Proceedings of the Institute of Acoustics

 

 

Assessing the coupling strength between subsystems in (hybrid deterministic-)statistical energy analysis

 

Edwin P. B. Reynders¹, KU Leuven – Department of Civil Engineering, Leuven, Belgium

Cédric Van Hoorickx, KU Leuven – Department of Civil Engineering, Leuven, Belgium

 

ABSTRACT

 

Statistical energy analysis (SEA) is a standard approach to high-frequency vibro-acoustic analysis that relies on a conceptual division of the system into subsystems that are assumed to carry a diffuse field and to be weakly coupled. The weak coupling assumption means that the exchange of energy between subsystems can be described in terms of their uncoupled free vibration modes. In this work, a criterion is derived for assessing the coupling strength in the general case where the subsystems are rigidly coupled and/or via deterministic linear dynamic components. The criterion is elaborated such that it can be directly evaluated from quantities that appear in the SEA power balance. In this process, the hybrid deterministic-SEA approach is employed such that subsystems and connections of arbitrary complexity can be tackled in a rigorous way. In one of its approximate forms, the proposed general coupling strength criterion reduces to the gamma criterion that has appeared in the literature for assessing some special cases of coupling. The criterion is validated with a numerical example involving two diffuse plate subsystems connected via a deterministic beam, whose dynamic behaviour influences the coupling strength.

 

1. INTRODUCTION

 

In statistical energy analysis (SEA) and related approaches to high-frequency vibro-acoustic analysis, the studied system is conceptually divided into subsystems that are assumed to carry a diffuse wave field. Two subsystems can be either directly coupled, via other subsystems, or via deterministic components, i.e., components that do not carry a diffuse wave field. The total energies in the subsystems are determined by a power flow analysis [1]. When setting up the power balances, it is additionally assumed that the dynamic response of each subsystem is governed by its uncoupled, local vibration modes [2]. The power balances can then be expressed in terms of the modal density or modal overlap of the uncoupled subsystems. This second assumption is usually referred to as weak coupling. It is opposed to strong coupling, which occurs when the vibro-acoustic behaviour of the overall system can be described only in terms of global vibration modes, of which the vibration energies are not situated predominantly in one of the subsystems [3].

 

Several quantitative criteria have been proposed to assess the coupling strength between subsystems. Some of these criteria have a relatively wide range of applicability but their evaluation is computationally involved, while others are relatively straightforward to evaluate but have been derived by considering specific subsystems and/or coupling components. For example, the maximum coupling eigenvalue criterion as proposed in [4] considers the coupling strength in terms of the maximum eigenvalue of the impedance matrix relating the interface degrees of freedom of two subsystems that are connected via a flexible coupling component [4]. If all eigenvalues are smaller than unity, then an analysis converges in which the displacements of the individual subsystems are first computed for the decoupled situation and next iteratively updated by considering vibration exchange between the subsystems via the coupling impedance matrix. This criterion is therefore generally applicable to subsystems connected via flexible coupling components, but the evaluation of the eigenvalues can be computationally costly when there are many interface degrees of freedom. As another example, the γ criterion, which equals the scaled ratio of the power transmission coefficient between both subsystems to the product of their modal overlaps, was originally derived by considering the transition from local to global modal behaviour of a system composed of two directly connected one-dimensional subsystems [5, 6], but it can be evaluated from quantities that appear in the SEA analysis itself. Therefore, it also lends itself well for analysing the influence of physical parameters on the coupling strength, such as the damping of the subsystems involved [7].

 

Substantial progress was made by Finnveden [8], who showed that the γ criterion remains a correct coupling strength evaluation criterion when its application range is extended towards general subsystems coupled by flexible springs. This was achieved by making a connection with the largest coupling eigenvalue criterion from [4] and demonstrating that the squared maximum coupling eigenvalue approximates γ for the considered connection type. However, the assessment of the coupling strength between general SEA subsystems that are coupled in an arbitrary way, has remained an open problem. The maximum eigenvalue of the coupling impedance matrix is not a suitable criterion when rigid or stiff connection components are present: this criterion would always indicate strong coupling [4], since the decoupled response of each subsystem, from which the analysis starts, is defined by blocking the interface degrees of freedom of all other subsystems. Furthermore, it is not immediately clear how the analysis can be extended to include general linear dynamic coupling components. It is also not immediately clear how the SEA coupling loss factor, which appears in the coupling strength analysis, can be evaluated for general subsystems and couplings.

 

In the present work, a methodology is presented that solves these issues. First, a new coupling strength criterion is developed in a purely deterministic setting. A coupling matrix is derived that connects the true subsystem interface displacements to the values that are obtained when neglecting the reaction forces of the other subsystems. It is demonstrated that weak coupling occurs when the largest eigenvalue amplitude of this coupling matrix is smaller than unity. Next, it is assumed that the subsystems carry a diffuse field and the analysis is extended to the corresponding random ensemble. To this end, the problem is formulated within the hybrid deterministic-statistical energy analysis (det SEA) framework [9]. The adoption of this framework also enables to compute the coupling loss factor between the subsystems in a rigorous and straightforward way, without making a formal distinction between resonant and non-resonant wave transmission.

 

The outline of the paper is as follows. In section 2, a deterministic built-up system is considered, for which a new maximum coupling eigenvalue criterion is proposed that can be used for any type of linear dynamic coupling component. A coupling criterion is furthermore obtained that can straightforwardly be evaluated from the system components’ dynamic stiffness matrices. These results are then used in section 3 to obtain a general coupling strength criterion for built-up systems which can contain both random, diffuse subsystems and deterministic components. In section 4, the methodology is validated with a numerical example of two diffuse plates coupled by a deterministic beam with distributed mass, stiffness and damping.

 

 

Figure 1: Illustrative example of a built-up system consisting of two subsystems (light gray) and a coupling component (dark gray). The interface degrees of freedom are indicated with green squares for q_1\2, with red circles for q_1∩2, and with blue triangles for q_2\1.

 

2. COUPLING STRENGTH BETWEEN DETERMINISTIC SUBSYSTEMS

 

The analysis presented in this section is concerned with the assessment of the coupling strength between deterministic subsystems of a built-up system. The subsystems can be directly coupled, via linear dynamic coupling components, or both. First, a coupling matrix is derived that connects the true subsystem interface displacements to the values that are obtained when neglecting the reaction forces of the other subsystems. Weak coupling occurs when the true displacements can be obtained from the decoupled values through a finite number of iterations involving the coupling matrix. It happens when the largest eigenvalue amplitude of the coupling matrix is smaller than unity; this is therefore used as coupling criterion. For simplicity the analysis is limited to two subsystems but the extension to an arbitrary number of subsystems is immediate. A subsequent analysis results in an explicit expression relating the coupling criterion between pairs of subsystems to their interface impedance matrices. This expression will be used in the next section, which concerns the coupling strength analysis between diffuse subsystems.

 

2.1. The coupling matrix of an arbitrary built-up system

 

Two subsystems are considered that are connected directly and/or via (general) coupling components (Figure 1). The interface displacement degrees of freedom (DOFs) are collected in the vector

 

 

where q_1\2 contains the interface DOFs belonging to subsystem 1 but not to subsystem 2, q_1∩2 contains the interface DOFs belonging to both subsystems 1 and 2, and q_2\1 contains the interface DOFs belonging to subsystem 2 but not to subsystem 1. The dimensions of these vectors are denoted as N_1\2 × 1, N_1∩2 × 1 and N_2\1 × 1, respectively. After condensing out all other degrees of freedom, the equations of motion of the built-up system can be expressed as

 

 

where D⁽¹⁾, D⁽²⁾ and D_d are the condensed dynamic stiffness matrices of subsystem 1, subsystem 2, and the coupling components, respectively, and f is the condensed load vector. Note that D⁽¹⁾ and D⁽²⁾ are partially empty, i.e., they have the following structure:

 

 

It follows immediately from Equation 2 that the reaction forces f⁽¹⁾ and f⁽²⁾ of subsystems 1 and 2, respectively, can be obtained from

 

 

Consider now the interface displacement degrees of freedom belonging to subsystem 1:

 

 

The number of elements of q₁ is denoted as N₁ =: N_1\2 + N_1∩2. I denotes an identity matrix while 0 is an empty matrix. By substitution of Equation 2 and subsequently Equation 4, the following expression is obtained:

 

 

It can be reformulated as

 

 

In this expression, q₁₀ contains the displacements of subsystem 1 that would be obtained when the reaction forces from subsystem 2 are neglected, while the second term contains the contribution of the reaction forces of subsystem 2 to the displacements of subsystem 1. These reaction forces will now be expressed in terms of the interface displacements belonging to subsystem 2, which are

 

 

and N₂ := N_1∩2 + N_2\1. With these definitions, one has that

 

 

where the final equality results from the sparse structure of D⁽²⁾ as detailed in Equation 3. Substitution into Equation 7 then leads to

 

 

Following the same lines, a similar decomposition can be achieved for the second subsystem:

 

 

Equation 10 and Equation 11 can be combined into a single expression:

 

 

where

 

 

If the inverse of I − K exists, it can be expressed as a Neumann series, such that

 

 

Given the structure of K, this means that the true interface displacements q₁ and q₂ can be obtained from the decoupled interface displacements q₁₀ and q₂₀, which neglect the reaction forces from the other subsystem, through a series of energy exchanges of decaying magnitude. The coupling is weak when the decay is fast. This occurs when the eigenvalue of K with the largest amplitude, λ_max (K), is substantially smaller than unity. The matrix K is therefore termed the coupling matrix. So the validity criterion for weak coupling is

 

 

This general coupling criterion bears a certain similarity with the criterion proposed in [4], but the definition of the coupling matrix is fundamentally different. By defining the uncoupled situation to be the one where the other subsystem has zero interface forces rather than zero interface displacements, the criterion does not break down when the connection between both subsystems becomes very stiff.

 

2.2. An explicit expression for the coupling strength

 

As discussed in section 1, the amplitude of the largest eigenvalue of the coupling matrix K can be used as a measure for the coupling strength. In the present section, an expression that explicitly relates this coupling criterion to the interface impedance matrices is derived. The analysis extends the approach of [4], which is restricted to spring-coupled subsystems such that the eigenvalues of the (differently defined) coupling matrix are real. For the general case treated here, the eigenvalues are complex. Further on in this paper, the derived expression will form the basis for the analysis of the coupling strength between two SEA systems, which are random rather than deterministic.

 

The present analysis starts by defining a transformed coupling matrix which has the same eigenvalues as the coupling matrix K:

 

 

By exploiting the structure of K from Equation 13 and the structure of D⁽¹⁾ and D⁽²⁾ from Equation 3, it can be readily demonstrated that

 

 

Note that the existence of ( D⁽ⁿ⁾)^1/2 is guaranteed from the fact that D⁽ⁿ⁾ is positive (semi-)definite. If the system is undamped, the transformed coupling matrix is real and symmetric and its eigenvalues and eigenvectors are real. Proportional viscous damping can be introduced by considering complex eigenvalues, while keeping the real eigenvectors obtained from the undamped eigenvalue problem [10], such that the eigenvalue problems for  and its complex conjugate ^∗ are

 

 

Pre-multiplying both sides of the second expression by the transformed coupling matrix leads to

 

 

The resulting separate eigenvalue problems

 

 

reveal that the eigenvalues of ₁₂^∗₂₁ equal the squared amplitude of the eigenvalues of the transformed coupling matrix K and therefore of the coupling matrix K. It follows from Equation 17 that

 

 

where the superscript H denotes complex conjugate transpose or Hermitian transpose.

 

Since the trace of a matrix equals the sum of its eigenvalues [11], one has that

 

 

The final equality results from the property that the matrix trace is invariant under cyclic permutations [11]. Weak coupling is guaranteed when γ_det ≪ 1. This quantity can also be expressed in terms of the impedance matrices Y^(k) := iω (D^(k))⁻¹ of the coupled subsystems by noting that:

 

 

Consequently,

 

 

where use was again made of the property that the trace is invariant under cyclic permutations [11].

 

3. COUPLING STRENGTH BETWEEN DIFFUSE SUBSYSTEMS

 

This section is concerned with the coupling between two subsystems that each carry a diffuse wave field. A diffuse wave field is a random field, so a random ensemble of coupled subsystems needs to be considered. The analysis of the exchange of energy between diffuse subsystems is termed statistical energy analysis (SEA). For stationary vibration at frequency ω, the time- and ensemble averaged power balance of a built-up system containing two weakly coupled diffuse subsystems takes the form

 

 

In this expression,  is the power input from external loading applied directly to subsystem i ; _i := E_i / n_i is the modal energy in subsystem i, i.e. its total energy E_i divided by its modal density n_i ; m_i := ωη_i n_i is the modal overlap of subsystem i with η_i its damping loss factor; and h denotes the power transfer coefficient or conductivity between both subsystems,

 

 

where P₁₂ is the net coupling power between subsystems 1 and 2. The modal density n_i is the average number of modes of decoupled subsystem i per radial frequency across the random ensemble, and similar for n_j. The coupling between both subsystems is often expressed in terms of the coupling loss factor η_ij instead of the power transfer coefficient h. Both are related via

 

 

The aim of this section is to assess the coupling strength between both diffuse subsystems in terms of quantities that directly relate to the SEA power balance Equation 25.

 

Again, a built-up system such as the one illustrated in Figure 1 is considered, but now subsystems 1 and 2 are random, diffuse subsystems. They can be connected directly or through a deterministic component d. For built-up systems consisting of both diffuse and deterministic components, the hybrid deterministic-statistical energy analysis (det-SEA) framework can be adopted. This framework is introduced first, whereafter it is used to derive a the coupling strength criterion.

 

3.1. The hybrid deterministic-statistical energy analysis approach

 

In the hybrid det-SEA approach [9], a built-up system is considered that consists of diffuse, random subsystems and deterministic components. In this setting, the dynamic stiffness matrices of the subsystems D^(k) which appear in Equation 2 are random matrices. They are decomposed into a deterministic, ensemble averaged part and a zero-mean random part  :

 

 

If a random subsystem can be described by its local modes, it can be shown that the ensemble mean of the dynamic stiffness matrix E [ D^(k) ] equals the direct field dynamic stiffness matrix [12], which represents the dynamic stiffness of the equivalent unbounded kth subsystem at the interface DOFs. The reverberant dynamic stiffness matrix represents the response of the reverberant field in the kth subsystem, caused by diffuse random wave scattering. Substitution of this decomposition into Equation 2 yields

 

 

where D_tot := D_d + Σ_k   is the deterministic total dynamic stiffness matrix and f(k) rev := −q the blocked reverberant force vector of the kth subsystem.

 

In this setting, the net coupling power P₁₂ which appears in Equation 26 equals the difference between the power transferred from 1 to 2 and the power transferred from 2 to 1 due to reverberant loading associated with the reverberant field in subsystem 1 and 2, respectively. The direct field transferred power follows from the displacement due to this reverberant loading from the kth subsystem and the resulting direct field forces of the jth subsystem

 

 

where

 

 

are the cross-spectrum of the blocked reverberant force associated with the reverberant field of the kth subsystem and the cross-spectrum of the resulting generalized displacement, respectively. In a weakly coupled subsystem, the cross-spectrum of the blocked reverberant force is related to the modal energy via the diffuse field reciprocity relation [13]:

 

 

Substitution into Equation 31 yields

 

 

where the last equality follows from the property that the trace is invariant under cyclic permutations [11] and the fact that the dynamic stiffness matrices are symmetric. From the definition of the direct field power transfer coefficient, Equation 26, it follows that

 

 

3.2. A coupling strength criterion for diffuse subsystems

 

For the random ensemble of built-up systems that is considered here, weak coupling holds when the criterion γ_det ≪ 1 that was derived in section 2 holds for each individual member of that ensemble. In other words,

 

 

with 𝕊 denoting the random ensemble. Considering Equation 24, it can reasonably be assumed that the maximum of γ_det occurs when the matrices |Y⁽¹⁾| and |Y⁽²⁾| are at resonance. To obtain a conservative estimate, the impedance matrices in the inversed factors of Equation 24 are replaced by the ensemble averaged or direct field impedance matrices:

 

 

with  representing the uncoupled mobility matrix for subsystem s at resonance. The mobility at resonance of a random field can be related to the real part of the direct field mobility through a quantity depending on the modal overlap factor m_s [14, 15]:

 

 

The second approximation, which is valid at low modal overlap, was derived by Skudrzyk [14]. The first approximation was derived by Langley [15] and assumes that the natural frequencies in the vicinity of the considered frequency are evenly spaced for every random ensemble member. Although this latter assumption is not correct for most systems and might result in an underestimation of the true mobility at resonance, the estimate is always larger than the direct field mobility (which can be regarded as the mean field mobility over the considered ensemble), since the approximation ensures that β_s ≥ 1. In contrast, with Skudrzyk’s approximation, values of β_s that are smaller than unity are possible, such that this approximation is less accurate for high modal overlap m_s. Substitution of Equation 39 into Equation 38 yields:

 

 

The quantity γ can also be expressed in terms of the direct field stiffness matrices of the coupled subsystems by noting that:

 

 

and therefore:

 

 

where use was made of the property that the trace is invariant under cyclic permutations [11] and the following property for symmetric matrices:

 

 

Comparing Equation 42 and Equation 36 reveals the following relation between γ and h:

 

 

where in the first and second approximation, Langley’s and Skudrzyk’s approximations for β_s in Equation 39 were used, respectively. Langley’s approximation is more accurate, especially at high modal overlap, and it will therefore be employed in the validation example of section 4.

 

In summary, it was demonstrated here that weak coupling holds between general and arbitrarily coupled diffuse subsystems when γ ≪ 1, where γ can be computed from the approximate expressions in Equation 44. These expressions contain the modal overlap factors and the direct field power transfer coefficient, which appear also in the SEA power balance Equation 25 itself. Thanks to the general validity of Equation 36, the direct field power transfer coefficient can be computed for any type of built-up system.

 

4. VALIDATION EXAMPLE: TWO PLATES COUPLED BY A BEAM

 

We consider a system of two plates coupled via a beam, such that the coupling strength between the plates is influenced by the dynamic behaviour of the beam (Figure 2). The two plates are made of timber with a Young’s modulus of 5 GPa, a Poisson coefficient of 0.3, a mass density of 525 kg/m³ and a damping loss factor of η = 0.01. The dimensions of both plates are 1.5 m × 3 m and their thickness is 2 cm. The beam is made of steel, having a Young’s modulus of 210 GPa, a Poisson coefficient of 0.3, a mass density of 7850 kg/m³, and a damping loss factor of η = 0.001. The beam has a length of 2 m and a solid rectangular cross section with a width of 8 cm and a height of 2 cm. Every plate is connected to the beam at 5 distinct points, resulting in a total of 10 point connections. The point connections are situated at a distance of 22.5 cm from each other. The outer connections are located at a distance of 5 cm from the nearest beam end. The spacing between both plates is negligibly small, although they are not directly connected. The beam is positioned centrally in the system, as indicated in Figure 2. A unit point force is applied at (x, y) = (0.8 m, 2.1 m). The two plates are subjected to wave scattering caused by a random mass distribution on the plates. On each plate, a random distribution of 50 point masses is present each having a mass equal to 1.5 % of the total mass of the plate.

 

 

Figure 2: Two plates coupled by a beam, excited by a point force F.

 

In what follows, two different models of this system are compared. The first model is constructed using the assumed-modes method, [16] where the analytical plate and beam modes are used as trial functions and substituted into Langrage’s equation of motion, just as the random point masses. The attachment locations of the point masses are then varied in a detailed Monte Carlo analysis. Note that in this first, detailed model, no assumption is made regarding the nature of the wave field in the plates nor of the coupling strength. The second model is a hybrid det-SEA model, in which the point masses are not modeled explicitly but they are assumed to cause a diffuse wave field in both plates. The plates are therefore modeled as SEA subsystems, while the beam is a deterministic coupling component.

 

While modal overlap determines mode mixing in every individual member of the random ensemble and is therefore a measure for how many modes are expected to contribute to the response at a given frequency, statistical overlap determines mode mixing across the random ensemble itself and is therefore a measure for the sensitivity of a natural frequency to small disturbances with a wave scattering effect [3]. Figure 3a contains the statistical overlap s for the plates as computed from [17, Eq. A.11]: s > 1 from about 105 Hz meaning that the two plates can be modelled as diffuse. Figure 3c displays the mean of the total energy in an uncoupled plate if excited by a unit point force at (x, y) = (0.8 m, 2.1 m). As the statistical overlap is larger than 1 for nearly the entire frequency range, the wave field in both plates is well approximated by a diffuse field, so on this point a good correspondence is expected between the detailed Monte Carlo and the hybrid det-SEA models.

 

 

Figure 3: (a) Statistical overlap s for the plate subsystems and (b) the coupling strength factor γ (solid line) for the assembled system. Mean of the total energy in (c) the excited uncoupled plates and (d) plates 1 (black lines) and 2 (gray lines) of the assembled system as a function of frequency. The dashed lines are the hybrid det-SEA results, while the solid lines are the results obtained with a detailed Monte Carlo approach.

 

Figure 3b displays γ (Equation 44) for the assembled structure. The coupling strength peaks at the natural frequencies of the beam, at which the power transfer coefficient has a maximum. It furthermore decreases with increasing frequency as the modal overlaps m_i of the two plates increase with increasing frequency. As a result, γ is clearly smaller than 1 at high frequencies, indicating weak coupling. At lower frequencies, however, γ is approximately equal to or larger than 1, indicating strong coupling.

 

The mean of the total energy in both plates, computed from both models, are compared in Figure 3d. At low frequencies, where strong coupling was predicted by the γ criterion, a poor prediction of the mean energy by the hybrid det-SEA is obtained. In contrast, a good correspondence is found at high frequencies, where the γ criterion predicts weak coupling. This numerical example demonstrates that (i) strong coupling may occur even when the wave fields in the subsystems can be considered diffuse, (ii) the γ criterion for assessing the coupling strength can be applied to complex connection structures, and (iii) the hybrid det-SEA approach makes it possible to obtain both the energetic response and its range of applicability with respect to the coupling strength.

 

5. CONCLUSIONS

 

Statistical energy analysis and related approaches assume that the diffuse subsystems are weakly coupled. In this paper, a coupling strength criterion has been derived for the general case where subsystems can be coupled both via rigid connections and via a linear dynamic system. The subsystems were first taken to be deterministic and next they were taken to be diffuse subsystems in a (hybrid deterministic-)SEA setting. In the latter case, the criterion was further elaborated such that it can be directly evaluated from quantities that appear in the SEA power balance. In one of its approximate forms, this general criterion was shown to be identical to the γ criterion that has appeared in the literature for the coupling strength assessment of some special cases. The adoption of the hybrid det-SEA framework also allows for a straightforward and rigorous computation of the direct field power transfer coefficient between two general subsystems. The findings were numerically validated with an example system consisting of two diffuse plates coupled via a deterministic beam.

 

ACKNOWLEDGMENTS

 

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 edwin.reynders@kuleuven.be