Vibro-acoustic analysis of a trapezoidal cavity with a clamped flexible wall

Abstract

The coupled model between trapezoidal cavity and its clamped flexible wall is developed using classical modal coupling theory. Based on the coupled model, the resonance frequencies of coupled system are obtained and compared with the corresponding uncoupled one. Meanwhile, the reason for the variation of resonance frequencies of coupled system modes is analyzed in detail. Then, the response of coupled system is investigated using the acoustic potential energy in the cavity and panel vibration kinetic energy when it is excited by an incident plane wave outside of the cavity. Coupling coefficient between trapezoidal cavity and its clamped flexible wall is proposed to assess the modal matching degree between them. It is shown that the coupling selection is not satisfied except in the axis direction which is parallel to the inclined wall. In addition, a rectangular cavity with a clamped flexible wall is also considered and compared with that of the trapezoidal one.

Keywords

Trapezoidal cavity clamped panel coupling coefficient resonance frequency

Introduction

The coupling of a rectangular cavity with a flexible wall has attracted a lot of research, such as the free vibration features,¹ the forced response,^2,3 the noise controlled in cavity,^4–9 etc. However, the cavity usually takes the form of irregular shape in industrial applications, for example the passenger compartment and aircraft cabin, etc.

For irregular cavity, the classical modal coupling method was usually used to analyze the vibro-acoustic characteristics. The mode shape can be obtained by the normal expanding of its rigid wall bounding regular cavity modes. Based on this analytical method, the modal properties of the acoustic field in irregular cavities were analyzed, for example car cabin,¹⁰ semicircle cavity, and aircraft cabin.¹¹ When the irregular cavity takes the form of trapezoidal with one inclined wall, the effect of elevation angle of tilted wall on the mode characteristics of rigid wall trapezoidal cavity was researched.^12,13 The analysis of mode decay times in trapezoidal cavity with an inclined wall was conducted only when the local reactive wall was considered.¹⁴ For the irregular cavity with a simply supported flexible wall, the coupling coefficient was used to reflect the coupling extent between irregular cavity and its flexible wall, and the effect of elevation angle on the coupling coefficient and forced response was analyzed.^13,15 The flexible wall on the cavity considered above was supposed simply supported along its edges. In addition to classical modal coupling method, a weak form variational-based method was used to construct the coupling model between cavity sound field and its flexible wall.¹⁶ Then, the resonance frequencies of coupled system with the flexible wall subjected to arbitrary restrained boundaries were computed, and the forced response of coupled system was compared with the results obtained by FEM method. But the coupling characteristics between irregular cavity and its flexible wall were still unknown.

The irregular cavity with clamped flexible wall is commonly encountered in industrial applications. However, the coupling characteristics between irregular cavity and flexible wall subjected to this restrained boundaries are quite different from other cases, for example simply supported one. In this paper, the classical modal coupling method is employed to analyze the response of structure-acoustic coupled system consisting of an irregular trapezoidal cavity with a clamped flexible wall; the coupling characteristics are also presented in detail.

The work presented here considers a theoretical investigation into the vibro-acoustic behavior of a trapezoidal cavity with a clamped flexible wall, and the results are compared with that of a rectangular one. The effect of different external excitations acting on the flexible wall on the response of the coupled system is also discussed. The acoustic potential energy and vibration kinetic energy are used to describe the response in the cavity and in the clamped panel, respectively. Modal coupling coefficient between trapezoidal cavity and clamped flexible wall is developed. Compared with the rectangular cavity, the variation of coupling coefficient and the resonance frequencies of panel- and cavity-controlled modes of coupled system are analyzed in detail.

Analytical model of a structural-acoustic coupled system

Consider the trapezoidal cavity with a homogeneous and isotropic flexible wall shown in Figure 1. The trapezoidal cavity has one tilted wall with the inclination angle $α$ . The flexible wall is located at $z = L_{z}$ and it is clamped along its four edges. The other five walls are rigid.

Figure 1.

Schematic illustration of trapezoidal cavity.

The vibration control differential equation of the flexible panel under a plane wave excitation $p^{e}$ is

D \nabla^{4} w + ρ h \frac{\partial^{2} w}{\partial t^{2}} = p - p^{e}

(1)

where

D

ρ

h

w

are flexible rigidity, density, thickness, and vibration displacement of the flexible panel, respectively.

p

is the sound pressure which acts on the panel surface inside the cavity and

p^{e}

is the sound wave outside of the trapezoidal cavity.

Here, the plane wave outside of the cavity is used to active the clamped flexible panel, and $p_{i n} = p^{e}$ . The incident plane wave is represented as $p_{i n} = p_{i n 0} \exp (j ω t - j k r)$ , where $p_{i n 0}$ is the complex pressure amplitude of the plane wave, $k$ is the wavenumber vector, and $r$ is the location vector to the observation point. Here, the sound pressure radiated toward outside of the cavity by the clamped flexible panel is neglected.

Using the classical modal coupling theory, the vibration displacement $w (x, y, t)$ of the flexible panel can be expanded as

w (x, y, t) = \sum_{i = 1}^{I} \sum_{j = 1}^{J} ϕ_{i j} (x, y) W_{i j} (t)

(2)

where

ϕ_{ij} (x, y)

is the mode shape function of the clamped panel and

W_{i j} (t)

is the displacement modal amplitude in time domain. (I, J) are the numbers of the terms to be kept after the truncation of the series.

The mode shape function $ϕ_{ij} (x, y)$ is given by¹⁷

ϕ_{i j} (x, y) = ϕ_{i} (x) ϕ_{j} (y)

(3)

ϕ_{i} (x) = cosh (\frac{λ_{i} x}{L_{x}}) - cos (\frac{λ_{i} x}{L_{x}}) - \frac{cosh (λ_{i}) - cos (λ_{i})}{sinh (λ_{i}) - sin (λ_{i})} (sinh (\frac{λ_{i} x}{L_{x}}) - sin (\frac{λ_{i} x}{L_{x}}))

(4)

ϕ_{j} (y) = cosh (\frac{λ_{j} y}{L_{y}}) - cos (\frac{λ_{j} y}{L_{y}}) - \frac{cosh (λ_{j}) - cos (λ_{j})}{sinh (λ_{j}) - sin (λ_{j})} (sinh (\frac{λ_{j} y}{L_{y}}) - sin (\frac{λ_{j} y}{L_{y}}))

(5)

The sound pressure $p$ in the trapezoidal cavity can be described by the homogenous sound wave equation as follows

\nabla^{2} p - \frac{1}{c_{0}^{2}} \frac{\partial^{2} p}{\partial t^{2}} = 0

(6)

where

c_{0}

is the sound velocity.

Similar to the previous investigations, the acoustic field in the trapezoidal cavity can be obtained by the rigid wall modes of its bounding rectangular cavity.^12,14 The dimensions of bounding rectangular cavity are specified as $L_{x}$ , $L_{y} + L_{z} \tan α$ , and $L_{z}$ , and it encloses exactly the trapezoidal cavity as shown in Figure 1. The mode shape of the bounding rectangular cavity can be obtained by

φ_{l m n} (x, y, z) = \cos (l π x / L_{x}) \cos (m π y / (L_{y} + L_{z} \tan α)) \cos (n π z / L_{z})

(7)

Equation (7) satisfies the wave equation for bounding cavity, which is described by equations (8) and (9)

\nabla^{2} φ_{l m n} + k_{l m n}^{2} φ_{l m n} = 0

(8)

k_{l m n}^{2} = {(l π / L_{x})}^{2} + {(m π / (L_{y} + L_{z} \tan α))}^{2} + {(n π / L_{z})}^{2}

(9)

where

k_{l m n}

is the wave number,

l

m

, and

n

are mode indices of cavity modes.

Utilizing the base function $φ_{l m n} (x, y, z)$ , the sound pressure $p$ in the trapezoidal cavity can be expanded as

p = \sum_{l = 1}^{L} \sum_{m = 1}^{M} \sum_{n = 1}^{N} P_{l m n} (t) φ_{l m n} (x, y, z)

(10)

where

P_{l m n}

is the modal amplitude of lmn th trapezoidal cavity mode. (L,M,N) are the numbers of the terms to be kept after the truncation of the series.

The boundary conditions of trapezoidal cavity take the form of

\frac{\partial p}{\partial n} = - ρ_{0} \frac{\partial^{2} w}{\partial t^{2}} on the interface between the cavity and flexible panel

(11)

\frac{\partial p}{\partial n} = 0 on the rigid wall

(12)

where

ρ_{0}

is the air density, and the unit normal vector of the surface of the trapezoidal cavity is

n

(positive outside).

The vibration velocity of the flexible panel can be obtained by $v = j ω w$ , and $v (x, y, t) = \sum_{i = 1}^{I} \sum_{j = 1}^{J} ϕ_{i j} (x, y) V_{i j} (t)$ . Here, $V$ is the modal amplitude of flexible panel. Introducing the modal loss factor $η_{i j}$ of panel, the vibration of flexible panel can be expanded through the mode shape function from equations (1) and (2)

(ω_{i j}^{2} + 2 j η_{i j} ω_{i j} ω - ω^{2}) V_{i j} M_{i j}^{p} = j ω \int_{A_{f}} p ϕ_{i j} d S - j ω \int_{A_{f}} p^{e} ϕ_{i j} d S

(13)

M_{i j}^{p} = ρ h \int_{A_{f}} ϕ_{i j} ϕ_{i j} d S

(14)

η_{i j} = 2.2 π / T_{i j} ω_{i j}

(15)

where

A_{f}

T_{i j}

M_{i j}^{p}

ω_{i j}

are the area, the ijth modal decay time, the generalized modal mass, and the natural frequency of the flexible panel, respectively.

When the clamped panel is excited by the incident plane wave $p_{i n}$ outside of the cavity, the global modal force can be obtained by

\begin{matrix} E_{i j} = \int_{A_{f}} p^{e} ϕ_{i j} d S = \int_{A_{f}} p_{i n} ϕ_{i j} d S = \frac{2 p_{i n 0}}{A_{f}} \int_{A_{f}} ϕ_{i j} (σ) \exp [- j (\frac{β_{1} x}{L_{x}}) - j (\frac{β_{2} y}{L_{y}})] \exp [j (\frac{β_{1}}{2}) + j (\frac{β_{2}}{2})] d σ \end{matrix}

(16)

where

β_{1} = k L_{x} \sin θ_{1} \cos θ_{2}

β_{2} = k L_{y} \sin θ_{1} \sin θ_{2}

θ_{1}

and

θ_{2}

are the elevation angle and azimuth angle of the plane wave,

σ

is the observation vector on the panel surface.

By multiplying equations (6) and (8) with $φ_{l m n}$ and $p$ , respectively, equation (16) can be obtained by subtracting the former from the latter and integrating over the trapezoidal cavity $V_{T}$

\int_{V_{T}} (φ_{l m n} \nabla^{2} p - p \nabla^{2} φ_{l m n}) d V = (k_{l m n}^{2} - k^{2}) \int_{V_{T}} (p φ_{l m n}) d V

(17)

Then, by introducing the modal loss factor $η_{l m n}$ of trapezoidal cavity sound field and using the boundary condition and mode shape expanding, equation (18) can be obtained as the following form

\begin{matrix} (ω_{l m n}^{2} + 2 j η_{l m n} ω_{l m n} ω - ω^{2}) \sum_{r = 1}^{L} \sum_{s = 1}^{M} \sum_{t = 1}^{N} M_{l m n, r s t}^{a} P_{r s t} = - ρ_{0} c_{0}^{2} A_{W} \sum_{r = 1}^{L} \sum_{s = 1}^{M} \sum_{t = 1}^{N} I_{l m n, r s t} P_{r s t} \\ - j ω ρ_{0}^{2} c_{0}^{2} A_{f} \sum_{i = 1}^{I} \sum_{j = 1}^{J} L_{l m n, i j} V_{i j} \end{matrix}

(18)

η_{l m n} = 2.2 π / T_{l m n} ω_{l m n}

(19)

M_{l m n, r s t}^{a} = ρ_{0} \int_{V_{T}} φ_{l m n} φ_{r s t} d V

(20)

I_{l m n, r s t} = \frac{1}{A_{W}} \int_{_{A_{_{W}}}} φ_{r s t} \frac{\partial φ_{l m n}}{\partial n} d S

(21)

L_{l m n, i j} = \frac{1}{A_{f}} \int_{A f} φ_{l m n} ϕ_{i j} d S

(22)

where

ω_{l m n}

l m n th

natural frequency of the bounding rectangular cavity,

M_{l m n, r s t}^{a}

is the general modal mass of the sound field in trapezoidal cavity which is calculated using its bounding rectangular cavity mode,

I_{l m n, r s t}

is the factor which is generated due to the inclined wall and it can be obtained after a coordination transformation from

Y

and

Z

Y'

and

Z′

(Figure 1),

A_{W}

is the surface area of inclined wall, and

L_{l m n, i j}

is the modal coupling coefficient on the interface between the bounding rectangular cavity and flexible panel.

Equation (13) can also be written as

(ω_{i j}^{2} + 2 j η_{i j} ω_{i j} ω - ω^{2}) V_{i j} M_{i j}^{p} = j ω A_{f} \sum_{l = 1}^{L} \sum_{m = 1}^{M} \sum_{n = 1}^{N} L_{l m n, i j} P_{l m n} - j ω E_{i j}

(23)

By combining equations (18) and (23), the matrix form can be obtained by introducing $γ = - j ω$

(γ^{2} M + γ L + S) X = γ F

(24)

Equation (24) can be written as

(γ G - [\begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix}]) (\begin{matrix} X \\ γ X \end{matrix}) = \tilde{F}

(25)

where

A_{11}

(I J + L M N) \times (I J + L M N)

zeros matrix,

G

and

A_{12}

are

2 (I J + L M N) \times (I J + L M N)

and

(I J + L M N) \times (I J + L M N)

unit matrix, respectively,

A_{21} = - M^{- 1} S

and

A_{22} = - M^{- 1} L

\tilde{F} = γ [\begin{matrix} F_{z} \\ M^{- 1} F \end{matrix}]

F = [\begin{matrix} F_{1} \\ F_{2} \end{matrix}]

F_{2} = {[E_{1}, E_{2} \dots, E_{I J}]}^{T}

F_{z}

and

F_{1}

are

(I J + L M N) \times 1

and

L M N \times 1

zeros matrix

M = [\begin{matrix} M_{111, 111}^{a} & \dots & M_{111, L M N}^{a} \\ ⋮ & ⋱ & ⋮ \\ M_{L M N, 111}^{a} & \dots & M_{L M N, L M N}^{a} \\ M_{11}^{p} \\ ⋱ \\ M_{I J}^{p} \end{matrix}]

(26)

L = [\begin{matrix} - 2 η_{111} ω_{111} M_{111, 111}^{a} & \dots & - 2 η_{111} ω_{111} M_{111, L M N}^{a} & - ρ_{0}^{2} c_{0}^{2} A_{f} L_{111, 11} & \dots & - ρ_{0}^{2} c_{0}^{2} A_{f} L_{111, I J} \\ ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ \\ - 2 η_{L M N} ω_{L M N} M_{L M N, 111}^{a} & \dots & - 2 η_{L M N} ω_{L M N} M_{L M N, L M N}^{a} & - ρ_{0}^{2} c_{0}^{2} A_{f} L_{L M N, 11} & \dots & - ρ_{0}^{2} c_{0}^{2} A_{f} L_{L M N, I J} \\ A_{f} L_{111, 11} & \dots & A_{f} L_{L M N, 11} & - 2 η_{11} ω_{11} M_{11}^{p} \\ ⋮ & ⋱ & ⋮ & ⋱ \\ A_{f} L_{111, I J} & \dots & A_{f} L_{L M N, I J} & - 2 η_{I J} ω_{I J} M_{I J}^{p} \end{matrix}]

(27)

S = [\begin{matrix} ω_{111}^{2} M_{111, 111}^{a} + ρ_{0} c_{0}^{2} A_{W} I_{111, 111} & \dots & ω_{111}^{2} M_{111, L M N}^{a} + ρ_{0} c_{0}^{2} A_{W} I_{111, L M N} \\ ⋮ & ⋱ & ⋮ \\ ω_{L M N}^{2} M_{L M N, 111}^{a} + ρ_{0} c_{0}^{2} A_{W} I_{L M N, 111} & \dots & ω_{L M N}^{2} M_{L M N, L M N}^{a} + ρ_{0} c_{0}^{2} A_{W} I_{L M N, L M N} \\ ω_{11}^{2} M_{11}^{P} \\ ⋱ \\ ω_{I J}^{2} M_{I J}^{P} \end{matrix}]

(28)

X = {[P_{1}, \dots P_{L M N}, V_{1}, \dots V_{I J}]}^{T}

(29)

The vibro-acoustic model is constructed through equations (24) or (25), taking into account the elevation angle of cavity and the full coupling between the trapezoidal cavity and its clamped flexible wall. If the external excitation does not exist in the coupled system in equation (24) or (25), equation (25) will be a standard eigenvalue matrix equation. There will be $2 (L M N + I J)$ eigenvalues $γ_{U}$ and $γ_{U}^{*}$ , and $γ_{U} = - η_{U} - j ω_{U}$ ( $U = 1, 2 \dots (L M N + I J)$ ). Then, the resonance frequency $f_{U}$ and the decay time $T_{U 60}$ of Uth acoustic mode can be obtained, $f_{U} = ω_{U} / 2 π$ , $T_{U 60} = 2.2 π / η_{U}$ . If $F \neq 0$ , the solution of the coefficient $X$ is the modal amplitude of coupled system through equation (24) or (25). Then, the vibration displacement of the flexible wall and the acoustic pressure in the cavity can be obtained from equations (2) and (10).

Results and discussion

Examples are conducted to illustrate the inclination angle of cavity on the response of the coupled system. The length and height of cavity are $L_{x} = 0.91 m$ and $L_{z} = 0.55 m$ . The widths of cavity at $z = 0$ and $z = L_{z}$ are $L_{y} = 0.78 + L_{z} \tan α m$ and $L_{y} = 0.78 m$ , respectively, as shown in Figure 1. The thickness of the clamped flexible wall is set as $h = 0.002 m$ . The panel material properties are taken as follows: the panel material is aluminum, with density $ρ = 2770 kg/ m^{3}$ , Young’s modulus $E = 71 Gpa$ , Poisson’s ratio $μ = 0.33$ . The modal loss factors of panel and cavity sound field are 0.003 and 0.001, respectively.

In order to analyze the effect geometry irregularity on the behavior of vibro-acoustic system, two elevation angles of inclined wall of trapezoidal cavity are considered, including the elevation angle ( $α = 10 °$ ) and a larger one ( $α = 45 °$ ). From previous works on irregular cavity,^11–15 the higher irregularity of cavity, the more boundary regular cavity modes are needed to reduce the truncation errors. In subsequent analysis, 588 bounding rectangular cavity modes and 100 flexible panel modes are used.

Modal coupling coefficient

Modal coupling coefficient is the space matching extent between cavity mode and flexible wall mode on the interfacing surface of cavity sound field and flexible wall. For the rectangular cavity, the coupling coefficient between cavity sound field and flexible wall was analyzed.¹ As for the irregular cavity, Li and Cheng^13,15 studied the coupling coefficients between modes of trapezoidal cavity and simply supported flexible wall. But for the case of clamped flexible wall, the coupling coefficient is still unknown. In this section, the modal coupling coefficient between trapezoidal cavity and clamped flexible wall is proposed. In addition, the variation of coupling selection between them with different elevation angle is also analyzed and compared with the case of regular cavity.

When the cavity is covered by a rigid wall, the eigenvalue equation for trapezoidal cavity can be obtained from equation (18) as

(ω_{l m n}^{2} - ω^{2}) \sum_{r = 1}^{L} \sum_{s = 1}^{M} \sum_{t = 1}^{N} M_{l m n, r s t}^{a} P_{r s t} = - ρ_{0} c_{0}^{2} A_{W} \sum_{r = 1}^{L} \sum_{s = 1}^{M} \sum_{t = 1}^{N} I_{l m n, r s t} P_{r s t}

(30)

A matrix form equation can be constituted by substituting $γ = - j ω$ into equation (30) and it can be shown that

(γ^{2} M_{a} + S_{a}) P_{a} = 0

(31)

M_{a} = [\begin{matrix} M_{111, 111}^{a} & \dots & M_{111, L M N}^{a} \\ ⋮ & ⋱ & ⋮ \\ M_{L M N, 111}^{a} & \dots & M_{L M N, L M N}^{a} \end{matrix}]

(32)

S_{a} = [\begin{matrix} ω_{111}^{2} M_{111, 111}^{a} + ρ_{0} c_{0}^{2} A_{W} I_{111, 111} & \dots & ω_{111}^{2} M_{111, L M N}^{a} + ρ_{0} c_{0}^{2} A_{W} I_{111, L M N} \\ ⋮ & ⋱ & ⋮ \\ ω_{L M N}^{2} M_{L M N, 111}^{a} + ρ_{0} c_{0}^{2} A_{W} I_{L M N, 111} & \dots & ω_{L M N}^{2} M_{L M N, L M N}^{a} + ρ_{0} c_{0}^{2} A_{W} I_{L M N, L M N} \end{matrix}]

(33)

P_{a} = {[P_{1}, \dots P_{L M N}]}^{T}

(34)

The eigenvalue and corresponding eigenvector of the trapezoidal cavity can be obtained by equation (31). For the lmnth eigenvalue, $γ_{l m n}^{2}$ , there is a corresponding set of modal pressure amplitude of rigid wall trapezoidal cavity, $P_{a, l m n} = {[P_{l m n, 1}, \dots P_{l m n, L M N}]}^{T}$ . Then, the mode shape of rigid wall trapezoidal cavity $φ_{l m n}^{α}$ can be achieved as

φ_{l m n}^{α} = \sum_{r = 1}^{L} \sum_{s = 1}^{M} \sum_{t = 1}^{N} P_{l m n, r s t}^{α} φ_{r s t}

(35)

Here, the lmnth modal amplitude of trapezoidal cavity $P_{l m n, r s t}$ is replaced by $P_{l m n, r s t}^{α}$ .

Then, the modal coupling coefficient $L_{l m n, i j}^{α}$ between trapezoidal cavity mode and clamped panel mode is developed as

\begin{array}{l} L_{l m n, i j}^{α} = \frac{1}{A_{f}} \int_{A_{f}} φ_{l m n}^{α} ϕ_{i j} d S = \frac{1}{A_{f}} \int_{A_{f}} \sum_{r = 1}^{L} \sum_{s = 1}^{M} \sum_{t = 1}^{N} P_{l m n, r s t}^{α} φ_{r s t} ϕ_{i j} d S \\ = \sum_{r = 1}^{L} \sum_{s = 1}^{M} \sum_{t = 1}^{N} P_{l m n, r s t}^{α} \frac{1}{A_{f}} \int_{A_{f}} φ_{r s t} ϕ_{i j} d S = \sum_{r = 1}^{L} \sum_{s = 1}^{M} \sum_{t = 1}^{N} P_{l m n, r s t}^{α} L_{r s t, i j} \end{array}

(36)

In order to analyze the effect of elevation wall on the coupling features between sound field in cavity and flexible wall, the coupling coefficients between clamped panel modes and trapezoidal cavity modes with two different flexible wall locations are presented in Tables 1 to 4. One is that the flexible wall is located at $z = L_{z}$ as shown in Figure 1, the other is located at $y = 0$ . Compared with the case of the rectangular one, the coupling coefficients between trapezoidal cavity and clamped flexible wall modes are changed. It is shown that:

Table 1.

Coupling coefficient between clamped panel and cavity modes, flexible wall at $z = L_{z}$ , $α = 10 °$ .

Mode	(0,1,0)	(0,0,1)	(0,1,1)	(0,2,1)	(1,0,0)	(1,1,0)	(1,0,1)	(1,1,1)	(2,0,0)
(1,1)	0.099	0.689	0.176	0.299	0.000	0.000	0.000	0.000	0.327
(1,2)	0.373	0.037	0.336	0.205	0.000	0.000	0.000	0.000	0.000
(1,3)	0.029	0.271	0.020	0.185	0.000	0.000	0.000	0.000	0.143
(1,4)	0.211	0.025	0.195	0.050	0.000	0.000	0.000	0.000	0.000
(1,5)	0.016	0.170	0.017	0.124	0.000	0.000	0.000	0.000	0.091
(2,1)	0.000	0.000	0.000	0.000	0.435	0.062	0.434	0.111	0.000
(2,2)	0.000	0.000	0.000	0.000	0.000	0.235	0.023	0.212	0.000
(2,3)	0.000	0.000	0.000	0.000	0.190	0.018	0.171	0.012	0.000
(2,4)	0.000	0.000	0.000	0.000	0.000	0.133	0.016	0.123	0.000
(3,1)	0.043	0.302	0.077	0.131	0.000	0.000	0.000	0.000	0.338
(3,2)	0.163	0.016	0.147	0.090	0.000	0.000	0.000	0.000	0.000
(3,3)	0.013	0.119	0.009	0.081	0.000	0.000	0.000	0.000	0.148
(3,4)	0.092	0.011	0.086	0.022	0.000	0.000	0.000	0.000	0.000

When the cavity is changed from rectangular to trapezoidal with $α = 10 °$ or $α = 45 °$ , the coupling selection between cavity mode and clamped flexible wall mode is still satisfied in the $x$ directions where the inclination wall is parallel to it.

When the elevation angle $α = 10 °$ or $α = 45 °$ , the coupling selection between cavity modes and clamped panel modes is not valid in $y$ - and $z$ -axis directions, and more clamped panel modes participate in the coupling. Even though the sum of mode indices of trapezoidal cavity and flexible wall is equal to even number in these directions, the coupling coefficient between them may be nonzero. For example, among the researched clamped modes (1,1),…,(3,4), four more, i.e. (1,2), (1,4), (3,2), and (3,4) and cavity mode (0,2,1) when the flexible wall located at $z = L_{z}$ have nonzero coupling coefficients for $α = 10 °$ and $α = 45 °$ compared with the rectangular cavity as shown in Tables 1 and 2. As for the flexible wall located at $y = 0$ , among the researched clamped modes (1,1),…,(3,4), five more, i.e. (1,1), (1,3), (1,5), (3,1), and (3,3) and cavity mode (0,2,1) have nonzero coupling coefficients for $α = 10 °$ and $α = 45 °$ compared with the rectangular cavity as shown in Tables 3 and 4.

Resonance frequency

Associated with the eigenvalue $γ_{l m n}^{2}$ from equation (31), the resonance frequency of the lmnth rigid wall trapezoidal cavity mode can then be obtained as $f_{l m n} = \sqrt{- γ_{l m n}^{2}} / (2 π)$ . As for the clamped flexible wall, the resonance frequency of it can be obtained from equation (37).¹⁷ The first 36 natural frequencies of uncoupled clamped panel and the first 18 uncoupled cavity modes are presented in Table 5. The location of flexible wall on the cavity considered here and subsequently located at $z = L_{z}$ is shown in Figure 1

f_{i j} = \frac{1}{2 π} \sqrt{\frac{D}{ρ h}} \sqrt{{(\frac{λ_{i}}{L_{x}})}^{4} + {(\frac{λ_{j}}{L_{y}})}^{4} + 2 {(\frac{λ_{i} λ_{j}}{L_{x} L_{y}})}^{2} \frac{κ_{i} κ_{j}}{ξ_{i} ξ_{j}}}

(37)

\begin{matrix} k_{i} = \frac{1}{4} (1 + D_{i}^{2}) \sinh (2 λ_{i}) - \frac{1}{2} D_{i} \cosh (2 λ_{i}) - \frac{1}{2} (1 - D_{i}^{2}) \sin (λ_{i}) \cos (λ_{i}) - D_{i} \cos^{2} (λ_{i}) - D_{i}^{2} λ_{i} + \frac{3}{2} D_{i} \end{matrix}

(38)

\begin{array}{l} ξ_{i} = \frac{1}{4} (1 + D_{i}^{2}) \sinh (2 λ_{i}) + \sinh (λ_{i}) [2 D_{i} \sinh (λ_{i}) - (1 - D_{i}^{2}) \cos (λ_{i})] - (1 + D_{i}^{2}) \sin (λ_{i}) \cosh (λ_{i}) \\ + \frac{1}{2} (1 - D_{i}^{2}) \sin (λ_{i}) \cos (λ_{i}) + λ_{i} - \frac{1}{2} D_{i} [1 + \cosh (2 λ_{i})] + D_{i} \cos^{2} (λ_{i}) \end{array}

(39)

D_{i} = \frac{\cosh (λ_{i}) - \cos (λ_{i})}{\sinh (λ_{i}) - \sin (λ_{i})}

(40)

Compared with the rectangular cavity, when $α = 0 °$ , some resonance frequencies of trapezoidal cavity modes are unchanged, for example: (1,0,0), (2,0,0), and (3,0,0), the others are changed with inclination angle, as shown in Table 5. The reason is that the bounding rectangular cavity mode of trapezoidal cavity $(*, 0, 0)$ where * is a nonnegative integer belongs to the perpendicular grazing mode, $I_{l m n, r s t} = 0$ and the elevation angle α has no effect on the resonance frequency of this type of trapezoidal cavity mode.¹² The variation of the resonance frequencies of some trapezoidal cavity modes, which the bounding rectangular cavity modes belong to nonperpendicular grazing modes $(*, 0, #)$ where # is positive integer, is very small when $α = 10 °$ , and is getting larger when $α = 45 °$ , for example cavity modes (0,0,1), (1,0,1), etc. As for cavity modes (0,1,0), (0,1,1), etc. the bounding rectangular cavity modes belong to nongrazing modes $(*, #, *)$ . Compared with the rectangular cavity, the resonance frequencies of these trapezoidal cavity modes are decreased. This is because that the dimension of cavity in y direction is altered, leading to increase of acoustic wavelengths $λ_{l m n} (f_{l m n} = c_{0} / λ_{l m n})$ .

For the coupled system which consists of a trapezoidal cavity with a clamped flexible wall, Table 6 shows the resonance frequencies of it which can be calculated from equation (25). As mentioned in Tables 1 and 2, the coupling coefficients between panel and cavity modes are changed with α. It means that the coupling extent between them may also be changed. As a result, the resonance frequencies of panel- and cavity-controlled modes, as shown in Table 6, are changed with α compared with corresponding uncoupled ones in Table 5. And the variation degree is determined by the coupling strength between panel and cavity modes. Focusing on the resonance frequencies of panel- or cavity-controlled modes, they are either bigger or smaller than corresponding uncoupled ones in Table 5. For example, the resonance frequencies of panel-controlled mode are (1,1), (2,1), etc. and cavity-controlled modes are (0,1,0), (1,1,0), etc. It is because that the resonance frequencies of panel- and cavity-controlled modes will appear as “push away” phenomena when they interact.¹

Table 2.

Coupling coefficient between clamped panel and cavity modes, flexible wall at $z = L_{z}$ , $α = 45 °$ .

Mode	(0,1,0)	(0,0,1)	(0,1,1)	(0,2,1)	(1,0,0)	(1,1,0)	(1,0,1)	(1,1,1)	(2,0,0)
(1,1)	0.550	0.199	0.040	0.077	0.000	0.000	0.000	0.000	0.321
(1,2)	0.208	0.302	0.044	0.002	0.000	0.000	0.000	0.000	0.002
(1,3)	0.192	0.113	0.022	0.203	0.000	0.000	0.000	0.000	0.141
(1,4)	0.121	0.133	0.054	0.045	0.000	0.000	0.000	0.000	0.000
(1,5)	0.124	0.075	0.023	0.127	0.000	0.000	0.000	0.000	0.090
(2,1)	0.000	0.000	0.000	0.000	0.427	0.346	0.125	0.025	0.000
(2,2)	0.000	0.000	0.000	0.000	0.003	0.131	0.190	0.027	0.000
(2,3)	0.000	0.000	0.000	0.000	0.188	0.121	0.071	0.014	0.000
(2,4)	0.000	0.000	0.000	0.000	0.000	0.076	0.084	0.034	0.000
(3,1)	0.241	0.087	0.018	0.034	0.000	0.000	0.000	0.000	0.333
(3,2)	0.091	0.132	0.019	0.001	0.000	0.000	0.000	0.000	0.002
(3,3)	0.084	0.049	0.010	0.089	0.000	0.000	0.000	0.000	0.146
(3,4)	0.053	0.058	0.023	0.020	0.000	0.000	0.000	0.000	0.000

Table 3.

Coupling coefficient between clamped panel and cavity modes, flexible wall at $y = 0$ , $α = 10 °$ .

Mode	(0,1,0)	(0,0,1)	(0,1,1)	(0,2,1)	(1,0,0)	(1,1,0)	(1,0,1)	(1,1,1)	(2,0,0)
(1,1)	0.766	0.125	0.186	0.202	0.000	0.000	0.000	0.000	0.327
(1,2)	0.056	0.432	0.448	0.565	0.000	0.000	0.000	0.000	0.000
(1,3)	0.339	0.076	0.088	0.151	0.000	0.000	0.000	0.000	0.143
(1,4)	0.033	0.238	0.245	0.313	0.000	0.000	0.000	0.000	0.000
(1,5)	0.216	0.050	0.057	0.097	0.000	0.000	0.000	0.000	0.091
(2,1)	0.000	0.000	0.000	0.000	0.435	0.482	0.078	0.117	0.000
(2,2)	0.000	0.000	0.000	0.000	0.000	0.035	0.272	0.282	0.000
(2,3)	0.000	0.000	0.000	0.000	0.190	0.213	0.048	0.055	0.000
(2,4)	0.000	0.000	0.000	0.000	0.000	0.021	0.150	0.154	0.000
(3,1)	0.335	0.055	0.082	0.089	0.000	0.000	0.000	0.000	0.338
(3,2)	0.025	0.189	0.196	0.247	0.000	0.000	0.000	0.000	0.000
(3,3)	0.148	0.033	0.038	0.066	0.000	0.000	0.000	0.000	0.148
(3,4)	0.014	0.104	0.107	0.137	0.000	0.000	0.000	0.000	0.000

Table 4.

Coupling coefficient between clamped panel and cavity modes, flexible wall at $y = 0$ , $α = 45 °$ .

Mode	(0,1,0)	(0,0,1)	(0,1,1)	(0,2,1)	(1,0,0)	(1,1,0)	(1,0,1)	(1,1,1)	(2,0,0)
(1,1)	0.865	0.869	0.440	0.583	0.000	0.000	0.000	0.000	0.330
(1,2)	0.134	0.433	0.620	0.318	0.000	0.000	0.000	0.000	0.021
(1,3)	0.452	0.632	0.421	0.272	0.000	0.000	0.000	0.000	0.164
(1,4)	0.117	0.328	0.434	0.214	0.000	0.000	0.000	0.000	0.018
(1,5)	0.336	0.495	0.394	0.416	0.000	0.000	0.000	0.000	0.109
(2,1)	0.000	0.000	0.000	0.000	0.439	0.544	0.547	0.277	0.000
(2,2)	0.000	0.000	0.000	0.000	0.027	0.084	0.272	0.390	0.000
(2,3)	0.000	0.000	0.000	0.000	0.219	0.285	0.398	0.265	0.000
(2,4)	0.000	0.000	0.000	0.000	0.024	0.074	0.207	0.273	0.000
(3,1)	0.379	0.380	0.193	0.255	0.000	0.000	0.000	0.000	0.342
(3,2)	0.058	0.189	0.271	0.139	0.000	0.000	0.000	0.000	0.021
(3,3)	0.198	0.277	0.184	0.119	0.000	0.000	0.000	0.000	0.170
(3,4)	0.051	0.144	0.190	0.094	0.000	0.000	0.000	0.000	0.019

Table 5.

Resonance frequencies (Hz) of uncoupled clamped panel and cavity modes.

Mode	Panel	Mode	Panel	Mode	Cavity $α = 0 °$	Cavity $α = 10 °$	Cavity $α = 45 °$
(1 1)	25.21	(5 3)	253.30	(0 0 0)	0.00	0.00	0.00
(2 1)	46.68	(4 4)	259.04	(0 1 0)	220.51	206.95	154.62
(1 2)	56.36	(6 1)	259.09	(0 0 1)	312.73	312.61	269.19
(2 2)	76.16	(2 5)	266.92	(0 1 1)	382.65	372.15	323.17
(3 1)	80.75	(6 2)	290.40	(0 2 0)	441.03	417.87	390.01
(1 3)	103.85	(3 5)	294.39	(0 2 1)	540.65	518.95	402.53
(3 2)	107.78	(5 4)	316.17	(1 0 0)	189.01	189.01	189.01
(2 3)	122.00	(6 3)	334.65	(1 1 0)	290.43	280.27	244.20
(4 1)	127.82	(4 5)	338.85	(1 0 1)	365.41	365.31	328.92
(3 3)	150.73	(1 6)	345.66	(1 1 1)	426.79	417.40	374.38
(4 2)	154.52	(2 6)	368.36	(1 2 0)	479.82	458.63	433.40
(1 4)	168.07	(5 5)	395.74	(1 2 1)	572.74	552.30	444.70
(2 4)	186.52	(3 6)	400.11	(2 0 0)	378.02	378.02	378.02
(5 1)	186.64	(6 4)	403.31	(2 1 0)	437.64	430.96	408.42
(4 3)	195.75	(4 6)	450.67	(2 0 1)	490.61	490.54	464.07
(5 2)	213.25	(6 5)	488.03	(2 1 1)	537.89	530.47	497.33
(3 4)	214.40	(5 6)	513.58	(2 2 0)	580.86	563.49	543.15
(1 5)	248.19	(7 1)	592.02	(3 0 0)	567.03	567.03	567.03

Table 6.

Resonance frequencies (Hz) of panel- or cavity-controlled coupled system modes.

Mode	Resonance frequency			Mode	Resonance frequency
Mode	$α = 0 °$	$α = 10 °$	$α = 45 °$	Mode	$α = 0 °$	$α = 10 °$	$α = 45 °$
(1,1)	33.91	33.45	31.76	(1,6)	343.87	343.56	344.35
(2 1)	45.89	45.89	45.91	(2,6)	367.01	366.77	366.99
(1 2)	55.54	55.52	55.53	(1,0,1)	368.29	368.21	329.87
(3 1)	80.72	80.65	80.45	(2,0,0)	379.01	378.95	378.75
(1 3)	103.54	103.48	103.32	(0,1,1)	385.15	375.06	325.98
(3 2)	106.91	106.88	106.87	(5,5)	395.26	395.21	395.24
(4 1)	126.89	126.57	126.61	(3,6)	399.20	398.88	398.42
(3 3)	149.93	149.89	149.75	(6,4)	402.73	402.34	402.65
(4 2)	153.89	153.52	153.53	(1,1,1)	428.92	419.82	377.03
(1 4)	166.56	166.46	167.68	(2,1,0)	438.67	431.85	409.11
(5 1)	186.05	186.01	186.10	(0,2,0)	441.95	418.60	390.66
(1,0,0)	189.80	189.69	189.56	(4,6)	450.43	449.58	450.33
(4,3)	196.35	195.92	195.65	(1,2,0)	480.31	459.22	434.05
(0,1,0)	223.71	206.83	155.05	(2,0,1)	492.04	491.96	464.82
(1,5)	246.86	246.83	246.89	(5,6)	512.86	512.74	512.98
(5,3)	252.70	252.68	252.69	(2,1,1)	539.52	532.14	498.60
(6,1)	258.75	258.49	258.80	(0,2,1)	541.88	520.47	404.70
(2,5)	265.83	265.77	265.85	(3,0,0)	567.78	567.74	567.58
(3,5)	293.10	293.06	293.15	(1,2,1)	574.09	553.84	445.42
(1,1,0)	294.35	280.45	244.42	(2,2,0)	580.91	563.79	543.65
(0,0,1)	316.13	316.00	270.58	(7,1)	591.81	591.28	591.36

In order to verify the accuracy of the solution, the free vibration characteristics of coupled system are calculated with more rigid wall rectangular enclosure modes $(L M N = 896)$ and flexible wall modes $(I J = 169)$ when $α = 45 °$ . The resonance frequencies of some cavity-controlled modes with different mode numbers are shown in Table 7. Compared with the solution for the combination of 896 rectangular cavity modes and 169 flexible wall modes, the solution for the resonance frequencies of cavity-controlled system modes using 588 rectangular cavity modes and 100 flexible wall modes meets the requirement.

Table 7.

Resonance frequencies (Hz) of some cavity-controlled coupled system modes, $α = 45 °$ .

Mode	$L M N = 588$ $I J = 100$	$L M N = 896$ $I J = 169$	Mode	$L M N = 588$ $I J = 100$	$L M N = 896$ $I J = 169$
(0 1 0)	155.05	155.05	(1 2 0)	434.05	434.04
(0 0 1)	270.58	270.58	(1 2 1)	445.42	445.39
(0 1 1)	325.98	325.97	(2 0 0)	378.75	378.74
(0 2 0)	390.66	390.64	(2 1 0)	409.11	409.11
(0 2 1)	404.70	404.68	(2 0 1)	464.82	464.82
(1 0 0)	189.56	189.56	(2 1 1)	498.60	498.60
(1 1 0)	244.42	244.42	(2 2 0)	543.65	543.63
(1 0 1)	329.87	329.87	(3 0 0)	567.58	567.58
(1 1 1)	377.03	377.03

IJ: (I, J) are the numbers of the terms to be kept after the truncation of the series; LMN: (L,M,N) are the numbers of the terms to be kept after the truncation of the series.

Forced response of the coupled system

When the clamped flexible wall on the trapezoidal cavity is excited by an incident plane wave outside of the cavity, the response of coupled system is solved using equation (25). The results are compared with the case of the regular cavity one. Here, the time-averaged acoustic potential energy and the time-averaged panel vibration kinetic energy are used to describe the response of cavity sound field and clamped panel.

The time-averaged acoustic potential energy $E_{a}$ and the time-averaged vibration kinetic energy $E_{p}$ of the flexible wall can be calculated by⁷

E_{a} = \frac{1}{4 ρ_{0} c_{0}^{2}} \int_{V_{T}} {| p (r, ω) |}^{2} d V = \frac{P^{H} M^{a} P}{4 ρ_{0}^{2} c_{0}^{2}}

(41)

E_{p} = \frac{ρ h}{4} \int_{A_{f}} {| ν (σ, ω) |}^{2} d S = \frac{V^{H} M^{p} V}{4}

(42)

where superscript H denotes the Hermitian transpose,

P = {[P_{1}, P_{2} \dots P_{L M N}]}^{T}

and

V = {[V_{1}, V_{2} \dots V_{I J}]}^{T}

M^{a}

and

M^{p}

are the general modal mass matrix of the cavity sound field and clamped panel, respectively.

Figure 2 presents the acoustic potential energy level value and panel vibration kinetic energy level value as a function of frequency (20–600 Hz) under a normally incident plane wave ( $θ_{1} = 180 °$ , $θ_{2} = 0 °$ ) excitation, and the amplitude of plane wave is 1 Pa. The elevation angle $α = 10 °$ and $α = 45 °$ are considered here. The peaks in acoustic potential energy level and panel vibration energy level versus frequency curves reflect the panel- or cavity-controlled coupled system modes.

Figure 2.

Response of coupled system with excitation of an incident plane wave ( $θ_{1} = 180 °$ , $θ_{2} = 0 °$ ). □: modes dominated by the clamped panel; ▽: modes dominated by the cavity when $α = 0 °$ ; △: modes dominated by the cavity when $α = 10 °$ ; o: modes dominated by the cavity when $α = 45 °$ . (a) Acoustic potential energy of the cavity (dB ref.=10⁻¹²J) and (b) vibration kinetic energy of the plate (dB ref.=10⁻⁹J).

When the cavity is changed from rectangular to trapezoidal with $α = 10 °$ or $α = 45 °$ , more cavity-controlled modes are evoked, for example (0,1,0), (0,1,1), etc. as shown in Figure 2(a). Meanwhile, the panel-controlled mode (3,4) is evoked when $α = 10 °$ , and the panel-controlled mode (1,4) is evoked when $α = 45 °$ . The reason is that the closeness between the natural frequency of cavity mode (0,1,0) ( $f_{010} = 206.95 Hz$ ) and that of the panel mode (3,4) ( $f_{34} = 214.40 Hz$ ) when $α = 10 °$ , and this cavity mode ( $f_{010} = 154.62 Hz$ ) and that of panel mode (1,4) ( $f_{14} = 168.07 Hz$ )when $α = 45 °$ , as shown in Table 5. In the meanwhile, the coupling coefficients between them are nonzero as shown in Tables 1 and 2. The frequencies of peaks which reflect the panel-controlled modes in the acoustic potential energy level curves are almost unchanged with different $α$ . Some cavity-controlled modes in which their bounding rectangular cavity modes belong to perpendicular grazing mode take the same form, for example (2,0,0), etc. The variation of the frequencies of peaks of some cavity-controlled modes in which their bounding rectangular cavity modes belong to nonperpendicular grazing modes is small when $α = 10 °$ and getting larger when $α = 45 °$ , for example (0,0,1). For the cavity-controlled modes in which their bounding rectangular cavity modes belong to nongrazing modes, the peaks of these coupled system modes are shifted to low frequency, and the variation is getting larger when $α$ increases, e.g. (0,2,1) and so on. As mentioned in Tables 5 and 6, the left shift of peaks of cavity-controlled coupled system modes are determined by three factors, including the kind of the bounding rectangular modes of cavity, the coupling strength between modes of cavity and flexible wall, and the elevation angle of cavity.

Figure 2(b) shows the panel vibration kinetic energy level with different elevation angles. When the cavity shape is changed from rectangular to trapezoidal with $α = 10 °$ or $α = 45 °$ , the additional cavity-controlled mode (0,1,0) is evoked. Meanwhile, the additional panel-controlled mode (3,4) and (1,4) is evoked when $α = 10 °$ and $α = 45 °$ , respectively. The reason is the same as mentioned above. When $α = 45 °$ , the additional cavity-controlled mode (0,1,1) is also evoked. Compared with the acoustic potential energy in the cavity sound field, the effect of elevation wall on the vibration kinetic energy in the panel is small.

Figure 3 shows the sound pressure level in the cavity at $x = L_{x} / 2$ when the excitation frequencies of plane wave are 542, 520, and 405 Hz. As shown in Table 6, they are corresponding to the resonance frequencies of cavity-controlled mode (0,2,1) when $α = 0 °$ , $α = 10 °$ , and $α = 45 °$ , respectively. Obviously, the (0,2,1) cavity-controlled mode is excited when $α = 0 °$ and $α = 10 °$ as shown in Figure 3(a) and (e). When the excitation frequency is 520 Hz with $α = 0 °$ and the excitation frequency is 542 Hz with $α = 10 °$ , the response of acoustic field cannot be recognized as shown in Figure 3(b) and (d). The response of acoustic field appears one nodal line in the direction of z-axis when the excitation frequency is 404 Hz with $α = 0 °$ and $α = 10 °$ . When $α = 45 °$ , the cavity deviates more from a rectangular shape, the response of acoustic field cannot be recognized, as shown in Figure 3(g) to (i).

Figure 3.

Acoustic pressure level (dB) distribution at the cross-section $x = L_{x} / 2$ of the cavity. (a) f ( $α = 0 °$ )=542 Hz, (b) f ( $α = 0 °$ )=520 Hz, (c) f ( $α = 0 °$ )=405 Hz, (d) f ( $α = 10 °$ )=542 Hz, (e) f ( $α = 10 °$ )=520 Hz, (f) f ( $α = 10 °$ )=405 Hz, (g) f ( $α = 45 °$ )=542 Hz, (h) f ( $α = 45 °$ )=520 Hz, and (i) f ( $α = 45 °$ )=405 Hz.

When the elevation angle and the azimuth angle of incident plane wave are $135 °$ and $45 °$ , the acoustic potential energy level and clamped panel vibration kinetic energy level are presented in Figure 4. Compared with the normal incident plane wave, more panel- and cavity-controlled modes are evoked when an elevation plane wave acts on the panel. Similar to the normal plane wave, the low-frequency clamped panel-controlled mode (1,1) dominates the response of coupled system as shown in Figure 4(a) and (b). When the elevation angle α becomes from $0 °$ to $10 °$ , the peaks of panel-controlled modes are almost unchanged. Some cavity-controlled modes, however, are shifted to low frequency, as shown in Table 6.

Figure 4.

The response of cavity with excitation of the incident plane wave ( $θ_{1} = 135 °$ , $θ_{2} = 45 °$ ). □: modes dominated by the clamped panel; ▽: modes dominated by the cavity when $α = 0 °$ ; △: modes dominated by the cavity when $α = 10 °$ . (a) Acoustic potential energy of the cavity (dB ref.=10⁻¹²J) and (b) vibration kinetic energy of the plate (dB ref.=10⁻⁹J).

Conclusions

A structural-acoustic coupling model for a trapezoidal cavity and its clamped flexible wall is proposed and the modal coupling coefficient between trapezoidal cavity sound field and clamped panel modes is developed. The coupling characteristics between cavity sound field and flexible wall, and the forced response of coupled system are then analyzed. In addition, the rectangular cavity is considered and compared with the case of trapezoidal cavity. The results show that:

Compared with rectangular cavity, there are more clamped panel modes participating in the coupling with trapezoidal cavity modes. The coupling selection between trapezoidal cavity modes and clamped panel modes is not satisfied except in the axis direction which is parallel to the inclined wall.

When cavity changes from regular shape to trapezoidal, the variation of resonance frequencies of panel-controlled modes is only determined by the coupling extent between these panel modes and cavity modes. However, the variation of resonance frequencies of cavity-controlled modes is not only determined by the coupling extent between these cavity modes and panel modes, but also depends on the elevation angle and the kind of the bounding rectangular mode of cavity.

The forced response of coupled system is conducted with the excitation of a plane wave. When the cavity becomes trapezoidal from rectangular, more clamped panel- and cavity-controlled modes are evoked. The peaks of some cavity-controlled modes of acoustic potential energy curves versus frequency are shifted to low frequency, particularly for the cavity mode whose bounding rectangular cavity mode belongs to nongrazing mode. Compared with the acoustic potential energy in the cavity, the vibration kinetic energy of the clamped panel is not sensitive of the cavity geometry shape. When the plane wave changes from normal incident to oblique incident, more clamped panel- and cavity-controlled modes are evoked.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors thank for the support of senior professional talent research funds of Jiangsu University (1291110064, 14JDG138), the key project of natural science in colleges and universities in Jiangsu Province (16KJB140002, 15KJB460005) and postdoctoral science foundation of Jiangsu Province (1501102C).

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