The six lowest natural frequencies for a cavity with perfectly rigid walls.
Combustion instability is often encountered in various power systems, a good understanding on the sound field in acoustic cavity as well as its coupling with boundary flexible structure will be of great help for the reliability design of such combustion system. An improved Fourier series method is presented for the acoustic/vibro-acoustic modelling of acoustic cavity as well as the panel-cavity coupling system. The structural-acoustic coupling system is described in a unified pattern using the energy principle. With the aim to construct the admissible functions sufficiently smooth for the enclosed sound space as well as the flexible boundary structure, the boundary-smoothed auxiliary functions are introduced to the standard multi-dimensional Fourier series. All the unknown coefficients and higher order variables are determined in conjunction with Rayleigh-Ritz procedure and differential operation term by term. Numerical examples are then presented to show the correctness and effectiveness of the current model. The model is verified through the comparison with those from analytic solution and other approaches. Based on the model established, the influence of boundary conditions on the acoustic and/or vibro-acoustic characteristics of the structural-acoustic coupling system is addressed and investigated.
- enclosed sound space
- acoustics analysis
- structural-acoustic coupling
- flexible boundary structure
Combustion instability is often encountered in various power systems, which will further cause the combustion noise or even the dynamic damage of combustion chamber structure . A good understanding on the vibro-acoustic coupling between the bounded flexible structure and the thermo-driven acoustic oscillation will be of great significance for the correct design of combustion system of various power plants. As an important part of such whole thermos-acoustic coupling system, the acoustics in cavity and its coupling with its flexible boundary also plays an important role. For many years, a lot of research effort has been devoted to the coupled structural-acoustic system.
The acoustic analysis in enclosed space is a classical research topic in acoustics community, and the rectangular cavity is widely used as the theoretical model. Morse and Bolt  first introduced the normal modes theory into room acoustics, and developed a non-linear transcendental characteristic equation through combining the assumed sound pressure modes with complex impedance boundary conditions on the walls. Maa  derived the transcendental equation for a rectangular room with non-uniform acoustical boundaries which took the same form as that of a uniform acoustic admittance case, while the impedance term was consequently non-uniform on certain wall. Recent studies have been mainly focused on developing more effective root searching algorithms for finding eigensolutions. For instance, Bistafa and Morrissey  compared two different numerical procedures: one is the Newton’s method and the other is referred to as the homotopic continuation technique based on the numerical integration of differential equations. The roots are searched for the cases from soft walls to the terminal impedance with small increments. They found that the latter procedure is much faster in finding all the possible roots. Naka et al.  utilized an interval Newton/generalized bisection (IN/GB) method to find the roots of the non-linear characteristic equation within any given interval for the modal analysis of rectangular room with arbitrary wall impedances.
In many occasions, the acoustic cavity is bounded by the flexible structure, and the interaction between the structural vibration and the acoustic cavity should be taken into account simultaneously for the determination of acoustic field characteristics. Among the existing studies, the most popular modelling approach is the so-called modal coupling theory  in which the structural modes
In this chapter, a unified structural-acoustic coupling analysis framework will be introduced for the representative rectangular cavity and its coupled panel structure. The fully coupling system is described in the framework of energy. The Fourier series with supplementary terms is constructed as the admissible functions, which are smoothed in the whole solving domain including the elastic structural and/or impedance acoustic boundary and coupling interface. All the unknown coefficients are solved in conjunction with Rayleigh-Ritz procedure. Since the field functions are sufficiently smooth, the corresponding high-order variables can be calculated straightforwardly.
2. Theoretical formulations
2.1. Acoustic cavity with impedance boundary condition
A rectangular acoustical cavity of dimensions
2.2. Improved Fourier series representation of admissible function
It is well known that the modal functions for rigid-walled rectangular cavity are simply the products of cosine functions in three dimensions. Based on the modal superposition principle, the corresponding sound pressure field inside the cavity can be generally expressed as a 3-D Fourier cosine series. However, such a Fourier series representation will become problematic when an impedance boundary condition is specified on one or more of the interior walls. This assertion is evident from Eq. (1) because the left side of the equation is identically equal to zero regardless of the actual value of the right side. This problem is mathematically related to the inability to converge of the traditional Fourier series on the boundaries of a domain under general boundary conditions. To overcome this difficulty, in this study, a 3-D version of an improved Fourier series representation previously developed for the in-plane vibrations of elastically restrained plates will be used to expand the sound pressure inside the cavity .
It is easy to verify that
In light of Eqs. (3)–(5), one can understand that the 2-D Fourier series expansions in Eq. (2) mathematically represent the possible non-zero (normal) derivatives of the acoustic pressure on each of the cavity walls, and the 3-D Fourier series a residual pressure field as if the impedance boundary conditions on the cavity walls were modified to being infinite rigid. Mathematically, it can be proved that the modified series solution converges faster and uniformly over the entire solution domain including the boundary walls [10, 11].
Since the pressure solution is constructed sufficiently smooth in the current formulation, the unknown expansion coefficients can be solved in a strong form by letting the series solution simultaneously satisfy both the governing differential equation (Helmholtz equation) inside the cavity and the boundary conditions, on the cavity walls on a point-wise basis. In such a case, because of the boundary conditions, the expansion coefficients for the 2-D series are not fully independent of those for the 3-D series. While such a procedure may be preferred in the context of ‘exact’ solution, an alternative procedure for obtaining a weak form of solution will be employed here because of its potential benefits in modelling complex acoustic systems consisting of many cavities. The corresponding Lagrangian for the rectangular cavity with arbitrary impedance boundary conditions can be written as
The total potential energy
The total kinetic energy
By using the relationship between the sound pressure and the particle velocity on impedance surface, the dissipated acoustic energy can be calculated from
where Z is the complex acoustic impedance of the wall surface, and
The work done by a sound source inside cavity can be represented as
Substituting Eqs. (7)–(12) into the acoustic Lagrangian, Eq. (6), and applying the Rayleigh-Ritz procedure against each of the unknown Fourier series coefficient, a system of linear algebra equations can be derived as
In order to determine the modal characteristics of the acoustic cavity, one needs to solve the characteristic equation by setting the external load vector
where , , and
2.3. Vibro-acoustic coupling of panel-cavity system
The above formulation is mainly about the modelling of pure acoustic cavity, in many situations, the cavity is bounded by the flexible structure, such as the combustion chamber. For such structural-acoustic coupling system, the vibration of flexible boundary structure and the acoustic filed will couple together. As a classical example, the rectangular panel-cavity is often used as the analysis example for the structural-acoustic coupling study.
As shown in Figure 2 , an elastically restrained plate is one of the surfaces enclosing a rectangular acoustical cavity (other five surfaces are assumed to be perfectly rigid for simplicity). Suppose that the plate is excited by a normal concentrated force
Although such structural-acoustic coupling system can be analysed by solving the governing equation and boundary conditions, simultaneously. Similar to the above acoustic analysis of enclosed sound space, the energy principle can also give the sufficiently accurate prediction of vibro-acoustic behaviour, when the admissible functions are constructed smooth enough. For the transverse vibration of a rectangular plate with general elastic boundary supports, its displacement function will be invariantly expanded into an improved Fourier series as .
For the panel cavity considered here, the main attention will be paid to the structural-acoustic continuity, with the other walls kept as rigid. The acoustic pressure filed function is constructed as 
The Lagrangian for the plate structure can be expressed as
The total potential and kinetic energy for the elastic plate can be explicitly expressed as
The Lagrangian for the acoustic cavity is
Once the Fourier coefficient vectors
3. Numerical examples and discussion
In this section, numerical examples will be presented to demonstrate the effectiveness and reliability of the proposed method, then based on model established, the vibro-acoustic behaviour of the cavity as well as its coupling system with the flexible panel will be studied. The first example involves a rectangular cavity with each of its walls being perfectly rigid. The related parameters are as follows: the dimensions are
Table 1 shows a comparison of the first six natural frequencies using the familiar analytic solution and from the current method. It should be mentioned that the current results were calculated by truncating the series expansions in Eq. (2) to
|Natural frequency (Hz)|
Now, place a point source of strength
Another extreme case of the non-dissipative boundary conditions is the so-called pressure release (or zero-pressure), which is described by infinitely small pure imaginary impedance on surface (
The sound pressure at (0.1, 5.96 and 2.02 m) is plotted in Figure 4 as a function of frequency. This problem was previously solved by using an equivalent source technique, and the result was also shown in Figure 4 as a reference. It is seen that the two predictions are in a good agreement. However, slight separation between them can be noticed as frequency increases. This is probably caused by the possible loss of the accuracy of the equivalent source technique due to the use of an insufficient number of equivalent sources at higher frequencies. Plotted in Figure 5 are sound pressure fields inside the cavity at 14 and 42 Hz, respectively. Since two frequencies are very close to the first two resonance frequencies (refer to Figure 4 ), the distributions essentially resemble the first and second acoustic mode shapes. It is observed from Figure 5 that the sound pressure decreases rapidly in approaching to (and eventually vanishes on) the wall of
To better understand the effect of impedance boundary conditions on the acoustical characteristic of an enclosed space, the frequency responses at the observing point (0.1, 5.96 and 2.02 m) are plotted in Figure 6 for a wide range of impedance values from
In the aforementioned examples, the results are mainly on the acoustic cavity analysis, as demonstrated in the above formulations, the current modelling framework can be used for the treatment of vibro-acoustics analysis of panel-cavity system by simply including the vibrational energy in the whole description. The model will be first valeted, and then, the main emphasis will be put on the model validation and the influence of structural boundary condition on the coupling characteristics of such panel-cavity system.
For the model verification on the modal parameter prediction, consider a problem previously studied in Ref.  where an acoustic cavity (
Table 2 shows the first 20 natural frequencies of the coupled panel-cavity system. The data from Ref.  were also listed there for comparison. A nice agreement can be observed between these two sets of results with the largest difference being less than 0.35%. In this example, the Fourier series is truncated to
|Mode order||Ref.  (Hz)||Current (Hz)||Difference (%)|
Next, we will examine the effect of cavity depth on the modal parameters. As in Ref. , the cavity dimensions are chosen as:
From the standpoint of structural vibration, the acoustic cavity may be approximately viewed as Winkler springs with a probably non-uniform stiffness distribution over the area of the panel, depending upon frequency. To understand its significance, the effects of varying edge restraining stiffnesses on the fundamental frequency of the coupled system are studied for a range of cavity depths. In this analysis, the copper panel is assumed to be uniformly supported along all four edges. Shown in Figure 7(a) are the results for a configuration in which, by keeping the rotational stiffness to zero, the stiffness for the translational spring is increased from zero (completely free) to infinity (simply supported). It is evident that reducing the cavity depth is equivalent to increasing the stiffness of the Winker springs. For small restraining stiffness,
By letting the translational springs be infinitely rigid, we now add rotational restraints to the edges. The fundamental frequency curves are plotted in Figure 7(b) for various combinations of cavity depths and spring stiffnesses. Again, it is seen that the decreasing the cavity depth is equivalent to increasing the stiffness of the Winkler springs and hence the fundamental frequency of the panel wall. The above results also indicate that there tends to exist stronger structural acoustic coupling for a thinner air gap. This statement may have a meaningful implication to the design of double-walled sound isolation.
We will now direct our attention to the vibro-acoustic responses of the coupled system. For validation, the cavity dimensions will be modified to:
As mentioned earlier, the conventional modal coupling theory suffers a velocity discontinuity problem at the fluid-structure interface, that is, the particle velocity on/near the interface cannot correctly calculated from the pressure gradient. However, this velocity continuity requirement is faithfully enforced in the current method. To illustrate this point, Figure 10(a) shows the velocity response at (3
An improved Fourier series method is presented for the acoustic/vibro-acoustic modelling of acoustic cavity as well as its coupling with flexible boundary structure. The coupled system is described in a unified pattern by using the energy description. With the aim to construct the structural-acoustic admissible functions smooth sufficiently in the whole solving domain, boundary-smoothed auxiliary functions are introduced to the standard multi-dimensional Fourier series on the system boundary as well as the coupling interface. In conjunction with Rayleigh-Ritz procedure, all the unknown coefficients can be easily derived, and the relevant higher order acoustic variables, such as energy power flow, can be determined straightforwardly.
The theoretical formulation is implemented in the Matlab environment. Numerical results are presented to illustrate the effectiveness and efficiency of the proposed model. The correctness and reliability is then verified by comparing with those from other method or numerical solution. Based on the model established, influence of boundary condition on the acoustic or structural-acoustic coupling characteristics is addressed and investigated in details. This work can present an efficient analysis tool for the acoustic or structural-acoustic analysis of the enclosed sound space and flexible structure. This work shows that the desired modal characteristics of coupling system can be obtained by adjusting boundary conditions properly.
This work was supported by the Fundamental Research Funds for the Central Universities (HEUCFP201758).
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