Research Article  Open Access
Bin Xiao, Yulong Zhou, Chao Gao, ShuangXia Shi, ZeXi Sun, Zongbiao Song, "Vibration Analysis of DrivingPoint System with Uncertainties Using Polynomial Chaos Expansion", Shock and Vibration, vol. 2020, Article ID 8074351, 14 pages, 2020. https://doi.org/10.1155/2020/8074351
Vibration Analysis of DrivingPoint System with Uncertainties Using Polynomial Chaos Expansion
Abstract
A vibration transfer analysis method based on polynomial chaos expansion (PCE) is proposed in this study and is used to analyze the stochastic dynamic compliance of uncertain systems with the Gaussian distribution. The random dynamic compliance is established by utilizing mode superposition on the system as the parameters of system uncertainties are regarded as input variables. Considering the asymptotic probability density function of mode shape, the dynamic compliance is decomposed into the mean of mode shape and the subsystem represented as an orthogonal polynomial expansion. Following this, the vibration transmission analysis approach is proposed for the random vibration. Results of a numerical simulation carried out employing the PCE approach show that broadband spectrum analysis is more effective than narrowband spectrum analysis because the former jump of the dynamic compliance amplitude is weakened. This proposed approach is valid and feasible, but since broadband spectrum analysis loses some important information about the random vibration, both the aforementioned processes need to simultaneously be applied to analyze the random vibration transmission of lowmedium frequency systems.
1. Introduction
The research of random dynamic system uncertainty originated from the stochastic finite element method (SFEM) applies the Monte Carlo simulation (MCS) method to the structural finite element analysis (FEA) process and repeats FEA to the model through specific samples [1, 2]; consequently, the characteristics of the statistical distribution of the random response are obtained. For determined structural dynamic systems, extensive research regarding vibration analysis has been carried out, such as decoupling analysis on the nonlinear system [3, 4], analysis on vibration characteristics and response of structures, and study on the vibration transmission path.
Furthermore, the roles of various boundary conditions [5, 6], for example, classical restraints, elastic supports and their combinations, and geometric properties [7], such as, cylindrical, conical, and spherical, have been taken into account in the research process. It is thus of great interest to understand thoroughly the vibration behaviors of the system. After more accurate and applicable methods for determined vibration analysis are developed, perturbation theory is applied to the transformation equation, including stochastic distribution parameters, stochastic elastic boundary support structures, and stochastic boundary condition problems, and then the stochastic problem is transformed into a series of deterministic differential equations and boundary conditions, and finally, the random vibration analysis is realized. Furthermore, in order to obtain the mean and standard deviations of the results, the uncertainty of parameters and nonparameters should be considered. Fortunately, the stochastic perturbation method is devoted to the theoretical aspects and computational implementation of the generalized stochastic perturbation technique. It is based on anyorder Taylor expansions of random variables and enables for determination of up to fourthorder probabilistic moments and characteristics of the physical system response, which provides grounding in the basic elements of statistics and probability and reliability engineering [8]. Therefore, Huang et al. [9] obtained the approximate statistical expressions of stochastic eigenvalues. Wu and Law [10] discussed the statistical prediction of the dynamic response of bridge structures under random moving loads. Xiao et al. [11] analyzed the random vibration characteristics of the Levinson beam on random elastic constraint conditions. Szafran et al. [12] presented a reliability estimation procedure for steel lattice telecommunication towers based on tensioned joint reliability and used the firstorder and secondorder reliability methods to confirm the elasticplastic range of joint reliability. Handa and Anderson [13] used the first and secondorder perturbation of the mean of stochastic variables to obtain statistical properties of the response when analyzing the fluctuation of stochastic variables in the system. On this basis, Chen et al. [14] proposed interval variables to describe various uncertain parameters of the system. Adhikari [15] presented variable matrix distributions of the mass, stiffness, and damping of the quantitative linear structure dynamics under uncertainty based on the optimal stochastic matrix theory [16]. Typically, the statistics of the system are obtained via MCS, SFEM, or polynomial chaos expansion (PCE). MCS is often used to obtain reference results. Although simulation techniques can be used for a wide range of structural dynamics problems, it is computationally intractable, particularly for largescale problems. It is equally evident that while numerical techniques based on Taylor series expansion are efficient, they do not yield sufficient accuracy when there are significant uncertainties.
Due to its accuracy and ability to provide powerful representations of vibration transfer capabilities of nonembedded PCE technology, it has become increasingly popular among researchers and has evolved into an attractive approach. Within the framework of orthogonal polynomial expansion, Pascual and Adhikari [17] proposed four kinds of solution forms: Rayleigh quotient, power method, inverse power method, and eigenvalue equation based on the mixed perturbation PCE method. Sarsri et al. [18] obtained the first two statistical moments of the frequency transfer function through chaos expansion and elaborated the coupling problems of the first and secondorder PCE; furthermore, more than three order statistics of the structural response were analyzed by Xiao and Li [19]. Sepahvand et al. [20] used the truncated PCE of arbitrary random basis to express the uncertain elastic modulus, eigenfrequency, and mode of an orthotropic plate in order to study the stochastic free vibration of the plate. Bahmyari et al. [21] combined the meshless Galerkin method with the generalized PCE to analyze the stochastic bending of moderately thick plates with elastic constraint edges. Najlawi et al. [22, 23] proposed a hybrid multiobjective imperialist competitive algorithm (MOICA) and MCS method for the multiobjective robust design optimization of a mechanism and further developed a hybrid MOICAPCE algorithm to be exploited for robust optimization [24]. In addition, Hadigol and Doostan [25] presented a hybrid sampling method that employs the socalled alphabetic optimality criteria used in the context of the optimal design of experiments in conjunction with coherenceoptimal samples, especially for highorder PCE.
The main objective of this study was to present an efficient and reliable method for evaluating the vibration transmission and response of a random dynamical system based on PCE of the dynamic compliance when subjected to pure harmonic excitation. For this purpose, mode superpositionbased dynamic compliance is introduced for the random dynamical system, which is used in extracting the mean mode when considering random mode elements with the lognormal asymptotic distribution. A new form of PCE is proposed to give the approximate expression of the dynamic compliance, which is combined with embedded Fourier–Hermite polynomial expansion. Thereafter, the proposed approach is established for discussing the vibration transmission and response of the random vibration system.
This paper is organized as follows. Section 2 briefly reviews the dynamic compliance and orthogonal polynomial expansion theory; thereafter, the PCE approach is presented. In Section 3, numerical simulation is used to demonstrate the PCE approach on both processes of broadband spectrum analysis and narrowband spectrum analysis, and analysis of random vibration transmission and response is implemented by comparing with MCS. Finally, in Section 4, the concluding remarks are given.
2. Theoretical Method
2.1. Dynamic Compliance
Let us consider a multidegreeoffreedom linear structural system with mass, damping, and stiffness matrices , , and , respectively. The equations of motion describing the forced vibration of a linear and damped discrete system arewhere is the nodal displacement vector and is the external harmonic excitation with angular frequency .
An undamped or proportional damped system is considered in this study, whose matrices , , and are real symmetric. Hence, the dynamic compliance matrix based on the mode superposition is defined bywhere is the vibration transmission from the node to the node and [26] is the mode impedance:where , , and are the mode mass, mode shape, and mode damping ratio with respect to the nthorder natural frequency , respectively.
However, since we are interested in the limit as , consider N_{e}order modal truncation; the real dynamic compliance of equation (2) for ,may become arbitrarily large only if the external excitation frequency becomes very close to some natural frequency .
The weight of eigenfunctions,is defined on . The eigenvector of the matrix, , at driving point x has to satisfy the orthogonality of eigenfunctions. A convenient way to characterize the statistics of is via the moments of arbitrary index :where is the probability density function (PDF), and is the dimension of the square matrices of the system. As a consequence, the weight is a random variable distributed in a Gaussian orthogonal ensemble (GOE) which obeys lognormal distribution [27]. The authors in [27] have given the correlation between the lognormal distribution and the multifractal algorithm as follows:
Hence, when , the equation holds: the spectrum of fractal dimensions ; satisfies probability distribution , i.e.,
If the external excitation frequency becomes very close to some natural frequency , then the corresponding term dominates over the others in equation (4), and the dynamic compliance can be approximated by this biggest term; therefore, in such case, the probability distribution of the dynamic compliance is presented as follows:
Otherwise, the dynamic compliance can be approximated bywhere is the mean of when all the random variables are mean values [28], with being the polynomial function.
2.2. Orthogonal Polynomial Expansion
2.2.1. Fourier–Hermite Polynomials
Consider a continuous, differentiable, realvalued, multivariate function that is defined on , where is an Mdimensional real vector space. A dimensional decomposition of , described bycan be viewed as a finite hierarchical expansion of an output function in terms of its input variables with increasing dimensions, where is a constant representing the mean response of the transfer function, and is an Svariate component function quantifying the cooperative effects of S input variables .
In order to generate a hierarchical and convergent sequence of approximations of , considering input variables following I.I.D Gaussian distribution, a general Svariate approximation of is obtained [29]:whereis a coefficient associated with the product of through basis functions expressed in terms of , where follows probability density function (PDF) , follows the joint PDF , and is the bth order Hermite polynomial, where .
2.2.2. DimensionReduction Integration for Calculating Coefficients
Following an early idea by Xu and Rahman [30], a lowervariate approximation of the Mvariate function is considered, which leads to a reduction in the dimensionality of the Mdimensional integral coefficients of the equation. Using Xu and Rahman’s multivariate function theorem [30], it can be shown that a special Rvariate approximation of , defined byconsists of all terms of the Taylor series of that have less than or equal to variables, where represents the th dimensional component function of , with and .
Substituting equation (14) into equation (13), the polynomial expansion coefficients becomeand the mean is
It is obvious that introducing the dimensionreduction integration for calculating coefficients is substantially simpler and more efficient than performing one Mdimensional integral, particularly when .
2.2.3. Gauss–Hermite Integration
The Gauss–Hermite multiple quadrature formulas for Hermite polynomials arewhere are the integral points and associated coefficient depends on the probability distribution of . Considering a special term , equation (17) can be transformed into the summation form and used to solve the polynomial coefficients and mean values efficiently. Correspondingly, in order to improve its calculation accuracy and efficiency, we can define .
2.3. Error Analysis
2.3.1. Mean
For the drivingpoint vibration analysis, i.e.,, the expectation of the dynamic compliance in equation (4) is obtained:
And the expectation of the dynamic compliance in equation (9) is
Therefore, equation (19) becomes
Comparing equation (19) with equation (20) and considering the relationship , the following relation can be obtained:
2.3.2. Standard Error
In the statistical analysis procedure of random variables , the standard error (SE) is defined as [18]
Supposing a quantity is a function of the measured variables, i.e.,, the error propagation equation is obtained as follows:
By using equations (23) and (24), the SE of the dynamic compliance in equation (4) can be obtained:
And then, the SE of dynamic compliance (10) can be obtained as follows:
Using equations (24) and (25), the SE level can be defined as
Then, the statistic of the PCEbased dynamic compliance can be yielded as the format of the SE level, i.e.,
It follows that the proposed PCE approach is reasonable and feasible, by equation (21), but its error in equation (27) is inevitable, according to equations (24) and (25). So, the curve shape distortion (CSD) is defined based on equation (21) aswhere and are the PDFs of the SE level by employing MCS and PCE.
3. Numerical Simulation
3.1. FEMBased Plate Model
Rectangular plates are widely used in our daily life; all kinds of modeling method have been proposed for the vibration characteristics of rectangular plates with system certainty [31]. In this paper, a rectangular plate is fixed at its left boundary and supported by springs at its right boundary. Considering the vibration transmission and response at a driving point, the coordinate of which is (0.8, 0.4), with singlepoint harmonic excitation, FEM modeling can be given, shown in Figure 1 with parameters listed in Table 1. The stiffness is the ith random variable with a total of , that is, , follows the PDF of a Gaussian distribution with and .

In order to demonstrate the efficiency of PCE, some benchmark tests are analyzed with random parameters and vertical vibration. The proposed PCE approach with univariate , bivariate , and trivariate approximations is elaborated and considered to be the main results of this study. For the sake of assessing the accuracy of the PCE method, a comparison MCS approach is used, with its results using samples, serving as reference results. The errors between the estimated and reference responses are defined in equation (27), and CSD is defined in equation (28).

3.2. NarrowBand Spectrum Analysis
(a)Employing mode analysis, the natural frequencies within were obtained, as listed in Table 2. Assuming that random variables are mean values, we obtained the mean model of the dynamic compliance, as shown in Figure 2. For comparison purposes, the mean of the dynamic compliance is given in Figure 3. The curves in Figures 2 and 3 are almost identical; these results coincided with equation (21), i.e., the mean model embodies the mean of the vibration transmission and response of the random system, thus implying that the PCE approach is valid and feasible for narrowband spectrum analysis. ,
(a)
(b)
(c)
(d)
(e)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
3.3. BroadBand Spectrum Analysis
In view of the platestructure FEM modeling shown in Figure 1, on the basis of narrowband spectrum analysis, the vibration transmission and response of the 1/3 octave (Oct) spectrum level are further analyzed using the proposed PCE approach.(a)Figure 7 shows the PDFs of the 1/3 Oct spectral vibration response with center frequency Hz. Results indicate that a higher Svariate approximation is better than the univariate one. There is no vertical vibration dominant mode (resonant frequency) within the frequency band of 20 Hz, whose result is significantly better than that of the 31.5 Hz frequencyband result, which does have a vertical vibration dominant mode (resonant frequency). A comparison of Figure 7 with Figures 5 and 6 implies that the PDF of a broadband spectrum converges towards a Gaussian distribution, and the jump of the dynamic compliance amplitude is weakened due to the 1/3 Oct broadband energy superposition, which makes the proposed PCE approach more effective.
(a)
(b)
4. Concluding Remarks
A vibration transmission analysis approach is presented based on the PCE of the random dynamic compliance for a system with uncertainties. The numerical illustrations demonstrated that the proposed PCE approach is efficient in comparison with MCS. Some conclusions were drawn as follows:(a)Compared to the MCS, the results of narrow and broadband spectrum analysis under uncertain dynamical systems based on the proposed PCE approach are reasonable and feasible. The jump of the dynamic compliance amplitude caused by the random fluctuation of resonant frequency is weakened because of broadband energy superposition.(b)Compared to MCS, the PDF curves obtained by PCE have CSD, which is a relative error of vibration analysis. The error cannot be eliminated because of modalmodeling asymptotic estimation, even if we increase the order of statistics, multirandom variable decoupling, and Hermite polynomial truncation in the orthogonal polynomial expansion. In particular, random fluctuation of the resonant frequency is generated by the system uncertainty, causing a jump in the dynamic compliance amplitude.(c)The estimated results will converge towards statistical energy analysis results along with an increase in mode overlap numbers. It is noteworthy that some important information about the narrowband spectrum will be lost when carrying out broadband spectrum analysis. Therefore, when applying the PCE approach to a random vibration system of lowmedium frequency, it is suggested that narrow and broadband spectrum analysis be implemented simultaneously.
However, the PDF estimation of the mode shape of the random vibration system, the jump in the dynamic compliance amplitude, and nonlinear problems need to be further investigated. Meanwhile, studies should focus on the PCE approach for the crosspoint vibration transmission of the dynamical system. In addition, since the system is discretized into FEA models, whose boundary conditions of stochastic elastic constraints are described by finite number of stiffness, PCE can be used for random vibration analysis of the system in this study. However, the problem that PCE is computationally intractable, particularly for numerous random stiffness, even invalid for continuous elastic restraint, offers many challenges; therefore, the methodology of PCE also needs to be discussed in the future.
Nomenclature
:  Order of the Hermite polynomial 
:  Standard error level 
:  Natural frequency 
:  Marginal distribution density 
:  Node number 
:  Number of variables truncated 
:  Dimensional decomposition 
:  Number of Hermite polynomials 
:  Modal order 
:  Moments of arbitrary index 
:  Weight of eigenfunctions 
:  Source point 
:  Random parameter 
:  Crosspoint 
:  Mean response of the transfer function 
:  Transfer function 
:  Rectangular plate width 
:  Mode damping 
:  Polynomial expansion coefficients 
:  Standard error 
:  Young’s modulus 
:  Expectation of the dynamic compliance 
:  External harmonic excitation 
:  Rectangular plate length 
:  Response displacement level 
:  Mode mass 
:  Dimension of the square matrices 
:  Dimension of the square matrices 
:  Modal truncation order 
:  Integral point 
:  Probability distribution 
:  Number of variables expanded for the transfer function 
:  Mdimensional real vector space 
:  Number of variables truncated 
:  Associated coefficient 
:  Mean of 
:  Node displacement vector 
:  Mode impedance 
:  Extreme point 
:  Rectangular plate thickness 
:  Mode damping ratio 
:  Poisson’s ratio 
:  Mass density 
:  Dynamic compliance matrix 
:  Mode shape 
:  Hermite polynomial 
:  Angular frequency 
:  Probability density function (PDF). 
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Acknowledgments
The authors gratefully acknowledge the support from the National Natural Science Foundation of China (Grant nos. 51475088 and 51705071) and Jilin Province Science and Technology Development Program of China (Grant nos. 20140101065JC, 20190201109JC, and 20190103058JH).
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