Proceedings of the Institute of Acoustics

 

 

Frequency response function statistics of diffuse systems

 

Cédric Van Hoorickx¹, KU Leuven, Leuven, Belgium

Edwin P. B. Reynders², KU Leuven, Leuven, Belgium

 

ABSTRACT

 

Vibro-acoustic analysis at high frequencies is challenging due to a large sensitivity to spatial variations and short wavelengths, rendering deterministic methods expensive. At these frequencies, the wave field is usually assumed diffuse. A realization of a diffuse field can be obtained considering that the mode shapes of the system are Gaussian random fields, and that its squared eigenfrequency spacings distribution conforms to the Gaussian orthogonal ensemble (GOE) eigenvalue spacings distribution. These diffuse field properties were for example used to extend statistical energy analysis (SEA) towards energetic variance prediction. However, energetic methods such as SEA lack the propagation of phase information, which means that, e.g., time-domain reconstruction is not possible. In this contribution, instead of total energies, expressions for the ensemble average and the cross-frequency covariance of frequency response functions are presented. These expressions are obtained from the generalized definition of a diffuse field, treating the eigenfrequencies of a diffuse subsystem as a collection of points randomly located on the frequency axis, making the analysis amenable to random point process theory. These expressions are numerically validated by comparison with computationally costly detailed models where random wave scatterers are modelled explicitly.

 

1. INTRODUCTION

 

Diffuse or reverberant field models are widely employed for vibro-acoustic analysis at high frequencies because of their computational efficiency and because they inherently account for uncertainty related to small spatial variations in geometry, material properties, or boundary conditions, which have a wave scattering effect. Conventionally, a diffuse field is defined to be a random field, composed of a large number of statistically independent plane waves, the spatial phase of which is uniformly distributed and independent from the amplitude [1]. Power flows between diffuse fields can be efficiently analyzed with, e.g., statistical energy analysis (SEA) [2].

 

However, in statistical energy-based approaches such as SEA, the response (displacement or pressure) fields are not modelled but only the related energies, which means that, e.g., time-domain reconstruction, which requires phase information, is not possible. A new method of analysis, termed the Gaussian Orthogonal Ensemble-Monte Carlo (GOE-MC) method, that overcomes this limitation was recently developed [3, 4]. It makes use of a generalized definition of a diffuse field, relying on the fact that the natural frequencies (or equivalently, the undamped eigenvalues) of a diffuse component with uncertain local wave scattering properties converge, as the frequency increases, to a specific joint probability distribution function after appropriate scaling: that of the eigenvalues of a Gaussian Orthogonal Ensemble (GOE) matrix [5] from random matrix theory. The mode shapes are independent Gaussian random wave fields. In the GOE-MC method, samples of the natural frequencies and mode shapes of the decoupled diffuse subsystems are directly obtained with a Gaussian random number generator, as they relate to a GOE matrix and a Gaussian random field, respectively.

 

Instead of relying on the generation of MC samples, the natural frequencies can be considered to form a sequence of random points on the frequency axis, thus making the problem amenable to analysis by random point process theory. The theory is phrased in terms of a set of functions g_s that describe the statistics of the placement of the random points. These are called correlation functions [6] or cumulant functions [7]. The latter terminology will be adopted here. With this theory, analytical expressions are obtained in this paper for frequency response function (FRF) statistics. These expressions do not assume a large distance between source and receiver and a large modal density, as previous derivations of FRF statistics did [8]. They can furthermore be used for time domain analysis or can be applied for the response analysis of built-up systems by using a generalized diffuse field reciprocity relation.

 

The outline of this paper is as follows. First, the frequency response function is introduced and the application of the random point process theory is described. This theory is then used in the following two sections to obtain expressions for the ensemble average and the frequency covariance of the FRF. The expressions are finally validated numerically by comparing the results with a GOE-MC model and a detailed model in which the wave scattering features are modelled explicitly.

 

2. RANDOM POINT PROCESS THEORY

 

The frequency response function (FRF) H(x, x′, ω) can be expressed as the following integral:

 

 

where c(x, x′, ω_n) = ϕ_n(x)ϕ_n(x′) with ϕ_n(x) the nth real mode shape component at x, ω_n the corresponding natural frequency, and:

 

 

with  where the approximation is valid for η → 0.

 

Stratonovich [6] has demonstrated that the cumulants of the process ξ(x, x′, ω′) can be written in terms of the cumulant functions g_s, which describe the statistics of the natural frequencies. The first function g₁ equals the modal density n and the first cumulant of ξ, i.e., its mean value, has the form:

 

 

Similarly, the second cumulant, i.e. the covariance function, is given by:

 

 

3. ENSEMBLE AVERAGE OF THE FRF

 

The ensemble average of the FRF follows from Equation 1 and Equation 4:

 

 

Substitution of Equation 2 yields:

 

 

where use was made of partial fraction decomposition and the fact that E [c(x, x′, ω′)] and n(ω′) are even functions. This integral can be elaborated using the Sokhotski–Plemelj theorem for an analytic function f :

 

 

where denotes the Cauchy principal value. As ω/a ≈ ω − i|ω|η/2, the ensemble average of the FRF becomes for η → 0+ and therefore a → 1:

 

 

Note that if we would assume that n(ω) and E [c(x, x′, ω)] were constant, only the imaginary part would be non-zero, yielding a result which was previously derived in [9].

 

The ensemble average of the FRF can be interpreted as follows. The diffuse field response can be divided into a direct part (which equals the free-space response) and a reverberant part. As the ensemble average of this reverberant part equals zero, the average of the diffuse field response equals the free-space response. The average of the FRF therefore equals the free-space Green’s function:

 

 

To conclude, the imaginary and real part of the free-space Green’s function equal

 

 

These expressions yield a relation between the real and imaginary part of the Green’s function:

 

 

which is nothing else than the Kramers-Kronig relation, connecting the real and imaginary parts of any complex function that is causal.

 

Equation 11 relates the imaginary part of the free-space Green’s function and the spatial mode correlation function. The latter can therefore be obtained as follows:

 

 

On the assumption that the mode shapes are Gaussian, we furthermore have that:

 

 

As the mode shapes are scaled to unit generalized mass, this becomes:

 

 

where E [c(x, x′, ω)] can be obtained from Equation 13 if points x and x′ are far apart. In a diffuse field, E [c(x, x′, ω)] is small, and E[c²(x, x′, ω)] ≈ 1/M ². At the excitation point, E [c(x, x, ω_n)] = E[ (x)] = 1/M and E [c²(x, x′, ω)] = 3/M ². This means that E[c²(x, x′, ω)] is approximately constant in the diffuse field and at the excitation point.

 

4. COVARIANCE OF THE FRF

 

4.1. General expressions

 

The covariance Cov (H(x, x′, ω₁), H(x, x′, ω₂)) equals:

 

 

with

 

 

In the diffuse field, away from the excitation point, E [(x, x′, ω′₁)] becomes small, which results in I₁ ≫ I₂. Close to the excitation point, E[(x, x′, ω′₁)] increases, and I₂ will also contribute to the covariance.

 

By using the same steps as in the previous section for the derivation of the ensemble average, i.e., substitution of Equation 2, partial fraction decomposition, and the use of the Sokhotski-Plemelj theorem (Equation 8), the following expression is obtained:

 

 

where the notation

 

 

is introduced with f(x) any continuous function and (f(x)) (ω) its Hilbert transform evaluated at x = ω. In Equation 20, A_12^∗ and A_2^∗1 are furthermore given by

 

 

Note furthermore that the dependency of the correlation function c with respect to the position of the excitation and receiver location x and x′ has been omitted and is from here on implicitly assumed. The integrals in the expression for I₂ (Equation 19) can be extended to negative values of ω^′₁ and ω^′₂ by introducing the Heaviside step function θ(x):

 

 

where the substitution ξ = ω′₂ − ω′₁has been used. Applying partial fraction decomposition and the Sokhotski-Plemelj theorem (Equation 8) twice, finally yields the following expression for I₂ if |ω₁|η₁ ≥ |ω₂|η₂:

 

 

In the other case, |ω₁|η₁ < |ω₂|η₂, the sign of the first and third term should be altered.

 

The second correlation function g₂(χ) can be obtained from the following inverse Fouriertransform:

 

 

A closed-form expression for the Fourier transform of the correlation function G₂(ζ) exists [10]:

 

 

and therefore the integral in Equation 25 can be solved using numerical integration or a Fast Fourier Transform (FFT) algorithm.

 

4.2. Constant correlation function and modal density

 

Assume now that the modal density n and the correlation function E [c_n] are approximately constant in the frequency range of interest. Equation 20 then becomes:

 

 

For a constant correlation function, Equation 24 equals (approximately, as θ(x) is not constant over the entire frequency domain):

 

 

with

 

 

where use is made of the definition of the Fourier transform of g₂(χ). The inner integral is the Hilbert transform , for which an analytical expression exists, resulting in:

 

 

Substitution of Equation 26 yields:

 

 

where

 

 

The integral can be evaluated using integration by parts [11], yielding:

 

 

with

 

 

To conclude, I₂ becomes:

 

 

and the covariance is obtained by summing Equation 27 and Equation 35.

 

For the special case where ω₁ = ω₂ > 0, we have that m₁₂|a₁|² = nω₁η₁,

 

 

and summing I₁ and I₂ yields the variance expression, which corresponds to the one obtained in [9].

 

5. NUMERICAL VALIDATION

 

Consider a rectangular aluminium plate subject to random wave scattering, as illustrated in Figure 1. The plate has dimensions L_x × L_y × t = 2.1 m × 1.9 m × 1.25 mm. The material properties are: a Young’s modulus of E = 71 GPa, a mass density of ρ = 2700 kg/m³, a Poisson’s ratio of ν = 0.3, and a damping loss factor of η = 0.01. The plate is subjected to wave scattering caused by 50 point masses, randomly distributed on the plate; each point mass equals 0.5 % of the total mass of the plate. Three models are compared to model the displacement in the plate: (1) a detailed MC model, in which the point masses are modelled explicitly, (2) a GOE-MC model [4], in which realizations of diffuse eigenfrequencies and modeshapes are generated in a MC framework, and (3) the statistical model presented in this paper. A unit force is applied at (x, y) = (0.84 m, 0.76 m), corresponding to location A in Figure 1. The response is studied at (x, y) = (1.12 m, 0.98 m) and (x, y) = (1.46 m, 0.43 m), corresponding to locations B and C in figure 1.

 

 

Figure 1: Aluminium plate with a random mass distribution excited by a point load.

 

Figure 2 displays the average of the FRF E [H]. A good agreement is found between the three models. The oscillations in the results of the detailed MC simulation are due to coherent reflections at the boundaries of the plate. The results of the statistical model equal the free-space Green’s function. Figure 3 displays the covariance of the FRF Cov (H(f )H(200 Hz). A good agreement is obtained between the three methods. Around the reference frequency 200 Hz, the correlation coefficient equals 1 and the covariance equals the variance. It can be noted that the FRF covariance decreases rapidly with increasing frequency distance to the reference frequency.

 

 

Figure 2: Real and imaginary part of the average of the FRF EH(f ) at locations (a) B and (b) C. Results for the detailed MC are plotted in blue, for the GOE-MC in red, and for the statistical model in black.

 

6. CONCLUSIONS

 

Random point process theory has been applied to obtain the ensemble average and the cross frequency covariance of the frequency response function (FRF) of diffuse systems. For the derivation, a generalized definition of a diffuse field is used, where the statistics of the local spacings between the eigenvalues are related to those of a Gaussian orthogonal ensemble (GOE) matrix. The obtained analytical expressions make fast computations possible and the obtained full response field makes it possible to perform time domain analysis. The expressions are numerically validated by comparison with a GOE-MC and a detailed MC model of an aluminium plate.

 

 

Figure 3: Real and imaginary part of the covariance of the FRF Cov (H(f )H(200 Hz)) at locations (a) B and (b) C. Results for the detailed MC are plotted in blue, for the GOE-MC in red, and for the statistical model in black.

 

ACKNOWLEDGEMENTS

 

The research presented in this paper has been performed within the frame of the VirBAcous project (project ID 714591) “Virtual building acoustics: a robust and efficient analysis and optimization framework for noise transmission reduction” funded by the European Research Council in the form of an ERC Starting Grant. The financial support is gratefully acknowledged.

 

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¹ cedric.vanhoorickx@kuleuven.be

² edwin.reynders@kuleuven.be