# A New Method for Reliability-Based Sensitivity Analysis of Dynamic Random Systems

Authors: Qianqian Wang; Changchun Wang; Hui Wang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

A novel numerical method for investigating time-dependent reliability and sensitivity issues of dynamic systems is proposed, which involves random structure parameters and is subjected to stochastic process excitation simultaneously. The Karhunen–Loève (K-L) random process expansion method is used to express the excitation process in the form of a series of deterministic functions of time multiplied by independent zero-mean standard random quantities, and the discrete points are made to be the same as Legendre integration points. Then, the precise Gauss–Legendre integration is used to solve the oscillation differential functions. Considering the independent relationship of the structural random parameters and the parameters of random process, the time-varying moments of the response are evaluated by the point estimate method. Combining with the fourth-moment method theory of reliability analysis, the dynamic reliability response can be evaluated. The dynamic reliability curve is useful for getting the weakness time so as to avoid breakage. Reliability-based sensitivity analysis gives the importance sort of the distribution parameters, which is useful for increasing system reliability. The result obtained by the proposed method is accurate enough compared with that obtained by the Monte Carlo simulation (MCS) method.


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## Body

## 1. Introduction

Random analysis of vibrating structures has received much more attention in recent years [1–4]. Mechanical structures always contain unavoidable uncertain factors such as geometrical errors, uncertainty of material properties, and manufacturing errors, which constitute the basic random parameters of the vibrating system. And the structures may be subjected to stochastic process excitation in the meanwhile, such as the excitation from the base, ocean wave, wind, and rough road. If the responses of the structures under random conditions exceed the allowable boundary, it will cause failures, e.g., base vibration, noise, and interference. The reliability analysis of random vibrating systems aims to analyze failure probability in the vibrating process especially at the unstable time when the machine is started. The reliability based-sensitivity analysis ranks the importance of the distribution characteristics of the random parameters contributing to reliability, which is of great significance for increasing the reliability value.For the time-independent random systems, traditional methodologies can be classified into two categories: the numerical Monte Carlo simulation method [5] and analytical reliability methods (e.g., the first-order reliability method, the second-order reliability method, and the advanced mean value method). However, the state functions of complex vibrating structures are always in the implicit form and difficult to obtain. In these conditions, the analytical reliability methods are seldom applicable. Therefore, more suitable methods are developed for implicit limit-state functions, e.g., the response surface method [6–9], the artificial neutral network method [10], and the point estimate method [11, 12]. Within all these methods, the point estimate method is convenient for implicit multivariate state functions [13–15].As the exciting force of the dynamic random system is a time-dependent random process, it is computationally expensive to obtain the time-varying reliability value in the time domain. Hu [16, 17] used an extreme value response method to solve the time-dependent reliability problems, and Wang et al. [18] used this method in kinematic reliability analysis. This extreme value method transforms the time-dependent problems to time-independent problems and simplifies the calculation. However, it ignores random response at each time point. Another method is decomposing the spectral representation of the random process using K-L expansion [19, 20], orthogonal series expansion, expansion optimal linear estimation, etc. Of all these expansion methods, K-L expansion is an efficient tool for the excitation process in the dynamic system.In the past literatures, a number of works [21, 22] have been performed on linear structures’ reliability analysis issues. The perturbation method, fourth-moment method combined with the Edgeworth series [23], improved response surface method combined with the trigonometric series expression of the stochastic process [24, 25], path integration method [26, 27], and a new probability density evolution method [2] for dynamic response analysis and reliability assessment were investigated in recent years. However, time-varying random responses of structures with random structure parameters and random process excitation are not properly investigated in all the literatures mentioned above.In the present study, an attempt has been made to develop a time-varying reliability and sensitivity analysis for a random parametric structural system under stationary or nonstationary excitation incorporating uncertainty in the structural parameters. An approach for evaluating the response of the random vibration is proposed by combining K-L expansion, Gauss–Legendre precise integration, and the dimension reduction point estimate method. And then, the reliability-based sensitivities with respect to the mean and variance of the random structural parameters are analyzed based on the system reliability utilizing the core function method. The process of reliability and sensitivity analysis of the random dynamic system is presented in Figure1. The main idea of the process is as follows:(1)
The Gaussian process is expanded by the K-L method to a series of time-dependent deterministic functions multiplied by Gaussian random parameters, which are dispersed at the Gauss–Legendre integral points.(2)
The precise integration is utilized to solve the differential equations of the random structures subjected to each expansion deterministic function.(3)
Combining with the point estimate method, the moments of each response under the series of deterministic functions are calculated.(4)
Considering the Gaussian random parameters produced in the K-L expansion, the moments of the structure’s random response are calculated.(5)
The moment method is used to estimate the reliability and sensitivities.Figure 1
Flow diagram of the solution process.


## 2. K-L Expansion of the Stochastic Process

A quite general and precise spectral representation utilized for the Gaussian random process is the K-L expansion of the covariance function. This expansion is available to stationary and nonstationary stochastic processes. The expansion of a stochastic process takes the following form:(1)ft=f0t+∑j=1∞ξjfjt,where the first term f0t is the mean of the random process. The second term is the summation of a set of orthogonal zero-mean random variables (ξ) with unit variance multiplied by a series of deterministic functions of time:(2)Eξj=0,Eξiξj=δij,where δij denotes Kronecker’s delta symbol. In practical applications, the continuous stochastic process ft is dispersed at time points tk = kΔt, k = 1, 2, …, n, and the n × n-dimensional covariance matrix Cfftk,tk′ of the dispersed stochastic process is obtained:(3)Cff=Eftk−f0tkftk′−f0tk′T.The dispersed valuesfjtkk=1L of the deterministic functions fjt in the K-L expansion terms are uniquely specified by the eigenvectors φj and eigenvalues λj of the symmetric covariance matrix C = [Cik] as follows:(4)fjtk=ϕkjλj,where ϕkj denotes the kth component of the jth eigenvector. When the expansion term equals the number of time points n, it is accurate for the K-L representation to describe the stochastic process. However, in practical applications, it is too complex for all terms to be considered. Assuming the eigenvalues are decreasingly ordered, and accepting a tolerance ε, the number N < n can be estimated accordingly. And then, equation (1) can be written as(5)f^tk=f0tk+∑j=1Nξjfjtk.If the dispersed time pointstk are replaced by the Gauss–Legendre integration points as described in Section 3, the dispersed exciting force functions can be obtained by equation (5), and they are applicable in the following precise integration.To quantify the accuracy of the truncated K-L expansion with numberN, the following variance and covariance errors are defined according to literatures [28, 29]:(6)εVar,Nt=Varft−f^tVarft=σ2t−∑j=1Nλjϕj2tσ2t,εCov,Nt,t′=Covf′t,f′t′−Covf^t,f^′t′Covf′t−f′t′,where σ2t is the variance of the random process and Cov⋅ is the covariance function. f′t=ft−f0t denotes fluctuations around the mean value.


## 3. Solution of Random Vibration

### 3.1. Precise Gauss–Legendre Integration Points [30–33]

A multi-degree-of-freedom linear random vibration system can be expressed by differential equations in the matrix form as follows:(7)MY¨+KY+CY˙=ft,(8)Ytt=0=Y˙t0,Y˙tt=0=Y¨t0,where M, C, and K are the mass, damping, and stiffness matrices, respectively; Yt, Y˙t, and Y¨t are the displacement, velocity, and acceleration vectors, respectively; and ft is the force vector.The state equation for the random vibration system can be written as(9)v˙=Hv+Ft,where(10)v=YY˙,H=0I−M−1K−M−1C,Ft=0M−1ft.The general solution of equation (7) is(11)vt=exptHv0+∫0texpt−τHFτdτ,where v0 is the initial value of the dynamic system.The interested time interval of the structural dynamic response is discredited at time pointstk = kΔt, k = 1, 2, …, q, as described in Section 2. The responses at time tk−1 and tk are written as(12)vtk−1=exptk−1Hv0+∫0tk−1exptk−1−τHFτdτ,(13)vtk=exptkHv0+∫0tkexptk−τHFτdτ.Noting thattk = tk−1 + Δt, the relationship between vtk−1 and vtk obtained from equations (12) and (13) is(14)vtk=Tvk−1+∫tk−1tkexptk−τHFτdτ,where T=expHΔt, and the accurate evaluation of T can be deduced from the following equation:(15)T=expHΔt=expHτ2m,in which τ = Δt/2m. Expanding expHτ by Taylor series, the following expression is obtained:(16)expHτ≈I+Hτ+Hτ22=I+T0,in which T0=Hτ⋅I+Hτ/2 and I is the unit matrix.It is further noted that(17)expHτ2=I+T02=I2+2I⋅T0+T02=I+T1,expHτ22=I+T12=I2+2I⋅T1+T12=I+T2,………expHτ2w=I+Tw−12=I2+2I⋅Tw−1+Tw−12=I+Tw.So the recurrence formula is concluded as(18)Tw=2Tw−1+Tw−1⋅Tw−1,w=1,2,…,m,and the solution of T can be deduced using the recurrence formula.The random load process vectorFt in equations (9)–(14) is represented by K-L expansion as(19)Ft=0M−1f0t+∑j=1Nξjfjt.Therefore, thejth term of Ft and its corresponding solution vkj at time tk can be written, respectively, as(20)Fjt=0M−1fjt,(21)vkj=Tvk−1j+∫tk−1tkexptk−τHFjτdτ,j=1,…,N.The integration of equation (21) can be calculated using the Gauss–Legendre numerical method, and the solution can be represented as(22)vkj=Tvk−1j+Δt2∑l=1MwleHΔt1−ζl/2Fjtk−1+Δt1+ζl2,where M is the number of Gaussian integral points and ζl and wl are the Gaussian integral point and the corresponding weight, respectively.Furthermore, for the linear systems, asvkj=Ykj,Y˙kjT, the response Y of the random vibrating structure can be expressed as(23)Yk=Yk0+∑j=1NξjYkj.


### 3.2. Statistical Moment of the Random Extreme Response by Dimension Reduction Point Estimate Method

In each term of random responseYj (j = 0, 1, …, N), the structural random parameters in vector X are time independent. Therefore, it is suitable to use the dimension reduction point estimate method to evaluate the moments of the jth response. The correlated nonnormal random parameters can be transformed to independent standard normal variables by the inverse normal transformation, which can be written as(24)X=T−1U,in which U is the n-dimensional independent standard normal vector and T−1⋅ is the inverse transforming function such as the Rosenblatt transformation, Nataf transformation, and third-moment transformation technique, which is determined by the known conditions [34].Then, the dimension reduction method at timetk is expressed as(25)Y^kjX=∑r=1nYkjT−1Ur−n−1Ykjμ,where μ represents the vector of mean values of the random variables, and(26)Ur=uμ1,uμ2,…,uμr−1,ur,uμr+1,…,uμn,in which ur is the only standard normal distributed random parameter and the other variables are set equal to mean values transformed into the standard normal space.ConsideringYkjT−1Ur=Yk,rj, the lth origin moment of the jth random response can be expressed by the binomial theorem as(27)EY^kjl=E∑r=1nYk,rj−n−1Ykjμl=E∑s=1lCls∑r=1nYk,rjs−n−1Ykjμl−s=∑s=1lClsE∑r=1nYk,rjs−n−1Ykjμl−s.Using the binomial theorem for times, thesth exponentiation in equation (27) is expressed as(28)E∑r=1nYk,rjs=E∑k1=0sCsk1∑r=1n−1Yk,rjk1Yk,njs−k1=E∑k1=0sCsk1∑k2=0k1Ck1k2∑r=1n−2Yk,rjk2Yk,n−1jk1−k2Yk,njs−k1=E∑k1=0sCsk1∑k2=0k1Ck1k2⋯∑kn=0kn−1Ckn−1knYk,1jknYk,2jkn−1−kn⋯Yk,njs−k1=∑k1=0sCsk1∑k2=0k1Ck1k2⋯∑kn=0kn−1Ckn−1knEYk,1jknEYk,2jkn−1−kn⋯EYk,njs−k1.It is obvious from the above expression that when the origin moments ofY are obtained, the statistical moment of the random extreme response can be evaluated afterwards. As Yk,rj is a function with only one random variable, it is expediently calculated by Gaussian numerical integration:(29)EYk,rjl=∫−∞∞Yk,rjlfururdur,where fur is the probability density function of the standard normal variable ur. The approach for estimating the statistical moment of the response function above is the so-called point estimate method, which is developed by Zhao [35].SupposingMlj=EY^kjl (j = 0, 1, …, N), the first four central moments of the random response can be derived from equations (30)–(33) as(30)EY^=EY0t+E∑j=1NξjYjt=EY0t+∑j=1NEξjEYjt=M10,(31)σ2Y^=EY−M102=EY0t2+∑j=1NEYjt2−M102=M20+∑j=1NM2j−M102,(32)V3Y^=EY−M103=EY0t+∑j=1NξjYjt−M103=M30−3M20M10+2M103,(33)V3Y^=EY−M104=EY0t+∑j=1NξjYjt−M104=M40−4M30M10+6M20M102−3M104+12M20∑j=1NM2j−6M102∑j=1NM2j+6∑j=1NM4j+∑i=1N∑j=1,j≠iNM2iM2j.


## 3.1. Precise Gauss–Legendre Integration Points [30–33]

A multi-degree-of-freedom linear random vibration system can be expressed by differential equations in the matrix form as follows:(7)MY¨+KY+CY˙=ft,(8)Ytt=0=Y˙t0,Y˙tt=0=Y¨t0,where M, C, and K are the mass, damping, and stiffness matrices, respectively; Yt, Y˙t, and Y¨t are the displacement, velocity, and acceleration vectors, respectively; and ft is the force vector.The state equation for the random vibration system can be written as(9)v˙=Hv+Ft,where(10)v=YY˙,H=0I−M−1K−M−1C,Ft=0M−1ft.The general solution of equation (7) is(11)vt=exptHv0+∫0texpt−τHFτdτ,where v0 is the initial value of the dynamic system.The interested time interval of the structural dynamic response is discredited at time pointstk = kΔt, k = 1, 2, …, q, as described in Section 2. The responses at time tk−1 and tk are written as(12)vtk−1=exptk−1Hv0+∫0tk−1exptk−1−τHFτdτ,(13)vtk=exptkHv0+∫0tkexptk−τHFτdτ.Noting thattk = tk−1 + Δt, the relationship between vtk−1 and vtk obtained from equations (12) and (13) is(14)vtk=Tvk−1+∫tk−1tkexptk−τHFτdτ,where T=expHΔt, and the accurate evaluation of T can be deduced from the following equation:(15)T=expHΔt=expHτ2m,in which τ = Δt/2m. Expanding expHτ by Taylor series, the following expression is obtained:(16)expHτ≈I+Hτ+Hτ22=I+T0,in which T0=Hτ⋅I+Hτ/2 and I is the unit matrix.It is further noted that(17)expHτ2=I+T02=I2+2I⋅T0+T02=I+T1,expHτ22=I+T12=I2+2I⋅T1+T12=I+T2,………expHτ2w=I+Tw−12=I2+2I⋅Tw−1+Tw−12=I+Tw.So the recurrence formula is concluded as(18)Tw=2Tw−1+Tw−1⋅Tw−1,w=1,2,…,m,and the solution of T can be deduced using the recurrence formula.The random load process vectorFt in equations (9)–(14) is represented by K-L expansion as(19)Ft=0M−1f0t+∑j=1Nξjfjt.Therefore, thejth term of Ft and its corresponding solution vkj at time tk can be written, respectively, as(20)Fjt=0M−1fjt,(21)vkj=Tvk−1j+∫tk−1tkexptk−τHFjτdτ,j=1,…,N.The integration of equation (21) can be calculated using the Gauss–Legendre numerical method, and the solution can be represented as(22)vkj=Tvk−1j+Δt2∑l=1MwleHΔt1−ζl/2Fjtk−1+Δt1+ζl2,where M is the number of Gaussian integral points and ζl and wl are the Gaussian integral point and the corresponding weight, respectively.Furthermore, for the linear systems, asvkj=Ykj,Y˙kjT, the response Y of the random vibrating structure can be expressed as(23)Yk=Yk0+∑j=1NξjYkj.


## 3.2. Statistical Moment of the Random Extreme Response by Dimension Reduction Point Estimate Method

In each term of random responseYj (j = 0, 1, …, N), the structural random parameters in vector X are time independent. Therefore, it is suitable to use the dimension reduction point estimate method to evaluate the moments of the jth response. The correlated nonnormal random parameters can be transformed to independent standard normal variables by the inverse normal transformation, which can be written as(24)X=T−1U,in which U is the n-dimensional independent standard normal vector and T−1⋅ is the inverse transforming function such as the Rosenblatt transformation, Nataf transformation, and third-moment transformation technique, which is determined by the known conditions [34].Then, the dimension reduction method at timetk is expressed as(25)Y^kjX=∑r=1nYkjT−1Ur−n−1Ykjμ,where μ represents the vector of mean values of the random variables, and(26)Ur=uμ1,uμ2,…,uμr−1,ur,uμr+1,…,uμn,in which ur is the only standard normal distributed random parameter and the other variables are set equal to mean values transformed into the standard normal space.ConsideringYkjT−1Ur=Yk,rj, the lth origin moment of the jth random response can be expressed by the binomial theorem as(27)EY^kjl=E∑r=1nYk,rj−n−1Ykjμl=E∑s=1lCls∑r=1nYk,rjs−n−1Ykjμl−s=∑s=1lClsE∑r=1nYk,rjs−n−1Ykjμl−s.Using the binomial theorem for times, thesth exponentiation in equation (27) is expressed as(28)E∑r=1nYk,rjs=E∑k1=0sCsk1∑r=1n−1Yk,rjk1Yk,njs−k1=E∑k1=0sCsk1∑k2=0k1Ck1k2∑r=1n−2Yk,rjk2Yk,n−1jk1−k2Yk,njs−k1=E∑k1=0sCsk1∑k2=0k1Ck1k2⋯∑kn=0kn−1Ckn−1knYk,1jknYk,2jkn−1−kn⋯Yk,njs−k1=∑k1=0sCsk1∑k2=0k1Ck1k2⋯∑kn=0kn−1Ckn−1knEYk,1jknEYk,2jkn−1−kn⋯EYk,njs−k1.It is obvious from the above expression that when the origin moments ofY are obtained, the statistical moment of the random extreme response can be evaluated afterwards. As Yk,rj is a function with only one random variable, it is expediently calculated by Gaussian numerical integration:(29)EYk,rjl=∫−∞∞Yk,rjlfururdur,where fur is the probability density function of the standard normal variable ur. The approach for estimating the statistical moment of the response function above is the so-called point estimate method, which is developed by Zhao [35].SupposingMlj=EY^kjl (j = 0, 1, …, N), the first four central moments of the random response can be derived from equations (30)–(33) as(30)EY^=EY0t+E∑j=1NξjYjt=EY0t+∑j=1NEξjEYjt=M10,(31)σ2Y^=EY−M102=EY0t2+∑j=1NEYjt2−M102=M20+∑j=1NM2j−M102,(32)V3Y^=EY−M103=EY0t+∑j=1NξjYjt−M103=M30−3M20M10+2M103,(33)V3Y^=EY−M104=EY0t+∑j=1NξjYjt−M104=M40−4M30M10+6M20M102−3M104+12M20∑j=1NM2j−6M102∑j=1NM2j+6∑j=1NM4j+∑i=1N∑j=1,j≠iNM2iM2j.


## 4. Reliability-Based Sensitivity Analysis for the Random Vibrating System

Consider a system with state functiongX, where the vector X represents the input random variables with a joint probability density function fXX. The probability of failure Pf=PgX<0 of the system is defined as(34)Pf=∫⋯∫gX<0fXXdX,where g < 0 indicates the failure state, g > 0 the safe state, and g = 0 the limit state.The effect of the distribution parameters on the reliability of the basic input variables is defined as the reliability-based sensitivity, which can be obtained by the methodology of derivation expressed as(35)∂Pf∂θ=∂∫⋯∫gX<0fXX,θdX∂θ=∫⋯∫gX<0∂fXX,θ∂θdX,where θ is the distribution parameter of the basic input variables such as the mean value and the variance, and the probability density function fXX of X and θ is expressed as fXX,θ.However, in many engineering systems, some input random variables may depend on time, and thus, the response can be described by a random process. In this case, a time-dependent analysis is necessary to reveal the reliability and reliability-based sensitivity characteristics of the random process system. Denoting the interested time interval as [tmin, tmax], the instantaneous probability of failure is(36)Pftmin,tmax=PgX,t<0,∃t∈tmin,tmax,where the probability of failure is calculated in a quasi-steady sense, by fixing time t and replacing the random process with a random variable.In the numeric solving process, the time interval [tmin, tmax] is always dispersed to q discrete points; the failure probability at each discrete point tii=1,…,q is defined by equation (34), which can be denoted as Pfi=PgX,ti<0; and the failure probability in the time interval is(37)Pftmin,tmax=⋃i=1qPfi,which requires to evaluate the union of failure value in all domains by solving q performance functions acting together.The limit-state function in the time-dependent problem can be written as(38)g=G−Y,where G is the threshold value and Y is the random response. The reliability can be investigated by the fourth-moment method [36].The first four central moments of the limit-state function is as equations (39)–(42), under the assumption that G and Y are independent:(39)Eg=EG−EY,(40)σ2g=σ2G+σ2Y,(41)V3g=V3G−V3Y,(42)V4g=V4G+V4Y+6σ2Gσ2Y,where E⋅ indicates the mathematical expectation, σ2⋅ indicates the variance, V3⋅ denotes the third central moment, and V4⋅ is the fourth central moment, and the random response Y can be substituted by Y^.The reliability index is defined as(43)β2=Egσg.The performance functiong can be standardized as(44)zu=g−Egσg.Letα3g=V3g/σ3g and α4g=V4g/σ4g, which denote skewness and kurtosis of the state function, respectively. The Edgeworth series can be used to approximate the distribution function of the standardized variable zu, which is expanded as(45)Fzu=Φzu−ϕzu16α3gH2zu+124α4g−3H3zu+172α3g3H5zu,where(46)H2x=x2−1,H3x=x3−3x,H5x=x5−10x2+15x,are the second-, third-, and fifth-order Hermite polynomials, respectively.The failure probability is(47)Pg≤0=Pzu≤β2.According to equation (45), the failure probability is expressed as(48)P=F−β2=Φ−β2−ϕβ216α3gH2−β2+124α4g−3H3−β2+172α3g3H5−β2,and the reliability is obtained as(49)R=1−P=1−F−β2.Because only the first four terms of the Edgeworth series are used, sometimes the reliabilityR > 1 may occur. When R is larger than 1, a revised formula is used to obtain the accurate solution [37]:(50)R∗β2≈Φβ2−β2572α3g2ϕβ2.The direct derivation method is used to evaluate the reliability-based sensitivities of the distribution parametersθk of the basic variables. Here, the distribution parameters include mean value and variance:(51)dRdθk=∂R∂β2∂β2∂Eg∂Eg∂EY^∂EY^∂θk+∂β2∂σg∂σg∂σY^∂σY^∂θk+∂R∂α3g∂α3g∂V3g∂V3g∂V3Y^∂V3Y^∂θk+∂α3g∂σg∂σg∂σY^∂σY^∂θk+∂R∂α4g∂α4g∂V4g∂V4g∂V4Y^∂V4Y^∂θk+∂α4g∂V4g∂V4g∂σY^∂σY^∂θk+∂α4g∂σg∂σg∂σY^∂σY^∂θk.Let∂Vl/∂θk denote the partial derivative of the lth center moment of Y^ with respect to θk, which can be calculated by the integral(52)∂Vl∂θk=∫−∞+∞⋯∫−∞+∞Y^l∂fXX,θ∂θkdX=EY^lfXX,θ∂fXX,θ∂θk=EY^lCθ,where Cθ=1/fX,θX∂fX,θX/∂θk is the kernel function, and equation (52) can be calculated by numerical integration together with equation (29).


## 5. Numerical Example

### 5.1. Reliability and Reliability-Based Sensitivity Analysis of a Multi-Degree-of-Freedom Truss Structure

A truss structure is considered in this example, as shown in Figure2, with parameters depicted in Table 1.Figure 2
Model of truss.Table 1
Parameters of the truss structure.
Parameters
Mean value
Distribution style
Cross-sectional area
2.5×10−3 m2
Lognormal distribution
Elastic modulus of materials
2.0 × 1011 pa
Lognormal distribution
Density
7.86 × 103 kg/m3
Lognormal distributionThe stimulating forceFt is supposed to be a stochastic process with constant autospectrum 2 × 105 N2/Hz, frequency domain 0–180 Hz, and mean value 5 × 106 N, and the power spectral density function and covariance function are as follows:(53)Sffω=S01+4ηg2ω/ωg21−ω/ωg22+4ηg2ω/ωg2,(54)Cffτ=πS02eηgωgτη1cosωdτ+η2sinωdτ,where ωd=ωg1−ηg2, η1=ωg1+4ηg2/ηg, η2=ωg1−4ηg2/1−ηg2, ωg=1.5rad/s, ηg=1.2, and S0=1e5m2/s3.Whenτ equals 0, the variance of the random process σ2t=πS0/2.First, the stiffness matrix, mass matrix, and damping matrix are established by the FEM, and then the vibration equation of this truss is developed. The vertical response at node 4 is considered for reliability analysis. Rosenblatt transformation is used here to transfer the lognormal normal distribution parameters to independent normal distribution parameters. Then, the proposed method and the MCS are constructed to estimate the reliability and reliability-based sensitivity of the parameters of the structure. The number of K-L expansion terms is selected to be 300, and the tolerance valuesεVar,N and εCov,N are about 10−15 and 10−10. They are accurate enough for calculation. The 100 realizations of random exciting force are simulated in Figure 3. The 100 realizations of response of the structure are shown in Figure 4.Figure 3
100 realizations of random exciting force.Figure 4
100 realizations of vertical response at node 4.The transient reliability response curve is shown in Figure5, and the accumulative reliability response curve is shown in Figure 6. The results obtained from the MCS method and the method proposed in this paper agree very well with each other, which indicates that the method proposed here has great computing accuracy and computing efficiency. The transient reliability curve shows that the reliability is minimum at 0.04 s after start, so it is the most dangerous time for the system. The transient curve and the accumulative curve are kept stable after the dangerous time. In the stable state, the reliability-based sensitivity of the parameters is obtained, as shown in Figure 7, and the positive influence of the mean value of the cross-sectional area and the negative influence of the variance of the cross-sectional area are outstanding.Figure 5
Transient reliability curve of response.Figure 6
Accumulative reliability curve of response.Figure 7
Reliability-based sensitivity of the random parameters.


### 5.2. System with Unknown Distribution Parameters

A damped dynamic vibration absorber model is shown in Figure8, which is used as the vibrating screen for instance. As illustrated in Figure 8, the system is composed of a vibrating system (mass m1, spring k1, and viscous damping c1) and a dynamic shock absorption system (mass m2, spring k2, and viscous damping c2), and the statistical moments of the random parameters of the system with unknown distributions are displayed in Table 2.Figure 8
Dynamic absorber model.Table 2
Stochastic characteristics of basic parameters.
Random parameters
m
1 (kg)
k
1 (N/m)
c
1 (N·s/m)
m
2 (kg)
k
2 (N/m)
c
2 (N·s/m)
ω
t (rad/s)
F
1 (N)
Average
500
1e6
1e5
50
1e7
1e5
50
8000
Variance
25
5e4
5e3
2.5
5e5
5e3
2.5
4e2
Third moment
2.55e3
2.10e3
2.21e6
3.26
2.32e6
2.21e6
1.6387
6.67e6
Fourth moment
1.22e6
1.96e19
1.13e10
120.26
1.98e23
1.13e10
121.75
7.85The different equation of motion is(55)m1x¨1+k1+k1x1−k2x2+c1+c2x˙1−c2x˙2=F1sinωt+Atft,m2x¨2−k2x1+k2x2−c2x˙1+c2x˙2=0.The stimulating forceft is the stochastic disturbance in work, which is a uniform modulated nonstationary stochastic process. The modulating function is(56)At=t2tb2,0≤t≤tb,1.0,tb≤t≤tc,e−ct−tc,t≥tc,where tb = 3 s, tc = 6 s, and c = 0.1572. The power spectral density function and the covariance function of a Gaussian stochastic process are expressed as equations (53) and (54), respectively, in which ωg=15rad/s, ηg=0.06, and S0=1e5m2/s3. Considering τ=t2−t1, the power spectral density function and covariance of the modulated process are(57)Sffω=At2S01+4ηg2ω/ωg21−ω/ωg22+4ηg2ω/ωg2,Cffτ=At1At2πS02eηgωgτη1cosωdτ+η2sinωdτ.The number of K-L expansion terms is selected to be 200, and the tolerance valuesεVar,N and εCov,N are about 10−13 and 10−8, which are accurate enough. The 100 realizations of the random exciting force are shown in Figure 9. The mean response of the structure is shown in Figure 10. The perturbation method proposed in paper [23] is used here to solve the problem in the system with unknown distribution random parameters. Given the first four moments of the random parameters, the statistical characteristic of the response can be calculated. And then, the reliability and reliability-based sensitivity can be obtained. The transient reliability and the accumulative reliability obtained by the method proposed in this paper comparing with the MCS method are shown in Figures 11 and 12. Different from the examples proposed above, the curves of the reliability cannot reach a stable state because of the system’s unstable characteristic.Figure 9
100 realizations of the exciting force.Figure 10
Mean response of the vibration absorber model.Figure 11
Transient reliability curve of response.Figure 12
Accumulative reliability curve of response.To investigate the influence of the random parameters on the reliability, the accumulative sensitivities are calculated which are shown in Figure13. According to Figure 13, the mean values of m1, k1, and ωt have a positive effect on the reliability significantly, while the mean value of c2 and the variance of m2, k2, c1, and ωt have a negative effect on the reliability significantly.Figure 13
Reliability-based sensitivity of the random parameters.


## 5.1. Reliability and Reliability-Based Sensitivity Analysis of a Multi-Degree-of-Freedom Truss Structure

A truss structure is considered in this example, as shown in Figure2, with parameters depicted in Table 1.Figure 2
Model of truss.Table 1
Parameters of the truss structure.
Parameters
Mean value
Distribution style
Cross-sectional area
2.5×10−3 m2
Lognormal distribution
Elastic modulus of materials
2.0 × 1011 pa
Lognormal distribution
Density
7.86 × 103 kg/m3
Lognormal distributionThe stimulating forceFt is supposed to be a stochastic process with constant autospectrum 2 × 105 N2/Hz, frequency domain 0–180 Hz, and mean value 5 × 106 N, and the power spectral density function and covariance function are as follows:(53)Sffω=S01+4ηg2ω/ωg21−ω/ωg22+4ηg2ω/ωg2,(54)Cffτ=πS02eηgωgτη1cosωdτ+η2sinωdτ,where ωd=ωg1−ηg2, η1=ωg1+4ηg2/ηg, η2=ωg1−4ηg2/1−ηg2, ωg=1.5rad/s, ηg=1.2, and S0=1e5m2/s3.Whenτ equals 0, the variance of the random process σ2t=πS0/2.First, the stiffness matrix, mass matrix, and damping matrix are established by the FEM, and then the vibration equation of this truss is developed. The vertical response at node 4 is considered for reliability analysis. Rosenblatt transformation is used here to transfer the lognormal normal distribution parameters to independent normal distribution parameters. Then, the proposed method and the MCS are constructed to estimate the reliability and reliability-based sensitivity of the parameters of the structure. The number of K-L expansion terms is selected to be 300, and the tolerance valuesεVar,N and εCov,N are about 10−15 and 10−10. They are accurate enough for calculation. The 100 realizations of random exciting force are simulated in Figure 3. The 100 realizations of response of the structure are shown in Figure 4.Figure 3
100 realizations of random exciting force.Figure 4
100 realizations of vertical response at node 4.The transient reliability response curve is shown in Figure5, and the accumulative reliability response curve is shown in Figure 6. The results obtained from the MCS method and the method proposed in this paper agree very well with each other, which indicates that the method proposed here has great computing accuracy and computing efficiency. The transient reliability curve shows that the reliability is minimum at 0.04 s after start, so it is the most dangerous time for the system. The transient curve and the accumulative curve are kept stable after the dangerous time. In the stable state, the reliability-based sensitivity of the parameters is obtained, as shown in Figure 7, and the positive influence of the mean value of the cross-sectional area and the negative influence of the variance of the cross-sectional area are outstanding.Figure 5
Transient reliability curve of response.Figure 6
Accumulative reliability curve of response.Figure 7
Reliability-based sensitivity of the random parameters.


## 5.2. System with Unknown Distribution Parameters

A damped dynamic vibration absorber model is shown in Figure8, which is used as the vibrating screen for instance. As illustrated in Figure 8, the system is composed of a vibrating system (mass m1, spring k1, and viscous damping c1) and a dynamic shock absorption system (mass m2, spring k2, and viscous damping c2), and the statistical moments of the random parameters of the system with unknown distributions are displayed in Table 2.Figure 8
Dynamic absorber model.Table 2
Stochastic characteristics of basic parameters.
Random parameters
m
1 (kg)
k
1 (N/m)
c
1 (N·s/m)
m
2 (kg)
k
2 (N/m)
c
2 (N·s/m)
ω
t (rad/s)
F
1 (N)
Average
500
1e6
1e5
50
1e7
1e5
50
8000
Variance
25
5e4
5e3
2.5
5e5
5e3
2.5
4e2
Third moment
2.55e3
2.10e3
2.21e6
3.26
2.32e6
2.21e6
1.6387
6.67e6
Fourth moment
1.22e6
1.96e19
1.13e10
120.26
1.98e23
1.13e10
121.75
7.85The different equation of motion is(55)m1x¨1+k1+k1x1−k2x2+c1+c2x˙1−c2x˙2=F1sinωt+Atft,m2x¨2−k2x1+k2x2−c2x˙1+c2x˙2=0.The stimulating forceft is the stochastic disturbance in work, which is a uniform modulated nonstationary stochastic process. The modulating function is(56)At=t2tb2,0≤t≤tb,1.0,tb≤t≤tc,e−ct−tc,t≥tc,where tb = 3 s, tc = 6 s, and c = 0.1572. The power spectral density function and the covariance function of a Gaussian stochastic process are expressed as equations (53) and (54), respectively, in which ωg=15rad/s, ηg=0.06, and S0=1e5m2/s3. Considering τ=t2−t1, the power spectral density function and covariance of the modulated process are(57)Sffω=At2S01+4ηg2ω/ωg21−ω/ωg22+4ηg2ω/ωg2,Cffτ=At1At2πS02eηgωgτη1cosωdτ+η2sinωdτ.The number of K-L expansion terms is selected to be 200, and the tolerance valuesεVar,N and εCov,N are about 10−13 and 10−8, which are accurate enough. The 100 realizations of the random exciting force are shown in Figure 9. The mean response of the structure is shown in Figure 10. The perturbation method proposed in paper [23] is used here to solve the problem in the system with unknown distribution random parameters. Given the first four moments of the random parameters, the statistical characteristic of the response can be calculated. And then, the reliability and reliability-based sensitivity can be obtained. The transient reliability and the accumulative reliability obtained by the method proposed in this paper comparing with the MCS method are shown in Figures 11 and 12. Different from the examples proposed above, the curves of the reliability cannot reach a stable state because of the system’s unstable characteristic.Figure 9
100 realizations of the exciting force.Figure 10
Mean response of the vibration absorber model.Figure 11
Transient reliability curve of response.Figure 12
Accumulative reliability curve of response.To investigate the influence of the random parameters on the reliability, the accumulative sensitivities are calculated which are shown in Figure13. According to Figure 13, the mean values of m1, k1, and ωt have a positive effect on the reliability significantly, while the mean value of c2 and the variance of m2, k2, c1, and ωt have a negative effect on the reliability significantly.Figure 13
Reliability-based sensitivity of the random parameters.


## 6. Conclusion

For the random parametric systems subjected to random process excitation, a method combining the stochastic process K-L expansion method, the precise Gauss–Legendre integration, the point estimate method, and the fourth-moment method is proposed in this paper. The precise integration is suitable for solving differential functions in a time-saving way and can combine with the K-L expansion method at the same dispersed time points. The combination of the two methods simplifies the calculation process and makes it realizable for estimate structure responses subjected to a large number of expansion exciting forces. A large number of Gaussian random parameters produced in the K-L expansion scarcely contributed to difficulties in the moment calculation according to equations (30)–(33). However, the method has its own deficiencies in the structure model, which is required to be linear to apply the principle of superposition.Therefore, the method proposed in this paper is recommended to be applied in the reliability problems of linear structures with time-independent and time-dependent random parameters, which is easy to implement and has high efficiency and accuracy.



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