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Abstract

The coupled acoustic field of fully elastic plate model is described by the modal analysis method. The acoustic potential energy resonance peaks of the fully elastic plate model are significantly more than that of the one elastic plate model due to the influence of the vibration of multi elastic plates. The acoustic field characteristics of the fully elastic plate model are analyzed when the primary excitation source is applied on the different elastic plates. The results show that the coupled acoustic field of the fully elastic plate model is dominated by the structural mode of the elastic plate with primary excitation, and the acoustic mode of the enclosure, and the structural-acoustic coupling between the plate and the enclosure; the structure modes of the other elastic plates have less effects on the acoustic field in the enclosure except the first ones of them.

Keywords

Acoustic field modeling acoustic characteristics enclosure multi elastic plates

Introduction

The noise field in an enclosure is the most representative acoustic fields in practice, such as aircraft,^1–3 ship cabin, automobile cab,^4,5 work shop, living room⁶ and so on.⁷ The initial research on active control of enclosed acoustic field began in the 1950s, but due to the limitation of the electronic technology, the active control technology of the enclosed acoustic field has made little progress within the next 30 years. Since the 1980s, with the development of the microprocessor technology, the realization of the active control system of enclosed acoustic field is possible, especially in the area of transportation. Therefore, the study of the active control of enclosed sound field is of great practical value.

In the study of the active noise control of an enclosure, the inner wall of the enclosure is generally treated by rigid or elastic walls. The enclosed acoustic field characteristics of elastic structures are not only related to the characteristics of the noise source and the geometric characteristics of the enclosure, but also directly related to the vibration state of the elastic wall. This is called “structural-acoustic coupling analysis” problem. At present, structural vibration and acoustic researchers are concerned about it.^8,9 The structural-acoustic coupling not only has important influence on the secondary source position, but also determines the active control mode selection.

Thus far, an enclosure consisting of one elastic plate and five rigid walls (termed as one elastic plate model in this paper)^10–18 is the main research model of the plate-cavity system. Dowell and Voss¹⁰ first study the response of plate-cavity coupling system. Further investigations by Guy,¹¹ etc. have improved theoretical methods and physical insight. Naryannan and Shanbhag^12,13 discuss the problem of acoustic transmission and structural response of a sandwich panel backed by a cavity in an acoustoelastic formulation. Pan and Bies^14,15 present a theoretical investigation into the effect of the interaction between an acoustic field and its boundaries upon the characteristics of the acoustic field in an enclosure. Peretti and Dowell¹⁶ analyze the influence of the elastic plate vibration on the acoustic field in an enclosure.

Also, some researchers study coupled acoustic field in an enclosure consisting of two flexible plates and four rigid walls.^19–21 Jin et al.¹⁹ present an investigation into the active control of acoustic transmission into a structural-acoustic coupled system. An enclosure with four acoustically rigid walls and two flexible plates is considered. Geng et al.²⁰ develop a new modeling method for the active minimization of noise within a three-dimensional irregular enclosure. The irregular enclosure is modeled with four rigid walls and two simply supported flexible panels. Cui and Chen²¹ present an investigation into the active control of acoustic radiation and transmission into a structural-acoustic coupled system. A rectangular enclosure involving two simply supported flexible plates is considered. But so far, the research on the acoustic field in a rectangular enclosure consisting of six elastic plates (termed as the fully elastic plate model in this paper) has not been reported.

The first problem encountered in active noise control is a determination of the primary acoustic field distribution, and then we can discuss how to use the secondary acoustic source to produce an “anti-acoustic field” which is matched with it. Thus, the accurate description and analysis of the noise field are of great importance. Secondly, the secondary acoustic source is not only related to the type of acoustic field, but also directly with the generation of primary acoustic field and the physical characteristics of peripheral structures. Hence, in order to accurately describe the coupled acoustic field characteristics, this paper presents the fully elastic plate model as shown in Figure 1. The acoustic pressure in the enclosure is expanded in terms of the normal modes of the rigid-walled cavity. The equations of motion of the elastic wall are also derived in terms of in vacuo structural normal modes. The effectiveness of the method by using the normal modes of the cavity with rigid walls has been verified by experiments.²² The primary excitation source is applied on the different elastic plates according to the three disturbing cases. By analyzing the primary acoustic field in the three different cases, the coupled acoustic field characteristics of the fully elastic plate model are obtained, which lay a theoretical foundation for the design of active structural acoustic. It is necessary to point out that the method in this paper is only applicable to the weak coupling problem. Therefore, the conclusions in this paper are generally used only in the case of simple shape, and good rigidity of the plate or deep cavity.

Figure 1.

Acoustic cavity model with six elastic plates.

Acoustic field model of fully elastic plates

Figure 1 shows panel-cavity system that is modeled in this analysis. A rectangular enclosure consisting of six simply supported elastic plates are labeled as Plate a, a′, b, b′, c and c′ and located at z = L_z, z = 0, y = L_y, y = 0, x = L_x and x = 0, respectively. The simply supported boundary conditions for these flexible plates are taken into account because a simply supported rectangular panel closely approximates many actual structural configurations. The enclosure is 0.868 m long (L_x), 1.15 m wide (L_y), and 1.0 m high (L_z). The elastic panels with thicknesses of 6 mm are all aluminum plates with the material properties of Young’s modulus E = 71 GPa, mass density ρ₁ = 2700 kg/m³, and Poisson’s ratio υ = 0.3. The acoustic speed is c = 344 m/s, and mass density of air is ρ = 1.21 kg/m³.

The characteristics of coupled acoustic field in the enclosure are discussed, respectively, according to three cases: (1) the primary excitation source F_a applied on the Plate a; (2) the primary excitation source F_b applied on the Plate b; (3) the primary excitation sources F_a and F_b applied simultaneously on the Plate a and Plate b, respectively. The forces F_a and F_b are located at σ ₀ = (0.3, 0.4, 1.0) m on the Plate a and σ ₁ = (0.5, 1.15, 0.3) m on the Plate b, respectively. The position of the force used in this paper is as far as possible to avoid the nodal line of normal modes, in order to excite as much as possible the structural modes. The amplitudes of the forces are both 1 N.

Excitation F_a applied on the plate a

Theoretical formulation

Consider only the excitation source F_a applied on the Plate a, i.e. F_b = 0. With the system described above, the acoustic pressure in the enclosure is governed by the inhomogeneous wave equation as

\nabla 2 p - \frac{1}{c 2} \frac{\partial 2 p}{\partial t 2} = ρ \frac{\partial 2 w_{a}}{\partial t 2} δ (z - L_{z}) + ρ \frac{\partial 2 w_{a'}}{\partial t 2} δ (z) + ρ \frac{\partial 2 w_{b}}{\partial t 2} δ (y - L_{y}) + ρ \frac{\partial 2 w_{b'}}{\partial t 2} δ (y) + ρ \frac{\partial 2 w_{c}}{\partial t 2} δ (x - L_{x}) + ρ \frac{\partial 2 w_{c'}}{\partial t 2} δ (x)

(1)

where p is the acoustic pressure,

\nabla 2

is the Laplacian operator, w_a,

w_{a'}

, w_b,

w_{b'}

, w_c,

w_{c'}

are the displacements of the plates a,

a'

, b,

b'

, c,

c'

, respectively. δ(.) is one-dimensional Dirac delta function of a coordinate normal to the surface.

The equations of motion of the Plate a can be expressed as

ρ_{a} h_{a} {\overset{··}{w}}_{a} + D_{a} \nabla 4 w_{a} = F_{a} δ' (σ_{a} - σ_{0}) - p (σ_{a}, t)

(2)

where ρ_a, h_a, D_a are the material density, the thickness, and the bending stiffness of Plate a,

D_{a} = E_{a} h_{a}^{3} / 12 (1 - υ_{a})

; here, E_a, v_a represent Young’s modulus and Poisson’s ratio of Plate a;

\nabla 4 = \nabla 2 \nabla 2

; σ _a denotes the coordinate on the surface of Plate a; the term p( σ _a , t) is the acoustic pressure on the inner surface of Plate a; δ′ (.) is the two-dimensional Dirac function.

The equation of motion of the rest of the elastic plates is expressed as

ρ_{k} h_{k} {\overset{··}{w}}_{k} + D_{k} \nabla 4 w_{k} = - p (σ_{k}, t), k = a', b, b', c, c'

(3)

where ρ_k, h_k, D_k are the material density, the thickness, and the bending stiffness of Plate k,

D_{k} = E_{k} h_{k}^{3} / 12 (1 - υ_{k})

; here, E_k, υ_k represent Young’s modulus and Poisson’s ratio of Plate k; σ _k denotes the coordinate on the surface of Plate k; the term p( σ _k , t) is the acoustic pressure on the inner surface of Plate k.

If the acoustic pressure in the enclosure and the structural vibration of plates are assumed to be described by the summation of N and M, S, L modes, respectively, the acoustic pressure and the normal structural vibration velocities of plates are given, respectively, by^23,24

p (r, t) = \sum_{n = 1}^{N} ψ_{n} (r) P_{n} (t)

(4)

u_{a} (σ_{a}, t) = \sum_{m = 1}^{M} ϕ_{m} (σ_{a}) V_{am} (t)

(5)

u_{a'} (σ_{a'}, t) = \sum_{m = 1}^{M} ϕ_{m} (σ_{a'}) V_{a' m} (t)

(6)

u_{b} (σ_{b}, t) = \sum_{s = 1}^{S} ϕ_{s} (σ_{b}) V_{bs} (t)

(7)

u_{b'} (σ_{b'}, t) = \sum_{s = 1}^{S} ϕ_{s} (σ_{b'}) V_{b' s} (t)

(8)

u_{c} (σ_{c}, t) = \sum_{l = 1}^{L} ϕ_{l} (σ_{c}) V_{cl} (t)

(9)

u_{c'} (σ_{c'}, t) = \sum_{l = 1}^{L} ϕ_{l} (σ_{c'}) V_{c' l} (t)

(10)

where

ψ_{n} (r) = \cos (n_{1} π x / L_{x}) \cos (n_{2} π y / L_{y}) \cos (n_{3} π z / L_{z})

is the uncoupled acoustic mode shape function, (n₁, n₂, n₃) are the modal indices of the nth cavity mode;

ϕ_{m} (σ_{a}) = ϕ_{m} (σ_{a'}) = \sin (m_{1} π x / L_{x}) \sin (m_{2} π y / L_{y})

ϕ_{s} (σ_{b}) = ϕ_{s} (σ_{b'}) = \sin (s_{1} π x / L_{x}) \sin (s_{2} π z / L_{z})

and

ϕ_{l} (σ_{c}) = ϕ_{l} (σ_{c'}) = \sin (l_{1} π y / L_{y}) \sin (l_{2} π z / L_{z})

are the uncoupled structural mode shape functions under simply supported boundary conditions, (m₁, m₂) are the modal indices of the mth mode of the plates a and

a'

, (s₁, s₂) are the modal indices of the sth mode of the plates b and

b'

, (l₁, l₂) are the modal indices of the lth mode of the plates c and

c'

; P_n(t), V_am(t),

V_{a' m} (t)

, V_bs(t),

V_{b' s} (t)

, V_cl(t) and

V_{c' l} (t)

are time-dependent modal amplitudes to be determined. According to the orthogonal properties of the uncoupled acoustic and structural mode shape functions, we have

M_{n} = \frac{1}{V} \int_{V} ψ_{n} (r) ψ_{n} (r) d V

(11)

M_{am} = ρ_{a} h_{a} \int_{A_{a}} ϕ_{m} (σ_{a}) ϕ_{m} (σ_{a}) d A

(12)

M_{a' m} = ρ_{a'} h_{a'} \int_{A_{a'}} ϕ_{m} (σ_{a'}) ϕ_{m} (σ_{a'}) d A

(13)

M_{bs} = ρ_{b} h_{b} \int_{A_{b}} ϕ_{s} (σ_{b}) ϕ_{s} (σ_{b}) d A

(14)

M_{b' s} = ρ_{b'} h_{b'} \int_{A_{b'}} ϕ_{s'} (σ_{b'}) ϕ_{s'} (σ_{b'}) d A

(15)

M_{cl} = ρ_{c} h_{c} \int_{A_{c}} ϕ_{l} (σ_{c}) ϕ_{l} (σ_{c}) d A

(16)

M_{c' l} = ρ_{c'} h_{c'} \int_{A_{c'}} ϕ_{l} (σ_{c'}) ϕ_{l} (σ_{c'}) d A

(17)

where M_n is the nth acoustic modal mass; M_am and

M_{a' m}

are the mth structural modal masses of plates a and

a'

, respectively; M_bs and

M_{b' s}

are the sth structural modal masses of plates b and

b'

; M_cl and

M_{c' l}

are the lth structural modal masses of plates c and

c'

; A_a,

A_{a'}

, A_b,

A_{b'}

, A_c and

A_{c'}

are the areas of plates a,

a'

, b,

b'

, c and

c'

, respectively; V is the volume of the enclosure. Note the relationship between displacement and velocity, i.e. u ( σ , t) = jωw ( σ , t). Considering the case for a harmonic external disturbance with frequency ω, utilizing equations (4) to (17), and taking into account the modal damping terms, the following equations for the structural-acoustic coupled system can be derived from equations (1) to (3) by Fourier transform

P_{n} (ω) = \frac{h_{n} (ω) ρ c 2}{{VM}_{n}} [A_{a} \sum_{m = 1}^{M} C_{nm}^{a} V_{am} (ω) + A_{a'} \sum_{m = 1}^{M} C_{nm}^{a'} V_{a' m} (ω) + A_{b} \sum_{s = 1}^{S} C_{ns}^{b} V_{bs} (ω) + A_{b'} \sum_{s = 1}^{S} C_{ns}^{b'} V_{b' s} (ω) + A_{c} \sum_{l = 1}^{L} C_{nl}^{c} V_{cl} (ω) + A_{c'} \sum_{l = 1}^{L} C_{nl}^{c'} V_{c' l} (ω)]

(18)

V_{am} (ω) = \frac{h_{am} (ω)}{M_{am}} [Q_{F_{a} m} (ω) - A_{a} \sum_{n = 1}^{N} C_{nm}^{a} P_{n} (ω)]

(19)

V_{a' m} (ω) = \frac{h_{a' m} (ω)}{M_{a' m}} [- A_{a'} \sum_{n = 1}^{N} C_{nm}^{a'} P_{n} (ω)]

(20)

V_{bs} (ω) = \frac{h_{bs} (ω)}{M_{bs}} [- A_{b} \sum_{n = 1}^{N} C_{ns}^{b} P_{n} (ω)]

(21)

V_{b' s} (ω) = \frac{h_{b' s} (ω)}{M_{b' s}} [- A_{b'} \sum_{n = 1}^{N} C_{ns}^{b'} P_{n} (ω)]

(22)

V_{cl} (ω) = \frac{h_{cl} (ω)}{M_{cl}} [- A_{c} \sum_{n = 1}^{N} C_{nl}^{c} P_{n} (ω)]

(23)

V_{c' l} (ω) = \frac{h_{c' l} (ω)}{M_{c' l}} [- A_{c'} \sum_{n = 1}^{N} C_{nl}^{c'} P_{n} (ω)]

(24)

where the acoustic mode resonance term is given by

h_{n} (ω) = {1 / (1 / T_{a} + j ω) n = 1 j ω / (ω_{n}^{2} + 2 j ξ_{n} ω_{n} ω - ω 2) n \neq 1

where T_a is the time constant of the first mode;²⁵ the structural mode resonance terms are given by

h_{am} (ω) = h_{a' m} (ω) = j ω / (ω_{am}^{2} + 2 j ξ_{am} ω_{am} ω - ω 2)

h_{bs} (ω) = h_{b' s} (ω) = j ω / (ω_{bs}^{2} + 2 j ξ_{bs} ω_{bs} ω - ω 2)

h_{cl} (ω) = h_{c' l} (ω) = j ω / (ω_{cl}^{2} + 2 j ξ_{cl} ω_{cl} ω - ω 2)

where ξ_n, ξ_am, ξ_bs and ξ_cl are the modal damping coefficients of fluid and plates, ω_n and ω_am, ω_bs, ω_cl are the natural angular frequencies of the uncoupled cavity and plates, respectively; the coupling coefficients between the acoustic mode and structural modes are given by

C_{nm}^{a} = (1 / A_{a}) \int_{A_{a}} ψ_{n} (z = L_{z}) ϕ_{m} (σ_{a}) d A

(25)

C_{nm}^{a'} = (1 / A_{a'}) \int_{A_{a'}} ψ_{n} (z = 0) ϕ_{m} (σ_{a'}) d A

(26)

C_{ns}^{b} = (1 / A_{b}) \int_{A_{b}} ψ_{n} (y = L_{y}) ϕ_{s} (σ_{b}) d A

(27)

C_{ns}^{b'} = (1 / A_{b'}) \int_{A_{b'}} ψ_{n} (y = 0) ϕ_{s} (σ_{b'}) d A

(28)

C_{nl}^{c} = (1 / A_{c}) \int_{A_{c}} ψ_{n} (x = L_{x}) ϕ_{l} (σ_{c}) d A

(29)

C_{nl}^{c'} = (1 / A_{c'}) \int_{A_{c'}} ψ_{n} (x = 0) ϕ_{l} (σ_{c'}) d A

(30)

The generalized modal force $Q_{F_{a} m}$ can be expressed as

Q_{F_{a} m} = \int_{A_{a}} F_{a} δ (σ_{a} - σ_{0}) ϕ_{m} (σ_{a}) d A

(31)

Combining equations (18) to (24), the acoustic and structural modal amplitude vectors can be expressed as

P = Z_{a} V_{a} + \sum_{k} Z_{k} V_{k} k = a', b, b', c, c'

(32)

V_{a} = Q_{F_{a}} - H_{a} P

(33)

V_{k} = - H_{k} P

(34)

where

Z_{a} = [\frac{h_{1} ρ c 2 A_{a}}{{VM}_{1}} C_{1}^{a}, \frac{h_{2} ρ c 2 A_{a}}{{VM}_{2}} C_{2}^{a}, \dots, \frac{h_{n} ρ c 2 A_{a}}{{VM}_{n}} C_{n}^{a}] T

(35)

Z_{k} = [\frac{h_{1} ρ c 2 A_{k}}{{VM}_{1}} C_{1}^{k}, \frac{h_{2} ρ c 2 A_{k}}{{VM}_{2}} C_{2}^{k}, \dots, \frac{h_{n} ρ c 2 A_{k}}{{VM}_{n}} C_{n}^{k}] T

(36)

H_{a} = [\frac{h_{a 1} A_{a}}{M_{a 1}} {C_{1}^{a'}} T, \frac{h_{a 2} A_{a}}{M_{a 2}} {C_{2}^{a'}} T, \dots, \frac{h_{am} A_{a}}{M_{am}} {C_{m}^{a'}} T] T

(37)

H_{a'} = [\frac{h_{a' 1} A_{a'}}{M_{a' 1}} {C_{1}^{a ″}} T, \frac{h_{a' 2} A_{a'}}{M_{a' 2}} {C_{2}^{a ″}} T, \dots, \frac{h_{a' m} A_{a'}}{M_{a' m}} {C_{m}^{a ″}} T] T

(38)

H_{b} = [\frac{h_{b 1} A_{b}}{M_{b 1}} {C_{1}^{b'}} T, \frac{h_{b 2} A_{b}}{M_{b 2}} {C_{2}^{b'}} T, \dots, \frac{h_{bs} A_{b}}{M_{bs}} {C_{s}^{b'}} T] T

(39)

H_{b'} = [\frac{h_{b' 1} A_{b'}}{M_{b' 1}} {C_{1}^{b ″}} T, \frac{h_{b' 2} A_{b'}}{M_{b' 2}} {C_{2}^{b ″}} T, \dots, \frac{h_{b' s} A_{b'}}{M_{b' s}} {C_{s}^{b ″}} T] T

(40)

H_{c} = [\frac{h_{c 1} A_{c}}{M_{c 1}} {C_{1}^{c'}} T, \frac{h_{c 2} A_{c}}{M_{c 2}} {C_{2}^{c'}} T, \dots, \frac{h_{cl} A_{c}}{M_{cl}} {C_{l}^{c'}} T] T

(41)

H_{c'} = [\frac{h_{c' 1} A_{c'}}{M_{c' 1}} {C_{1}^{c ″}} T, \frac{h_{c' 2} A_{c'}}{M_{c' 2}} {C_{2}^{c ″}} T, \dots, \frac{h_{c' l} A_{c'}}{M_{c' l}} {C_{l}^{c ″}} T] T

(42)

Q_{F_{a}} = [h_{a 1} Q_{F_{a} 1} / M_{a 1}, h_{a 2} Q_{F_{a} 2} / M_{a 2}, \dots, h_{am} Q_{F_{a} m} / M_{am}] T

(43)

where

C_{n}^{a}

and

C_{n}^{k}

are the row vectors of coupling coefficients matrices

C a

(it’s elements are described by equation (25)) and

C k

(it’s elements are described by equations (26) to (30)), respectively;

C_{m}^{a'}

C_{m}^{a ″}

C_{s}^{b'}

C_{s}^{b ″}

C_{l}^{c'}

C_{l}^{c ″}

are the column vectors of coupling coefficients matrices

C a

C a'

C b

C b'

C c

C c'

, respectively. Combining equations (32) to (34), the amplitude of acoustic pressure can be obtained as

P = (I + Z_{a} H_{a} + \sum_{k} Z_{k} H_{k}) - 1 Z_{a} Q_{F_{a}}

(44)

Thus, the acoustic pressure and acoustical potential energy in the enclosure can be expressed as

p (r, ω) = ψ T P

(45)

E_{p} = \frac{1}{4 ρ c 2} \int_{V} | p (r, ω) | 2 d V = \frac{V}{4 ρ c 2} P H Λ P

(46)

where Λ is the (N × N) diagonal matrix with each diagonal term consisting of M_n and the superscript H indicates complex conjugate transpose.

Coupled acoustic field characteristics

As far as we know, the research about two adjacent coupled plates in dynamics is very few. Yao et al.^26,27 analyze the structural acoustic coupling characteristics of a rectangular enclosure consisting of the two simply supported flexible plates and four rigid plates. A general formulation considering the full coupling between the plates and cavity is developed by using Hamiltonian function and Rayleigh-Ritz method. The analytical results and experimental results coincide well with Kim and Brennan’s²⁵ and Yao et al.^26,27 as well as the numerical solution obtained by FEM and BEM. At the same time, the results indicate that the coupling of the two adjacent plates is very weak. In this paper, the coupling between the flexible plates is omitted.

In addition, to check the validity of the formulation derivation, simulations generated are compared with the experimental results of Kim and Brennan.²⁵ The model has the same parameters as the model of Kim, including the simply supported boundary conditions for plate. Plate a is only driven by a disturbance point force. The results of simulation using the author’s and Kim’s method are perfectly equivalent and coincided well with Kim and Brennan’s experimental results, as shown in Figures 2 and 3.

Figure 2.

Sound pressure responses by using the author’s and Kim’s methods, respectively. (The simulated results are perfectly equivalent.)

Figure 3.

Experimental (—) and simulated (- - -) responses to a point force excitation of the structural-acoustic system (see Figure 7(b), page 109 in Kim and Brennan²⁵).

Among the six elastic plates, the natural frequencies of Plate a and Plate

a'

, Plate b and Plate

b'

, Plate c and Plate

c'

are the same, respectively, see Table 1. In the simulation study, the mode orders of Plate a(

a'

), Plate b(

b'

), Plate c(

c'

) and the enclosure are taken the first 14, 13, 17 and 12 orders, respectively. The modal damping coefficients of fluid in enclosure and panels are taken to be ξ_n = 0.005 and ξ_am = ξ_bs = ξ_cl = 0.002, respectively, which are usually determined based upon empirical data. The coupled acoustic field in the enclosure is compared between the fully elastic plates model and one elastic plate model, as shown in Figure 4. Figure 4 shows that the resonance peaks of the acoustic potential energy and some resonant frequencies in the fully elastic plates model are changed little compared to those in one elastic plate model. But as a whole, the large potential energy resonance peaks in the fully elastic plates model are still related to the natural frequencies of the Plate a (such as 30.5, 63.6, 88.7, 121.9, 177.2, 254.6, 295.9 Hz) and the enclosure (such as 149.6, 172.0, 248.3, 299.1 Hz). Among them, the 10th (254.6 Hz) and 12th (295.9 Hz) structural modes of the Plate a strongly couple with the 6th (248.3 Hz) and 8th (299.1 Hz) acoustic modes of the enclosure, respectively, which also causes the larger resonance peaks. In addition, the potential energy resonance peaks of the fully elastic plates model are obviously greater than that of one elastic plate model, such as the resonance peaks at 28, 41, 68, 77, 102, 135, 151, 159, 234, 236 Hz, etc. This is mainly caused by the structural modes of the Plates b (

b'

) (such as the modes of the 1st, 2nd, 4th, 5th etc.) and the Plates c (

c'

) (such as the modes of the 1st, 3rd, 7th, 11th etc.). But only the first modes of the Plates b (

b'

) and Plates c (

c'

) make a significant contribution to the coupled acoustic field in the enclosure; the other modes on the contribution of the acoustic field are relatively small as shown in Figure 4. Hence, the coupled acoustic field of the fully elastic plates model is dominated by the structural modes (natural frequencies) of the Plate a (excited by F_a) and acoustic modes (natural frequencies) of the enclosure, which is basically the same as that of one elastic plate model.

Table 1.

The modal indices and uncoupled natural frequencies of the plates and enclosure.

Plates a ( $a'$ )		Plates b ( $b'$ )		Plates c ( $c'$ )		Enclosure
mode	f (Hz)	mode	f (Hz)	mode	f (Hz)	mode	f (Hz)
(1,1)	30.5	(1,1)	34.0	(1,1)	25.7	(0,0,0)	0
(1,2)	63.6	(1,2)	77.9	(2,1)	58.9	(0,1,0)	149.6
(2,1)	88.7	(2,1)	92.3	(1,2)	69.6	(0,0,1)	172.0
(1,3)	118.9	(2,2)	136.1	(2,2)	102.7	(1,0,0)	198.2
(2,2)	121.9	(1,3)	151.0	(3,1)	114.2	(0,1,1)	227.9
(2,3)	177.2	(3,1)	189.3	(1,3)	142.7	(1,1,0)	248.3
(3,1)	185.8	(2,3)	209.3	(3,2)	158.0	(1,0,1)	262.4
(1,4)	196.4	(3,2)	233.2	(2,3)	175.9	(0,2,0)	299.1
(3,2)	218.9	(1,4)	253.4	(4,1)	191.6	(1,1,1)	302.0
(2,4)	254.6	(3,3)	306.3	(3,3)	231.2	(0,0,2)	344.0
(3,3)	274.2	(2,4)	311.7	(4,2)	235.4	(0,2,1)	345.1
(1,5)	295.9	(4,1)	325.2	(1,4)	245.1	(1,2,0)	358.8
(4,1)	321.6	(4,2)	369.1	(2,4)	278.2
(3,4)	351.6			(5,1)	291.1
				(4,3)	308.6
				(3,4)	333.5
				(5,2)	335.0

Figure 4.

Comparison of coupled acoustic fields in the fully elastic and one elastic plate models.

Excitation F_b applied on the plate b

Theoretical formulation

Consider only the excitation source F_b applied on the Plate b, i.e. F_a = 0. The control equation of the cavity pressure is the same as equation (1). The equations of motion of Plate b can be expressed as

ρ_{b} h_{b} {\overset{··}{w}}_{b} + D_{b} \nabla 4 w_{b} = F_{b} δ (σ_{b} - σ_{1}) - p (σ_{b}, t)

(47)

The equation of motion of the rest of the elastic plates is expressed as

ρ_{k'} h_{k'} {\overset{··}{w}}_{k'} + D_{k'} \nabla 4 w_{k'} = - p (σ_{k'}, t), k' = a, a', b', c, c'

(48)

The generalized modal force $Q_{F_{b} s}$ can be expressed as

Q_{F_{b} s} = \int_{A_{b}} F_{b} δ (σ_{b} - σ_{1}) ϕ_{s} (σ_{b}) d A

(49)

According to the theoretical derivation of section ‘Theoretical formulation’ and equations (32) to (34), the following formulas can be derived as

P' = Z_{b} V_{b} + \sum_{k'} Z_{k'} V_{k'}

(50)

V_{b} = Q_{F_{b}} - H_{b} P'

(51)

V_{k'} = - H_{k'} P'

(52)

Combining equations (50) to (52), the acoustic modal amplitude vector can be written as

P' = (I + Z_{b} H_{b} + \sum_{k'} Z_{k'} H_{k'}) - 1 Z_{b} Q_{F_{b}}

(53)

Then the acoustic pressure and acoustical potential energy in the enclosure can be expressed as

p' (r, ω) = ψ T P'

(54)

E_{p}' = \frac{1}{4 ρ c 2} \int_{V} | p' (r, ω) | 2 d V = \frac{V}{4 ρ c 2} P' H Λ P'

(55)

Coupled acoustic field characteristics

Figure 5 shows the coupled acoustic field in the fully elastic plate and one elastic plate models. Figure 5 shows that the potential energy resonance peaks of the fully elastic plates model are obviously greater than that of one elastic plate model. Meantime, some resonant frequencies in the fully elastic plates model are changed little compared to those in one elastic plate model. This is similar to the case of the primary excitation force applied on the Plate a. In the fully elastic plates model, the large acoustic potential energy peaks are related to the natural frequencies of the Plate b (such as 34.0, 77.9, 189.3, 233.2, 253.4 Hz) and the enclosure (such as 149.6, 172.0, 248.3 Hz), as shown in Figure 5. In addition, although the first modes of the Plates a ( $a'$ ) and Plates c ( $c'$ ) make a significant contribution to the coupled acoustic field in the enclosure, the contribution of other modes on the acoustic field is very small relatively those of Plate b and the enclosure. Hence, when the primary excitation is applied on the Plate b, the acoustic field in the enclosure is dominated by the structural modes of the Plate b and the acoustic modes of the enclosure, which is also consistent with the case of the primary excitation applied on the Plate a.

Figure 5.

Acoustic potential energy in the fully elastic plate and one elastic plate models when the excitation applied on the Plate b.

Two excitations applied simultaneously on the plates a and b

Theoretical formulation

The primary excitation sources F_a and F_b are applied simultaneously on the Plate a and Plate b, respectively. The control equation of the acoustic pressure is the same as equation (1). The equations of motion of the Plate a and Plate b are expressed by equations (2) and (47), respectively. The equation of motion of the rest of the elastic plates is expressed as

ρ_{k ″} h_{k ″} {\overset{··}{w}}_{k ″} + D_{k ″} \nabla 4 w_{k ″} = - p (σ_{k ″}, t), k ″ = a', b', c, c'

(56)

The structural modal amplitude vectors of the Plate a and Plate b are seen in equations (33) and (51), respectively. Then by the theoretical derivation of section ‘Theoretical formulation’ and equations (32) to (34), the modal amplitude vectors of the cavity pressure and the rest plates’ vibration velocity can be expressed as

P ″ = Z_{a} V_{a} + Z_{b} V_{b} + \sum_{k ″} Z_{k ″} V_{k ″}

(57)

V_{k ″} = - H_{k ″} P ″

(58)

By equations (33), (51), (57) and (58), the mode amplitude of the acoustic pressure is

P ″ = (I + Z_{a} H_{a} + Z_{b} H_{b} + \sum_{k ″} Z_{k ″} H_{k}) - 1 (Z_{a} Q_{F_{a}} + Z_{b} Q_{F_{b}})

(59)

Then the acoustic pressure and acoustical potential energy in the enclosure can be expressed as

p ″ (r, ω) = ψ T P ″

(60)

E_{p} ″ = \frac{1}{4 ρ c 2} \int_{V} | p ″ (r, ω) | 2 d V = \frac{V}{4 ρ c 2} P ″ H Λ P ″

(61)

Coupled acoustic field characteristics

Figure 6 shows the coupled acoustic field in the enclosure. The large acoustic potential energy peaks are related to the natural frequencies of the Plate a (such as 30.5, 177.2, 196.4, 254.6, 295.5 Hz), the Plate b (such as 34.0, 77.9, 151.0, 189.3, 233.2 Hz) and the enclosure (such as 149.6, 172.0, 198.2, 227.9, 248.3 Hz), as shown in Figure 6. In addition, the contribution of each mode of the Plate c ( $c'$ ) to the acoustic field is relatively small except the first one. Hence, when the excitation sources F_a and F_b are applied simultaneously on the Plate a and Plate b, respectively, the acoustic field in the enclosure is dominated by the structural modes of the Plate a and b, and the acoustic modes of the enclosure. Synthesizing the first two cases, the following conclusions can be drawn: the acoustic field in the enclosure is dominated by the structural modes of the elastic plate excited by the primary excitation and the acoustic modes of the enclosure; only the first structural modes of the other elastic plates make a significant contribution to the coupled acoustic field in the enclosure.

Figure 6.

Acoustic potential energy in the fully elastic plate model when the two excitations applied simultaneously on the Plate a and b.

Conclusions

In this paper, the coupled acoustic field of the fully elastic plates model is modeled by the modal superposition method. The acoustic field characteristics of the fully elastic plates and one elastic plate model are compared and analyzed. The simulation results show that the potential energy resonance peaks of the fully elastic plates model are significantly more than that of one elastic plate model. Then the acoustic field characteristics of the fully elastic plate model are analyzed with the three different disturbing cases. The study shows that the high potential energy resonance peaks are dominated by the structural modes of the elastic plate excited by the primary excitation source and the acoustic modes of the enclosure; only the first structural modes of the other elastic plates make a significant contribution to the coupled acoustic field in the enclosure, the contributions of the rest of the structure modes on the acoustic field are smaller.

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: This work was supported by the Natural Science Foundation of Ningbo, China (Grant No. 2016A610107).

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Modelling and analysis of acoustic field in a rectangular enclosure bounded by elastic plates under the excitation of different point force

Abstract

Keywords

Introduction

Acoustic field model of fully elastic plates

Excitation Fa applied on the plate a

Theoretical formulation

Coupled acoustic field characteristics

Excitation Fb applied on the plate b

Theoretical formulation

Coupled acoustic field characteristics

Two excitations applied simultaneously on the plates a and b

Theoretical formulation

Coupled acoustic field characteristics

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References

Excitation F_a applied on the plate a

Excitation F_b applied on the plate b