- Email: [email protected]
BEM
Accepted Manuscript Vibro-Acoustic Analysis under Stationary and Non-Stationary Random Excitations with KLE/FEM/BEM
Yanbin Li, Sameer B. Mulani, Qingguo Fei, Shaoqing Wu, Peng Zhang
PII: DOI: Reference:
S1270-9638(17)30394-2 http://dx.doi.org/10.1016/j.ast.2017.03.011 AESCTE 3946
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
3 May 2016 28 December 2016 5 March 2017
Please cite this article in press as: Y. Li et al., Vibro-Acoustic Analysis under Stationary and Non-Stationary Random Excitations with KLE/FEM/BEM, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.03.011
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Vibro-Acoustic Analysis under Stationary and NonStationary Random Excitations with KLE/FEM/BEM
Yanbin Li1, Sameer B. Mulani2, Qingguo Fei1, Shaoqing Wu1, Peng Zhang1 1. Department of Mechanical Engineering, Southeast University, Nanjing, Jiangsu 210096, China 2. The University of Alabama, Tuscaloosa, Alabama 35487, USA
Corresponding author: Qingguo Fei Department of Engineering Mechanics, Southeast University, Nanjing, Jiangsu 210096, China Email: [email protected] Phone: (+86) 02583790168
1
Abstract: An algorithm that integrates Karhunen-Loeve expansion (KLE), finite element method (FEM), and boundary element method (BEM) is proposed to carry out a vibro-acoustic analysis under stationary and non-stationary random excitations which are uncorrelated or correlated. In the KLE, a set of orthogonal basis functions is employed to discretize the autocovariance function of the excitations and obtain the eigenvalues and eigenfunctions of the auto-covariances. The KLE for correlated random excitation relies on the expansions of correlated random variables. During the response calculation, the FEM and BEM are employed to obtain structural and acoustic responses. A circular piston in an infinite baffle and a stiffened panel excited by stationary or non-stationary random processes are used to illustrate the proposed algorithm’s accuracy. Results show that the statistics of vibro-acoustic response are accurately obtained with the proposed method and this method is also applicable for the vibro-acoustic analysis of more complex structures under various types of random excitations.
Keywords: Vibro-acoustic; Non-stationary; Random Vibration; Karhunen-Loeve Expansion; FEM; BEM
2
1. Introduction Aerospace structures often subject to severe environments, such as the combination of heat, vibration, and noise, etc. Vibro-acoustic response due to dynamic loads may endanger the survivability of the payload and the vehicles electronic equipment, and consequently the success of the mission [1]. Hence, a vibro-acoustic analysis is drawing an increasing attention in recent years to improve the design of aerospace structures. The excitations on such kind of structures are often very complex with characteristics of random, non-Gaussian and nonstationary. In engineering practice, the stochastic excitation is often assumed to be stationary, Gaussian, and spatially uncorrelated. To conduct a vibro-acoustic analysis of structures with simplified load models may result in inaccurate results. The development of vibro-acoustic analysis under non-stationary excitation is inhibited by both the extensive computational cost and the inherent mathematical intricacies. With the relentless progress in high-performance computing, computational methods are being increasingly used to perform a vibro-acoustic analysis. The vibro-acoustic analysis is carried out by using energy methods [2] and discrete methods [3-6]. The discrete methods are mainly based on finite element method (FEM) and boundary element method (BEM), such as FEM-FEM [3] and FEM-BEM [4-6]. For the FEM-FEM, the structural and acoustic fields are both modeled by finite elements. When the acoustic field is modeled by 3D solid elements, the computational cost can be extremely high. For the FEM-BEM method, the structural and acoustic fields are respectively simulated by finite elements and boundary elements where the degrees-of-freedom of the boundary element model and the dimensionality of vibro-acoustic analysis are effectively reduced as compared to FEM-FEM approach. On the other hand, a vibro-acoustic analysis is usually conducted by using coupled or uncoupled approaches. In the uncoupled vibro-acoustic analysis [3-5], the mutual interaction between the structural and the acoustic components is
3
considered to be weak or may be neglected, i.e. only a one-way interaction is considered. In the coupled vibro-acoustic analysis [6], the mutual interaction is no longer negligible and all components must be regarded as parts of the coupled system, instead of considering the structural components as independent excitations for the acoustic components or vice versa. In the present work, the vibro-acoustic analysis is performed using uncoupled approach where the structural response is calculated using FEM and the acoustic response is obtained using BEM. Stationary random excitations are described by the methods in time or frequency domain respectively using the autocorrelation function and power spectral density (PSD) function. Since it is easier to solve random vibration problem in the frequency domain, PSD function is usually adopted to represent the stationary random excitation and response [7-9]. The usage probability density function (PDF) makes sense for the non-stationary random excitation because it changes over time. To capture the relation between PDFs at different times, covariance definition of an excitation is used to define a non-stationary random process. So, covariance function is usually used to define the random processes in the time domain. When calculating the response of a dynamical system under non-stationary random excitation, the first step is to discretize the auto-covariance functions of excitations [10]. Efforts have been made in the covariance decomposition which is usually adopted in the Karhunen-Loeve expansion (KLE) and Polynomial Chaos expansion (PCE) [11]. The KLE is a useful and efficient tool for discretizing second-order random processes with known covariance function [12-13] by solving the Fredholm integral equation. Recently, an algorithm using orthogonal decomposition to discretize the auto-covariance function with KLE was proposed by Mulani et al. [14, 15] and Phoon et al. [16]. However, it often yields negative eigenvalues of the autocovariance function based upon the selection of the type of basis functions. Therefore, it is critical to select suitable basis functions for the KLE. In previous work [17], different basis
4
functions were investigated, for example: (1) the global trigonometric basis functions which requires the user to have some a priori knowledge of the system response [14], and (2) the piecewise linear or higher order polynomial Lagrange interpolation functions [17, 18] which neither yield negative nor infinite-value eigenvalues for the auto-covariance function, nor any substantial inaccuracies. After the decomposing of auto-covariance, the response of the system is often obtained by using the KLE under non-stationary and non-Gaussian excitation. For the complex system, the Monte-Carlo method [19] is employed to obtain the response. In previous works [14, 15, 17, 18], the random vibration analysis under stationary and nonstationary random excitation was carried out by using KLE and FEM. However, the random vibration analysis in previous works mainly focuses on the calculation of structural response, the acoustic response of a dynamical system under non-stationary excitations was not intensely studied. Meanwhile, the KLE-based vibro-acoustic analysis has not been applied to complex structures. To develop a method suitable for the vibro-acoustic analysis of a complex system under stationary and non-stationary random excitations, an algorithm that integrates KLE, FEM and BEM is proposed. In conjunction with the FEM and BEM models to the KLE, the proposed algorithm can be not only used to carry out both the structural and acoustic responses but also applied for the vibro-acoustic analysis of large and complex structures under stationary and non-stationary random excitations. Meanwhile, the proposed algorithm can increase the effectiveness and computational efficiency of the vibro-acoustic analysis, while ensuring the analysis accuracy. The outline of this work is as follows: In Section 2, the theory of proposed algorithm is addressed. The proposed method is further validated in Section 3 by a circular piston in infinite baffle and a stiffened panel problems. Finally, conclusions are drawn in the last Section 4.
5
2. Theoretical Derivation 2.1. Karhunen-Loeve Expansion A time-invariant linear second-order system excited by forces can be written as: M x + Cx + Kx = fi (t , ω )
(1)
where M, C, and K are mass, damping and stiffness properties of the system, fi(t, Ȧ) are discrete random forces of excitation. If excitations are distributed random forces Fi(t, Ȧ), then the distributed forces can be converted into discrete forces fi(t, Ȧ) as [20, 21]: NE
f i (t , ω ) = ¦ T ³ NFi (t , ω ) d Ω i =1
Ω
(2)
where NE is the finite element number, T is the transfer matrix that form element coordinates to global coordinates, N is the shape functions of a system, ȍ is the area over which the force is applied. While the discrete random excitations are studied, it is advantageous to study the interaction of multiple random excitations and extract as much information as possible from the relationship of the multiple random processes [7]. The relationship among multi-random processes is achieved through a covariance and correlation analysis. The covariance of random processes, fi(s, Ȧ) and fj(t, Ȧ) with mean values ȝi and ȝj respectively is given as:
CFi Fj ( s, t ) = E[( f i ( s, ω ) − μi )( f j (t, ω ) − μ j )]
(3)
where E[ ] denotes the statistical expectation operator. If a process fi*(t, Ȧ) is substituted for fj(t, Ȧ) in Eq. (3)where fi*(t, Ȧ) is simply a time-shifted version of fi(s, Ȧ), then Eq. (3) is known as auto-covariance CFi Fi ( s, t ) . The auto-covariance reflects the relationship between of two discrete excitations. Due to the auto-covariance of the excitations are un-correlated to spatial parameters, thus the distribution of the random excitations are spatially uniform in the spatial domain.
6
The KLE provides a way to discretize the auto-covariance function of excitation such that it can be used to obtain the response of system [22]. Using KLE, the random processes can be written as a combination of the mean forcing component μ F( i ) and a set of random orthogonal components: n
f i ( t , ω ) = μ F( i ) ( t ) + ¦ λk( i ) φk( i ) ( t )ȟ (ki ) (ω )
(4)
k =1
where λk( i ) and the corresponding φk( i ) (t ) are respectively the eigenvalues and eigenfunctions of the auto-covariance function CFi Fi ( s, t ) , ξk( i ) (ω ) represent sets of random variables, n is truncated number of terms in KLE and its choice strongly depends on the desired accuracy and the auto-covariance function. Because the eigenvalues λk( i ) decay monotonically with increasing values of its index, an appropriate measure of KLE truncation error is given as [23]: n § ∞ · err = ¨ ¦ λk − ¦ λk ¸ k =1 © k =1 ¹
∞
¦λ
k
(5)
k =1
The eigenvalues λk( i ) and corresponding eigenfunctions φk( i ) (t ) of the auto-covariance function can be found by solving a Fredholm equation of the second kind [24]:
³
tmax
tmin
CFi Fi (t1 , t2 )φk( i ) (t1 ) d t1 = λk( i )φk( i ) (t2 )
(6)
where tmin and tmax are the initial and final times, respectively. The procedure for solving these problems is numerically outlined in Ghanem and Spanos [22]. The eigenfunctions φk( i ) (t ) can be further expanded as: n
φk(i ) (t ) = ¦θ (jki ) h(j i ) (t ) j =1
7
(7)
where θ (jki ) are the basis function participation factors and h(ji ) (t ) are the user-defined basis functions. The result of solving Eq. (6) using user-defined basis functions is a matrix eigenvalue problem, as shown in Eq. (8): GD = ȁQD
(8)
where: Gij = ³
tmax
Qij = ³
tmax
tmin
tmin
³
tmax
tmin
CFi Fi (t1 , t2 )hi (t2 ) h j (t1 )dt1dt2
(9)
hi (t )h j (t ) d t
ȁ (kmi ) = δ km λk( i )
In the authors’ previous work [14, 15, 17, 18], it has been proved that the use of simple piecewise constants h(ji ) (t ) as basis functions neither yields negative nor infinite eigenvalues, and also makes the calculation of the eigenvalues and eigenfunctions of auto-covariance very easy and computationally inexpensive. The basis functions of simple piecewise constants are more generalized where no prior knowledge of the decomposition of the covariance function is required for the analysis of simple or complex systems. Therefore, in this research simple piecewise constants as shown in Eq. (10) are adopted as the basis functions.
1 t ∈ [t j −1 , t j ] h j (t ) = ® ¯0 t ∈ other
(10)
When the random processes fi(t, Ȧ) are uncorrelated, ξk( i ) (ω ) represent a set of independent identically distributed random variables having a mean value equal to zero and a variance equal to one. However, when the random processes fi(t, Ȧ) are correlated, the correlation of random variables ξk( i ) (ω ) is given as follows: L(kmij ) = E ª¬ξ k( i )ξ m( j ) º¼
8
(11)
If the random processes have correlation only in a time domain, the cross-covariance functions only depend on time parameters s and t. If the random processes are correlated in both time and spatial domains, the cross-covariance functions depend on not only time parameters s and t but also spatial parameters l1 and l2. Using KLE and the definition of correlation in Eq. (11), the cross-covariance functions can be obtained as follows: n
n
ij ) C Fi F j ( s , t , l1 , l2 ) = E ª¬ f i ( s, ω ) f j (t , ω ) º¼ = ¦ ¦ L(km λk( i ) λm( j ) φk( i ) ( s )φm( j ) (t )
(12)
k =1 m =1
ij ) The correlation L(km in Eq. (11) can be obtained by projecting the cross-covariance
CFi Fj ( s, t ) onto the eigenfunction set of each random process, which yields [23, 25]: L(kmij ) =
1
λ λ (i ) k
( j) m
T
T
0
0
³ ³
C Fi F j ( s, t , l1 , l2 )φk( i )φm( j ) dsdt
(13)
Let L be the block matrix:
ªI « 21 L L=« « « n1 ¬L
L12 L1n º » I L2 n » » » I ¼ Ln 2
(14)
where I is the identity matrix and Lij is the matrix defined in Eq. (13). The correlated random variables ξk( i ) (ω ) with correlation matrix L can be obtained from a transformation of uncorrelated random variables as follows [26]:
ȟ = ij Ȗȗ
(15)
where ȗ is a vector of independent standard random variables having a mean value equal to zero and a variance equal to one, ij and Ȗ are eigenvalues and eigenvectors of the correlation matrix L, respectively.
2.2. Structural Response under Random Excitations Using the definition of KL expansion of the forcing function as given in Eq. (4), the Eq. (1) can be written as: 9
n
M x + C x + K x = μ F( i ) ( t ) + ¦ λk( i ) ȟ (ki ) (ω )φk( i ) ( t )
(16)
k =1
The structural response at the p’th location due to two contributions are evaluated as: (a) the response due to the eigenvector φi( d ) (t ) is designated as m(pid ) (t ) ; (b) the response due to the mean of the excitation μ F (t ) and the initial conditions xk (0) and xk (0) is designated as vk (t ) . Therefore, the structural response at the p’th location is obtained as: d
n
k =1
i1 =1
d
d
n
n
i2 =1
id =1
(2) (2) (2) (d ) (d ) (d ) x p ( t ) = ¦ vk (t ) + ¦ λi1(1) ξi1(1) m (1) pi1 ( t ) + ¦ λi2 ξ i2 m pi2 ( t ) + + ¦ λid ξ id m pid ( t ) n
(17)
= ¦ vk (t ) + ¦¦ λ ξ m (t ) k =1
k =1 ik =1
(k ) (k ) ik ik
(k ) pik
where d represents the number of random processes, p accounts for the structural DOF being considered. In this work, the finite element method (FEM) is adopted to obtain the m(pii ) (t ) and vk (t ) , which is a more generic method for large and complex systems. The random variables ξk( i ) (ω ) used in KLE have a useful property when constructing the variance of the response; their expectation is given as:
i= j δ E ª¬ξk(i ) , ξm( j ) º¼ = ® ˄kmij˅ ¯ Lkm i ≠ j
(18)
where įkm is the Kronecker delta. Using the structural response of displacement in Eq. (17) and the property of random variables as shown in Eq. (18), the auto-covariance Rx p x p ( t1 , t2 ) and the cross-covariance
Rx p x p ( t1 , t2 ) of structural response can be obtained as:
10
d
d
d
k =1
k =1
n
Rx p x p ( t1 , t2 ) = ¦ ª¬vk (t1 ) − μvk (t1 ) º¼ ¦ ª¬vk (t2 ) − μvk (t2 ) º¼ + ¦¦ λi(k k ) m(pikk) ( t1 ) m(pikk) ( t2 ) + d −1
ª
d
n
n
¦ ¦ «¦ ¦ L
λi( k ) λi( k
( k1k2 ) ik1ik2
¬«
k1 =1 k2 = k1 +1 ik1 =1 ik2 =1
k =1 ik =1
1
k1
d
d
k =1
k =1
2)
k2
(m
( k1 ) pik1
2
2
1
k2
k2
k1
d
n
Rx p xq ( t1 , t2 ) = ¦ ª¬vk (t1 ) − μvk (t1 ) º¼ ¦ ª¬vk (t2 ) − μvk (t2 ) º¼ + ¦¦ λ m d −1
ª
d
n
n
¦ ¦ «¦ ¦ L ¬«
k1 =1 k2 = k1 +1 ik1 =1 ik2 =1
k =1 ik =1
λi( k ) λi( k
( k1k2 ) ik1ik2
1
k1
2)
k2
(m
( k1 ) pik1
º
( t1 ) m(pik ) ( t2 ) + m(pik ) ( t1 ) m(pik ) ( t2 ) )» (k ) ik
(k ) pik
¼»
(19)
( t1 ) m ( t2 ) + (k ) qik
º
( t1 ) mqi( k ) ( t2 ) + m(pik ) ( t1 ) mqi( k ) ( t2 ) )» 2
2
1
k2
k2
k1
¼»
where μvk represents the mean of structural response xp(t) as follows:
μv ( t ) = E ª¬ x p ( t ) º¼ = E ª¬ v p ( t ) º¼ = μv ( t ) k
(20)
p
Then the variance of the structural responses become as follows: d
d
2
n
d −1
σ x2 ( t ) = ¦ σ v2 + ¦¦ λi( k ) ª¬ m(pik ) ( t ) º¼ + ¦ p
k
k =1 d
k =1 ik =1 d
k
k
d
ª
n
n
¦ « ¦ ¦ 2L «¬
k1 =1 k2 = k1 +1 ik1 =1 ik2 =1
( k1k2 ) ik1ik2
λi( k ) λi( k ) m(pik ) ( t ) m(pik 1
k1
2
k2
1
2)
k1
k2
k
k =1
k =1 ik =1
k
k
(21)
k
ª n n (k ) (k ) (k ) (k ) (k ) (k ) (k k ) « ¦ ¦ 2 Lik11ik22 λik1 1 λik2 2 m pik11 ( t ) mqik22 ( t ) + m pik22 ( t ) mqik11 ( t ) ¦ ¦ k1 =1 k2 = k1 +1 « ¬ik1 =1 ik2 =1 d −1
»¼
n
σ x2 ( t ) = ¦ σ v2 + ¦¦ λi( k ) m(pik ) ( t ) mqi( k ) ( t ) + pq
º
( t )»
d
(
º
)»» ¼
where σ v2k represents the variance of vi due to the mean value of the forcing function and the initial conditions.
2.3. Acoustic Response under Random Excitations In this research, structural vibration is considered to be independent of the surrounding acoustic medium. Such an assumption is valid when the structure is immersed in light fluids such as air. Once the structural response is known, the boundary element method is utilized to compute the acoustic sound pressure ps due to the velocity boundary condition vn. In this approach, the Helmholtz integral equation given by Seybert et al. [27] is used to obtain the pressure as given follow:
11
ª º § e −iZR · ª ∂ § 1 · º ∂ § e −iZR · + = − ωρ 1 dS p ( r ) p ( r ) i « ¨ ¸ ¨ ¸ vn (r0 ) » dS S 0 « ³s ∂n ¨ 4π R ¸ » s ³ S ∂n © 4π R ¹ © ¹ ¼ ¬ © 4π R ¹ ¬ ¼
(22)
where S represents the vibrating surface, Z is the acoustic wave number, /n implies partial differentiation with respect to the surface normal, r and r0 represent locations on the vibrating surface, and R is the magnitude of the distance between r and r0, ɏ is the density of air. After properly accounting for singularities encountered during integration [28, 29], and non-unique solutions associated with interior characteristic frequencies [30], Eq. (22) can be written in matrix form as: A { g k( i ) (t )} = B {velk( i ) (t )}
(23)
where A and B are the acoustic system matrices. Using FEM, the structural velocity velk( i ) (t ) at all nodes are obtained by applying the eigenfunction φk(i ) (t ) as excitation to the structure. The acoustic response at the q’th location due to two contributions are also evaluated as: (a) the response due to the structural velocity velk( d ) (t ) is designated as gqi( d ) (t ) ; (b) the response due to the mean of the velocity and the initial conditions xk (0) and xk (0) is designated as wk (t ) . Therefore, the acoustic response of sound pressure at the q’th location is obtained as: d
n
k =1
i1 =1
d
d
n
n
i2 =1
id =1
preq ( t ) = ¦ wk (t ) + ¦ λi1(1) ξi1(1) g qi(1)1 (t ) + ¦ λi(2) ξi(2) g qi(2)2 (t ) + + ¦ λi(dd ) ξi(dd ) g qi( dd) (t ) 2 2 n
= ¦ wk (t ) + ¦¦ λ ξ k =1
k =1 ik =1
(k ) ik
(k ) ik
(24)
(k ) qik
g (t )
where q represents the acoustic DOF being considered. In a similar fashion as the structural response, the auto-covariance Rpreq preq ( t1 , t2 ) and 2 variance σ pre ( t ) of sound pressure response can be obtained as follows: q
12
d
d
d
k =1
k =1
n
R preq preq ( t1 , t2 ) = ¦ ª¬ wk (t1 ) − μ prek (t1 ) º¼ ¦ ª¬ wk (t2 ) − μ prek (t2 ) º¼ + ¦¦ λi(k k ) g qi( kk) ( t1 ) g qi( kk) ( t2 ) + k =1 ik =1
ª n n (k k ) (k ) (k ) (k ) (k ) (k ) (k ) « ¦ ¦ Lik11ik22 λik1 1 λik2 2 g qik11 ( t1 ) g qik22 ( t2 ) + g qik22 ( t1 ) g qik11 ( t2 ) ¦ ¦ k1 =1 k2 = k1 +1 « ¬ ik1 =1 ik2 =1 d −1
d
(
d
d
2
n
d −1
2 σ pre ( t ) = ¦σ w2 + ¦¦ λi( k ) ª¬ g qi( k ) ( t )º¼ + ¦ q
k =1
k
k =1 ik =1
k
k
d
ª
n
n
¦ « ¦ ¦ 2L «¬
k1 =1 k2 = k1 +1 ik1 =1 ik2 =1
( k1k2 ) ik1ik2
λi( k ) λi( k ) g qi( k ) ( t ) g qi( k 1
k1
2
k2
1
2)
k1
k2
)
º » »¼ º
( t )» »¼
(25)
(26)
where μ prek represents the mean of acoustic response preq(t), σ w2k is the variance of wk due to the mean value of the velocity and the initial conditions. From Eqs. (19), (21), (25), and (26), it can be seen that the auto-covariance and variance of the vibro-acoustic response depend upon the eigenvalues of the excitation’s auto-covariance function, the deterministic structural response due to the eigenvectors, the deterministic acoustic response due to the structural velocity, the deterministic structural and acoustic response due to mean and initial conditions of the excitation, and the correlation of the correlated random variables. The vibro-acoustic response is irrelevant to the distribution of the random excitation. Thus, the proposed algorithm can be applied to either Gaussian or non-Gaussian excitation. In this research, the random excitations are assumed to the Gaussian processes.
2.4. Implementation Procedure An algorithm is proposed to carry out vibro-acoustic analysis excited by stationary and non-stationary random excitations. The implementation procedure is as follows: (1) Define the geometry and material properties of structure, (2) Create the finite element model of the structure and the boundary element model of the acoustic model, (3) Define or obtain the auto-covariance and cross-covariance functions of the random excitation,
13
(4) Choose the piecewise constant functions as the basis function. Then the eigenvalues
λk( i ) and eigenfunctions φk(i ) of auto-covariance function CF F are obtained by solving a i
i
i i
Fredholm equation of the second kind, (5) Correlation matrix L of random excitation is obtained by using Eqs. (13) and (14) , then the correlated random variables ξ (jki i) are obtained by Eq. (15), (6) The structural responses of displacement m(pii ) (t ) at concerned node and velocity
velk( i ) (t ) at all nodes due to eigenfuctions φk(ii ) are obtained by using FEM, (7) The acoustic response of sound pressure gqi( d ) (t ) at interested node due to velocity
velk( i ) (t ) at all nodes is calculated by BEM, (8) Finally, the structural and acoustic responses are obtained using Eqs.(19) to (21), and Eqs. (25) to (26), respectively.
3. Validation and Application In this section, vibro-acoustic analysis under stationary and non-stationary random excitations for two vibrating structures are presented. The first vibro-acoustic system is used to validate the proposed algorithm. This simple system represents a single degree of freedom of piston placed in an infinite baffle. The second system represented by a stiffened panel is employed to demonstrate the general applicability of the proposed algorithm.
3.1. Validation of the Methodology using a Baffled Piston The piston in an infinite baffle illustrated in Fig. 1(a) is a single degree of freedom system. The characteristics of the piston are provided in Table 1. To calculate the radiated sound pressure due to the vibration of the circular piston in the direction of the radius, the vibroacoustic model of the baffled circular piston is established based on the structural finite element method and the acoustic boundary element method. The structural FEM is modeled
14
using three-noded triangular linear shell elements, as shown in Fig. 1(b). The acoustic BEM is also modeled using three-noded triangular elements with the same size to the structural finite element model. It is assumed that the initial displacement and the velocity are deterministic, and assumed to be zero. The excitations of the random force as shown in Fig. 1 have two different types: (i) F(t) has the exponential auto-covariance as given in Eq. (28), and (ii) F(t) has the exponentially modulated auto-covariance as given in Eq. (29). n
F ( t ) = F0 sin(ω0t ) + ¦ λi φi ( t )ξ i (ω )
(27)
i =1
CFi Fi (t1 , t2 ) =
CFi Fi (t1 , t2 ) =
Di
τi
Di
τi
exp( −
t1 − t2
τi
exp( −t1 − t2 ) exp( −
)
t1 − t2
τi
(28)
)
(29)
where F0 is the magnitude of the harmonic load, Ȧ0 is the frequency of the harmonic load in radians, Di and IJi are the correlation amplitude and correlation length, respectively. To validate the proposed method, the results obtained by proposed numerical method are compared with the results obtained by using analytical method. The analytical displacement and velocity of a structure are obtained as given in Wirsching et al. [7], and the analytical response of the piston is obtained by two steps: (a) the gqi(i ) (t ) due to the velocity velqk( i ) (t ) is obtained using the Eq. (30) as [31]:
g qi(i ) (t ) = i ρ cZη 2velqk(i ) (t )
J1 (Zη sin θ ) e−iZr Zη sin θ r
(30)
where c is the speed of sound, c = 343 m/s; ߟ is the piston diameter; J1 represents Bessel function of the first kind of order one. (b) the stochastic response of radiated sound pressure at the concerned node is obtained by substituting the Eq. (30) into Eqs. (25) and (26).
15
A convergence criterion of truncated error in KLE by using a summation of eigenvalues as given in Eq. (5) is firstly studied for both exponential and exponentially modulated autocovariances. It can be seen that the truncated error decay monotonically with the increasing terms of KLE as shown in Fig. 2. And 40 KLE terms for both exponential and exponentially modulated auto-covariance are required to carry out the vibro-acoustic analysis. The clocktime required for vibro-acoustic analysis with the proposed approach is 3.42s and 2.64s for the exponential and exponentially modulated auto-covariance excitations with Inter(R) Core(TM) i7-6700 CPU 3.41GHz and 8GM RAM, respectively. The means and standard deviations of the structural velocity for baffled piston and radiated sound pressure at point (0.61, 0, 0) m under the exponential and exponentially modulated covariance excitations are shown in Figs. 3 to 5, the corresponding PDFs at 0.04s and 0.8s are shown in Fig. 6, and the corresponding auto-covariances are shown in Figs. 7 and 8. Results show that even when the excitations are stationary for the exponential covariance, the structural and acoustic responses are non-stationary before about 0.4 s due to the effect of initial conditions. As the excitation is non-stationary for exponentially modulated covariance, the responses of structure and acoustic are nonstationary during the entire duration. All PDFs of the responses are Gaussian because the excitations applied to the linear system are Gaussian. Meanwhile, the structural and acoustic responses obtained by proposed KLE/FEM/BEM algorithm have an excellent agreement with the analytical results, which confirms the accuracy of the proposed algorithm.
3.2. Application on a Stiffened Panel To further demonstrate the general applicability of the proposed algorithm, a complex structure, stiffened panel [3], as shown in Fig. 9(a) is used. Stiffened panels are widely used 16
in engineering structures, such as bridges, automobiles, vehicles, and vessels, etc. due to their very high stiffness to weight ratio. Stiffened panel usually consists of a basic structure and local reinforcement elements called stiffeners are added to improve the static and dynamic characteristics of the structure. These stiffened panels are used as parts of fuselage, submarine and ship panels where the turbulent boundary layer (TBL) is acting on the outside surface of the panel. Physical and geometric characteristics of the stiffened panel are provided in Table 2. It is assumed that the mean value of the excitation and the initial conditions are deterministic, and assumed to be zero. Point A (0.89, 0.89, 0) and point B (0.58, 0.58, 0.58) are selected for monitoring the structural and acoustic responses, respectively. The finite element model is established by using 1080 four-noded quadrilateral elements in which there are 7626 degrees of freedom. The boundary element models is built using 1800 three-noded triangular elements. Vibration and noise generated by TBL excitation have been and continues to be of interest in many engineering applications. Much previous works have been done to describe the characteristics of TBL. For this research, the definition of the auto power spectral density function of TBL is used from the tests and empirical formulation. The Ray W. Herrick Laboratory of Purdue University [32] created a TBL excitation with wind tunnel tests and fit a curve using the test results. The auto-spectral density function SFi Fi of TBL is given as follows:
0.0004W −0.66 ° S Fi Fi ° 0.0012 =® 2 3 ρ hU 0 ° 0.0001W −2.2 °¯ 24.5W −9.1
0.06 ≤ W ≤ 0.17 0.17 ≤ W ≤ 0.32 0.32 ≤ W ≤ 6.0
(31)
W ≥ 6.0
where W=Ȧh/U0, Ȧ is the circular frequency in rad/sec, h is the fence height in the experiment conducted by Han and Bernhard [32], U0 is the free stream velocity. From Eq.
17
(31), the power spectral density (PSD) of the TBL can be obtained, as shown in Fig. 10. Then, the magnitude of the auto-covariance parameter, Di=1650.96 Pa2 is calculated by integrating the PSD of TBL. The TBL excitation is correlated both in time and spatial domain in this work. The Corcos model [33] is one of the classical models for describing cross-correlation of the TBL. Based on the Corcos model, the cross-covariance of the TBL is assumed as follows:
§ § ζ · ζ · CFi Fl (ζ 1 , ζ 2 , t1 , t2 ) = CFi Fi (t1 , t2 ) exp ¨¨ −γ 1 1 ¸¸ exp ¨¨ −γ 2 2 ¸¸ Uc ¹ Uc ¹ © ©
(32)
where ζ1 and ζ 2 are the separation distances in the streamwise and spanwise directions; Ȗ1 and Ȗ2 are the decay rates, Uc is the turbulent convection speed, Uc=0.65U0, U0 is the mean flow velocity. The parameters of the TBL excitation used in this manuscript are given in Table 3. Taking the exponential covariance for example, the cross-covariance can be obtained as follows:
CFi Fj (ζ 1 , ζ 2 , t1 , t2 ) =
§ t −t exp ¨ − 1 2 ¨ τ ij τ ij © Dij
· § § ζ · ζ · ¸¸ exp ¨¨ −γ 1 1 ¸¸ exp ¨¨ −γ 2 2 ¸¸ Uc ¹ Uc ¹ © © ¹
(33)
3.2.1. Stationary Covariance Excitation To check the generality of developed algorithm, the stiffened panel is excited by an exponential covariance TBL excitation. The exponential auto-covariance and crosscovariance functions are given by Eq. (28) and Eq. (33), respectively. The correlation lengths IJi and IJij are set equal to 0.1 and 0.2, respectively. The convergence of truncated error of the KLE is also studied, and 40 number of KLE terms is required for the exponential autocovariance, as shown in Fig. 2. The clock-time required for the response analysis with the proposed approach is 1110.9s for the stationary excitations with Inter(R) Core(TM) i7-6700 CPU 3.41GHz and 8GM RAM.
18
As the cross-covariance described in Eq. (31), the random excitation is correlated both in time and space, which is the form of an exponential correlation along the streamwise and spanwise directions. For the vibro-acoustic analysis of complex structures under multiple correlated random excitations, the amount of computing data would become enormous if the spatial correlations for all nodes are considered. From the cross-covariance of the random excitation, the correlation of the excitation decays exponentially with the separation distances ȗ1 and ȗ2. When the separation distances are sufficiently small, the correlations of the excitation between separation distances tend to 1, which means that the excitations are spatially uncorrelated. To capture the correlation of pressure excitation efficiently, the stiffened panel is divided into several subpanels, as shown in Fig. 11; e.g. xl, xu, yl, yu are the bounds of a subpanel. Subscript l and u are used for the lower and upper bounds in the x and y directions, respectively. When the size of the subpanel is small enough, the spatial correlation of the panel is captured accurately, which can significantly improve calculation efficiency. However, the process of partitioning the domain cannot be completely arbitrary. The division in subpanels must be fine enough so as to obtain accurate results. To find an optimal number of subpanels, a convergence of response with a different number of subpanels is carried out. Therefore, the stiffened panel is divided into different subpanels using 2×2 to 5×5 grids. The standard deviations of the displacement at point A and the radiated sound pressure at point B with different subpanels are shown in Fig. 12. To describe the effect of the subpanels’ number on the vibro-acoustic response, Fig. 12 also gives the results without subpanels where the 1×1 subpanel is used. Results show that the standard deviations of the displacement and radiated sound pressure converge to a relatively stable value with the increase of the number of subpanels. Based on a combination of computational efficiency and sufficient accuracy, the number of subpanels 5×5 is adequate for the subsequent vibro-acoustic analysis.
19
In order to study the effect of the correlation on structural and acoustic responses, the vibro-acoustic analyses are carried out under three different excitations: (i) perfectly correlated excitations where the correlation between the random processes is unity, (ii) uncorrelated excitations where the correlation between the random processes is zero, and (iii) partially correlated excitations that the correlation interval between the random processes is between -0.065 and 0.830 obtained by Eq. (13). From the cross-covariance of the random excitation, which is the form of an exponential correlation, the correlation of the excitation which decays exponentially along the x and y directions is not a constant. The standard deviation of the displacement at points A and radiated sound pressure at point B under the three different excitations are shown in Fig. 13. The corresponding PDFs of the responses in the 0.04s and 0.5s are shown in Fig. 14, and the corresponding covariances of the displacement and radiated sound pressure are presented in Fig. 15. Results show that the proposed method is also suitable for the vibro-acoustic analysis of complex system under uncorrelated and correlated random processes. Both the structural and acoustic responses excited by partially correlated random excitation are bounded by the responses excited by perfectly correlated and uncorrelated random processes. Therefore, it is imperative to consider the correlation of random excitations in the structural optimization or fatigue calculations. 3.2.2. Non-Stationary Covariance Excitation To check the applicability of the developed algorithm to any type of auto-covariance excitation, a Bessel auto-covariance excitation is utilized to define the excitation. The Bessel functions have been used in many engineering applications such as heat transfer, acoustics, radio physics, hydrodynamics, atomic and nuclear physics, and so on [34]. The Bessel autocovariance and cross-covariance excitation functions are given in Eq. (34) and Eq. (35) respectively:
20
CFi Fi (t1 , t2 ) =
CFi Fj (ζ 1 , ζ 2 , t1 , t2 ) =
Dij
τ ij
J0 (
Di
τi
J0(
t1 − t2
τi
t1 − t2
τi
)
§ § ζ · ζ · ) exp ¨¨ −γ 1 1 ¸¸ exp ¨¨ −γ 2 2 ¸¸ Uc ¹ Uc ¹ © ©
(34)
(35)
The Bessel auto-covariance function is plotted in Fig. 16. The correlation lengths IJi and IJij are set equal to 0.1 and 0.2, respectively. The convergence of truncated error of the KLE for the Bessel auto-covariance is also studied as shown in Fig. 2, and the number of terms for KLE equal to five is used for the analysis. The clock-time cost for the response analysis with the proposed approach is 487.4s for the non-stationary excitations with Inter(R) Core(TM) i76700 CPU 3.41GHz and 8GM RAM. The convergence of the structural and acoustic responses with different number of division subpanels is also carried out firstly. The standard deviation of the structural displacement at point A and radiated sound pressure at point B with different number of subpanels are shown in Fig. 17. Results show that both the structural and acoustic responses converge to a relatively stable value with the increase of the number of subpanels. Therefore, the number of subpanels 5×5 is also sufficient to carry out the vibro-acoustic analysis for the Bessel covariance excitation. The standard deviation of the structural displacement at point A and radiated sound pressure at point B under the perfectly correlated excitation, uncorrelated excitation, and partially correlated excitation are shown in Fig. 18. In the partially correlated excitation, the correlation interval between the random processes which decays exponentially along the x and y directions is between 0 and 0.99. The corresponding PDFs of the responses in the 0.1s and 0.5s are shown in Fig. 14, and the corresponding auto-covariances of the displacement and radiated sound pressure due to partially correlated Bessel covariance excitation are
21
plotted in Fig. 19. Results show that the proposed methods can also be applied to the Bessel auto-covariance excitation.
4. Conclusions A methodology is presented in which the Karhunen-Loeve expansion (KLE), finite element method (FEM) and boundary element method (BEM) are integrated to conduct the vibro-acoustic analysis under stationary and non-stationary random excitations. Both the uncorrelated and correlated random excitations are considered. The eigenvalues and eigenfunctions of the auto-covariance are obtained by using KLE where a set of orthogonal basis functions are used to discretize the auto-covariance. The KLE for multi-correlated random processes relies on expansions in terms of correlated sets of random variables. During the stochastic response analysis, the FEM and BEM are employed to obtain structural and acoustic responses, respectively. In numerical simulations, a circular piston in infinite baffle excited by exponential and exponentially modulated covariance is firstly presented to validate the proposed algorithm, and a convergence study on the truncation error of KLE is also presented. The simulation results have an excellent agreement with analytical results. Furthermore, the algorithm is applied to a stiffened panel problem having the correlated random excitations whose covariances are characterized by using exponential and Bessel functions, respectively. The vibro-acoustic analysis is carried out with three types of excitations: perfectly correlated, partial correlated and uncorrelated. Results show that the proposed methodology is capable of solving a vibro-acoustic problem of complex structures under various type of random excitations. The structural or acoustic responses due to partially correlated random processes are bounded by the responses obtained by perfectly correlated and uncorrelated random excitations.
22
Acknowledgments The work described in this paper was supported by China Scholarship Council (CSC), a Program for New Century Excellent Talents in University (NCET-11-0086), research grant by the National Natural Science Foundation of China (11572086 & 11402052), Natural Science Foundation of Jiangsu Province (BK20140616); and a Graduate Research and Innovation projects in Jiangsu Province (CXZZ13_0084). We gratefully acknowledge the contribution and help of Dr. Rakesh K. Kapania (Virginia Polytechnic Institute and State University, Blacksburg, USA) on this research.
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[32] F. Han, R. J. Bernhard, and L. G. Mongeau, Prediction of flow-induced structural vibration and sound radiation using energy flow analysis. Journal of Sound and Vibration. 227 (4) (1999) 685-709. doi: 10.1006/jsvi.1998.3013 [33] G. M. Corcos, The structure of the turbulent pressure field in boundary-layer flows. Journal of Fluid Mechanics. 18 (3) (1964) 353-378. doi: 10.1017/S002211206400026X [34] L. Devroye, Simulating bessel random variables, Statistics and Probability Letters. 57 (3) (2002) 249257. doi: 10.1016/S0167-7152(02)00055-X
26
Table 1 Characteristics of the baffled circular piston Table 2 The dimensions and material properties of the stiffened panel Table 3 Numerical values of the parameters used in the TBL mode
27
Table 1 Characteristics of the baffled circular piston Parameters
Values
Mass
8 kg
Stiffness
4.39×105 N/m
Damping
130 N s/m
Diameter
0.1 m
Frequency of the harmonic load
10 Hz
Amplitude of the harmonic load
200 N
Correlation amplitude
5000 N2
Correlation length
0.2
Table 2 The dimensions and material properties of the stiffened panel Parameters
Values
Length of the panel
1.1684 m
Width of the panel
1.1684 m
Thickness of the panel
0.005 m
Height of the stiffener
0.0577 m
Thickness of the stiffener
0.003 m
Distance of the stiffener from panel edge
0.400 m
Young’s modulus
73.08×109 N/m2
Poisson’s ratio
0.33
Density
2.70×103 kg/m3
Damping loss factor
0.02
28
Table 3 Numerical values of the parameters used in the TBL model Parameters
Values
Free stream velocity (U0)
35.75 m/s
Density of air (ȡ)
1.21 kg/m3
Decay rate, streamwise (Ȗ1)
0.16
Decay rate, spanwise (Ȗ2)
0.46
Fence height in the experiment (h)
0.0254 m
29
I.Figure Captions Fig. 1 The configuration and finite/boundary element model of a baffled circular piston (a) Baffled circular piston configuration (b) The finite or boundary element model Fig. 2 Truncation errors of KLE for exponential, exponentially modulated and Bessel autocovariances Fig. 3 Mean of the displacement and sound pressure (a) Mean of the displacement (b) Mean of the sound pressure Fig. 4 Standard deviations of the displacement under exponential and exponentially modulated covariance excitations (a) Exponential covariance excitation (b) Exponentially modulated covariance excitation Fig. 5 Standard deviations of the sound pressure under exponential and exponentially modulated covariance excitations (a) Exponential covariance excitation (b) Exponentially modulated covariance excitation Fig.6 PDFs of the displacement and pressure under exponential and exponentially modulated covariance excitations (a) PDFs of the displacement (b) PDFs of the pressure Fig. 7 Auto-covariances of the displacement under exponential and exponentially modulated covariance excitations (a) Exponential covariance excitation (b) Exponentially modulated covariance excitation Fig. 8 Auto-covariances of the sound pressure under the exponential and exponentially modulated covariance excitations (a) Exponential covariance excitation (b) Exponentially modulated covariance excitations Fig. 9 The finite element and boundary element model of the stiffened panel (a) Finite element model (b) Boundary element model Fig. 10 PSD of the TBL excitation
30
Fig. 11 Subpanels to capture the correlation of a stiffened panel under TBL excitation Fig. 12 Standard deviations of the displacement and sound pressure with different number of subpanels under exponential covariance excitation (a) Standard deviations of the displacement (b) Standard deviations of the sound pressure Fig. 13 Standard deviations of the displacement and sound pressure under perfectly correlated, partially correlated, and uncorrelated exponential covariance excitations (a) Standard deviations of the displacement (b) Standard deviations of the sound pressure Fig. 14 PDFs of the displacement and pressure under exponential and Bessel covariance excitations (a) PDFs of the displacement (b) PDFs of the pressure Fig. 15 Auto-covariances of the displacement and sound pressure under partially correlated exponential covariance excitation (a) Auto-covariance of the displacement (b) Auto-covariance of the sound pressure Fig. 16 Bessel auto-covariance excitation Fig. 17 Standard deviations of the displacement and sound pressure with different number of subpanels under Bessel covariance excitation (a) Standard deviations of the displacement (b) Standard deviations of the sound pressure Fig. 18 Standard deviations of the displacement and sound pressure under perfectly correlated, partially correlated, and uncorrelated Bessel covariance excitations (a) Standard deviations of the displacement (b) Standard deviations of the sound pressure Fig. 19 Auto-covariance of the displacement and sound pressure under partially correlated Bessel covariance excitation (a) Auto-covariance of the displacement (b) Auto-covariance of the sound pressure
31
y
k
x
F
m c (a) Baffled circular piston configuration
(b) The finite or boundary element model Fig. 1 The configuration and finite/boundary element model of a baffled circular piston
Truncation Errors (%)
100 80
Exponential ( i = 0.1) Exponential ( i = 0.2)
60
Exponentially Modulated ( i = 0.2) Bessel ( i = 0.1)
40 20 0
0
10
20
30 40 KLE Terms
50
60
Fig. 2 Truncation errors of KLE for exponential, exponentially modulated and Bessel autocovariances
32
10-4
8
Analytical Numerical
6 4 2 0 -2 -4 -6
0
0.2
0.4
0.6 Time (s)
Mean of the Pressure (Pa)
(a) Mean of the displacement
(b) Mean of the sound pressure Fig. 3 Mean of the displacement and sound pressure
33
0.8
1
10-4
7
Analytical Numerical
6 5 4 3 2 1 0
0
0.2
0.4
0.6
0.8
1
Time (s)
Standard Deviation of the Displacement (m)
(a) Exponential covariance excitation
10-4
7
Analytical Numerical
6 5 4 3 2 1 0
0
0.2
0.4
0.6
0.8
1
Time (s)
(b) Exponentially modulated covariance excitation Fig. 4 Standard deviations of the displacement under exponential and exponentially modulated covariance excitations
34
Standard Deviation of the Pressure (Pa)
10-3
8
Analytical Numerical
6
4
2
0
0
0.2
0.4 0.6 Time (s)
0.8
Standard Deviation of the Pressure (Pa)
(a) Exponential covariance excitation
10-3
8
Analytical Numerical
6
4
2
0
0
0.2
0.4 0.6 Time (s)
0.8
1
(b) Exponentially modulated covariance excitation Fig. 5 Standard deviations of the sound pressure under exponential and exponentially modulated covariance excitations
35
(a) PDFs of the displacement
(b) PDFs of the pressure Fig.6 PDFs of the displacement and pressure under exponential and exponentially modulated covariance excitations
36
(a) Exponential covariance excitation
(b) Exponentially modulated covariance excitation Fig. 7 Auto-covariances of the displacement under exponential and exponentially modulated covariance excitations
37
(a) Exponential covariance excitation
(b) Exponentially modulated covariance excitations Fig. 8 Auto-covariances of the sound pressure under the exponential and exponentially modulated covariance excitations
(0, 0)
38
A
(a) Finite element model
(b)Boundary element model Fig. 9 The finite element and boundary element model of the stiffened panel
0
2
-
PSD(Pa /rad*s 1)
10
10
10
-2
-4
0
2000
4000 6000 8000 10000 12000 Frequency(rad/s)
Fig. 10 PSD of the TBL excitation
39
ߦଶ ߦଵ
ݕ௨ ݕ
y x
ݔ
ݔ௨
Fig. 11 Subpanels to capture the correlation of a stiffened panel under TBL excitation
40
Standard Deviation of the Displacement (m)
10-4
3
1×1 2×2 3×3 4×4 5×5
2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3 0.4 Time (s)
0.5
0.6
0.7
Standard Deviation of the Pressure (Pa)
(a) Standard deviations of the displacement
3.5 1x1 2x2 3x3 4x4 5x5
3 2.5 2
1 0.8 0.6 0.4
1.5
0.2
1
0 0.175
0.5 0
0
0.1
0.2
0.18
0.3 0.4 Time (s)
0.185
0.5
0.19
0.195
0.6
0.7
(b) Standard deviations of the sound pressure Fig. 12 Standard deviations of the displacement and sound pressure with different number of subpanels under exponential covariance excitation
41
Standard Deviation of the Displacement (m)
10-4
3
Perfectly Correlated Uncorrelated Partially Correlated
2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3 0.4 Time (s)
0.5
0.6
0.7
Standard Deviation of the Pressure (Pa)
(a) Standard deviations of the displacement
3.5
1
3
0.8 0.6
2.5
0.4 0.2
2
0.17
1.5
0.18
0.19
0.2
0.21
0.22
0.23
0.24
Perfectly Correlated Partially Correlated Uncorrelated
1 0.5 0
0
0.1
0.2
0.3 0.4 Time (s)
0.5
0.6
0.7
(b) Standard deviations of the sound pressure Fig. 13 Standard deviations of the displacement and sound pressure under perfectly correlated, partially correlated, and uncorrelated exponential covariance excitations
42
(a) PDFs of the displacement
(b) PDFs of the pressure Fig. 14 PDFs of the displacement and pressure under exponential and Bessel covariance excitations
43
(a) Auto-covariance of the displacement
(b) Auto-covariance of the sound pressure Fig. 15 Auto-covariances of the displacement and sound pressure under partially correlated exponential covariance excitation
44
Standard Deviation of the Displacement (m)
Fig. 16 Bessel auto-covariance excitation
10-4
3
1×1 2×2 3×3 4×4 5×5
2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3 0.4 Time (s)
45
0.5
0.6
0.7
Standard Deviation of the Pressure (Pa)
(a) Standard deviations of the displacement
3.5 1x1 2x2 3x3 4x4 5x5
3 2.5 2
1 0.8 0.6 0.4
1.5
0.2
1
0 0.17
0.18
0.19
0.2
0.5 0
0
0.1
0.2
0.3 0.4 Time (s)
0.5
0.6
0.7
(b) Standard deviations of the sound pressure Fig. 17 Standard deviations of the displacement and sound pressure with different number of
Standard Deviation of the Displacement (m)
subpanels under Bessel covariance excitation 10-4
3
Perfectly Correlated Uncorrelated Partially Correlated
2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3 0.4 Time(s)
(a) Standard deviations of the displacement
46
0.5
0.6
0.7
Standard Deviation of the Pressure (Pa)
3.5 1
3
0.8 0.6
2.5
0.4 0.2
2
0.17
1.5
0.18
0.19
0.2
0.21
0.22
0.23
0.24
Perfectly Correlated Partially Correlated Uncorrelated
1 0.5 0
0
0.1
0.2
0.3 0.4 Time(s)
0.5
0.6
0.7
(b) Standard deviations of the sound pressure Fig. 18 Standard deviations of the displacement and sound pressure under perfectly correlated, partially correlated, and uncorrelated Bessel covariance excitations
(a) Auto-covariance of the displacement
47
(b) Auto-covariance of the sound pressure Fig. 19 Auto-covariance of the displacement and sound pressure under partially correlated Bessel covariance excitation
48
Yanbin Li, Sameer B. Mulani, Qingguo Fei, Shaoqing Wu, Peng Zhang
PII: DOI: Reference:
S1270-9638(17)30394-2 http://dx.doi.org/10.1016/j.ast.2017.03.011 AESCTE 3946
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
3 May 2016 28 December 2016 5 March 2017
Please cite this article in press as: Y. Li et al., Vibro-Acoustic Analysis under Stationary and Non-Stationary Random Excitations with KLE/FEM/BEM, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.03.011
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Vibro-Acoustic Analysis under Stationary and NonStationary Random Excitations with KLE/FEM/BEM
Yanbin Li1, Sameer B. Mulani2, Qingguo Fei1, Shaoqing Wu1, Peng Zhang1 1. Department of Mechanical Engineering, Southeast University, Nanjing, Jiangsu 210096, China 2. The University of Alabama, Tuscaloosa, Alabama 35487, USA
Corresponding author: Qingguo Fei Department of Engineering Mechanics, Southeast University, Nanjing, Jiangsu 210096, China Email: [email protected] Phone: (+86) 02583790168
1
Abstract: An algorithm that integrates Karhunen-Loeve expansion (KLE), finite element method (FEM), and boundary element method (BEM) is proposed to carry out a vibro-acoustic analysis under stationary and non-stationary random excitations which are uncorrelated or correlated. In the KLE, a set of orthogonal basis functions is employed to discretize the autocovariance function of the excitations and obtain the eigenvalues and eigenfunctions of the auto-covariances. The KLE for correlated random excitation relies on the expansions of correlated random variables. During the response calculation, the FEM and BEM are employed to obtain structural and acoustic responses. A circular piston in an infinite baffle and a stiffened panel excited by stationary or non-stationary random processes are used to illustrate the proposed algorithm’s accuracy. Results show that the statistics of vibro-acoustic response are accurately obtained with the proposed method and this method is also applicable for the vibro-acoustic analysis of more complex structures under various types of random excitations.
Keywords: Vibro-acoustic; Non-stationary; Random Vibration; Karhunen-Loeve Expansion; FEM; BEM
2
1. Introduction Aerospace structures often subject to severe environments, such as the combination of heat, vibration, and noise, etc. Vibro-acoustic response due to dynamic loads may endanger the survivability of the payload and the vehicles electronic equipment, and consequently the success of the mission [1]. Hence, a vibro-acoustic analysis is drawing an increasing attention in recent years to improve the design of aerospace structures. The excitations on such kind of structures are often very complex with characteristics of random, non-Gaussian and nonstationary. In engineering practice, the stochastic excitation is often assumed to be stationary, Gaussian, and spatially uncorrelated. To conduct a vibro-acoustic analysis of structures with simplified load models may result in inaccurate results. The development of vibro-acoustic analysis under non-stationary excitation is inhibited by both the extensive computational cost and the inherent mathematical intricacies. With the relentless progress in high-performance computing, computational methods are being increasingly used to perform a vibro-acoustic analysis. The vibro-acoustic analysis is carried out by using energy methods [2] and discrete methods [3-6]. The discrete methods are mainly based on finite element method (FEM) and boundary element method (BEM), such as FEM-FEM [3] and FEM-BEM [4-6]. For the FEM-FEM, the structural and acoustic fields are both modeled by finite elements. When the acoustic field is modeled by 3D solid elements, the computational cost can be extremely high. For the FEM-BEM method, the structural and acoustic fields are respectively simulated by finite elements and boundary elements where the degrees-of-freedom of the boundary element model and the dimensionality of vibro-acoustic analysis are effectively reduced as compared to FEM-FEM approach. On the other hand, a vibro-acoustic analysis is usually conducted by using coupled or uncoupled approaches. In the uncoupled vibro-acoustic analysis [3-5], the mutual interaction between the structural and the acoustic components is
3
considered to be weak or may be neglected, i.e. only a one-way interaction is considered. In the coupled vibro-acoustic analysis [6], the mutual interaction is no longer negligible and all components must be regarded as parts of the coupled system, instead of considering the structural components as independent excitations for the acoustic components or vice versa. In the present work, the vibro-acoustic analysis is performed using uncoupled approach where the structural response is calculated using FEM and the acoustic response is obtained using BEM. Stationary random excitations are described by the methods in time or frequency domain respectively using the autocorrelation function and power spectral density (PSD) function. Since it is easier to solve random vibration problem in the frequency domain, PSD function is usually adopted to represent the stationary random excitation and response [7-9]. The usage probability density function (PDF) makes sense for the non-stationary random excitation because it changes over time. To capture the relation between PDFs at different times, covariance definition of an excitation is used to define a non-stationary random process. So, covariance function is usually used to define the random processes in the time domain. When calculating the response of a dynamical system under non-stationary random excitation, the first step is to discretize the auto-covariance functions of excitations [10]. Efforts have been made in the covariance decomposition which is usually adopted in the Karhunen-Loeve expansion (KLE) and Polynomial Chaos expansion (PCE) [11]. The KLE is a useful and efficient tool for discretizing second-order random processes with known covariance function [12-13] by solving the Fredholm integral equation. Recently, an algorithm using orthogonal decomposition to discretize the auto-covariance function with KLE was proposed by Mulani et al. [14, 15] and Phoon et al. [16]. However, it often yields negative eigenvalues of the autocovariance function based upon the selection of the type of basis functions. Therefore, it is critical to select suitable basis functions for the KLE. In previous work [17], different basis
4
functions were investigated, for example: (1) the global trigonometric basis functions which requires the user to have some a priori knowledge of the system response [14], and (2) the piecewise linear or higher order polynomial Lagrange interpolation functions [17, 18] which neither yield negative nor infinite-value eigenvalues for the auto-covariance function, nor any substantial inaccuracies. After the decomposing of auto-covariance, the response of the system is often obtained by using the KLE under non-stationary and non-Gaussian excitation. For the complex system, the Monte-Carlo method [19] is employed to obtain the response. In previous works [14, 15, 17, 18], the random vibration analysis under stationary and nonstationary random excitation was carried out by using KLE and FEM. However, the random vibration analysis in previous works mainly focuses on the calculation of structural response, the acoustic response of a dynamical system under non-stationary excitations was not intensely studied. Meanwhile, the KLE-based vibro-acoustic analysis has not been applied to complex structures. To develop a method suitable for the vibro-acoustic analysis of a complex system under stationary and non-stationary random excitations, an algorithm that integrates KLE, FEM and BEM is proposed. In conjunction with the FEM and BEM models to the KLE, the proposed algorithm can be not only used to carry out both the structural and acoustic responses but also applied for the vibro-acoustic analysis of large and complex structures under stationary and non-stationary random excitations. Meanwhile, the proposed algorithm can increase the effectiveness and computational efficiency of the vibro-acoustic analysis, while ensuring the analysis accuracy. The outline of this work is as follows: In Section 2, the theory of proposed algorithm is addressed. The proposed method is further validated in Section 3 by a circular piston in infinite baffle and a stiffened panel problems. Finally, conclusions are drawn in the last Section 4.
5
2. Theoretical Derivation 2.1. Karhunen-Loeve Expansion A time-invariant linear second-order system excited by forces can be written as: M x + Cx + Kx = fi (t , ω )
(1)
where M, C, and K are mass, damping and stiffness properties of the system, fi(t, Ȧ) are discrete random forces of excitation. If excitations are distributed random forces Fi(t, Ȧ), then the distributed forces can be converted into discrete forces fi(t, Ȧ) as [20, 21]: NE
f i (t , ω ) = ¦ T ³ NFi (t , ω ) d Ω i =1
Ω
(2)
where NE is the finite element number, T is the transfer matrix that form element coordinates to global coordinates, N is the shape functions of a system, ȍ is the area over which the force is applied. While the discrete random excitations are studied, it is advantageous to study the interaction of multiple random excitations and extract as much information as possible from the relationship of the multiple random processes [7]. The relationship among multi-random processes is achieved through a covariance and correlation analysis. The covariance of random processes, fi(s, Ȧ) and fj(t, Ȧ) with mean values ȝi and ȝj respectively is given as:
CFi Fj ( s, t ) = E[( f i ( s, ω ) − μi )( f j (t, ω ) − μ j )]
(3)
where E[ ] denotes the statistical expectation operator. If a process fi*(t, Ȧ) is substituted for fj(t, Ȧ) in Eq. (3)where fi*(t, Ȧ) is simply a time-shifted version of fi(s, Ȧ), then Eq. (3) is known as auto-covariance CFi Fi ( s, t ) . The auto-covariance reflects the relationship between of two discrete excitations. Due to the auto-covariance of the excitations are un-correlated to spatial parameters, thus the distribution of the random excitations are spatially uniform in the spatial domain.
6
The KLE provides a way to discretize the auto-covariance function of excitation such that it can be used to obtain the response of system [22]. Using KLE, the random processes can be written as a combination of the mean forcing component μ F( i ) and a set of random orthogonal components: n
f i ( t , ω ) = μ F( i ) ( t ) + ¦ λk( i ) φk( i ) ( t )ȟ (ki ) (ω )
(4)
k =1
where λk( i ) and the corresponding φk( i ) (t ) are respectively the eigenvalues and eigenfunctions of the auto-covariance function CFi Fi ( s, t ) , ξk( i ) (ω ) represent sets of random variables, n is truncated number of terms in KLE and its choice strongly depends on the desired accuracy and the auto-covariance function. Because the eigenvalues λk( i ) decay monotonically with increasing values of its index, an appropriate measure of KLE truncation error is given as [23]: n § ∞ · err = ¨ ¦ λk − ¦ λk ¸ k =1 © k =1 ¹
∞
¦λ
k
(5)
k =1
The eigenvalues λk( i ) and corresponding eigenfunctions φk( i ) (t ) of the auto-covariance function can be found by solving a Fredholm equation of the second kind [24]:
³
tmax
tmin
CFi Fi (t1 , t2 )φk( i ) (t1 ) d t1 = λk( i )φk( i ) (t2 )
(6)
where tmin and tmax are the initial and final times, respectively. The procedure for solving these problems is numerically outlined in Ghanem and Spanos [22]. The eigenfunctions φk( i ) (t ) can be further expanded as: n
φk(i ) (t ) = ¦θ (jki ) h(j i ) (t ) j =1
7
(7)
where θ (jki ) are the basis function participation factors and h(ji ) (t ) are the user-defined basis functions. The result of solving Eq. (6) using user-defined basis functions is a matrix eigenvalue problem, as shown in Eq. (8): GD = ȁQD
(8)
where: Gij = ³
tmax
Qij = ³
tmax
tmin
tmin
³
tmax
tmin
CFi Fi (t1 , t2 )hi (t2 ) h j (t1 )dt1dt2
(9)
hi (t )h j (t ) d t
ȁ (kmi ) = δ km λk( i )
In the authors’ previous work [14, 15, 17, 18], it has been proved that the use of simple piecewise constants h(ji ) (t ) as basis functions neither yields negative nor infinite eigenvalues, and also makes the calculation of the eigenvalues and eigenfunctions of auto-covariance very easy and computationally inexpensive. The basis functions of simple piecewise constants are more generalized where no prior knowledge of the decomposition of the covariance function is required for the analysis of simple or complex systems. Therefore, in this research simple piecewise constants as shown in Eq. (10) are adopted as the basis functions.
1 t ∈ [t j −1 , t j ] h j (t ) = ® ¯0 t ∈ other
(10)
When the random processes fi(t, Ȧ) are uncorrelated, ξk( i ) (ω ) represent a set of independent identically distributed random variables having a mean value equal to zero and a variance equal to one. However, when the random processes fi(t, Ȧ) are correlated, the correlation of random variables ξk( i ) (ω ) is given as follows: L(kmij ) = E ª¬ξ k( i )ξ m( j ) º¼
8
(11)
If the random processes have correlation only in a time domain, the cross-covariance functions only depend on time parameters s and t. If the random processes are correlated in both time and spatial domains, the cross-covariance functions depend on not only time parameters s and t but also spatial parameters l1 and l2. Using KLE and the definition of correlation in Eq. (11), the cross-covariance functions can be obtained as follows: n
n
ij ) C Fi F j ( s , t , l1 , l2 ) = E ª¬ f i ( s, ω ) f j (t , ω ) º¼ = ¦ ¦ L(km λk( i ) λm( j ) φk( i ) ( s )φm( j ) (t )
(12)
k =1 m =1
ij ) The correlation L(km in Eq. (11) can be obtained by projecting the cross-covariance
CFi Fj ( s, t ) onto the eigenfunction set of each random process, which yields [23, 25]: L(kmij ) =
1
λ λ (i ) k
( j) m
T
T
0
0
³ ³
C Fi F j ( s, t , l1 , l2 )φk( i )φm( j ) dsdt
(13)
Let L be the block matrix:
ªI « 21 L L=« « « n1 ¬L
L12 L1n º » I L2 n » » » I ¼ Ln 2
(14)
where I is the identity matrix and Lij is the matrix defined in Eq. (13). The correlated random variables ξk( i ) (ω ) with correlation matrix L can be obtained from a transformation of uncorrelated random variables as follows [26]:
ȟ = ij Ȗȗ
(15)
where ȗ is a vector of independent standard random variables having a mean value equal to zero and a variance equal to one, ij and Ȗ are eigenvalues and eigenvectors of the correlation matrix L, respectively.
2.2. Structural Response under Random Excitations Using the definition of KL expansion of the forcing function as given in Eq. (4), the Eq. (1) can be written as: 9
n
M x + C x + K x = μ F( i ) ( t ) + ¦ λk( i ) ȟ (ki ) (ω )φk( i ) ( t )
(16)
k =1
The structural response at the p’th location due to two contributions are evaluated as: (a) the response due to the eigenvector φi( d ) (t ) is designated as m(pid ) (t ) ; (b) the response due to the mean of the excitation μ F (t ) and the initial conditions xk (0) and xk (0) is designated as vk (t ) . Therefore, the structural response at the p’th location is obtained as: d
n
k =1
i1 =1
d
d
n
n
i2 =1
id =1
(2) (2) (2) (d ) (d ) (d ) x p ( t ) = ¦ vk (t ) + ¦ λi1(1) ξi1(1) m (1) pi1 ( t ) + ¦ λi2 ξ i2 m pi2 ( t ) + + ¦ λid ξ id m pid ( t ) n
(17)
= ¦ vk (t ) + ¦¦ λ ξ m (t ) k =1
k =1 ik =1
(k ) (k ) ik ik
(k ) pik
where d represents the number of random processes, p accounts for the structural DOF being considered. In this work, the finite element method (FEM) is adopted to obtain the m(pii ) (t ) and vk (t ) , which is a more generic method for large and complex systems. The random variables ξk( i ) (ω ) used in KLE have a useful property when constructing the variance of the response; their expectation is given as:
i= j δ E ª¬ξk(i ) , ξm( j ) º¼ = ® ˄kmij˅ ¯ Lkm i ≠ j
(18)
where įkm is the Kronecker delta. Using the structural response of displacement in Eq. (17) and the property of random variables as shown in Eq. (18), the auto-covariance Rx p x p ( t1 , t2 ) and the cross-covariance
Rx p x p ( t1 , t2 ) of structural response can be obtained as:
10
d
d
d
k =1
k =1
n
Rx p x p ( t1 , t2 ) = ¦ ª¬vk (t1 ) − μvk (t1 ) º¼ ¦ ª¬vk (t2 ) − μvk (t2 ) º¼ + ¦¦ λi(k k ) m(pikk) ( t1 ) m(pikk) ( t2 ) + d −1
ª
d
n
n
¦ ¦ «¦ ¦ L
λi( k ) λi( k
( k1k2 ) ik1ik2
¬«
k1 =1 k2 = k1 +1 ik1 =1 ik2 =1
k =1 ik =1
1
k1
d
d
k =1
k =1
2)
k2
(m
( k1 ) pik1
2
2
1
k2
k2
k1
d
n
Rx p xq ( t1 , t2 ) = ¦ ª¬vk (t1 ) − μvk (t1 ) º¼ ¦ ª¬vk (t2 ) − μvk (t2 ) º¼ + ¦¦ λ m d −1
ª
d
n
n
¦ ¦ «¦ ¦ L ¬«
k1 =1 k2 = k1 +1 ik1 =1 ik2 =1
k =1 ik =1
λi( k ) λi( k
( k1k2 ) ik1ik2
1
k1
2)
k2
(m
( k1 ) pik1
º
( t1 ) m(pik ) ( t2 ) + m(pik ) ( t1 ) m(pik ) ( t2 ) )» (k ) ik
(k ) pik
¼»
(19)
( t1 ) m ( t2 ) + (k ) qik
º
( t1 ) mqi( k ) ( t2 ) + m(pik ) ( t1 ) mqi( k ) ( t2 ) )» 2
2
1
k2
k2
k1
¼»
where μvk represents the mean of structural response xp(t) as follows:
μv ( t ) = E ª¬ x p ( t ) º¼ = E ª¬ v p ( t ) º¼ = μv ( t ) k
(20)
p
Then the variance of the structural responses become as follows: d
d
2
n
d −1
σ x2 ( t ) = ¦ σ v2 + ¦¦ λi( k ) ª¬ m(pik ) ( t ) º¼ + ¦ p
k
k =1 d
k =1 ik =1 d
k
k
d
ª
n
n
¦ « ¦ ¦ 2L «¬
k1 =1 k2 = k1 +1 ik1 =1 ik2 =1
( k1k2 ) ik1ik2
λi( k ) λi( k ) m(pik ) ( t ) m(pik 1
k1
2
k2
1
2)
k1
k2
k
k =1
k =1 ik =1
k
k
(21)
k
ª n n (k ) (k ) (k ) (k ) (k ) (k ) (k k ) « ¦ ¦ 2 Lik11ik22 λik1 1 λik2 2 m pik11 ( t ) mqik22 ( t ) + m pik22 ( t ) mqik11 ( t ) ¦ ¦ k1 =1 k2 = k1 +1 « ¬ik1 =1 ik2 =1 d −1
»¼
n
σ x2 ( t ) = ¦ σ v2 + ¦¦ λi( k ) m(pik ) ( t ) mqi( k ) ( t ) + pq
º
( t )»
d
(
º
)»» ¼
where σ v2k represents the variance of vi due to the mean value of the forcing function and the initial conditions.
2.3. Acoustic Response under Random Excitations In this research, structural vibration is considered to be independent of the surrounding acoustic medium. Such an assumption is valid when the structure is immersed in light fluids such as air. Once the structural response is known, the boundary element method is utilized to compute the acoustic sound pressure ps due to the velocity boundary condition vn. In this approach, the Helmholtz integral equation given by Seybert et al. [27] is used to obtain the pressure as given follow:
11
ª º § e −iZR · ª ∂ § 1 · º ∂ § e −iZR · + = − ωρ 1 dS p ( r ) p ( r ) i « ¨ ¸ ¨ ¸ vn (r0 ) » dS S 0 « ³s ∂n ¨ 4π R ¸ » s ³ S ∂n © 4π R ¹ © ¹ ¼ ¬ © 4π R ¹ ¬ ¼
(22)
where S represents the vibrating surface, Z is the acoustic wave number, /n implies partial differentiation with respect to the surface normal, r and r0 represent locations on the vibrating surface, and R is the magnitude of the distance between r and r0, ɏ is the density of air. After properly accounting for singularities encountered during integration [28, 29], and non-unique solutions associated with interior characteristic frequencies [30], Eq. (22) can be written in matrix form as: A { g k( i ) (t )} = B {velk( i ) (t )}
(23)
where A and B are the acoustic system matrices. Using FEM, the structural velocity velk( i ) (t ) at all nodes are obtained by applying the eigenfunction φk(i ) (t ) as excitation to the structure. The acoustic response at the q’th location due to two contributions are also evaluated as: (a) the response due to the structural velocity velk( d ) (t ) is designated as gqi( d ) (t ) ; (b) the response due to the mean of the velocity and the initial conditions xk (0) and xk (0) is designated as wk (t ) . Therefore, the acoustic response of sound pressure at the q’th location is obtained as: d
n
k =1
i1 =1
d
d
n
n
i2 =1
id =1
preq ( t ) = ¦ wk (t ) + ¦ λi1(1) ξi1(1) g qi(1)1 (t ) + ¦ λi(2) ξi(2) g qi(2)2 (t ) + + ¦ λi(dd ) ξi(dd ) g qi( dd) (t ) 2 2 n
= ¦ wk (t ) + ¦¦ λ ξ k =1
k =1 ik =1
(k ) ik
(k ) ik
(24)
(k ) qik
g (t )
where q represents the acoustic DOF being considered. In a similar fashion as the structural response, the auto-covariance Rpreq preq ( t1 , t2 ) and 2 variance σ pre ( t ) of sound pressure response can be obtained as follows: q
12
d
d
d
k =1
k =1
n
R preq preq ( t1 , t2 ) = ¦ ª¬ wk (t1 ) − μ prek (t1 ) º¼ ¦ ª¬ wk (t2 ) − μ prek (t2 ) º¼ + ¦¦ λi(k k ) g qi( kk) ( t1 ) g qi( kk) ( t2 ) + k =1 ik =1
ª n n (k k ) (k ) (k ) (k ) (k ) (k ) (k ) « ¦ ¦ Lik11ik22 λik1 1 λik2 2 g qik11 ( t1 ) g qik22 ( t2 ) + g qik22 ( t1 ) g qik11 ( t2 ) ¦ ¦ k1 =1 k2 = k1 +1 « ¬ ik1 =1 ik2 =1 d −1
d
(
d
d
2
n
d −1
2 σ pre ( t ) = ¦σ w2 + ¦¦ λi( k ) ª¬ g qi( k ) ( t )º¼ + ¦ q
k =1
k
k =1 ik =1
k
k
d
ª
n
n
¦ « ¦ ¦ 2L «¬
k1 =1 k2 = k1 +1 ik1 =1 ik2 =1
( k1k2 ) ik1ik2
λi( k ) λi( k ) g qi( k ) ( t ) g qi( k 1
k1
2
k2
1
2)
k1
k2
)
º » »¼ º
( t )» »¼
(25)
(26)
where μ prek represents the mean of acoustic response preq(t), σ w2k is the variance of wk due to the mean value of the velocity and the initial conditions. From Eqs. (19), (21), (25), and (26), it can be seen that the auto-covariance and variance of the vibro-acoustic response depend upon the eigenvalues of the excitation’s auto-covariance function, the deterministic structural response due to the eigenvectors, the deterministic acoustic response due to the structural velocity, the deterministic structural and acoustic response due to mean and initial conditions of the excitation, and the correlation of the correlated random variables. The vibro-acoustic response is irrelevant to the distribution of the random excitation. Thus, the proposed algorithm can be applied to either Gaussian or non-Gaussian excitation. In this research, the random excitations are assumed to the Gaussian processes.
2.4. Implementation Procedure An algorithm is proposed to carry out vibro-acoustic analysis excited by stationary and non-stationary random excitations. The implementation procedure is as follows: (1) Define the geometry and material properties of structure, (2) Create the finite element model of the structure and the boundary element model of the acoustic model, (3) Define or obtain the auto-covariance and cross-covariance functions of the random excitation,
13
(4) Choose the piecewise constant functions as the basis function. Then the eigenvalues
λk( i ) and eigenfunctions φk(i ) of auto-covariance function CF F are obtained by solving a i
i
i i
Fredholm equation of the second kind, (5) Correlation matrix L of random excitation is obtained by using Eqs. (13) and (14) , then the correlated random variables ξ (jki i) are obtained by Eq. (15), (6) The structural responses of displacement m(pii ) (t ) at concerned node and velocity
velk( i ) (t ) at all nodes due to eigenfuctions φk(ii ) are obtained by using FEM, (7) The acoustic response of sound pressure gqi( d ) (t ) at interested node due to velocity
velk( i ) (t ) at all nodes is calculated by BEM, (8) Finally, the structural and acoustic responses are obtained using Eqs.(19) to (21), and Eqs. (25) to (26), respectively.
3. Validation and Application In this section, vibro-acoustic analysis under stationary and non-stationary random excitations for two vibrating structures are presented. The first vibro-acoustic system is used to validate the proposed algorithm. This simple system represents a single degree of freedom of piston placed in an infinite baffle. The second system represented by a stiffened panel is employed to demonstrate the general applicability of the proposed algorithm.
3.1. Validation of the Methodology using a Baffled Piston The piston in an infinite baffle illustrated in Fig. 1(a) is a single degree of freedom system. The characteristics of the piston are provided in Table 1. To calculate the radiated sound pressure due to the vibration of the circular piston in the direction of the radius, the vibroacoustic model of the baffled circular piston is established based on the structural finite element method and the acoustic boundary element method. The structural FEM is modeled
14
using three-noded triangular linear shell elements, as shown in Fig. 1(b). The acoustic BEM is also modeled using three-noded triangular elements with the same size to the structural finite element model. It is assumed that the initial displacement and the velocity are deterministic, and assumed to be zero. The excitations of the random force as shown in Fig. 1 have two different types: (i) F(t) has the exponential auto-covariance as given in Eq. (28), and (ii) F(t) has the exponentially modulated auto-covariance as given in Eq. (29). n
F ( t ) = F0 sin(ω0t ) + ¦ λi φi ( t )ξ i (ω )
(27)
i =1
CFi Fi (t1 , t2 ) =
CFi Fi (t1 , t2 ) =
Di
τi
Di
τi
exp( −
t1 − t2
τi
exp( −t1 − t2 ) exp( −
)
t1 − t2
τi
(28)
)
(29)
where F0 is the magnitude of the harmonic load, Ȧ0 is the frequency of the harmonic load in radians, Di and IJi are the correlation amplitude and correlation length, respectively. To validate the proposed method, the results obtained by proposed numerical method are compared with the results obtained by using analytical method. The analytical displacement and velocity of a structure are obtained as given in Wirsching et al. [7], and the analytical response of the piston is obtained by two steps: (a) the gqi(i ) (t ) due to the velocity velqk( i ) (t ) is obtained using the Eq. (30) as [31]:
g qi(i ) (t ) = i ρ cZη 2velqk(i ) (t )
J1 (Zη sin θ ) e−iZr Zη sin θ r
(30)
where c is the speed of sound, c = 343 m/s; ߟ is the piston diameter; J1 represents Bessel function of the first kind of order one. (b) the stochastic response of radiated sound pressure at the concerned node is obtained by substituting the Eq. (30) into Eqs. (25) and (26).
15
A convergence criterion of truncated error in KLE by using a summation of eigenvalues as given in Eq. (5) is firstly studied for both exponential and exponentially modulated autocovariances. It can be seen that the truncated error decay monotonically with the increasing terms of KLE as shown in Fig. 2. And 40 KLE terms for both exponential and exponentially modulated auto-covariance are required to carry out the vibro-acoustic analysis. The clocktime required for vibro-acoustic analysis with the proposed approach is 3.42s and 2.64s for the exponential and exponentially modulated auto-covariance excitations with Inter(R) Core(TM) i7-6700 CPU 3.41GHz and 8GM RAM, respectively. The means and standard deviations of the structural velocity for baffled piston and radiated sound pressure at point (0.61, 0, 0) m under the exponential and exponentially modulated covariance excitations are shown in Figs. 3 to 5, the corresponding PDFs at 0.04s and 0.8s are shown in Fig. 6, and the corresponding auto-covariances are shown in Figs. 7 and 8. Results show that even when the excitations are stationary for the exponential covariance, the structural and acoustic responses are non-stationary before about 0.4 s due to the effect of initial conditions. As the excitation is non-stationary for exponentially modulated covariance, the responses of structure and acoustic are nonstationary during the entire duration. All PDFs of the responses are Gaussian because the excitations applied to the linear system are Gaussian. Meanwhile, the structural and acoustic responses obtained by proposed KLE/FEM/BEM algorithm have an excellent agreement with the analytical results, which confirms the accuracy of the proposed algorithm.
3.2. Application on a Stiffened Panel To further demonstrate the general applicability of the proposed algorithm, a complex structure, stiffened panel [3], as shown in Fig. 9(a) is used. Stiffened panels are widely used 16
in engineering structures, such as bridges, automobiles, vehicles, and vessels, etc. due to their very high stiffness to weight ratio. Stiffened panel usually consists of a basic structure and local reinforcement elements called stiffeners are added to improve the static and dynamic characteristics of the structure. These stiffened panels are used as parts of fuselage, submarine and ship panels where the turbulent boundary layer (TBL) is acting on the outside surface of the panel. Physical and geometric characteristics of the stiffened panel are provided in Table 2. It is assumed that the mean value of the excitation and the initial conditions are deterministic, and assumed to be zero. Point A (0.89, 0.89, 0) and point B (0.58, 0.58, 0.58) are selected for monitoring the structural and acoustic responses, respectively. The finite element model is established by using 1080 four-noded quadrilateral elements in which there are 7626 degrees of freedom. The boundary element models is built using 1800 three-noded triangular elements. Vibration and noise generated by TBL excitation have been and continues to be of interest in many engineering applications. Much previous works have been done to describe the characteristics of TBL. For this research, the definition of the auto power spectral density function of TBL is used from the tests and empirical formulation. The Ray W. Herrick Laboratory of Purdue University [32] created a TBL excitation with wind tunnel tests and fit a curve using the test results. The auto-spectral density function SFi Fi of TBL is given as follows:
0.0004W −0.66 ° S Fi Fi ° 0.0012 =® 2 3 ρ hU 0 ° 0.0001W −2.2 °¯ 24.5W −9.1
0.06 ≤ W ≤ 0.17 0.17 ≤ W ≤ 0.32 0.32 ≤ W ≤ 6.0
(31)
W ≥ 6.0
where W=Ȧh/U0, Ȧ is the circular frequency in rad/sec, h is the fence height in the experiment conducted by Han and Bernhard [32], U0 is the free stream velocity. From Eq.
17
(31), the power spectral density (PSD) of the TBL can be obtained, as shown in Fig. 10. Then, the magnitude of the auto-covariance parameter, Di=1650.96 Pa2 is calculated by integrating the PSD of TBL. The TBL excitation is correlated both in time and spatial domain in this work. The Corcos model [33] is one of the classical models for describing cross-correlation of the TBL. Based on the Corcos model, the cross-covariance of the TBL is assumed as follows:
§ § ζ · ζ · CFi Fl (ζ 1 , ζ 2 , t1 , t2 ) = CFi Fi (t1 , t2 ) exp ¨¨ −γ 1 1 ¸¸ exp ¨¨ −γ 2 2 ¸¸ Uc ¹ Uc ¹ © ©
(32)
where ζ1 and ζ 2 are the separation distances in the streamwise and spanwise directions; Ȗ1 and Ȗ2 are the decay rates, Uc is the turbulent convection speed, Uc=0.65U0, U0 is the mean flow velocity. The parameters of the TBL excitation used in this manuscript are given in Table 3. Taking the exponential covariance for example, the cross-covariance can be obtained as follows:
CFi Fj (ζ 1 , ζ 2 , t1 , t2 ) =
§ t −t exp ¨ − 1 2 ¨ τ ij τ ij © Dij
· § § ζ · ζ · ¸¸ exp ¨¨ −γ 1 1 ¸¸ exp ¨¨ −γ 2 2 ¸¸ Uc ¹ Uc ¹ © © ¹
(33)
3.2.1. Stationary Covariance Excitation To check the generality of developed algorithm, the stiffened panel is excited by an exponential covariance TBL excitation. The exponential auto-covariance and crosscovariance functions are given by Eq. (28) and Eq. (33), respectively. The correlation lengths IJi and IJij are set equal to 0.1 and 0.2, respectively. The convergence of truncated error of the KLE is also studied, and 40 number of KLE terms is required for the exponential autocovariance, as shown in Fig. 2. The clock-time required for the response analysis with the proposed approach is 1110.9s for the stationary excitations with Inter(R) Core(TM) i7-6700 CPU 3.41GHz and 8GM RAM.
18
As the cross-covariance described in Eq. (31), the random excitation is correlated both in time and space, which is the form of an exponential correlation along the streamwise and spanwise directions. For the vibro-acoustic analysis of complex structures under multiple correlated random excitations, the amount of computing data would become enormous if the spatial correlations for all nodes are considered. From the cross-covariance of the random excitation, the correlation of the excitation decays exponentially with the separation distances ȗ1 and ȗ2. When the separation distances are sufficiently small, the correlations of the excitation between separation distances tend to 1, which means that the excitations are spatially uncorrelated. To capture the correlation of pressure excitation efficiently, the stiffened panel is divided into several subpanels, as shown in Fig. 11; e.g. xl, xu, yl, yu are the bounds of a subpanel. Subscript l and u are used for the lower and upper bounds in the x and y directions, respectively. When the size of the subpanel is small enough, the spatial correlation of the panel is captured accurately, which can significantly improve calculation efficiency. However, the process of partitioning the domain cannot be completely arbitrary. The division in subpanels must be fine enough so as to obtain accurate results. To find an optimal number of subpanels, a convergence of response with a different number of subpanels is carried out. Therefore, the stiffened panel is divided into different subpanels using 2×2 to 5×5 grids. The standard deviations of the displacement at point A and the radiated sound pressure at point B with different subpanels are shown in Fig. 12. To describe the effect of the subpanels’ number on the vibro-acoustic response, Fig. 12 also gives the results without subpanels where the 1×1 subpanel is used. Results show that the standard deviations of the displacement and radiated sound pressure converge to a relatively stable value with the increase of the number of subpanels. Based on a combination of computational efficiency and sufficient accuracy, the number of subpanels 5×5 is adequate for the subsequent vibro-acoustic analysis.
19
In order to study the effect of the correlation on structural and acoustic responses, the vibro-acoustic analyses are carried out under three different excitations: (i) perfectly correlated excitations where the correlation between the random processes is unity, (ii) uncorrelated excitations where the correlation between the random processes is zero, and (iii) partially correlated excitations that the correlation interval between the random processes is between -0.065 and 0.830 obtained by Eq. (13). From the cross-covariance of the random excitation, which is the form of an exponential correlation, the correlation of the excitation which decays exponentially along the x and y directions is not a constant. The standard deviation of the displacement at points A and radiated sound pressure at point B under the three different excitations are shown in Fig. 13. The corresponding PDFs of the responses in the 0.04s and 0.5s are shown in Fig. 14, and the corresponding covariances of the displacement and radiated sound pressure are presented in Fig. 15. Results show that the proposed method is also suitable for the vibro-acoustic analysis of complex system under uncorrelated and correlated random processes. Both the structural and acoustic responses excited by partially correlated random excitation are bounded by the responses excited by perfectly correlated and uncorrelated random processes. Therefore, it is imperative to consider the correlation of random excitations in the structural optimization or fatigue calculations. 3.2.2. Non-Stationary Covariance Excitation To check the applicability of the developed algorithm to any type of auto-covariance excitation, a Bessel auto-covariance excitation is utilized to define the excitation. The Bessel functions have been used in many engineering applications such as heat transfer, acoustics, radio physics, hydrodynamics, atomic and nuclear physics, and so on [34]. The Bessel autocovariance and cross-covariance excitation functions are given in Eq. (34) and Eq. (35) respectively:
20
CFi Fi (t1 , t2 ) =
CFi Fj (ζ 1 , ζ 2 , t1 , t2 ) =
Dij
τ ij
J0 (
Di
τi
J0(
t1 − t2
τi
t1 − t2
τi
)
§ § ζ · ζ · ) exp ¨¨ −γ 1 1 ¸¸ exp ¨¨ −γ 2 2 ¸¸ Uc ¹ Uc ¹ © ©
(34)
(35)
The Bessel auto-covariance function is plotted in Fig. 16. The correlation lengths IJi and IJij are set equal to 0.1 and 0.2, respectively. The convergence of truncated error of the KLE for the Bessel auto-covariance is also studied as shown in Fig. 2, and the number of terms for KLE equal to five is used for the analysis. The clock-time cost for the response analysis with the proposed approach is 487.4s for the non-stationary excitations with Inter(R) Core(TM) i76700 CPU 3.41GHz and 8GM RAM. The convergence of the structural and acoustic responses with different number of division subpanels is also carried out firstly. The standard deviation of the structural displacement at point A and radiated sound pressure at point B with different number of subpanels are shown in Fig. 17. Results show that both the structural and acoustic responses converge to a relatively stable value with the increase of the number of subpanels. Therefore, the number of subpanels 5×5 is also sufficient to carry out the vibro-acoustic analysis for the Bessel covariance excitation. The standard deviation of the structural displacement at point A and radiated sound pressure at point B under the perfectly correlated excitation, uncorrelated excitation, and partially correlated excitation are shown in Fig. 18. In the partially correlated excitation, the correlation interval between the random processes which decays exponentially along the x and y directions is between 0 and 0.99. The corresponding PDFs of the responses in the 0.1s and 0.5s are shown in Fig. 14, and the corresponding auto-covariances of the displacement and radiated sound pressure due to partially correlated Bessel covariance excitation are
21
plotted in Fig. 19. Results show that the proposed methods can also be applied to the Bessel auto-covariance excitation.
4. Conclusions A methodology is presented in which the Karhunen-Loeve expansion (KLE), finite element method (FEM) and boundary element method (BEM) are integrated to conduct the vibro-acoustic analysis under stationary and non-stationary random excitations. Both the uncorrelated and correlated random excitations are considered. The eigenvalues and eigenfunctions of the auto-covariance are obtained by using KLE where a set of orthogonal basis functions are used to discretize the auto-covariance. The KLE for multi-correlated random processes relies on expansions in terms of correlated sets of random variables. During the stochastic response analysis, the FEM and BEM are employed to obtain structural and acoustic responses, respectively. In numerical simulations, a circular piston in infinite baffle excited by exponential and exponentially modulated covariance is firstly presented to validate the proposed algorithm, and a convergence study on the truncation error of KLE is also presented. The simulation results have an excellent agreement with analytical results. Furthermore, the algorithm is applied to a stiffened panel problem having the correlated random excitations whose covariances are characterized by using exponential and Bessel functions, respectively. The vibro-acoustic analysis is carried out with three types of excitations: perfectly correlated, partial correlated and uncorrelated. Results show that the proposed methodology is capable of solving a vibro-acoustic problem of complex structures under various type of random excitations. The structural or acoustic responses due to partially correlated random processes are bounded by the responses obtained by perfectly correlated and uncorrelated random excitations.
22
Acknowledgments The work described in this paper was supported by China Scholarship Council (CSC), a Program for New Century Excellent Talents in University (NCET-11-0086), research grant by the National Natural Science Foundation of China (11572086 & 11402052), Natural Science Foundation of Jiangsu Province (BK20140616); and a Graduate Research and Innovation projects in Jiangsu Province (CXZZ13_0084). We gratefully acknowledge the contribution and help of Dr. Rakesh K. Kapania (Virginia Polytechnic Institute and State University, Blacksburg, USA) on this research.
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[32] F. Han, R. J. Bernhard, and L. G. Mongeau, Prediction of flow-induced structural vibration and sound radiation using energy flow analysis. Journal of Sound and Vibration. 227 (4) (1999) 685-709. doi: 10.1006/jsvi.1998.3013 [33] G. M. Corcos, The structure of the turbulent pressure field in boundary-layer flows. Journal of Fluid Mechanics. 18 (3) (1964) 353-378. doi: 10.1017/S002211206400026X [34] L. Devroye, Simulating bessel random variables, Statistics and Probability Letters. 57 (3) (2002) 249257. doi: 10.1016/S0167-7152(02)00055-X
26
Table 1 Characteristics of the baffled circular piston Table 2 The dimensions and material properties of the stiffened panel Table 3 Numerical values of the parameters used in the TBL mode
27
Table 1 Characteristics of the baffled circular piston Parameters
Values
Mass
8 kg
Stiffness
4.39×105 N/m
Damping
130 N s/m
Diameter
0.1 m
Frequency of the harmonic load
10 Hz
Amplitude of the harmonic load
200 N
Correlation amplitude
5000 N2
Correlation length
0.2
Table 2 The dimensions and material properties of the stiffened panel Parameters
Values
Length of the panel
1.1684 m
Width of the panel
1.1684 m
Thickness of the panel
0.005 m
Height of the stiffener
0.0577 m
Thickness of the stiffener
0.003 m
Distance of the stiffener from panel edge
0.400 m
Young’s modulus
73.08×109 N/m2
Poisson’s ratio
0.33
Density
2.70×103 kg/m3
Damping loss factor
0.02
28
Table 3 Numerical values of the parameters used in the TBL model Parameters
Values
Free stream velocity (U0)
35.75 m/s
Density of air (ȡ)
1.21 kg/m3
Decay rate, streamwise (Ȗ1)
0.16
Decay rate, spanwise (Ȗ2)
0.46
Fence height in the experiment (h)
0.0254 m
29
I.Figure Captions Fig. 1 The configuration and finite/boundary element model of a baffled circular piston (a) Baffled circular piston configuration (b) The finite or boundary element model Fig. 2 Truncation errors of KLE for exponential, exponentially modulated and Bessel autocovariances Fig. 3 Mean of the displacement and sound pressure (a) Mean of the displacement (b) Mean of the sound pressure Fig. 4 Standard deviations of the displacement under exponential and exponentially modulated covariance excitations (a) Exponential covariance excitation (b) Exponentially modulated covariance excitation Fig. 5 Standard deviations of the sound pressure under exponential and exponentially modulated covariance excitations (a) Exponential covariance excitation (b) Exponentially modulated covariance excitation Fig.6 PDFs of the displacement and pressure under exponential and exponentially modulated covariance excitations (a) PDFs of the displacement (b) PDFs of the pressure Fig. 7 Auto-covariances of the displacement under exponential and exponentially modulated covariance excitations (a) Exponential covariance excitation (b) Exponentially modulated covariance excitation Fig. 8 Auto-covariances of the sound pressure under the exponential and exponentially modulated covariance excitations (a) Exponential covariance excitation (b) Exponentially modulated covariance excitations Fig. 9 The finite element and boundary element model of the stiffened panel (a) Finite element model (b) Boundary element model Fig. 10 PSD of the TBL excitation
30
Fig. 11 Subpanels to capture the correlation of a stiffened panel under TBL excitation Fig. 12 Standard deviations of the displacement and sound pressure with different number of subpanels under exponential covariance excitation (a) Standard deviations of the displacement (b) Standard deviations of the sound pressure Fig. 13 Standard deviations of the displacement and sound pressure under perfectly correlated, partially correlated, and uncorrelated exponential covariance excitations (a) Standard deviations of the displacement (b) Standard deviations of the sound pressure Fig. 14 PDFs of the displacement and pressure under exponential and Bessel covariance excitations (a) PDFs of the displacement (b) PDFs of the pressure Fig. 15 Auto-covariances of the displacement and sound pressure under partially correlated exponential covariance excitation (a) Auto-covariance of the displacement (b) Auto-covariance of the sound pressure Fig. 16 Bessel auto-covariance excitation Fig. 17 Standard deviations of the displacement and sound pressure with different number of subpanels under Bessel covariance excitation (a) Standard deviations of the displacement (b) Standard deviations of the sound pressure Fig. 18 Standard deviations of the displacement and sound pressure under perfectly correlated, partially correlated, and uncorrelated Bessel covariance excitations (a) Standard deviations of the displacement (b) Standard deviations of the sound pressure Fig. 19 Auto-covariance of the displacement and sound pressure under partially correlated Bessel covariance excitation (a) Auto-covariance of the displacement (b) Auto-covariance of the sound pressure
31
y
k
x
F
m c (a) Baffled circular piston configuration
(b) The finite or boundary element model Fig. 1 The configuration and finite/boundary element model of a baffled circular piston
Truncation Errors (%)
100 80
Exponential ( i = 0.1) Exponential ( i = 0.2)
60
Exponentially Modulated ( i = 0.2) Bessel ( i = 0.1)
40 20 0
0
10
20
30 40 KLE Terms
50
60
Fig. 2 Truncation errors of KLE for exponential, exponentially modulated and Bessel autocovariances
32
10-4
8
Analytical Numerical
6 4 2 0 -2 -4 -6
0
0.2
0.4
0.6 Time (s)
Mean of the Pressure (Pa)
(a) Mean of the displacement
(b) Mean of the sound pressure Fig. 3 Mean of the displacement and sound pressure
33
0.8
1
10-4
7
Analytical Numerical
6 5 4 3 2 1 0
0
0.2
0.4
0.6
0.8
1
Time (s)
Standard Deviation of the Displacement (m)
(a) Exponential covariance excitation
10-4
7
Analytical Numerical
6 5 4 3 2 1 0
0
0.2
0.4
0.6
0.8
1
Time (s)
(b) Exponentially modulated covariance excitation Fig. 4 Standard deviations of the displacement under exponential and exponentially modulated covariance excitations
34
Standard Deviation of the Pressure (Pa)
10-3
8
Analytical Numerical
6
4
2
0
0
0.2
0.4 0.6 Time (s)
0.8
Standard Deviation of the Pressure (Pa)
(a) Exponential covariance excitation
10-3
8
Analytical Numerical
6
4
2
0
0
0.2
0.4 0.6 Time (s)
0.8
1
(b) Exponentially modulated covariance excitation Fig. 5 Standard deviations of the sound pressure under exponential and exponentially modulated covariance excitations
35
(a) PDFs of the displacement
(b) PDFs of the pressure Fig.6 PDFs of the displacement and pressure under exponential and exponentially modulated covariance excitations
36
(a) Exponential covariance excitation
(b) Exponentially modulated covariance excitation Fig. 7 Auto-covariances of the displacement under exponential and exponentially modulated covariance excitations
37
(a) Exponential covariance excitation
(b) Exponentially modulated covariance excitations Fig. 8 Auto-covariances of the sound pressure under the exponential and exponentially modulated covariance excitations
(0, 0)
38
A
(a) Finite element model
(b)Boundary element model Fig. 9 The finite element and boundary element model of the stiffened panel
0
2
-
PSD(Pa /rad*s 1)
10
10
10
-2
-4
0
2000
4000 6000 8000 10000 12000 Frequency(rad/s)
Fig. 10 PSD of the TBL excitation
39
ߦଶ ߦଵ
ݕ௨ ݕ
y x
ݔ
ݔ௨
Fig. 11 Subpanels to capture the correlation of a stiffened panel under TBL excitation
40
Standard Deviation of the Displacement (m)
10-4
3
1×1 2×2 3×3 4×4 5×5
2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3 0.4 Time (s)
0.5
0.6
0.7
Standard Deviation of the Pressure (Pa)
(a) Standard deviations of the displacement
3.5 1x1 2x2 3x3 4x4 5x5
3 2.5 2
1 0.8 0.6 0.4
1.5
0.2
1
0 0.175
0.5 0
0
0.1
0.2
0.18
0.3 0.4 Time (s)
0.185
0.5
0.19
0.195
0.6
0.7
(b) Standard deviations of the sound pressure Fig. 12 Standard deviations of the displacement and sound pressure with different number of subpanels under exponential covariance excitation
41
Standard Deviation of the Displacement (m)
10-4
3
Perfectly Correlated Uncorrelated Partially Correlated
2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3 0.4 Time (s)
0.5
0.6
0.7
Standard Deviation of the Pressure (Pa)
(a) Standard deviations of the displacement
3.5
1
3
0.8 0.6
2.5
0.4 0.2
2
0.17
1.5
0.18
0.19
0.2
0.21
0.22
0.23
0.24
Perfectly Correlated Partially Correlated Uncorrelated
1 0.5 0
0
0.1
0.2
0.3 0.4 Time (s)
0.5
0.6
0.7
(b) Standard deviations of the sound pressure Fig. 13 Standard deviations of the displacement and sound pressure under perfectly correlated, partially correlated, and uncorrelated exponential covariance excitations
42
(a) PDFs of the displacement
(b) PDFs of the pressure Fig. 14 PDFs of the displacement and pressure under exponential and Bessel covariance excitations
43
(a) Auto-covariance of the displacement
(b) Auto-covariance of the sound pressure Fig. 15 Auto-covariances of the displacement and sound pressure under partially correlated exponential covariance excitation
44
Standard Deviation of the Displacement (m)
Fig. 16 Bessel auto-covariance excitation
10-4
3
1×1 2×2 3×3 4×4 5×5
2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3 0.4 Time (s)
45
0.5
0.6
0.7
Standard Deviation of the Pressure (Pa)
(a) Standard deviations of the displacement
3.5 1x1 2x2 3x3 4x4 5x5
3 2.5 2
1 0.8 0.6 0.4
1.5
0.2
1
0 0.17
0.18
0.19
0.2
0.5 0
0
0.1
0.2
0.3 0.4 Time (s)
0.5
0.6
0.7
(b) Standard deviations of the sound pressure Fig. 17 Standard deviations of the displacement and sound pressure with different number of
Standard Deviation of the Displacement (m)
subpanels under Bessel covariance excitation 10-4
3
Perfectly Correlated Uncorrelated Partially Correlated
2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3 0.4 Time(s)
(a) Standard deviations of the displacement
46
0.5
0.6
0.7
Standard Deviation of the Pressure (Pa)
3.5 1
3
0.8 0.6
2.5
0.4 0.2
2
0.17
1.5
0.18
0.19
0.2
0.21
0.22
0.23
0.24
Perfectly Correlated Partially Correlated Uncorrelated
1 0.5 0
0
0.1
0.2
0.3 0.4 Time(s)
0.5
0.6
0.7
(b) Standard deviations of the sound pressure Fig. 18 Standard deviations of the displacement and sound pressure under perfectly correlated, partially correlated, and uncorrelated Bessel covariance excitations
(a) Auto-covariance of the displacement
47
(b) Auto-covariance of the sound pressure Fig. 19 Auto-covariance of the displacement and sound pressure under partially correlated Bessel covariance excitation
48