Complex modal analysis of structural vibration and acoustic radiation of plates

Abstract

Inhomogeneous damping distribution leads to the occurrence of complex modes of structures. Complex modes' vibration and acoustic radiation characteristics are different from real modes. This paper studies the complex modal characteristics of inhomogeneously damped plates for forced vibration. We used the state space method to obtain the complex eigenvalues and eigenvectors of the vibration system and calculated the relationship between modal coordinates and frequency under unit force. The results show that the absolute values of the modal coordinates of two conjugate complex eigenvectors are different, and the structure's vibration is dominated by the eigenvectors corresponding to eigenvalues with positive imaginary parts near natural frequencies. When investigating the structural sound radiation in the vibration mode space, each mode's acoustic radiation is not independent but coupled with each other. The influence of the coupling between complex modes on the calculation results of the acoustic radiation power of the plate is studied in this paper. The results show that the coupling effect between modes cannot be ignored when calculating the acoustic radiation power based on complex modes. Finally, the error of calculating acoustic power using the decoupling approximation instead of the complex mode method is analyzed.

Introduction

In engineering industries such as ship, automobile, and aerospace, more and more attention has been paid to the vibration and noise of structures.^1,2 It is one of the essential measures to reduce vibration and sound radiation by laying damping materials on the surface of the systems to absorb vibration energy. For example, the common damping composite structures include free damping layer structures and constrained damping layer structures. To reduce weight and save cost, researchers often apply damping materials to locally critical areas of structural surfaces, hoping to achieve a satisfactory reduction of vibration and noise with less additional mass rather than laying damping on the entire surfaces. For example, some scholars have studied the optimization assignment method of damping^3–6 and the vibro-acoustic calculation method of locally damped structures. ²

Inhomogeneous damping could lead to the occurrence of complex modes of systems. ⁷ Complex modes can also arise from gyroscopic effects, aerodynamic effects, wave propagation, and nonlinear structural behavior. ⁸ The most intuitive difference between real modes and complex modes is shown in mode shapes. In real modes, each element in the eigenvector is a real number, the phase difference between two elements is either 0° or 180°, and all points in the mode shape reach their equilibrium positions simultaneously. In complex modes, however, each element in the eigenvector is a complex number, the phase difference between two elements is no longer only 0° or 180°, and all points in the mode shape no longer reach their equilibrium positions simultaneously. Additionally, an N-degree-of-freedom vibration system has N eigenvectors in the case of real modes. However, the system has N pairs (2N) of conjugate eigenvectors in the case of complex modes. In this paper, we discuss complex modes caused by inhomogeneous damping, let $\mathbf{M}$ , $\mathbf{C}$ , $\mathbf{K}$ be the mass matrix, damping matrix, and stiffness matrix. It is well known that the system has real modes when $\mathbf{C}=0$ or $\mathbf{C}$ satisfies the proportional damping condition $\mathbf{C}=\alpha \mathbf{M}+\beta \mathbf{K}$ . Caughey and O’Kelly ⁹ extended the range of the damping matrix $\mathbf{C}$ for real modes and pointed out that the necessary and sufficient condition for a system to have real modes is $\mathbf{C}{\mathbf{M}}^{-1}\mathbf{K}=\mathbf{K}{\mathbf{M}}^{-1}\mathbf{C}$ . We can see that a system has real modes only when the damping matrix $\mathbf{C}$ meets certain conditions, and the real mode is a simplified model that exists only in theory. For actual structures, the modes obtained through testing are all complex modes.^10,11 When the damping matrix $\mathbf{C}$ does not satisfy the real mode condition, the vibration system cannot be decoupled by real mode transformation, and the modes of the system are complex.

There are two main methods for calculating the complex modes of a system: one is to use the state space method, ¹² and the other is to directly solve the eigenvalues and eigenvectors of the matrix $\left[\mathbf{M}{\lambda }^2+\mathbf{C}\lambda +\mathbf{K}\right]$ . ¹³ Compared with complex modes, the concept of real modes is more intuitive to understand, and the related research is more extensive and in-depth. Therefore, some scholars hope to simplify complex mode problems into real mode ones for easy analysis. For example, Ibrahim, ¹⁰ Chen et al., ¹¹ and Fuellekrug ¹⁴ studied the methods for extracting real modes from complex modes obtained by testing. Decoupling approximation is a widely used simplified method, which directly ignores the off-diagonal elements in the modal damping matrix to decouple the system, then theories of real modes can be utilized to obtain the modal parameters or vibration response of the system approximately. Shahruz^15,16 derived tight upper bounds on the norm of errors and found that decoupling the system by neglecting the off-diagonal elements minimizes the error upper bound among all approximating diagonal matrices. Knowles ¹⁷ showed that the difference (matrix norm) between the original equation and the equation obtained by the decoupling approximation is minimized. Morzfeld ¹⁸ pointed out that small off-diagonal elements in the modal damping matrix are insufficient to ensure small errors due to the decoupling approximation.

When structures are forced to vibrate, the sound radiation of each vibration mode is not independent, and there is a coupling between them. Some scholars studied the effect of modal interaction on acoustic radiation in the condition of real modes. For example, Keltie and Peng ¹⁹ investigated the acoustic power emitted by one-dimensional lightly damped plates. They said the coupling effect cannot be ignored under off-resonant excitation at low frequency but can be neglected at resonant excitation. A similar conclusion is drawn in Ref. 20, in which the research object is a point-excited rectangular plate. Cunefare ²¹ took beams as research objects and found that the coupling between vibration modes greatly influences the total sound power well below the coincidence wave number ratio. A little different from the conclusions in Refs. 19,20, Li and Gibeling ²² studied the mutual-radiation resistances of simply-supported rectangular plated and said that the effects of cross-modal coupling cannot always be ignored at resonance frequencies, otherwise, the sound power may be over- or under-estimated. According to their results, the notable error only occurs at individual resonant frequencies among nearly twenty resonant frequencies, and the error value is no more than 3 dB, it can be said that the difference in the conclusions is mainly due to the different tolerance of error. Therefore, when the requirement of calculation accuracy is not extremely strict, it is feasible to say that mutual radiation mainly works at the off-resonant region of low frequency and has little influence on acoustic radiation at resonance frequencies in the case of real modes. Self-radiation efficiency and mutual-radiation efficiency reflect the ability of a mode itself and the interaction between two modes to radiate sound, respectively. Unruh^7,23,24 applied inhomogeneous damping to plates to generate complex modes and studied the self-radiation efficiencies and mutual-radiation efficiencies of complex modes. In Ref. 7, Unruh found that the characteristics of acoustic radiation of complex modes are greatly different from real modes. For a uniform rectangular plate, in the case of real modes, the coupling effect only exists in modes of the same type. For example, odd-even modes can only be coupled to odd-even modes and not to odd-odd, even-even, or even-odd modes. In the case of complex modes, however, modes of different types also interact with each other. From the theory of waves, complex modes are dominated by the superposition of traveling waves and standing waves. In Refs. 23,24, the author found that the traveling wave component significantly increases the self-radiation efficiencies of even-order modes and slightly decreases the self-radiation efficiencies of odd-order modes. Wei and Li ²⁵ presented a method to calculate the mutual-radiation efficiencies of complex modes, which can be well combined with finite element method and boundary element method software.

As mentioned above, the system has N pairs of conjugate complex eigenvectors in the case of complex modes, which differs from the system of real modes. However, the relationship between conjugate eigenvectors and the influence of the coupling between conjugate modes on the total sound power are not considered in the studies of complex modes mentioned above. Herein, vibration and acoustic radiation characteristics of conjugate complex eigenvectors are investigated. In this paper, based on the theory of complex modes, the difference between the contributions of a pair of conjugate modes to the total vibration is studied, and the influence of the coupling between complex modes, including the coupling between conjugate modes, on the radiated sound power of the plate is analyzed. Moreover, the error of the acoustic power calculated by the decoupling approximation instead of the complex mode method is discussed.

To our best knowledge, this is the first paper to point out that there exists a coupling effect of acoustic radiation between a pair of conjugate complex modes. This coupling effect could not be ignored when calculating the total radiated sound power.

Theory

Calculation of complex modes

For a system with N degrees of freedom, the equation of motion is

$\mathbf{M}\stackrel{..}{\mathbf{x}}+\mathbf{C}\stackrel{.}{\mathbf{x}}+\mathbf{K}\mathbf{x}=\mathbf{f}\left(\mathit{t}\right),$

(1)

where $\mathbf{M}$ , $\mathbf{C}$ , $\mathbf{K}$ are mass matrix, damping matrix, and stiffness matrix of the system, $\mathbf{x}$ is the displacement vector of the system, $\mathbf{f}\left(t\right)$ is the excitation force vector. When $\mathbf{C}$ does not satisfy the Caughey and O’Kelly condition, equation (1) cannot be decoupled by the real mode transformation in N-dimensional space. Here, the state space method is used to solve the system's response. One advantage of this method is that the coefficient matrices are real and symmetric, so the commands for solving eigenvalues and eigenvectors in the mathematical software (such as EIG and EIGS commands in MATLAB) can be directly utilized to obtain the complex modes. ¹² The equation of motion expressed in the state space is

$\left[\begin{array}{ll}\mathbf{C}& \mathbf{M}\\ \mathbf{M}& \mathbf{0}\end{array}\right]\left\{\begin{array}{l}\stackrel{.}{\mathbf{x}}\\ \stackrel{..}{\mathbf{x}}\end{array}\right\}+\left[\begin{array}{ll}\mathbf{K}& \mathbf{0}\\ \mathbf{0}& -\mathbf{M}\end{array}\right]\left\{\begin{array}{l}\mathbf{x}\\ \stackrel{.}{\mathbf{x}}\end{array}\right\}=\left\{\begin{array}{l}\mathbf{f}\left(t\right)\\ \mathbf{0}\end{array}\right\},$

(2)

or written as

$\mathbf{A}\stackrel{.}{\overline{\mathbf{x}}}+\mathbf{B}\overline{\mathbf{x}}=\overline{\mathbf{f}}\left(t\right),$

(3)

in which

$\mathbf{A}=\left[\begin{array}{ll}\mathbf{C}& \mathbf{M}\\ \mathbf{M}& \mathbf{0}\end{array}\right]\in R^{2N\times 2N},\mathbf{B}=\left[\begin{array}{ll}\mathbf{K}& \mathbf{0}\\ \mathbf{0}& -\mathbf{M}\end{array}\right]\in R^{2N\times 2N},$

$\overline{\mathbf{x}}=\left\{\begin{array}{l}\mathbf{x}\\ \stackrel{.}{\mathbf{x}}\end{array}\right\}\in R^{2N},\overline{\mathbf{f}}\left(t\right)=\left\{\begin{array}{l}\mathbf{f}\left(t\right)\\ \mathbf{0}\end{array}\right\}\in R^{2N},$

where the superscript “ ¯ ” denotes the vector in the state space, and the italic R represents real numbers. The eigenvalue problem corresponding to equation (3) is

$\lambda \mathbf{A}\overline{\varphi }+\mathbf{B}\overline{\varphi }=0,$

(4)

here $\lambda \in C$ is the complex eigenvalue, $\overline{\varphi }\in C^{2N}$ is the corresponding complex eigenvector, and the italic C denotes complex numbers. By solving equation (4), we can obtain N pairs of conjugate complex eigenvalues ${\lambda }_j$ , ${\lambda }_j^{*}$ and eigenvectors ${\overline{\varphi }}_j$ , ${{\overline{\varphi }}_j}^{*}$ , the superscript “*” denotes the conjugate complex. ${\overline{\varphi }}_j$ and ${{\overline{\varphi }}_j}^{*}$ can be written as

${\overline{\varphi }}_j=\left[\begin{array}{l}{\varphi }_j\\ {\lambda }_j{\varphi }_j\end{array}\right],{{\overline{\varphi }}_j}^{*}=\left[\begin{array}{l}{\varphi }_j^{*}\\ {\lambda }_j^{*}{\varphi }_j^{*}\end{array}\right]\left(j=\mathrm{1,2,3},\cdot \cdot \cdot ,N\right)$

(5)

in which the complex mode ${\varphi }_j\in C^N$ can be expressed as

${\varphi }_j={\left[\begin{array}{llll}{r}_{j1}{e}^{\mathrm{i}\cdot {\alpha }_{j1}}& r_{j2}{e}^{\mathrm{i}\cdot {\alpha }_{j2}}& ...& r_{jN}{e}^{\mathrm{i}\cdot {\alpha }_{jN}}\end{array}\right]}^{\mathrm{T}}$

(6)

here $r_{jk}$ and ${\alpha }_{jk}$ are real numbers. When $j\ne k$ , ${\overline{\varphi }}_j$ and ${\overline{\varphi }}_k$ satisfy the orthogonality condition

${\overline{\varphi }}_j^{\mathrm{T}}\mathbf{A}{\overline{\varphi }}_k=0,{\overline{\varphi }}_j^{\mathrm{T}}\mathbf{B}{\overline{\varphi }}_k=0$

(7)

Calculate the complex modes and then normalize them by

${\overline{\varphi }}_j^{\mathrm{T}}\mathbf{A}{\overline{\varphi }}_j={\gamma }_j,$

(8)

here we choose ${\gamma }_j=2{\lambda }_j$ , thus, when the damping matrix $\mathbf{C}=0$ , equation (8) can be reduced to ${\varphi }_j^{\mathrm{T}}\mathbf{M}{\varphi }_j=1$ , namely, the common normalized form of undamped real modes. $\overline{\mathbf{x}}$ can be represented by a linear superposition of ${\overline{\varphi }}_j$

$\overline{\mathbf{x}}\left(\omega \right)={\displaystyle \sum }_{j=1}^{2N}{\beta }_j\left(\omega \right)\cdot {\overline{\varphi }}_j,$

(9)

where ${\beta }_j\left(\omega \right)$ is the displacement modal coordinate, when the system is under forced steady-state vibration, the expression is

${\beta }_j\left(\omega \right)=\frac{{\overline{\varphi }}_j^T\cdot \overline{\mathbf{f}}\left(\omega \right)}{{\gamma }_j\left(\mathrm{i}\omega -{\lambda }_j\right)}$

(10)

Expand the matrix ${\overline{\varphi }}_j$ and $\overline{\mathbf{f}}\left(\omega \right)$ , then take the first N terms in the column vector $\overline{\mathbf{x}}\left(\omega \right)$ , the displacement response of the system's forced steady-state vibration can be expressed as

$\mathbf{x}\left(\omega \right)={\displaystyle \sum }_{j=1}^N\left[\frac{{\varphi }_j^{\mathrm{T}}\cdot \mathbf{f}\left(\omega \right)}{{\gamma }_j\left(\mathrm{i}\omega -{\lambda }_j\right)}{\varphi }_j+\frac{{{\varphi }_j^{*}}^{\mathrm{T}}\cdot \mathbf{f}\left(\omega \right)}{{\gamma }_j^{*}\left(\mathrm{i}\omega -{\lambda }_j^{*}\right)}{\varphi }_j^{*}\right]$

(11)

Calculation of radiated sound power

The radiated sound power P of a structure is given by

$P=\frac{1}{2}{\int }_{\mathrm{S}}\text{Re}\left[p\left(x,y,z\right)v_{\mathrm{n}}^{*}\left(x,y,z\right)\right]\text{dS}$

(12)

where $p\left(x,y,z\right)$ is the sound pressure of the surface, $v_{\mathrm{n}}^{*}\left(x,y,z\right)$ is the conjugate complex of normal velocities of the surface, $\text{Re}\left[\right]$ denotes the real part. The relationship between sound pressure $\mathbf{p}$ and ${\mathbf{v}}_{\mathrm{n}}$ is

$\mathbf{p}=\mathbf{Z}{\mathbf{v}}_{\mathrm{n}},$

(13)

in which $\mathbf{Z}$ is the acoustic impedance matrix. Equation (12) can be rewritten in the form of matrices as

$P=\frac{1}{2}\text{Re}\left[{\mathbf{v}}_{\mathrm{n}}^{\mathrm{H}}\mathbf{Y}\mathbf{Z}{\mathbf{v}}_{\mathrm{n}}\right]={\mathbf{v}}_{\mathrm{n}}^{\mathrm{H}}\mathbf{R}{\mathbf{v}}_{\mathrm{n}}$

(14)

where the superscript "^H" denotes the complex conjugate transpose, $\mathbf{Y}={\int }_{\mathrm{S}}{\mathbf{L}}^{\mathrm{T}}\mathbf{L}\text{dS}$ and $\mathbf{L}$ is the vector of interpolation functions. $\mathbf{R}=\left(\mathbf{Y}/2\right)\text{Re}\left[\mathbf{Z}\right]$ is defined as radiation resistance matrix, it is real, symmetric, and positive definite. ²⁶ ${\mathbf{v}}_{\mathrm{n}}$ can be expressed by the matrix of eigenvectors $\mathbf{\Phi }$ and velocity modal coordinates $\mathbf{a}$ as

${\mathbf{v}}_{\mathrm{n}}=\mathbf{\Phi }\mathbf{a}$

(15)

where ${\mathbf{v}}_{\mathbf{n}}\in C^N$ , $\mathbf{\Phi }=\left[{\varphi }_1,{\varphi }_1^{*},\cdot \cdot \cdot ,{\varphi }_N,{\varphi }_N^{*}\right]\in C^{N\times 2N}$ , $\mathbf{a}\in C^{2N}$ . equation (14) can be rewritten as

$P={\mathbf{a}}^{\mathrm{H}}\mathbf{T}\mathbf{a}={\displaystyle \sum }_{i=1}^{2N}{\displaystyle \sum }_{j=1}^{2N}{T}_{ij}\left({\mathbf{a}}_i^{*}{\mathbf{a}}_j\right)={\displaystyle \sum }_{i=1}^{2N}{\displaystyle \sum }_{j=1}^{2N}{P}_{ij},$

(16)

in which $\mathbf{T}={\mathbf{\Phi }}^{\mathrm{H}}\mathbf{R}\mathbf{\Phi }$ is the power transfer matrix. ²⁰ According to Refs. 7,26, in the condition of real modes, $\mathbf{R}\in R^{N\times N}$ , $\mathbf{\Phi }\in R^{N\times N}$ , $\mathbf{T}\in R^{N\times N}$ , $\mathbf{T}$ is real and symmetric; And in the situation of complex modes, $\mathbf{R}\in R^{N\times N}$ , $\mathbf{\Phi }\in C^{N\times 2N}$ , $\mathbf{T}\in C^{2N\times 2N}$ , $\mathbf{T}$ is Hermitian. It can be seen from equation (16) that there is a coupling between vibration modes when radiating sound power. If we only consider the sound power of two different modes, assuming that their velocity modal coordinates are $a_1$ and $a_2$ , respectively, the sound power can be expressed as

$P=a_1^{*}{T}_{11}{a}_1+a_2^{*}{T}_{22}{a}_2+a_1^{*}{T}_{12}{a}_2+a_2^{*}{T}_{21}{a}_1,$

(17)

where $a_1^{*}{T}_{11}{a}_1$ and $a_2^{*}{T}_{22}{a}_2$ are real numbers, and they are the self-radiated sound power of the two modes; $a_1^{*}{T}_{12}{a}_2$ and $a_2^{*}{T}_{21}{a}_1$ are conjugate complex, and they are the sound power generated by the coupling of the two modes, the contribution of the coupling effect to the total sound power ⁷ is

$P_m=a_1^{*}{T}_{12}{a}_2+a_2^{*}{T}_{21}{a}_1=2\text{Re}\left[a_1^{*}{T}_{12}{a}_2\right].$

(18)

Equation (18) shows that only the coupling term's real part (resistive sound power) will affect the total sound power. Even if the imaginary part (reactive sound power) exists, it will not influence the total radiated sound power, and the imaginary part is reactive power.

To obtain the radiation resistance matrix $\mathbf{R}$ , the elementary radiator approach summarized in Refs. 7,20,26 is used in this paper. According to the method, a baffled plate could be divided into $I$ small pistons of the same size, and each piston is assumed to be vibrating harmonically with its central velocity. If the area of each piston is $S_E$ , the sound power radiated by the structure can be expressed as

$P=\left(S_E/2\right)\text{Re}\left[{\mathbf{v}}_{\mathrm{n}}^{\mathrm{H}}\mathbf{p}\right].$

(19)

And $\mathbf{R}$ is a matrix of $I\times I$ elements, it is given by

$\mathbf{R}=\frac{{\omega }^2{\rho }_0{S}_E^2}{4\pi c}\left[\begin{array}{llll}1& \frac{\sin \left(k{r}_{12}\right)}{k{r}_{12}}& \cdot \cdot \cdot & \frac{\sin \left(k{r}_{1I}\right)}{k{r}_{1I}}\\ \frac{\sin \left(k{r}_{21}\right)}{k{r}_{21}}& 1& \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\sin \left(k{r}_{I1}\right)}{k{r}_{I1}}& \cdot \cdot \cdot & \cdot \cdot \cdot & 1\end{array}\right],$

(20)

where $k=\omega /c$ , $r_{ij}$ is the distance between two pistons, so $r_{ij}=r_{ji}$ , then $\mathbf{R}$ could be calculated easily for planar structures.

The decoupling approximation

The decoupling approximation is a widely used simplified method to obtain the vibration response of non-classically damped systems. References 15–18 investigated the error of the vibration response calculated by decoupling approximation. For the vibration system in equation (1), the damping matrix $\mathbf{C}$ is not considered, and the undamped system satisfies the eigenvalue problem

$\mathbf{K}{\psi }_j={\omega }_j^2\mathbf{M}{\psi }_j,$

(21)

where each element in ${\psi }_j$ is a real number. Write the calculated N eigenvalues and N eigenvectors in matrix form as

$\mathbf{\Omega }\hspace{0.17em}=\text{diag}\left[\begin{array}{llll}{\omega }_1^2& {\omega }_2^2& ...& {\omega }_N^2\end{array}\right],$

(22)

$\mathbf{\Psi }=\left[\begin{array}{llll}{\psi }_1& {\psi }_2& ...& {\psi }_N\end{array}\right].$

(23)

The orthogonality of the real modes to $\mathbf{M}$ and $\mathbf{K}$ can be expressed as

${\mathbf{\Psi }}^{\mathrm{T}}\mathbf{M}\mathbf{\Psi }=\mathbf{I},{\mathbf{\Psi }}^{\mathrm{T}}\mathbf{K}\mathbf{\Psi }=\mathbf{\Omega },$

(24)

where $\mathbf{I}$ is the identity matrix. Let $\mathbf{x}=\mathbf{\Psi }\mathbf{q}$ , then equation (1) can be transformed into

$\stackrel{..}{\mathbf{q}}+\mathbf{D}\stackrel{.}{\mathbf{q}}+\mathbf{\Omega }\mathbf{q}=\mathbf{g}\left(t\right),$

(25)

here $\mathbf{D}={\mathbf{\Psi }}^{\mathrm{T}}\mathbf{C}\mathbf{\Psi }$ , $\mathbf{g}\left(t\right)={\mathbf{\Psi }}^{\mathrm{T}}\mathbf{f}\left(t\right)$ . $\mathbf{D}$ is not a diagonal matrix, so there is a coupling between the principal vibrations. Ignore the off-diagonal terms, equation (25) can be approximated as

$\stackrel{..}{\mathbf{q}}+{\mathbf{D}}_{\mathrm{d}}\stackrel{.}{\mathbf{q}}+\mathbf{\Omega }\mathbf{q}=\mathbf{g}\left(t\right),$

(26)

where ${\mathbf{D}}_{\mathrm{d}}=\text{diag}\left[d_{11},d_{22},\cdot \cdot \cdot ,d_{\text{NN}}\right]$ , $d_{ij}$ is the element in matrix $\mathbf{D}$ . There is no coupling between the principal vibrations in equation (26), the calculation result of equation (26) is regarded as the solution of equation (25), and this is the decoupling approximation.

Numerical results

Forced vibration characteristics of complex modes

Firstly, we study the forced vibration characteristics of inhomogeneously damped plates. The research object is a rectangular aluminum plate, its length l = 348 mm, width w = 304.8 mm, and thickness h = 0.762 mm. Aluminum material properties (ρ = 2700 kg/m³, E = 72.0 GPa, ν = 0.3) are defined. The edges of the plate are clamped. Here, the vibration characteristics of four different local damping distributions are analyzed and compared. The damping distributions are shown in Figure 1, the damping ratio $\xi =0$ is defined for the light-colored areas, and the damping ratio $\xi =0.2$ is defined for the dark-colored areas. For each damping distribution, the dark-colored area is 1/4 of the total area. The finite element models of the plates are established in ANSYS, the element type is Shell 181, and the number of elements is 64 × 64 × 1. Commercial finite element software such as ANSYS currently does not support structural harmonic response analysis based on the complex modal superposition method. For the computation in this paper, the $\mathbf{M}$ , $\mathbf{C}$ , $\mathbf{K}$ matrices of the model are first extracted from the ANSYS software. Subsequently, utilizing the relevant equations in the theory section, a program is developed within MATLAB software to compute the vibration and acoustic radiation of the structure. The first 13 natural frequencies and the corresponding mode shapes of the undamped plate, as well as the plates with local damping in Figure 1, are shown in Table 1.

Figure 1.

Distribution of inhomogeneous damping: (a) distribution A, (b) distribution B, (c) distribution C, and (d) distribution D.

Table 1.

The natural frequencies and the corresponding mode shapes of the plates.

Mode		1	2	3	4	5	6	7	8	9	10	11	12	13
		(1.1)	(2.1)	(1.2)	(2.2)	(3.1)	(1.3)	(3.2)	(2.3)	(4.1)	(3.3)	(4.2)	(1.4)	(2.4)
Frequency/Hz	Undamped	65.2	122.1	143.3	195.8	213.3	264.0	283.2	314.0	337.1	398.3	404.4	426.3	475.0
	A	64.9	123.3	141.9	196.1	213.7	264.7	284.5	316.2	337.0	402.7	402.5	429.4	481.1
	B	65.3	122.3	143.6	196.4	213.8	265.0	284.0	316.3	337.5	400.1	405.2	429.7	484.6
	C	65.3	121.2	144.0	195.5	213.9	265.9	284.4	314.2	337.6	400.0	407.2	433.5	471.4
	D	64.0	122.0	143.3	196.2	213.8	265.1	284.2	315.9	337.1	400.4	407.2	428.1	477.2

Figure 2 shows the complex mode shapes of mode (2,4) for the four damping distributions, and the Ref shows the real mode shape for the undamped plate. It can be seen from the figure that the inhomogeneous damping generates the complex modes, the phases of the mode shapes are no longer 0° or 180°, and the complex mode shapes are different for different damping distributions. In addition, for real modes, there is one real mode shape corresponding to a resonance frequency, while for complex modes, there are two conjugate complex mode shapes corresponding to a resonance frequency. The real part and the amplitude of the conjugate complex mode shapes are the same, but the imaginary part and the phase are opposite. In the case of complex modes, the vibration response and the acoustic radiation of a structure are the results of the combined action of N pairs of conjugate (2N) complex modes.

Figure 2.

Mode shapes of mode (2,4) for different damping distributions.

As we know, the eigenvector corresponding to the natural frequency may contribute greatly to the vibration response near the resonance frequency in the condition of real modes. However, for complex modes, there is a pair of conjugate complex eigenvectors corresponding to a natural frequency. It has been briefly discussed in Ref. 12 that their contributions to the vibration response near the resonance frequency are different. Here, for inhomogeneously damped plates, the contributions of eigenvectors of complex modes to the total vibration response are studied under the steady-state vibration. We take the distribution A in Figure 1 as the research object and investigate the relationship between the modal coordinates ${\beta }_j\left(\omega \right)$ and frequency under a single-point normal force. The first five natural frequencies and corresponding eigenvalues of distribution A are shown in Table 2.

Table 2.

The first five natural frequencies and the corresponding eigenvalues of distribution A.

Order	Frequency/Hz	Eigenvalue	Notation
1	64.92	−45 + 408i	1+
		−45 – 408i	1-
2	123.31	−40 + 775i	2+
		−40 – 775i	2-
3	141.89	−63 + 891i	3+
		−63 – 891i	3-
4	196.14	−57 + 1232i	4+
		−57 – 1232i	4-
5	213.66	−47 + 1342i	5+
		−47 – 1342i	5-

The amplitude of the excitation force is 1 N at each frequency, and the two positions of the force are shown in Figure 3. The coordinate of point Ⅰ is (L/4, W/4), the coordinate of point Ⅱ is (L*3/8, W*3/8), and each point position corresponds to a calculation condition. The relationships between the absolute values of displacement modal coordinates ${\beta }_j\left(\omega \right)$ and frequency are shown in Figure 4, where we list the displacement modal coordinates corresponding to the first five pairs of conjugate eigenvectors. For the convenience of understanding, the notations in Table 2 will be directly used to refer to the corresponding eigenvectors in the following paragraphs. For example, “1+” represents the corresponding eigenvector of “-45 + 408i,” “1-” represents the corresponding eigenvector of “-45-408i,” and the rest are analogous.

Figure 3.

The positions of the excitation force.

Figure 4.

The absolute values of displacement modal coordinates of distribution A: (a) condition Ⅰ and (b) condition Ⅱ.

It is shown in Figure 4 that the contributions of two conjugate eigenvectors to the total vibration response are different under forced steady-state vibration. Suppose a pair of conjugate eigenvalues are ${\lambda }_{n+}=-a+b\mathrm{i}$ and ${\lambda }_{n-}=-a-b\mathrm{i}$ , their eigenvectors are ${\varphi }_{n+}$ and ${\varphi }_{n-}$ , and the corresponding displacement modal coordinates are ${\beta }_{n+}$ and ${\beta }_{n-}$ , respectively. As the frequency increases, for ${\lambda }_{n+}$ (the eigenvalue with a positive imaginary part), $\left|{\beta }_{n+}\right|$ reaches a maximum near its natural frequency, namely, the vibration response of the structure is mainly contributed by ${\varphi }_{n+}$ near its natural frequency. As the frequency continues to increase, $\left|{\beta }_{n+}\right|$ decreases gradually. However, for ${\lambda }_{n-}$ (the eigenvalue with a negative imaginary part), $\left|{\beta }_{n-}\right|$ continuously decreases as the frequency increases, and $\left|{\beta }_{n-}\right|$ is always less than $\left|{\beta }_{n+}\right|$ . The contribution of the eigenvector ${\varphi }_{n-}$ to the total vibration is still small even near the natural frequency of ${\lambda }_{n-}$ . In other words, the vibration becomes stronger near the natural frequency because the contribution of the eigenvector ${\varphi }_{n+}$ to the total vibration response increases, and it does not depend much on the eigenvector ${\varphi }_{n-}$ .

To explain the above results, we can see equation (10). Term $\left(\mathrm{i}\omega -{\lambda }_j\right)$ exists in the denominator of equation (10), the physical meaning of $\left|i\omega -{\lambda }_j\right|$ is the distance between the two points in the complex coordinate system, as shown in Figure 5. As ω gradually increases from zero, when $\omega =b$ , $\left|i\omega -{\lambda }_{n+}\right|$ reaches its minimum value, the curve of $\left|{\beta }_{n+}\right|$ has a peak. However, as ω increases, $\left|i\omega -{\lambda }_{n-}\right|$ increases monotonically, which causes $\left|{\beta }_{n-}\right|$ to decrease monotonically with frequency. It is also shown in Figure 5 that when $\omega =0$ , $\left|i\omega -{\lambda }_{n+}\right|$ is equal to $\left|i\omega -{\lambda }_{n-}\right|$ ; When $\omega \ne 0$ , $\left|i\omega -{\lambda }_{n+}\right|$ is always less than $\left|i\omega -{\lambda }_{n-}\right|$ . In the condition of the unit force, for equation (10), $\left|{\varphi }_j^{\mathrm{T}}\cdot \mathbf{f}\left(\omega \right)\right|=\left|{\varphi }_j^{*\mathrm{T}}\cdot \mathbf{f}\left(\omega \right)\right|$ , $\left|{\gamma }_j\right|=\left|{\gamma }_j^{*}\right|$ , so $\left|{\beta }_{n+}\right|$ is always larger than $\left|{\beta }_{n-}\right|$ .

Figure 5.

The physical meaning of |iω-λ_n+| and |iω-λ_n-|.

The effect of the coupling between complex modes on sound power

Here, we study the effect of the coupling between complex modes on radiated sound power. Firstly, validation of the acoustic power calculation results is conducted. An acoustics-structure coupled model is established within ANSYS, as illustrated in Figure 6. Vibrational responses and acoustic pressures at various plate nodes are computed in ANSYS. Utilizing equation (12) and performing integration yields the radiated acoustic power of the plate, hereinafter referred to as the software results. Figures 7 and 8 compare the acoustic power obtained using the methodology presented in this paper with the software results. It is evident from the figures that the approach proposed in this paper is reliable.

Figure 6.

Simplified diagram of finite element calculation model.

Figure 7.

Comparison of calculation results (Condition Ⅰ): (a) damping distribution A, (b) damping distribution B, (c) damping distribution C, and (d) damping distribution D.

Figure 8.

Comparison of calculation results (Condition Ⅱ): (a) damping distribution A, (b) damping distribution B, (c) damping distribution C, and (d) damping distribution D.

The acoustic power results calculated by equation (16) with and without considering the coupling are compared in Figures 9 and 10. The unit excitation force is at point Ⅰ in Figure 9, and at point Ⅱ in Figure 10. In the figures $P={\displaystyle \sum }_{i=1}^N{\displaystyle \sum }_{j=1}^N{P}_{ij}$ and $P_{Vd}={\displaystyle \sum }_{i=1}^N{P}_{ii}$ , the first 24 pairs of conjugate complex modes are used, namely $N=48$ . The density of the acoustic medium ρ₀ = 1.21 kg/m³, and the acoustic velocity c = 344 m/s.

Figure 9.

Sound power with and without the coupling effect considered (Condition Ⅰ): (a) damping distribution A, (b) damping distribution B, (c) damping distribution C, and (d) damping distribution D.

Figure 10.

Sound power with and without the coupling effect considered (Condition Ⅱ): (a) damping distribution A, (b) damping distribution B, (c) damping distribution C, and (d) damping distribution D.

As shown in Figures 9 and 10, the coupling between complex modes has different effects on the total sound power under various damping distributions. Meanwhile, the effects are different under various excitation conditions. For Figure 9(b), the effect of the interaction on sound power mainly works in the off-resonant region and has little influence in the resonant region. For Figures 9(a), (c) and (d), the coupling of modes has little influence on the sound power at most natural frequencies, but there is a significant difference between the acoustic power near the second (about 122 Hz) and third (about 143 Hz) natural frequencies of the structures; For Figures 10(a)–(d), the coupling effect has obvious influence on the sound power in the whole band of low frequency, and the coupling of complex modes cannot be ignored when calculating the acoustic power.

Under the four damping distributions, mode 1+ and mode 1- are a pair of conjugate modes, they have a strong coupling of sound radiation with each other, and they also have a strong coupling with other odd-odd modes. The interaction has a strong ability to radiate sound, which significantly influences the total acoustic power. By comparing Figure 4(a) and (b), we can see that the absolute values of modal coordinates of mode 1+ and mode 1- under condition Ⅱ are larger than condition Ⅰ, which causes the coupling between mode 1+ and mode 1-, as well as the coupling of the two modes with other odd-odd modes, of condition Ⅱ to be stronger than the coupling of condition Ⅰ. This is why the calculation errors in Figure 10 are greater than those in Figure 9 without considering the coupling. Table 3 lists the acoustic power radiated by modes 1+, 1-, 5+, and 5- of damping distribution A under condition Ⅰ at 214 Hz (the natural frequency corresponding to modes 5+ and 5-), the radiated sound power of the structure at 214 Hz is mainly generated by the joint action of these modes. Table 4 lists the acoustic power of damping distribution A under condition Ⅱ. The diagonal terms in the tables are the values of the self-radiated sound power of each mode. The off-diagonal terms are the power generated by the coupling between the two modes calculated by equation (18). In Table 3, the radiated sound power is mainly contributed by mode 5+ itself. The sound power generated by the coupling effect is minimal compared with the self-radiated power of mode 5+ (1.69E-03 W), so ignoring the coupling effect between modes has little influence on the total sound power. In Table 4, the sum of the self-radiated power of mode 1+ and mode 1- is 8.13E-04 W and the sound power generated by their coupling is −6.74E-04 W. The absolute value of the sound power generated by the coupling of mode 1+ and mode 1- not only exceeds the sound power radiated by the two modes (1.39 E-04 W) but exceeds the total radiated sound power of the structure at 214 Hz (5.02E-04 W). Therefore, the coupling effect between the first pair of conjugate modes cannot be ignored. In addition, the sound power generated by the coupling of mode 5+ and mode 1+ is 2.08E-04 W, the sound power generated by the coupling of mode 5+ and mode 1- is −1.35E-04 W, the absolute values are pretty great compared with the total radiated sound power 5.02E-04 W, and they cannot be neglected.

Table 3.

The sound power of damping distribution A at 214 Hz (condition Ⅰ)/W.

	1+	1-	5+	5-
1+	1.04E-04	–	–	–
1-	−1.10E-04	2.97E-05	–	–
5+	2.56E-04	−7.77E-05	1.69E-03	–
5-	−1.46E-05	7.67E-06	−2.21E-05	5.19E-07
Total sound power at 214 Hz: 1.90E-03

Table 4.

The sound power of damping distribution A at 214 Hz (condition Ⅱ)/W.

	1+	1-	5+	5-
1+	6.32E-04	–	–	–
1-	−6.74E-04	1.81E-04	–	–
5+	2.08E-04	−1.35E-04	2.17E-04	–
5-	1.17E-05	−6.42E-06	5.10E-06	6.66E-08
Total sound power at 214 Hz: 5.02E-04

The study in Ref. 7 shows that there is a coupling of acoustic radiation between different types of modes for a rectangular plate in the condition of complex modes. From this example, we can see that in the situation of complex modes, there is also a coupling of acoustic radiation between a pair of conjugate complex modes, and this coupling may be quite strong. In addition, when the complex mode method is used to calculate the total radiated sound power, the coupling effect between the modes should not be ignored arbitrarily, even at resonance frequencies. Otherwise, the calculation may lead to large errors.

The error of the acoustic power calculated by the decoupling approximation

The four damping distributions in Figure 1 under condition Ⅱ are taken as examples for calculation. Firstly, the real modes of the systems are obtained by equations (21) and (26) is used to calculate the approximate vibrational response by neglecting the off-diagonal terms of modal damping matrix $\mathbf{D}$ . Then, equation (14) is utilized to obtain the radiated acoustic power of the system. The first 24 real modes are used to calculate the vibration response and sound power by the decoupling approximation. Figure 11 shows the results of acoustic power calculated by the complex mode method and the decoupling approximation. In Figure 11, $P$ is the exact solution obtained by the complex mode method and $P_{da}$ is the approximate solution obtained by the decoupling approximation.

Figure 11.

Acoustic power calculated by complex mode method and the decoupling approximation: (a) damping distribution A, (b) damping distribution B, (c) damping distribution C, and (d) damping distribution D.

As shown in Figure 11, for distributions B and D, the sound power calculated by the decoupling approximation agrees with the exact solution obtained by the complex mode method. For distribution A, the results of decoupling approximation have obvious errors at many frequency points compared with the precise solution, and the error reaches a maximum of 1.8 dB at 211 Hz. For distribution C, the difference between the two curves is obvious: the peak of sound power at 121 Hz is missed in the resulting curve of decoupling approximation, the value of acoustic power at 121 Hz is even less than the value of the lower peak at 144 Hz, and the error of sound power level reached 3.6 dB at 121 Hz. In addition, for distribution C, the peak of the exact solution in the 400–500 Hz range is 435 Hz, while the peak of decoupling approximation is 425 Hz, and the frequency point error reaches 10 Hz.

It can be seen that the sound power results obtained by the decoupling approximation may not only differ from the exact solution but also have some problems, such as omitting the peak of acoustic power and great calculation error of the frequency point where the peak of acoustic power is located. Therefore, the decoupling approximation should be avoided to calculate the structure's radiated sound power when the calculation results' accuracy requirement is high.

Conclusions

This paper mainly investigates the vibration and acoustic radiation characteristics of conjugate complex eigenvectors of plates under forced steady-state vibration. Additionally, the error of the acoustic power calculated by the decoupling approximation instead of the complex mode method is discussed. The main conclusions are:

(1) The contributions of two conjugate complex eigenvectors to the total vibration response are different under a unit normal force. The vibration is dominated by the eigenvector corresponding to the eigenvalue with a positive imaginary part near the natural frequency. The contribution of the eigenvector corresponding to the eigenvalue with a negative imaginary part to the total vibration decreases with the increase in frequency.

(2) When plates have real modes with homogeneous damping, Refs. 19,20 have investigated their sound radiation. The results show that the coupling effect cannot be ignored under off-resonant excitation at low frequency but can be neglected at resonant excitation. For real modes, acoustic coupling can only occur between modes of different orders. In this paper, we found that the coupling between a pair of conjugate complex modes can also contribute to sound radiation when employing the complex mode approach to calculate the acoustic radiation from plates. The phenomenon is different from real mode analysis. Furthermore, as shown in Table 4, the coupling between conjugate modes can sometimes be significantly strong. Therefore, neglecting the coupling between modes may result in substantial errors, even at resonance frequencies, when utilizing complex mode analysis for acoustic radiation calculations.

(3) The acoustic power, which is calculated using the approximate vibration response obtained by decoupling approximation, may have errors compared with the exact solution. There may also be some problems, such as omitting the peak of sound power and large calculation errors of the frequency point where the peak of acoustic power is located.