# Bayesian Uncertainty Identification of Model Parameters for the Jointed Structures with Nonlinearity

Authors: Zhanpeng Shen; Xinen Liu; Chaoping Zang; Shaoquan Hu
Journal: Shock and Vibration
 (2021)

---

## Abstract

Jointed structures in engineering naturally perform with some of nonlinearity and uncertainty, which significantly affect the dynamic characteristics of the structural system. In this paper, the method of Bayesian uncertainty identification of model parameters for the jointed structures with local nonlinearity is proposed. Firstly, the nonlinear stiffness and damping of the joints under the random excitation are represented with functions of excitation magnitude in terms of the equivalent linearization. The process of uncertainty identification is separated from the representation of local nonlinearity. In this way, the dynamic behavior of the joints is penetratingly characterized instead of ascribing the nonlinearity to uncertainty. Secondly, a variable-expanded Bayesian (VEB) method is originally proposed to identify the mixed of aleatory and epistemic uncertainties of model parameters. Different from traditional Bayesian identification, the aleatory uncertainties of model parameters are identified as one of the most important parts rather than only measurement noise of output. Notablely, a series of intermediate variables are introduced to expand the parameter with aleatory uncertainty in order to overcome the difficulty of establishing the likelihood function. Moreover, a 3-DOF numerical example is illustrated with case studies to verify the proposed method. The influence of observed sample size and prior distribution selection on the identification results is tested. Furthermore, an engineering example of the jointed structure with rubber isolators is performed to show the practicability of the proposed method. It is indicated that the computational model updated with the accurately identified parameters with both nonlinearity and uncertainty has shown the excellent predictive capability.


---

## Body

## 1. Introduction

Engineering structures with joints usually behave with some of the nonlinear vibration characteristics. These nonlinear phenomena are usually concentrated in the local joints of the parts, such as bolt flanges, interference/clearance fit, cushion, and vibration isolator. It may lead to complex nonlinear responses such as natural frequency shift, phase distortion, frequency response jump, and other phenomena [1–3] with the increase of excitation levels. Though calculation of the direct nonlinear responses has been studied for many years, it is still in difficulty for dealing with engineering structures with a huge number of degrees of freedom (DOFs), especially under random excitations. The equivalent linearization to deal with nonlinearity is currently adopted because of the maturity and convenience to calculate the random vibration response [4, 5]. Friis et al. [4] summarized three commonly used equivalent methods with numerical examples, including the energy balance method, the intensity balance method, and the direct linear fit in the least squares sense. Alibrandi and Mosalam [6] proposed three linear equivalence methods for reliability evaluation of random vibration structures, that is, Gaussian linear equivalent linearization, tail equivalent linearization, and tail probability equivalent linearization. Other researchers have also done many studies about equivalent linearization under random vibration [7, 8]. However, the authors insist that the equivalent linearization model should address the nonlinear features such as the fact that the equivalent stiffness and damping are represented with functions of excitation magnitudes. Chen et al. [9], respectively, established the polynomial models for equivalent stiffness and equivalent damping varying with excitation magnitudes in the study of a rubber vibration isolator. The polynomial coefficients are analytically derived according to the sin-sweep vibration results in different test cases.From another point of view, the model parameters inevitably show some of uncertainties no matter linear or nonlinear model to be built up [10, 11]. For example, the modulus of some polymer materials like rubber could vary 20% among the same manufacture batch. In fact, the equivalent stiffness and damping of the component joints are also uncertain, because of the joint interface consisting of a large number of microconvex bodies with random sizes and positions. The uncertainties in model parameters may be characterized as either aleatory uncertainties, which are irreducible variabilities inherent in nature, or epistemic uncertainties, which are reducible uncertainties [11, 12] resulting from a lack of knowledge or intentional simplifications. As opposed to model parameter uncertainty, model form uncertainty [13, 14] is also epistemic and worthy of note, but it is not the focus of this paper. Model parameters are normally uncertain and therefore uncertainty also exists in the dynamic response. Moreover, it is usually much greater than measurement noise. As a result, the uncertainties as well as the optimal estimations for parameters should be identified in order to predict the uncertainty of the response.Though there have been lots of literatures to study the nonlinear dynamic behaviors of the jointed structure such as the vibration isolator [15–17], the uncertainties of model parameters are not often identified. Kerschen et al. [3] have made a detailed summary of parameter identification methods for nonlinear structures and divided them into seven categories. Therein, structural model updating [18, 19] is more convenient than other methods and recommendable for the complex structure due to large number of DOFs in the finite element model. Moreover, among various model updating methods, Bayesian inference is a very popular probabilistic identification method, which has been widely used in both linear and nonlinear structures [20–27]. Compared with the deterministic method based on optimization, Bayesian method can more flexibly deal with modelling uncertainties and measurement noise.However, most of the current Bayesian parameter identification methods focus on the optimal estimation, and the main uncertainty is considered to be the measurement noise. The probability distribution functions (PDFs) of the model parameters with aleatory uncertainty are scarcely obtained due to the high cost of repeated experiments to adequately describe the aleatory uncertainties, though statistical variance is identified to address uncertainty in some good way. The other difficulty is to establish the likelihood function since the uncertainty of experimental data comes from both the measurement noise and the uncertainty of parameters. It indicates that the experimental data of output cannot be generally assumed as the normal distributed and the likelihood function cannot be directly analytical expressed.In this paper, a new VEB method is originally proposed to identify the PDFs of model parameters with the mixed aleatory and epistemic uncertainties based on the experimental data. A series of intermediate variables with the same prior distribution are introduced, and the model parameters with aleatory uncertainty are tactfully transformed in order to establish the new likelihood function. Combined with the equivalent linearization under different magnitudes of random excitations, the local nonlinearity is also characterized together with the uncertainties for jointed structures. Compared with traditional Bayesian identification, uncertainties of model parameters are identified based on repeated experimental data rather than only measurement noise of output. It is highlighted herein that the influences of parameters’ uncertainties (e.g., elastic modulus of the rubbers in isolators) on structural responses are usually critical in engineering rather than measurement noise. Compared with interval analysis or other variance-based methods, more detailed information of the uncertainties could be achieved in terms of PDF and the identification results could be quickly updated with the additional useful experimental data. Moreover, the nonlinearity of the joints is separated from the uncertainty, leading to the fact that the dynamic behavior is penetratingly characterized instead of ascribing the nonlinearity to uncertainty. Thus, the computational model updated with this method has a high predictive capability for the jointed structures.The paper is organized as follows. Section2 provides a brief introduction to the local nonlinearity modelling of the jointed structures. Section 3 then introduces the Bayesian identification of the uncertainties for model parameters including the VEB method in detail. Section 4 studies a virtual 3-DOF spring mass system with 3 different cases to illustrate and verify the proposed method. Moreover, Section 5 provides an engineering example of the jointed structures with rubber isolators to show the practicability of the proposed method. Finally, Section 6 ends with our main conclusions.


## 2. Local Nonlinearity Modelling of the Jointed Structures

The jointed structures could be regarded as a combination of many components. Generalized spring-damping elements can be adopted to characterize the joints between the boundary and components or among different components. Some of elements are linear while some are nonlinear as shown in Figure1. It depends on the mechanical behavior of the joints.Figure 1
Illustration of the nonlinear spring-damping elements for the jointed structures.For the nonlinear system, the motion equation can be written as(1)My¨+Cy˙+Ky+fnly,y˙=xt,where x(t) denotes the excitation vector varying with time and y(t) denotes the displacement solution. M, C, and K are, respectively, the mass, damping, and stiffness matrices of the system, and fnl is the nonlinear term. For the weak nonlinearity, equivalent linearization is often adopted in engineering. The equivalent equation can be written as(2)My¨+Ceqy˙+Keqy=xt,where Ceq and Keq are equivalent damping and equivalent stiffness matrices, respectively. Compared with C and K, only a few elements of Ceq and Keq change because the nonlinearity of the jointed structures only appears in a few local positions.Due to random excitation input, the nonlinearity of the jointed structures is averaged or linearized in a sense [28]. The frequency response function (FRF) of the nonlinear system under the random excitation is defined by (3). It appears to be linear in terms of its shape with eigenfrequency drifting as shown in Figure 2(a):(3)Hnrω=SyxωSxxω,where Syxω and Sxxω are the cross- and auto-spectrum densities, respectively. It should be noted that the FRF of the nonlinear system under the random excitation (i.e., Hnrω) shows clear distinction from that under the sin-sweep excitation. It mainly depends on the input power spectral density (i.e., the vibration magnitude), and it approaches the linear FRF as the power spectral density approaches zero.Figure 2
Equivalent linearization of nonlinear spring-damping element with different vibration magnitudes. (a) FRFs and (b) equivalent stiffness and equivalent damping.
(a)(b)Taking modal parameters (i.e., natural frequency and damping ratio) from the FRFs with different vibration magnitudes as the target, the equivalent stiffness and damping of the nonlinear elements could be obtained, respectively, as shown in the following:(4)keqA=keq1A,keq2A,…,keq,sAT,(5)ceqA=ceq1A,ceq2A,…,ceq,sAT,where s denotes the number of nonlinear elements and A denotes the magnitude of the random excitation. According to the FRF of a nonlinear system with a signal DOF under different magnitudes of random excitation as shown in Figure 2(a), the equivalent stiffness and damping of the nonlinear spring-damping element are shown in Figure 2(b), respectively. It should be noted that the FRF in Figure 2 appears undistorted and seems like a linear FRF just with frequency drift. The reason is not soft nonlinearity but is the fact that the randomness of the amplitude and phase of the excitation signal creates a “linearized” or “averaged” FRF.


## 3. Bayesian Identification of the Model Parameter Uncertainties

Compared with the determinate parameter identification, Bayesian identification provides not only the optimal estimations of the model parameters but also the probability distribution to describe their uncertainties. Generally, the structural response of interest (e.g., natural frequency or modal damping ratio) can be expressed in terms of components as follows:(6)zi=zix;θ,α,i=1,2,…,q,where q is the number of output responses, x is the controllable and determinate input vector (e.g., the random vibration magnitude), θ denotes the aleatory uncertain parameter set, and α denotes the epistemic uncertain parameter set.It is noted thatθ has observable randomness and its uncertainty is inherent and irreducible. However, α has an unknown but unique truth value, whose uncertainty can be reduced with the addition of useful information. These two types of parameters are independent with each other. In addition, the prior distributions of the aleatory uncertain parameter could be expressed as πθθ|α1, where the hyperparameter set α1 has also epistemic uncertainty and does not appear in the model.Ignoring the model form error, experimental error is assumed to follow the normal distribution with zero mean, and then the experimental observation of each output can be expressed as(7)z˜i=zix;θ,α+εi,εi∼N0,σi2,i=1,2,…,q.Denoting then groups of test data set as D=xk,z˜k,k=1,2,…,n, there are unknown values of θk and εk corresponding to each set of test data xk,zk; that is,(8)z˜ki=zixk;θk,α+εki,i=1,2,…,q,k=1,2,…,n.


### 3.1. Brief Introduction of Traditional Bayesian Identification of Model Parameters

Traditional Bayesian parameter identification, namely, Bayesian model updating, deals with only the epistemic uncertain variables. Measurement error is assumed to be aleatory uncertainty, and each of the model parameters is assumed to have an unknown unique true value. The belief in the true value is represented with probability distribution in Bayesian method.Ignoring the aleatory uncertain variablesθ in (6)∼(8), the posterior distribution of parameters α is obtained according to test data set D by using the Bayesian formula:(9)pα|D=LD|απαα∫LD|απααdα∝LD|απαα,where παα is the prior PDF of α and L(•) is the likelihood function. The denominator in (9) is an integral constant ensuring that the posterior PDF integrates to 1.Assuming that the test data of output are independent identically distributed (i.i.d) samples and test error follows normal distribution, the likelihood function for a single output can be constructed as(10)LiD|α,σi=∏k=1n1σi2πexp−12σi2z˜ki−zixk;α2=12πn/2σinexp−12σi2∑k=1nz˜ki−zixk;α2.If Bayesian identification is performed simultaneously using multivariate outputs, one definition of the likelihood function is shown in the following:(11)LD|α,σ=∏i=1qηiLiD|α,σi,where ηi is the weighted coefficient and σ=σ1,σ2,…,σq is the test error vector.


### 3.2. Basic Theory of VEB Identification

Due to aleatory uncertainty in model parameters, uncertainties in output data do not just come from the measurement error. The whole observed data cannot be generally assumed as the normal distributed. Moreover, if the model outputs do not linearly vary with aleatory uncertain parameters or these parameters are not conjugate distributed (e.g., normally distributed), the likelihood function cannot be analytically expressed directly. Furthermore, the calculation amount of the likelihood function value in terms of numerical expression is usually too huge to implement in Bayesian method. As a result, the traditional Bayesian identification encounters generally serious difficulty when the aleatory uncertainty in model parameters cannot be neglected.Assuming then sets of test data are independent with each other, a series of intermediate variables, namely, θ1,θ2,…,θn, with the same prior distributions are introduced to temporarily replace the aleatory uncertain parameters θ. Each vector θk of the intermediate variables corresponds to every sample set of the aleatory uncertain parameters. Each intermediate variable is regarded as the denotation of the true value of the corresponding test sample with only epistemic uncertainty. Thus, each set of the output test data z˜k is randomly distributed around zkx;θk,α and the test error could be assumed as a normal random variable with zero mean. Hence, the likelihood function could be constructed as(12)LiD|θk:k=1…n,α,σi=12πn/2σinexp−12σi2∑k=1nz˜ki−zixk;θk,α2,LD|θk:k=1…n,α,σ=∏i=1qηiLiD|θk:k=1…n,α,σi,where the test data set D is shown in (8).Then, according to Bayesian theorem, the posterior joint PDF for all parameters can be expressed as(13)pθk:k=1…n,α,α1,σ|D∝LD|θk:k=1…n,α,σ∏k=1nπθθk|α1πααπα1α1πσσ,where πθθk|α1, παα, and πσσ, respectively, denote the prior PDFs for intermediate variable θk, epistemic uncertain parameter α and the experimental standard deviation σ. πα1α1 denotes the prior PDF for the hyperparameter α1 addressing the distribution of θk.Integrating the intermediate variables, the posterior joint PDF is rewritten as(14)pα,α1,σ|D=∫θkpθk:k=1…q,α,α1,σ|D∏k=1qdθk.By marginalizing equation (14), the posterior PDF for any epistemic uncertain parameter can be obtained.The posterior PDFs for aleatory uncertain parameters can be acquired by integrating the hyperparameters as shown in (15). Essentially, the result is weighted average by the posterior probability of the hyperparameters:(15)pθ|D=∫α1πθθ|α1pα1|Ddα1=∫α,α1,σπθθ|α1pα,α1,σ|Ddαdα1dσ.


### 3.3. Algorithm Implementation of VEB Identification

Due to the mathematical difficulties of the high-dimensional integration, Bayesian posterior distribution is generally estimated by an effective stochastic simulation method called Markov chain Monte Carlo (MCMC) [29–31] instead of the direct integration. It generates a long sequence of samples from the complex shaped unknown posterior distributions. Metropolis–Hasting algorithm is a classical useful algorithm to generate the MCMC samples.Moreover, further management needs to be performed because of the asymptotic convergence and autocorrelation for the MCMC samples in sequence. The head part of the sample sequence which has not been convergent is removed. The whole sequence is diluted by extracting a sample at a certain interval to ensure that the autocorrelation can be ignored. Then, the useful MCMC samples are obtained to characterize the posterior distribution of each variable.The sample data are usually arranged into a matrix, where each column corresponds to each parameter to be identified. It is convenient to obtain the marginal distribution of each parameter or study the correlation between parameters base on the sample matrix. It is also very easy to get the posterior joint PDF or marginal PDFs forα,α1,σ by sample matrix instead of (14). The corresponding columns of intermediate variables are just discarded from the full matrix, and the remaining sample data are employed to address the joint PDF.In order to acquire the accurate posterior distribution, the sample sequence should be long enough after head removing and dilution. In other words, a huge amount of MCMC samples is required to calculate. In general, the computational efficiency depends on the calculation cost of outputs represented by (6), which are used for likelihood function calculation. Unfortunately, structural outputs are usually expensively computed by solving the finite element model in engineering. Thus, efficient meta-model with high accuracy is required to build up, such as polynomial response surface, Kriging model, artificial neural network, and polynomial chaotic expansion.The flowchart is shown in Figure3 to illustrate the implementation of VEB identification. First of all, the uncertain parameters to be identified should be clearly classified according to their attributes. Then, nonlinear modelling, probabilistic modelling. and modal testing are carried out. The process of MCMC is performed to obtain the posterior samples of all parameters. Consequently, the samples should be verified to satisfy the requirement of asymptotic convergence and autocorrelation. Finally, the posterior PDF of the parameters could be gained from the reduced MCMC samples to represent both the nonlinearity and uncertainties.Figure 3
The flowchart of the VEB identification method.Frankly speaking, VEB identification costs more computational amount than traditional Bayesian method, because the variables to be identified are expanded and the convergence rate of MCMC slows down. Denoting the numbers of aleatory and epistemic uncertain parameters byr and s, respectively, nr + s variables (including intermediate variables) need to generate samples in MCMC processing, where n denotes the number of test groups for aleatory uncertain parameters. Thus, the variables would expand more serious as the number of test groups increases. Moreover, it needs longer time and more computational amount to get convergent MCMC samples using. However, the computational cost of VEB identification is generally acceptable and controllable. On one hand, the number n would not be too large due to expensive experimental cost. On the other hand, the advantage of Bayesian inference method lies in making full use of prior information and less test data to obtain accurate estimates. It prefers identifying the parameters and updating the results again and again according to the current small amount of new test data, rather than waiting for allocating a large number of test data and identifying them in one time.


### 3.4. Output Prediction

For structural analysis in engineering, our purpose of parameters identification is usually predicting the output accurately under different inputs. The advantage of uncertainty identification in this paper is that probability distribution of predictive output could also be addressed for further study.Ignoring the experimental errors in prediction, the predictive output vector is expressed asz′=z′x′,θ,α and its probability distribution is determined by the distributions of parameters θ and α. Thus, the PDF for z′ could be addressed with its samples, which could be calculated by (6) according to the posterior MCMC samples for θ and α.With consideration of experimental errors in prediction, the predictive output vector is expressed as(16)z∗=z′x′,θ,α+ε.The conditional PDF of the predictive outputs could be written as(17)pzi∗|θ,α,σi=12π1/2σiexp−12σi2zi∗−zi′x′,θ,α2,i=1,2,…,q.From Bayesian theory, the posterior PDF of predictive outputs is expressed as(18)pzi′|D=∫θ,α,α1,σpzi′|θ,α,σipα,α1,σi|Dπθθ|α1dθdαdα1dσi,i=1,2,…,q.


## 3.1. Brief Introduction of Traditional Bayesian Identification of Model Parameters

Traditional Bayesian parameter identification, namely, Bayesian model updating, deals with only the epistemic uncertain variables. Measurement error is assumed to be aleatory uncertainty, and each of the model parameters is assumed to have an unknown unique true value. The belief in the true value is represented with probability distribution in Bayesian method.Ignoring the aleatory uncertain variablesθ in (6)∼(8), the posterior distribution of parameters α is obtained according to test data set D by using the Bayesian formula:(9)pα|D=LD|απαα∫LD|απααdα∝LD|απαα,where παα is the prior PDF of α and L(•) is the likelihood function. The denominator in (9) is an integral constant ensuring that the posterior PDF integrates to 1.Assuming that the test data of output are independent identically distributed (i.i.d) samples and test error follows normal distribution, the likelihood function for a single output can be constructed as(10)LiD|α,σi=∏k=1n1σi2πexp−12σi2z˜ki−zixk;α2=12πn/2σinexp−12σi2∑k=1nz˜ki−zixk;α2.If Bayesian identification is performed simultaneously using multivariate outputs, one definition of the likelihood function is shown in the following:(11)LD|α,σ=∏i=1qηiLiD|α,σi,where ηi is the weighted coefficient and σ=σ1,σ2,…,σq is the test error vector.


## 3.2. Basic Theory of VEB Identification

Due to aleatory uncertainty in model parameters, uncertainties in output data do not just come from the measurement error. The whole observed data cannot be generally assumed as the normal distributed. Moreover, if the model outputs do not linearly vary with aleatory uncertain parameters or these parameters are not conjugate distributed (e.g., normally distributed), the likelihood function cannot be analytically expressed directly. Furthermore, the calculation amount of the likelihood function value in terms of numerical expression is usually too huge to implement in Bayesian method. As a result, the traditional Bayesian identification encounters generally serious difficulty when the aleatory uncertainty in model parameters cannot be neglected.Assuming then sets of test data are independent with each other, a series of intermediate variables, namely, θ1,θ2,…,θn, with the same prior distributions are introduced to temporarily replace the aleatory uncertain parameters θ. Each vector θk of the intermediate variables corresponds to every sample set of the aleatory uncertain parameters. Each intermediate variable is regarded as the denotation of the true value of the corresponding test sample with only epistemic uncertainty. Thus, each set of the output test data z˜k is randomly distributed around zkx;θk,α and the test error could be assumed as a normal random variable with zero mean. Hence, the likelihood function could be constructed as(12)LiD|θk:k=1…n,α,σi=12πn/2σinexp−12σi2∑k=1nz˜ki−zixk;θk,α2,LD|θk:k=1…n,α,σ=∏i=1qηiLiD|θk:k=1…n,α,σi,where the test data set D is shown in (8).Then, according to Bayesian theorem, the posterior joint PDF for all parameters can be expressed as(13)pθk:k=1…n,α,α1,σ|D∝LD|θk:k=1…n,α,σ∏k=1nπθθk|α1πααπα1α1πσσ,where πθθk|α1, παα, and πσσ, respectively, denote the prior PDFs for intermediate variable θk, epistemic uncertain parameter α and the experimental standard deviation σ. πα1α1 denotes the prior PDF for the hyperparameter α1 addressing the distribution of θk.Integrating the intermediate variables, the posterior joint PDF is rewritten as(14)pα,α1,σ|D=∫θkpθk:k=1…q,α,α1,σ|D∏k=1qdθk.By marginalizing equation (14), the posterior PDF for any epistemic uncertain parameter can be obtained.The posterior PDFs for aleatory uncertain parameters can be acquired by integrating the hyperparameters as shown in (15). Essentially, the result is weighted average by the posterior probability of the hyperparameters:(15)pθ|D=∫α1πθθ|α1pα1|Ddα1=∫α,α1,σπθθ|α1pα,α1,σ|Ddαdα1dσ.


## 3.3. Algorithm Implementation of VEB Identification

Due to the mathematical difficulties of the high-dimensional integration, Bayesian posterior distribution is generally estimated by an effective stochastic simulation method called Markov chain Monte Carlo (MCMC) [29–31] instead of the direct integration. It generates a long sequence of samples from the complex shaped unknown posterior distributions. Metropolis–Hasting algorithm is a classical useful algorithm to generate the MCMC samples.Moreover, further management needs to be performed because of the asymptotic convergence and autocorrelation for the MCMC samples in sequence. The head part of the sample sequence which has not been convergent is removed. The whole sequence is diluted by extracting a sample at a certain interval to ensure that the autocorrelation can be ignored. Then, the useful MCMC samples are obtained to characterize the posterior distribution of each variable.The sample data are usually arranged into a matrix, where each column corresponds to each parameter to be identified. It is convenient to obtain the marginal distribution of each parameter or study the correlation between parameters base on the sample matrix. It is also very easy to get the posterior joint PDF or marginal PDFs forα,α1,σ by sample matrix instead of (14). The corresponding columns of intermediate variables are just discarded from the full matrix, and the remaining sample data are employed to address the joint PDF.In order to acquire the accurate posterior distribution, the sample sequence should be long enough after head removing and dilution. In other words, a huge amount of MCMC samples is required to calculate. In general, the computational efficiency depends on the calculation cost of outputs represented by (6), which are used for likelihood function calculation. Unfortunately, structural outputs are usually expensively computed by solving the finite element model in engineering. Thus, efficient meta-model with high accuracy is required to build up, such as polynomial response surface, Kriging model, artificial neural network, and polynomial chaotic expansion.The flowchart is shown in Figure3 to illustrate the implementation of VEB identification. First of all, the uncertain parameters to be identified should be clearly classified according to their attributes. Then, nonlinear modelling, probabilistic modelling. and modal testing are carried out. The process of MCMC is performed to obtain the posterior samples of all parameters. Consequently, the samples should be verified to satisfy the requirement of asymptotic convergence and autocorrelation. Finally, the posterior PDF of the parameters could be gained from the reduced MCMC samples to represent both the nonlinearity and uncertainties.Figure 3
The flowchart of the VEB identification method.Frankly speaking, VEB identification costs more computational amount than traditional Bayesian method, because the variables to be identified are expanded and the convergence rate of MCMC slows down. Denoting the numbers of aleatory and epistemic uncertain parameters byr and s, respectively, nr + s variables (including intermediate variables) need to generate samples in MCMC processing, where n denotes the number of test groups for aleatory uncertain parameters. Thus, the variables would expand more serious as the number of test groups increases. Moreover, it needs longer time and more computational amount to get convergent MCMC samples using. However, the computational cost of VEB identification is generally acceptable and controllable. On one hand, the number n would not be too large due to expensive experimental cost. On the other hand, the advantage of Bayesian inference method lies in making full use of prior information and less test data to obtain accurate estimates. It prefers identifying the parameters and updating the results again and again according to the current small amount of new test data, rather than waiting for allocating a large number of test data and identifying them in one time.


## 3.4. Output Prediction

For structural analysis in engineering, our purpose of parameters identification is usually predicting the output accurately under different inputs. The advantage of uncertainty identification in this paper is that probability distribution of predictive output could also be addressed for further study.Ignoring the experimental errors in prediction, the predictive output vector is expressed asz′=z′x′,θ,α and its probability distribution is determined by the distributions of parameters θ and α. Thus, the PDF for z′ could be addressed with its samples, which could be calculated by (6) according to the posterior MCMC samples for θ and α.With consideration of experimental errors in prediction, the predictive output vector is expressed as(16)z∗=z′x′,θ,α+ε.The conditional PDF of the predictive outputs could be written as(17)pzi∗|θ,α,σi=12π1/2σiexp−12σi2zi∗−zi′x′,θ,α2,i=1,2,…,q.From Bayesian theory, the posterior PDF of predictive outputs is expressed as(18)pzi′|D=∫θ,α,α1,σpzi′|θ,α,σipα,α1,σi|Dπθθ|α1dθdαdα1dσi,i=1,2,…,q.


## 4. Numerical Examples

### 4.1. Example Statement

A 3-DOF system with 4 linear springs and a nonlinear spring is oscillated under base excitation as shown in Figure4. The virtual modal test of the system is accomplished, respectively, under random excitation with 6 acceleration levels, including 0.1 g, 1 g, 2 g, 4 g, 6 g, and 8 g. Since the stiffness of the nonlinear spring k4 varies with the vibration magnitudes, the FRF shifts as the excitation acceleration increases as shown in Figure 5.Figure 4
A 3-DOF oscillating system with linear and nonlinear springs.Figure 5
Nonlinear feature of the 3-DOF system: (a) spring stiffnessk4 and (b) FRF vary with excitation magnitude.
(a)(b)In order to verify the effectiveness and practicability of the proposed method, three cases were studied as shown in Table1. The mass of each block is 0.1 kg, while stiffness parameters k1 and k3 are equal to 1000 N/m and 2000 N/m, respectively, for all cases. The outputs are 3 natural frequencies, while controllable input variable is the acceleration magnitude of base excitation. The example focuses on the identification of the other spring stiffness parameters, that is, k2, k4, and k5.Table 1
Comparison of the example cases.
ItemCase 1Case 2Case 3Nonlinear/linearNonlinearLinearNonlinearUncertaintyEpistemicAleatoryEpistemic and aleatoryParameters (N/m)k1100010001000k22000Normal dist.Normal dist.k3200020002000k4ax2+bx+c2000ax2+bx+cc∼normal dist.k51000Gamma dist.Gamma dist.In case 1, the nonlinear stiffnessk4 is modelled with the quadratic polynomial, and the coefficients {a, b, c} need to be identified. The traditional Bayesian identification is adopted with the consideration of measurement noise, and the uncertainties of model parameter cannot be identified. In case 2, linear stiffness k2 and k5 with aleatory uncertainties are identified using VEB method. Symmetrical and asymmetrical PDFs are set for k2 and k5, respectively, that is, normal and gamma distributions. In case 3, both nonlinear stiffness k4 and linear stiffness k2 and k5 are identified together with their uncertainty using VEB method. Specifically, the constant term c of the polynomial model for nonlinear stiffness k4 is assumed as an aleatory uncertain parameter with normal distribution.The natural frequencies from virtual modal test are generated from (19), where f˜i and fi denote ith experimental and computational frequency and x denotes the base excitation magnitude. The measurement noise ε is normally distributed with zero mean and 0.02 Hz standard deviation:(19)f˜i=fix;k2,k4,k5+ε,i=1,2,3.


### 4.2. Case Study 1: Nonlinear Stiffness Identification with consideration of Measurement Noise

The quadratic polynomial model for the stiffness of nonlinear spring verse excitation magnitude is established as(20)k4=ax2+bx+c.The parameter set to be identified isα=a,b,c,σ in this case study, where σ is the standard deviation of the test error. Traditional Bayesian identification is employed to obtain the posterior PDFs for a,b,c,σ according to virtual experimental data at 6 excitation levels as shown in Figure 6. In fact, the second natural frequency is useless for nonlinearity identification, because the change is not obvious among different vibration levels. The uniform PDF in a wide range is taken as the prior distribution of the parameters.Figure 6
Boxplots of 50 sets of virtual experimental natural frequencies.
(a)(b)(c)The optimal identification values of the parameters are listed and compared with the true values in Table2. The posterior PDFs of the parameters are shown in Figure 7. The sequence of MCMC samples for parameter a are shown in Figure 8 as an example to demonstrate that the convergence and autocorrelation are satisfied.Table 2
Parameter identification results in case study 1.
ParameterTrue valueIdentification valuen = 1n = 5n = 50a1515.2521.68%15.1460.97%14.984−0.11%b−250−252.701.08%−251.690.68%−249.81−0.08%c20002002.90.15%2002.70.14%1999.7−0.02%σ0.020.01905−4.77%0.020040.20%0.020542.68%Figure 7
The posterior PDFs of the parameters according to experimental data with different sizes.
(a)(b)(c)(d)Figure 8
The sequence of MCMC samples for parametera to verify the convergence and autocorrelation.It is indicated that the Bayesian identification results are very close to corresponding true values. Moreover, the posterior PDF becomes more and more sharp and the optimal estimation is generally closer to the true value as the amount of experimental data increases as shown in Table2 and Figure 7. However, nothing is obtained for the aleatory uncertainty of model parameters. The epistemic uncertainties described in Figure 7 just come from the measurement noise and the identification error.


### 4.3. Case Study 2: Linear Stiffness Identification Together with Its Aleatory Uncertainties

Ignoring the nonlinear springk4, the linear model parameters, that is, spring stiffness k2 and k5, are set as the normal and gamma distribution, respectively, as shown in the following:(21)k2∼Nμ2,σ22,k5=1000+δ,δ∼Γd,e.Virtual experimental frequency samples and the corresponding samples of spring stiffness are scattered as shown in Figure9. In order to study the effect of the experimental data amount, 3, 5, and 10 sets of experimental samples are provided for VEB identification stage by stage. Because 1st frequency is barely affected by k2 and k5, the experimental samples of 2nd and 3rd frequencies are utilized for VEB identification. In this case study, the aleatory uncertain parameter and the hyperparameter sets are θ=k2,k5 and α2=μ2,σ2,d,e, respectively, while epistemic uncertain parameter set is empty.Figure 9
Virtual experimental frequency samples for case study 2 (a) and corresponding samples of spring stiffness (b).As mentioned above, the advantage of Bayesian parameter identification lies in the full use of prior information, which can greatly reduce the demand for experimental sample size. Three forms of priors for hyperparameters are provided including uniform distribution, flat distribution, and sharp distribution, together with the comparison of their posterior PDFs, as shown in Figure10. Similarly, the comparison for the aleatory uncertain parameters is shown in Figure 11. Identification errors are given in Table 3. 10 sets of experimental samples are provided for VEB identification herein.Figure 10
Comparison of the posterior PDFs for each hyperparameter according to different priors.Figure 11
Identification results of PDFs for spring stiffness according to different priors.Table 3
Identification errors of the hyperparameters for aleatory uncertain variables.
Subcaseμ2 (%)σ2 (%)d (%)e (%)Different priorsUniform−1.208.93−2.7613.48Flat−1.161.37−2.521.86Sharp−0.610.19−1.18−0.22Different sample sizesn = 10−1.161.37−2.521.86n = 5−1.212.27−12.662.00n = 3−1.24−2.72−9.012.38The effect of different test sample sizes on identification results is also studied with flat prior distribution for the hyperparameters. The comparison is shown in Figure12 and Table 3.Figure 12
Influence comparison of experimental data amount on identification of aleatory uncertain parameter.As the form of prior PDFs for the hyperparameters changes from “flat” to “sharp” (“uniform” is the flattest one), the epistemic uncertainty of prior information is essentially reduced. Thus, the posterior PDFs for the hyperparameters become sharper and the highest probability estimation is closer to true value as shown in Figure10 and Table 3. The probability distribution of the model parameter becomes closer to the true one as shown in Figure 11. Though the identification results become more accurate as the increase of the experimental sample size as shown in Figure 12, the influence of prior distribution seems much more obvious. It also reminds us that prior distribution should be carefully assigned. Improper sharp prior distributions may mislead the identification results.It should be noted that the standard deviation of measurement noise is not identified in this case study. The author insists that measurement error should be obtained through instrument calibration as a known parameter rather than identified together with other parameters. In fact, it could confuse the identification results of aleatory uncertain parameters.


### 4.4. Case Study 3: Identification of Both Nonlinear and Linear Stiffness Together with Its Uncertainty

Both the nonlinear stiffness (k4) and linear stiffness (k2 and k5) are identified in this case study as well as their uncertainty. The constant term c expressed in (20) is set as the normal distribution to address the aleatory uncertainty of nonlinear stiffness k4. Virtual experimental samples for frequencies are generated in 20 groups under 6 different excitation magnitudes. The 20 groups of samples imply the aleatory uncertainty of parameters. The boxplots of test data and corresponding samples for aleatory uncertain parameters are shown in Figure 13. In this case study, the aleatory and epistemic uncertain parameter sets are θ=k2,k5,c and α=a,b, respectively, while hyperparameter set is α2=μ2,σ2,d,e,μc,σc.Figure 13
Boxplots of virtual frequency data for case study 3 and the projective plots of the corresponding 20 samples for aleatory uncertain parameters.Adopting uniform prior distributions, the posterior PDFs of epistemic uncertain parameters and hyperparameters are shown in Figure14. Both the posterior mean and highest posterior density estimation are, respectively, listed as the identification results shown in Table 4. These two kinds of identification results show minor difference, while the latter is slightly more accurate. Since the former is more robust in general, the posterior mean is recommended as the optimal estimation in this paper.Figure 14
Posterior PDFs of the epistemic uncertain parameters and hyperparameters.Table 4
Identification errors of the epistemic uncertain parameters and hyperparameters.
ParameterTrue valuePosterior meanHighest posterior density estimationValueRelative error (%)ValueRelative error (%)a1514.86−0.9314.88−0.77b−250−248.74−0.51−248.79−0.48μ220002008.280.412007.790.39σ23033.9313.1031.906.34d22.2211.142.073.30e2021.055.2420.693.43μc20002003.440.172003.750.19σc4039.36−1.6038.00−4.99According to every set of possible values for hyperparameters, plausible PDFs of spring stiffness are obtained as plotted with grey lines in Figure15. According to the posterior mean of the hyperparameter, the optimal PDFs of spring stiffness are obtained as plotted with black lines to compare with true distribution plotted with green dotted lines in Figure 15. Furthermore, the expectation and 95% confidence interval (CI) for k4 in specified input could also be gained from the corresponding optimal PDF. The optimal expectation and bounds of 95% CI for k4 varying with excitation magnitude are plotted with red solid and blue dashed lines, respectively, in Figure 15.Figure 15
Identification results of PDFs for equivalent spring stiffness. Plausible PDFs are plotted with grey lines, and the optimal PDF is plotted with black lines.One of the important purposes of parameter identification is highlighted to improve the model accuracy and prediction capability. The natural frequencies of the system are predicted by the updated model as shown in Figure16. The expectations varying with excitation magnitude are plotted with blue solid lines. The 95% and 99% CIs are plotted with green and purple stripes. Moreover, the PDFs of frequencies at any specified excitation could be obtained as plotted with black dashed lines. It is demonstrated that shifting down of the 1st and 3rd frequencies are captured with the increase of excitation magnitude, and the uncertainties of the frequencies are also represented with PDFs or CIs for decision-making. By contrast, only optimal estimation plotted with red solid line in Figure 15 and expectation prediction plotted with blue solid line in Figure 16 could be accurately obtained in traditional Bayesian method.Figure 16
Prediction stripe with probability distribution for natural frequencies versus excitation magnitude.Furthermore, 100 sets of experimental samples for frequencies atx = 1, 3, 5 and 7, respectively, are generated and marked with red “+” in Figure 16 to validate the prediction stripes. The result shows that the 95% CI covers most of the validation points and the 99% CI covers almost all of them. It is indicated that the model updated by the proposed method has excellent predictive capability.


## 4.1. Example Statement

A 3-DOF system with 4 linear springs and a nonlinear spring is oscillated under base excitation as shown in Figure4. The virtual modal test of the system is accomplished, respectively, under random excitation with 6 acceleration levels, including 0.1 g, 1 g, 2 g, 4 g, 6 g, and 8 g. Since the stiffness of the nonlinear spring k4 varies with the vibration magnitudes, the FRF shifts as the excitation acceleration increases as shown in Figure 5.Figure 4
A 3-DOF oscillating system with linear and nonlinear springs.Figure 5
Nonlinear feature of the 3-DOF system: (a) spring stiffnessk4 and (b) FRF vary with excitation magnitude.
(a)(b)In order to verify the effectiveness and practicability of the proposed method, three cases were studied as shown in Table1. The mass of each block is 0.1 kg, while stiffness parameters k1 and k3 are equal to 1000 N/m and 2000 N/m, respectively, for all cases. The outputs are 3 natural frequencies, while controllable input variable is the acceleration magnitude of base excitation. The example focuses on the identification of the other spring stiffness parameters, that is, k2, k4, and k5.Table 1
Comparison of the example cases.
ItemCase 1Case 2Case 3Nonlinear/linearNonlinearLinearNonlinearUncertaintyEpistemicAleatoryEpistemic and aleatoryParameters (N/m)k1100010001000k22000Normal dist.Normal dist.k3200020002000k4ax2+bx+c2000ax2+bx+cc∼normal dist.k51000Gamma dist.Gamma dist.In case 1, the nonlinear stiffnessk4 is modelled with the quadratic polynomial, and the coefficients {a, b, c} need to be identified. The traditional Bayesian identification is adopted with the consideration of measurement noise, and the uncertainties of model parameter cannot be identified. In case 2, linear stiffness k2 and k5 with aleatory uncertainties are identified using VEB method. Symmetrical and asymmetrical PDFs are set for k2 and k5, respectively, that is, normal and gamma distributions. In case 3, both nonlinear stiffness k4 and linear stiffness k2 and k5 are identified together with their uncertainty using VEB method. Specifically, the constant term c of the polynomial model for nonlinear stiffness k4 is assumed as an aleatory uncertain parameter with normal distribution.The natural frequencies from virtual modal test are generated from (19), where f˜i and fi denote ith experimental and computational frequency and x denotes the base excitation magnitude. The measurement noise ε is normally distributed with zero mean and 0.02 Hz standard deviation:(19)f˜i=fix;k2,k4,k5+ε,i=1,2,3.


## 4.2. Case Study 1: Nonlinear Stiffness Identification with consideration of Measurement Noise

The quadratic polynomial model for the stiffness of nonlinear spring verse excitation magnitude is established as(20)k4=ax2+bx+c.The parameter set to be identified isα=a,b,c,σ in this case study, where σ is the standard deviation of the test error. Traditional Bayesian identification is employed to obtain the posterior PDFs for a,b,c,σ according to virtual experimental data at 6 excitation levels as shown in Figure 6. In fact, the second natural frequency is useless for nonlinearity identification, because the change is not obvious among different vibration levels. The uniform PDF in a wide range is taken as the prior distribution of the parameters.Figure 6
Boxplots of 50 sets of virtual experimental natural frequencies.
(a)(b)(c)The optimal identification values of the parameters are listed and compared with the true values in Table2. The posterior PDFs of the parameters are shown in Figure 7. The sequence of MCMC samples for parameter a are shown in Figure 8 as an example to demonstrate that the convergence and autocorrelation are satisfied.Table 2
Parameter identification results in case study 1.
ParameterTrue valueIdentification valuen = 1n = 5n = 50a1515.2521.68%15.1460.97%14.984−0.11%b−250−252.701.08%−251.690.68%−249.81−0.08%c20002002.90.15%2002.70.14%1999.7−0.02%σ0.020.01905−4.77%0.020040.20%0.020542.68%Figure 7
The posterior PDFs of the parameters according to experimental data with different sizes.
(a)(b)(c)(d)Figure 8
The sequence of MCMC samples for parametera to verify the convergence and autocorrelation.It is indicated that the Bayesian identification results are very close to corresponding true values. Moreover, the posterior PDF becomes more and more sharp and the optimal estimation is generally closer to the true value as the amount of experimental data increases as shown in Table2 and Figure 7. However, nothing is obtained for the aleatory uncertainty of model parameters. The epistemic uncertainties described in Figure 7 just come from the measurement noise and the identification error.


## 4.3. Case Study 2: Linear Stiffness Identification Together with Its Aleatory Uncertainties

Ignoring the nonlinear springk4, the linear model parameters, that is, spring stiffness k2 and k5, are set as the normal and gamma distribution, respectively, as shown in the following:(21)k2∼Nμ2,σ22,k5=1000+δ,δ∼Γd,e.Virtual experimental frequency samples and the corresponding samples of spring stiffness are scattered as shown in Figure9. In order to study the effect of the experimental data amount, 3, 5, and 10 sets of experimental samples are provided for VEB identification stage by stage. Because 1st frequency is barely affected by k2 and k5, the experimental samples of 2nd and 3rd frequencies are utilized for VEB identification. In this case study, the aleatory uncertain parameter and the hyperparameter sets are θ=k2,k5 and α2=μ2,σ2,d,e, respectively, while epistemic uncertain parameter set is empty.Figure 9
Virtual experimental frequency samples for case study 2 (a) and corresponding samples of spring stiffness (b).As mentioned above, the advantage of Bayesian parameter identification lies in the full use of prior information, which can greatly reduce the demand for experimental sample size. Three forms of priors for hyperparameters are provided including uniform distribution, flat distribution, and sharp distribution, together with the comparison of their posterior PDFs, as shown in Figure10. Similarly, the comparison for the aleatory uncertain parameters is shown in Figure 11. Identification errors are given in Table 3. 10 sets of experimental samples are provided for VEB identification herein.Figure 10
Comparison of the posterior PDFs for each hyperparameter according to different priors.Figure 11
Identification results of PDFs for spring stiffness according to different priors.Table 3
Identification errors of the hyperparameters for aleatory uncertain variables.
Subcaseμ2 (%)σ2 (%)d (%)e (%)Different priorsUniform−1.208.93−2.7613.48Flat−1.161.37−2.521.86Sharp−0.610.19−1.18−0.22Different sample sizesn = 10−1.161.37−2.521.86n = 5−1.212.27−12.662.00n = 3−1.24−2.72−9.012.38The effect of different test sample sizes on identification results is also studied with flat prior distribution for the hyperparameters. The comparison is shown in Figure12 and Table 3.Figure 12
Influence comparison of experimental data amount on identification of aleatory uncertain parameter.As the form of prior PDFs for the hyperparameters changes from “flat” to “sharp” (“uniform” is the flattest one), the epistemic uncertainty of prior information is essentially reduced. Thus, the posterior PDFs for the hyperparameters become sharper and the highest probability estimation is closer to true value as shown in Figure10 and Table 3. The probability distribution of the model parameter becomes closer to the true one as shown in Figure 11. Though the identification results become more accurate as the increase of the experimental sample size as shown in Figure 12, the influence of prior distribution seems much more obvious. It also reminds us that prior distribution should be carefully assigned. Improper sharp prior distributions may mislead the identification results.It should be noted that the standard deviation of measurement noise is not identified in this case study. The author insists that measurement error should be obtained through instrument calibration as a known parameter rather than identified together with other parameters. In fact, it could confuse the identification results of aleatory uncertain parameters.


## 4.4. Case Study 3: Identification of Both Nonlinear and Linear Stiffness Together with Its Uncertainty

Both the nonlinear stiffness (k4) and linear stiffness (k2 and k5) are identified in this case study as well as their uncertainty. The constant term c expressed in (20) is set as the normal distribution to address the aleatory uncertainty of nonlinear stiffness k4. Virtual experimental samples for frequencies are generated in 20 groups under 6 different excitation magnitudes. The 20 groups of samples imply the aleatory uncertainty of parameters. The boxplots of test data and corresponding samples for aleatory uncertain parameters are shown in Figure 13. In this case study, the aleatory and epistemic uncertain parameter sets are θ=k2,k5,c and α=a,b, respectively, while hyperparameter set is α2=μ2,σ2,d,e,μc,σc.Figure 13
Boxplots of virtual frequency data for case study 3 and the projective plots of the corresponding 20 samples for aleatory uncertain parameters.Adopting uniform prior distributions, the posterior PDFs of epistemic uncertain parameters and hyperparameters are shown in Figure14. Both the posterior mean and highest posterior density estimation are, respectively, listed as the identification results shown in Table 4. These two kinds of identification results show minor difference, while the latter is slightly more accurate. Since the former is more robust in general, the posterior mean is recommended as the optimal estimation in this paper.Figure 14
Posterior PDFs of the epistemic uncertain parameters and hyperparameters.Table 4
Identification errors of the epistemic uncertain parameters and hyperparameters.
ParameterTrue valuePosterior meanHighest posterior density estimationValueRelative error (%)ValueRelative error (%)a1514.86−0.9314.88−0.77b−250−248.74−0.51−248.79−0.48μ220002008.280.412007.790.39σ23033.9313.1031.906.34d22.2211.142.073.30e2021.055.2420.693.43μc20002003.440.172003.750.19σc4039.36−1.6038.00−4.99According to every set of possible values for hyperparameters, plausible PDFs of spring stiffness are obtained as plotted with grey lines in Figure15. According to the posterior mean of the hyperparameter, the optimal PDFs of spring stiffness are obtained as plotted with black lines to compare with true distribution plotted with green dotted lines in Figure 15. Furthermore, the expectation and 95% confidence interval (CI) for k4 in specified input could also be gained from the corresponding optimal PDF. The optimal expectation and bounds of 95% CI for k4 varying with excitation magnitude are plotted with red solid and blue dashed lines, respectively, in Figure 15.Figure 15
Identification results of PDFs for equivalent spring stiffness. Plausible PDFs are plotted with grey lines, and the optimal PDF is plotted with black lines.One of the important purposes of parameter identification is highlighted to improve the model accuracy and prediction capability. The natural frequencies of the system are predicted by the updated model as shown in Figure16. The expectations varying with excitation magnitude are plotted with blue solid lines. The 95% and 99% CIs are plotted with green and purple stripes. Moreover, the PDFs of frequencies at any specified excitation could be obtained as plotted with black dashed lines. It is demonstrated that shifting down of the 1st and 3rd frequencies are captured with the increase of excitation magnitude, and the uncertainties of the frequencies are also represented with PDFs or CIs for decision-making. By contrast, only optimal estimation plotted with red solid line in Figure 15 and expectation prediction plotted with blue solid line in Figure 16 could be accurately obtained in traditional Bayesian method.Figure 16
Prediction stripe with probability distribution for natural frequencies versus excitation magnitude.Furthermore, 100 sets of experimental samples for frequencies atx = 1, 3, 5 and 7, respectively, are generated and marked with red “+” in Figure 16 to validate the prediction stripes. The result shows that the 95% CI covers most of the validation points and the 99% CI covers almost all of them. It is indicated that the model updated by the proposed method has excellent predictive capability.


## 5. Engineering Example

The jointed structure suspended by elastics is randomly vibrated by a shaker at the bottom as shown in Figure17. The thick disk and the cylindrical mass block are connected by four rubber isolators to reduce the vibration of the block, which represents some important elements to be protected. The modal tests are performed using 10 groups of different rubber isolators under random force excitation with 5 different levels. The experimental samples of axial natural frequency are plotted in Figure 18. It is indicated that the frequency shifts down obviously with the increase of excitation force. Moreover, the aleatory uncertainty of the isolator stiffness due to the notable difference of the rubber material is also significant and should be quantified.Figure 17
Experimental setup of the jointed structure.Figure 18
The natural frequency results from modal test using 10 groups of different isolators under 5 different excitation magnitudes.A nonlinear probabilistic model of equivalent spring stiffnesske varying with excitation magnitude is established by first-order approximation, as shown in the following equation:(22)ke=pF+k0,k0∼Nμ0,σ02,where F is the effective force of random excitation, k0 is the underlying linear stiffness of the structure, and the parameter p denotes the gradient of equivalent stiffness with excitation. k0 is normal distributed with unknown hyperparameters μ0 and σ0, and p is the epistemic uncertain variable to be identified.The finite element model of the jointed structures is established, and the rubber isolators are simulated by spring element with equivalent stiffnesske. Polynomial response surface is also built up as the surrogate model for low computational cost. Thus, the model parameters, as well as their uncertainties, are identified by VEB method according to experimental frequency data. The posterior PDFs of uncertain model parameters are shown in Figure 19. Accordingly, the optimal estimations of the parameters are shown in Table 5.Figure 19
Identification results of PDFs for model parameters.Table 5
Identification results for the engineering example.
ParameterOptimal valuep (μm−1)−15.804μ0 (kN∙mm−1)3.6647σ0 (kN∙mm−1)0.1957The prediction results including the expectation and 95% CI of the structural natural frequency are shown in Figure20. It should be noted that the frequency expectation plotted with solid red line indicates the prediction of determinate nonlinear model, while the PDF at any specified force indicates the uncertainty prediction of the structural natural frequency.Figure 20
Prediction of the frequency with nonlinearity and uncertainty.Though the determinate nonlinear model could address the trend of frequency shift with effective force, the quantitative uncertainty of the frequency is unknown for decision-making. If the nonlinearity is not characterized in the model, the uncertainty range of the frequency would be very wide in order to cover all the test data. Thus, the uncertainty range may be useless for decision-making. However, combining the nonlinear prediction with the uncertainty, the 95% CI shows excellent capability of prediction as shown in Figure20 with light purple stripe. It completely covers the test samples and accurately describes the nonlinearity and uncertainty, which demonstrates significant practicability for engineering structures.


## 6. Conclusions

In order to address both the nonlinearity and uncertainties of jointed structures under random excitation, equivalent linearization and Bayesian identification method are used in this paper. A novel method of VEB identification is originally proposed to obtain the optimal estimation and the probability distribution of model parameters, especially for the parameters with aleatory uncertainty. The numerical example with 3 case studies is performed to illustrate and verify the proposed method. It is demonstrated that the model parameters identified by VEB method are very close to their true values, and they would be much closer as the supplement of more experimental data or more accurate priors. The engineering example of the jointed structure with rubber isolators is also carried out to show the practicability and prediction capability. The nonlinear stiffness varying with the inputs is well identified together with its uncertainty in terms of PDF. Moreover, the outputs are also accurately predicted with the uncertain stripes varying with the inputs. It is indicated that both the nonlinear feature and the uncertainties of the output are characterized to support decision-making in our method, which has significant practicability for analysis of engineering structures. The limitation of this method also shows that the convergence rate of MCMC would obviously slow down with the increasing of the number of model parameters. As a result, the identification should be avoided for high-dimensional (i.e., more than 20) parameters especially with aleatory uncertainties.



---

*Source: 2638995-2021-11-22.xml*