone and random excitation and the relevant parameters controlling the non-linearity were discussed. In Ref. 2 a study of harmonic interaction effects using two-tone excitations was made and in Ref. 3 the study was extended to multi-tone excitation for different types of perforates. In the linear case the impedance is independent of the sound field but when the sound pressure level is high the perforate impedance will be dependent on the acoustic particle velocity in the holes. For pure tone excitation it is obvious that the impedance will be controlled by the acoustic particle velocity at that frequency. If the acoustic excitation is random or periodic with multiple harmonics the impedance at a certain frequency may depend on the particle velocity at other frequencies. The results of Ref. 1-3 show that the total rms-value of the particle velocity in the holes seems the relevant parameter controlling the non-linearity. In this paper the experimental techniques discussed in Ref. 1 to 3 are further developed for studying this problem. A study is made of using non- linear system identification techniques for this purpose.
Many investigations of nonlinear effects occurring when high amplitude sound waves are incident on perforated plates or orifice plates have been published, see e.g., Ref. 4-9. In many of the early works a standing wave tube with single frequency excitation was used. Ingård and Ising
6in their classical study on nonlinearity of orifices used a differential pressure measurements and a hotwire probe in the hole, still with pure tone excitation. Maa
10-12, has studied linear and nonlinear behaviour of so-called micro-perforates. It is generally agreed that the non-linear losses are associated with vortex shedding at the outlet side of the orifice or perforate openings
13-16. Tam et al
14-16, have studied slit resonators using direct numerical simulation in
14-15.
Non-linear system identification techniques have not previously been applied to the perforate impedance problem. The idea of treating, a squared, nonlinear path as a separate non-linear input after which system identification is performed as for a linear two input one output system was first introduced by and Bendat and Piersol
17-18. The general methodology, for arbitrary nonlinear systems, as used in this paper was first published by Rice and Fitzpatrick
19. The techniques have later been summarized by Bendat
20. An example of a more recent work applying a modified version of the technique to mechanical system is Ref. 21.
1
Associate Professor, MWL, KTH Aeronautical and Vehicle Engineering, SE 10044 Stockholm, Sweden.
I
13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference) AIAA 2007-3530
Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
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II. Models for Perforate Impedance
In Ref. 7-8 a semi-empirical model for perforate resistance and reactance was presented. It summarises the previously published models and adds a few improvements to the resistance and reactance end corrections and the resistive and reactive terms associated with nonlinearities and grazing flow. Removing the terms associated with grazing and through flow which are not relevant for the present study gives
( ) ( ) ( )
n D
re D
c v C kd
kd f J
F F
t C jk
2 1 2 1
1 1
Re
2 22 1
int
⎟⎟ ⋅
⎠
⎜⎜ ⎞
⎝ + ⎛ −
⎥⎦ ⎤
⎢⎣ ⎡ −
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛ ⎥
⎦
⎢ ⎤
⎣
⎡ +
= ′
σ σ σ
μ δ μ
θ σ , (1)
( ) 0 . ( ) 5 1 2 1 3
Im
2 22 int
n D
D
v c f C
F d F
t C
jk ⎟⎟ ⎠ ⋅ ⋅
⎜⎜ ⎞
⎝
− ⎛ −
⎟⎟ ⎠
⎜⎜ ⎞
⎝
⎛ ⎥
⎦
⎢ ⎤
⎣
⎡ +
= ′
Χ σ
σ μ
μ
σ , (2)
where θ is the normalized resistance and Χ is the normalized reactance, k is the wave number, σ is the porosity (percentage open area), C
Dis the discharge coefficient, t is the plate thickness, μ is the adiabatic dynamic viscosity,
μ
μ = 2 , 179 is the dynamic viscosity close to a conducting wall, ν = μ/ρ is the kinematic viscosity, J is the Bessel function, d is the hole diameter, c is the speed of sound, v
nis the peak value of the acoustic particle velocity incident on the sample. The rest of the parameters are defined as
ν ω
K = − j , (3)
( ) ( ) ( 2 )
2 1 4
0 1
Kd J Kd
Kd F J
− ⋅
μ = , (4)
3 2
16000 200
2 .
0 d d d
re
= + +
δ , (5)
3 int
= 1 − 1 . 47 σ + 0 . 47 σ
f . (6)
The third term in (1) and the second term in (2) represent the nonlinear contributions proportional to the acoustic particle velocity. The second term in (1) is usually small compared to the first linear term and can be neglected.
These equations will be used for comparison with the experimental results.
Comparison will also be made with the model suggested by Maa
9-11for micro-perforates
c v t
d k k
d t c
p n
p 2
2
2
4
32 2 / 32 1
σ σ
θ ν +
⎥ ⎥
⎦
⎤
⎢ ⎢
⎣
⎡ + +
= , (7)
⎥ ⎥
⎥
⎦
⎤
⎢ ⎢
⎢
⎣
⎡
⎟⎟ ⎠
⎞
⎜⎜ ⎝
⎛ +
⎟ +
⎟
⎠
⎞
⎜ ⎜
⎝
⎛ + +
= Χ
− −1
2 12
2
1 85 . 2 0
9
1 c
v t
k d c
t
p nσ σ
ω , (8)
where
j k
pdK
= −
2 . (9)
It can be seen from (1) and (7) that the perforate resistance is proportional to the peak particle velocity in the holes
v
n. This is because it can be expected that the main non-linearity will be caused by the flow constriction caused by
American Institute of Aeronautics and Astronautics 3
the facing sheet. For a single constriction under stationary flow conditions the pressure difference over the constriction is
( 2 ) ( ) ( ) )
( t
0v t v t
p = ρ
Δ . (10)
It is shown in Ref. 20 that, under the assumption that v(t) follows a zero mean Gaussian distribution, the third- order polynomial least-squares approximation to ( ρ
02 ) v ( t ) v ( t ) is
( ) ( )
[ 2 ( ) 2 3 ( ) ]
) 2
( t
0v t v
3t
p = ρ σ
vπ + π σ
vΔ , (11)
where σ
vis the standard deviation of v(t). Taking the Fourier transform of Eq. (11) gives
( ) ( )
[ 2 ( ) 2 3 ( ) ]
) 2
( f
0V f V
3f
P = ρ σ
vπ + π σ
vΔ , (12)
where Δ P(f), V(f) and V
3(f) are the Fourier transforms of Δ p(t), v(t) and v
3(t). The original “square-law system with sign” ( ρ
02 ) v ( t ) v ( t ) could therefore to the third order be replaced by a linear system in parallel with a cubic system.
III. Non-linear system identification
The main idea behind the non-linear system identification techniques described in References 17-21 is that the non- linear path can be treated as a separate input followed by a linear system. This requires that we know the type of non-linearity. Another possibility, as described in Ref. 19, is that we test different powers of the input x
i(t ) and check which gives the best fit to our experimental data. In the present case we will assume that the type of non- linearity is known from the models described in the previous section. If the particle velocity (v(t)) is considered as the input and the acoustic pressure (p(t)) as the output from the system, the non-linear model can be described by Figure 1. The frequency response function Z(f) is the linear impedance while A(f) corresponds to the non-linear terms in (1), (2), (7) and (8) if x ( ) ( ) ( ) v ( ) t = v t v t and n(t) is uncorrelated noise at the output.
Figure 1: Non-linear model.
Σ Z(f)
A(f) x(v(t))
v(t)
p(t)
+
n(t)
American Institute of Aeronautics and Astronautics 4
The model in Fig. 1 is now replaced by the model shown in Figure 2.
Figure 2: Revised non-linear model.
The non-linear input is described by X(V(f)) which denotes the Fourier transform of x(t). It should be noted that when calculating the nonlinear function x(t) an anti-aliasing filter should be applied before performing the Fourier transform and calculation of auto-spectra and cross-spectra. The linear systems Z
0(f) and A(f) are now determined using V(f) and X(f) as inputs to two separate linear systems. It is possible to make the inputs of Fig. 2 independent of each other by removing all parts of X(f) which are correlated with V(f) giving the system shown in Fig. 3.
Figure 3: Revised non-linear model with uncorrelated inputs.
From measurement of v(t) and p(t) followed by the calculation of non-linear input data x(t) the following auto and cross spectra can be calculated: G
vv(f), G
pp(f), G
xx(f), G
vp(f), G
pv(f), G
xp(f). G
px(f), G
xu(f) and G
ux(f). The coherence function between v(t) and x(t) is
) ( ) (
) ( ) ) (
2
(
f G f G
f G f f G
vv xx
xv vx
vx
⋅
= ⋅
γ , (13)
and the uncorrelated input Y(f) is calculated from
( 1 ( ) )
) ( )
( f G f
2f
G
yy=
xx⋅ − γ
vx. (14)
The cross spectrum between Y(f) and P(f) is calculated from
) (
) ( ) ) (
( )
( G f
f G f f G
G f G
vv vp xv
xp yp
− ⋅
= . (15)
Σ Z
0(f)
A(f) V(f)
P(f) X(V(f))
Z
0(f)
A(f) V(f)
P(f) Y(f)
+
N(f)
+
N(f)
American Institute of Aeronautics and Astronautics 5
The system data is now estimated from
) (
) ) (
( G f
f f G
Z
vv vp
o
= , (16)
) (
) ) (
( G f
f f G
A
yy
=
yp. (17)
The original model linear impedance is then obtained from
( ) A ( ) f
f G
f f G
Z f Z
uu ux
o
( )
) ) (
( −
= . (18)
The coherence functions between V(f) and P(f) and Y(f) and P(f) can be calculated from
) ( ) (
) ( ) ) (
2
(
f G f G
f G f f G
vv pp
pv vp
vp
⋅
= ⋅
γ , (19)
) ( ) (
) ( ) ) (
2
(
f G f G
f G f f G
yy pp
py yp
yp
⋅
= ⋅
γ . (20)
They will tell us how much inputs V(f) and Y(f) contribute to the output signal. If the sum of these two coherence functions is close to unity the chosen model gives a good fit to the experimental data. In Ref. 21 it is suggested that it is not necessary to do the signal conditioning described in (13) – (15) in order to perform the non-linear system identification. Instead an input cross-spectral matrix is formed which in our two input – one output model is
( ) ( ) ( ) ( ) ⎥ ⎦ ⎤
⎢ ⎣
= ⎡
f G f G
f G f f G
G
xx vx
xv vv
II
( ) , (21)
and an input – output cross-spectral matrix is also formed
( ) ( )
[
vp xp f]
Ip
f G f G
G ( ) = . (22)
The system data is then calculated from
( ) ( )
[ Z f A f ] G ( ) f G ( ) f f
H ( ) = =
Ip⋅
II−1. (23)
This amounts to performing the same calculations leading to (17) and (18). It is however claimed in Ref. 21 that
this formalism is better if we want to extend the analysis to problems with multiple non-linear inputs and multiple
outputs. A multiple coherence function is also defined as
American Institute of Aeronautics and Astronautics 6
) (
) ( ) ) (
(
1 2
f G
G f G f f G
pp
H Ip II
Ip m
⋅
= ⋅
−
γ , (24)
where superscript H stands for a transposed and complex conjugated matrix. The multiple coherence according to (24) turns out to be the same as the sum of the component coherence functions given in (19) and (20).
IV Experimental technique
An impedance tube as shown in Fig. 4 was used for the experimental tests. The sample was placed in a holder at the end of the duct and measurements were made with and without the sample. The measurements performed are very similar to those made in Refs. 1-3 but in this case time domain data was collected using random excitation. The measured pressures were Fourier transformed and the pressure and particle velocity was calculated at the sample cross section assuming linear plane wave propagation in the duct. The particle velocity and pressure as a function of time was then calculated using an inverse Fourier transform. This data could then be used as input to the non-linear system identification.
Pos. 1 Pos. 2
Mic b Mic a
x
s l
Figure 4. Sketch and photo of impedance tube.
Another possibility tested was using a Microflown p-u probe to simultaneously measure pressure and particle velocity at the same cross section instead of calculating the from the two pressure measurements.
Tests were made for a perforate sample with 3.2 % porosity, hole diameter 2 mm and hole thickness 1.5 mm.
American Institute of Aeronautics and Astronautics 7
V. Experimental results and discussion
Figure 5 shows the real part of normalised impedance (Z
0(f)) estimated from (16) and the linear part estimated from (18), for different levels of excitation described by the sound pressure level and r.m.s.-value for the particle velocity at the sample. It can be clearly seen that we are able to remove the non-linear part of the impedance and to get a reasonably got agreement between the linear impedance estimates obtained from (18) for different levels of excitation.
50 100 150 200 250 300 350 400 450
-0.1 0 0.1 0.2 0.3 0.4
f [Hz]
Real(Z)
a) Linear part
50 100 150 200 250 300 350 400 450
-0.1 0 0.1 0.2 0.3 0.4
f [Hz]
Real(Z)
b) Including both linear and non-linear parts
Figure 5: Real part of normalized impedance: black 123 dB, 0.092 m/s; blue 113 dB, 0.032 m/s; red 104 dB, 0.012 m/s, green 93.2 dB 0.0043 m/s.
In the results presented in Fig. 5 it was assumed that the non-linear model was x ( ) ( ) ( ) v ( ) t = v t v t . If we instead assume that x ( ) ( ) v ( ) t = v t
3we get the result shown in Fig. 6. We can see that we are not getting as good an estimate of the linear part of the impedance in this case.
50 100 150 200 250 300 350 400 450
-0.1 0 0.1 0.2 0.3 0.4
f [Hz]
Real(Z)
Figure 6: Real part of normalized impedance for 123 dB sound pressure level and 0.092 m/s particle velocity at the sample: black – linear estimate using x ( ) ( ) ( ) v ( ) t = v t v t , red – linear estimate using
( ) ( ) ( ) v t v t
3x = , black dashed – linear and non-linear parts.
American Institute of Aeronautics and Astronautics 8
Figure 7 shows the coherence function estimates obtained using (19), (20) and (24). It can be seen that the linear path gives the largest contribution to the output in this case indicating that non-linearity is quite weak. It can also be seen the total coherence according to (24) is close to unity showing that the model fits the experimental data well.
50 100 150 200 250 300 350 400 450
0 0.2 0.4 0.6 0.8 1
f [Hz]
Coherence
Figure 7: Coherence function estimates for 123 dB sound pressure level and 0.092 m/s particle velocity at the sample: black – linear path (19), blue dashed – non-linear path (20), red dashed – total (24).
It is also of interest to compare the experimental results with theory according to (1), (2) and (7), (8). Figure 8 shows real part of the measured linear normalized impedance compared to the linear part from (1) and (7). It can be seen that the agreement is reasonably good.
.
50 100 150 200 250 300 350 400 450
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
f [Hz]
Real(Z)
Figure 8: Real part of measured normalized impedance: black 123 dB, 0.092 m/s; blue 113 dB, 0.032 m/s; red
104 dB, 0.012 m/s, green 93.2 dB 0.0043 m/s; black dashed theory according to (1); black dashed-dotted
theory according to (7).
American Institute of Aeronautics and Astronautics 9
Figure 9 shows a comparison between the measured non-linear system frequency response function A(f) and the corresponding non-linear terms from (1), (2) and (7). It can be seen that the agreement is fairly good.
50 100 150 200 250 300 350 400 450
0 0.5 1 1.5 2 2.5 3 3.5 4
f [Hz]
Real(A)
a) Real part.
50 100 150 200 250 300 350 400 450
-2 -1.5 -1 -0.5 0 0.5 1
f [Hz]
Imag(A)
b) Imaginary part
Figure 9: A(f) for 123 dB sound pressure level and 0.092 m/s particle velocity; full line - measurement, dashed line - theory according to (1), (2); dashed-dotted line - theory according to (7).
VI. Conclusions
Non-linear system identification techniques have been tested for obtaining linear impedance and non-linear acoustic
properties for perforates. It has been shown that the technique works and gives results in reasonable agreement with
theory. The technique described here provides an easy and quick technique or simultaneous measurement of linear
and non-linear acoustic properties using random excitation.
American Institute of Aeronautics and Astronautics 10
References
1
Bodén, H., “Determination of Flow Resistance from Acoustic In-Duct Measurements”, AIAA Paper 2004- 2841.
2
Bodén, H., “Experimental investigation of harmonic interaction effects for perforates”, AIAA Paper 2005- 2896.
3
Bodén, H., Ying, G. and Tözün, H.B., “Experimental investigation of nonlinear acoustic properties for perforates”, AIAA Paper 2005- 2896.
4
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5
Ingård, U. and Labate, S., "Acoustic circulation effects and the nonlinear impedance of orifices", Journal of the Acoustical Society of America, Vol. 22, 1950, pp. 211-219.
6
Ingård, U. and Ising, H., "Acoustic nonlinearity of an orifice", Journal of the Acoustical Society of America, Vol. 42, 1967, pp. 6-17.
7
Melling, T.H., “The acoustic impedance of perforates at medium and high sound pressure levels”, Journal of Sound and Vibration, Vol. 29, No. 1, 1973, pp. 1-65.
8
Elnady, T. and Bodén, H., “On semi-empirical liner impedance modeling with grazing flow”, AIAA Paper, AIAA 2003- 3304, May 2003.
9
Elnady, T., “Modelling and characterization of perforates in lined ducts and mufflers (Paper III)”, PhD Thesis, Department of Aeronautical and Vehicle Engineering, KTH, Stockholm, Sweden, 2004.
10
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11
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12
Maa, D.-Y., “Theory and design of microperforated-panel sound-absorbing construction” Sci. Sin. XVIII, 55-71 1975.
13
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15
Tam, C.K.W., Kurbatski, K.A., Ahuja K.K.and Gaeta Jr., R.J. “A Numerical and Experimental Investigation of the Dissipation Mechanisms of Resonant Acoustic Liners”, Journal of Sound and Vibration, Vol. 245, No. 3, 2001, pp. 545-557.
16
Tam, C.K.W., Ju, H., Jones, M.G, Watson, W.R. and Parrott, T.L. “A Computational and Experimental Study of Slit Resonators”, Journal of Sound and Vibration, Vol. 284, No. 3-5, 2005, pp. 947-984.
17
Bendat, J.S. and Piersol, A.G. “Spectral analysis of non-linear systems involving square-law operations”, Journal of Sound and Vibration, Vol. 81, pp. 199-213.
18
Bendat, J.S. and Piersol, A.G. “Decomposition of wave forces into linear and non-linear components”, Journal of Sound and Vibration, Vol. 106, pp. 391-408.
19
Rice, H.J. and Fitzpatrick, J.A. “A generalized technique for spectral analysis of non-linear systems”, Mechanical Systems and Signal Processing, Vol. 2, No. 2, pp. 195-207.
20
Magnevall, M. and Ahlin, K.A. “On nonlinear parameter estimation”, Proceedings of ISMA, 2006, Paper 505.
21
