Entropy propagation analysis in stochastic structural dynamics: application to a beam with uncertain cross sectional area

This paper investigates the impact of different probabilistic models of uncertain parameters on the response of a dynamical structure. The probabilistic models of the uncertain parameters are constructed using the maximum entropy principle, where different information is considered, such as bounds, mean value, etc. Nested probabilistic models are constructed with increasing information; as the information given increases, the level of entropy of the input model decreases. The response of the linear dynamical model is given in the frequency domain, and the propagation of the input uncertainty throughout the computational model is analyzed in terms of Shannon’s entropy. Low and high frequencies are analyzed because uncertainties propagate differently depending on the frequency band. A beam discretized by means of the finite element method with random cross sectional area (random field) is the application analyzed.     

Reliability based optimum design of Tuned Mass Damper in seismic vibration control of structures with bounded uncertain parameters

A reliability based optimization of Tuned Mass Damper (TMD) parameters in seismic vibration control under bounded uncertain system parameters is presented. The study on TMD with random parameters in a probabilistic framework is noteworthy. But, it cannot be applied when the necessary information about parameters uncertainties is limited. In such cases, the interval method is a viable alternative. Applying matrix perturbation theory through a first order Taylor series expansion about the mean values of the uncertain parameters’ conservative dynamic response bounds are obtained assuming a small degree of parameter uncertainty. The first-passage probability of failure of the system is taken as the performance objective. Using the interval extension of the performance objective, the vibration control problem under bounded uncertainties is transformed to the appropriate deterministic optimization problems yielding the lower and upper bound solutions. A numerical study is performed to elucidate the effect of parameters’ uncertainties on the TMD parameters’ optimization and the safety of the structure.     

Dynamic response of structures with uncertain parameters: A comparative study of probabilistic and fuzzy sets models

This paper presents the results of a study on the response of structures with uncertain properties such as mass, stiffness and damping. The effect of the uncertain parameters on the response and the effect of the modelling of the uncertainties on the response are investigated. In particular, two types of uncertainties are distinguished: random and fuzzy uncertainties. Two kinds of models are studied: probabilistic and fuzzy set models. The two approaches to uncertainty modelling are compared with regard to their impacts on the analysis and on the uncertain structural response obtained. The study considers free vibration, forced vibration with deterministic excitation, and forced vibration with Gaussian white noise excitation. It is concluded that, in general, fuzzy models are much easier to implement and the associated analysis easier to perform than their probabilistic counterparts. When the available data on the structural parameters are crude and do not support a rigorous probabilistic model, the fuzzy set approach should be considered in view of its simplicity.     

Free vibration and reliability of composite cantilevers featuring uncertain properties

The problem of free vibration and reliability of cantilever composite beams featuring structural uncertainties is analyzed. The random structural uncertainties involve material properties, thickness and fiber orientation of the individual constituent laminae. Such uncertainties undoubtedly affect the achievable performance as well as their structural reliabilities. In order to investigate the effects of random structural uncertainties on free vibration problem, a stochastic eigenvalue problem of self-adjoint systems is formulated to provide first and second moments of eigenvalues, i.e., their mean and variance. In this context, a stochastic finite element method based on the mean-centered-second-moment method and first-order perturbation technique are employed during the probabilistic discretization of uncertain distributed-parameter structural systems.Sensitivity and reliability analyses for the uncertain beam when subjected to an external oscillatory load are performed. In addition, in order to mitigate the detrimental effects of uncertainties and so, to render the structure more robust to such effects, the structural tailoring technique is implemented and its beneficial effects are revealed.     

Generalized Wishart distribution for probabilistic structural dynamics

An accurate and efficient uncertainty quantification of the dynamic response of complex structural systems is crucial for their design and analysis. Among the many approaches proposed, the random matrix approach has received significant attention over the past decade. In this paper two new random matrix models, namely (1) generalized scalar Wishart distribution and (2) generalized diagonal Wishart distribution have been proposed. The central aims behind the proposition of the new models are to (1) improve the accuracy of the statistical predictions, (2) simplify the analytical formulations and (3) improve computational efficiency. Identification of the parameters of the newly proposed random matrix models has been discussed. Closed-form expressions have been derived using rigorous analytical approaches. It is considered that the dynamical system is proportionally damped and the mass and stiffness properties of the system are random. The newly proposed approaches are compared with the existing Wishart random matrix model using numerical case studies. Results from the random matrix approaches have been validated using an experiment on a vibrating plate with randomly attached spring-mass oscillators. One hundred nominally identical samples have been created and separately tested within a laboratory framework. Relative merits and demerits of different random matrix formulations are discussed and based on the numerical and experimental studies the recommendation for the best model has been given. A simple step-by-step method for implementing the new computational approach in conjunction with general purpose finite element software has been outlined.     

Reliability analysis of structures based on a probability‐uncertainty hybrid model

The traditional reliability analysis method based on probabilistic method requires probability distributions of all the uncertain parameters. However, in practical applications, the distributions of some parameters may not be precisely known due to the lack of sufficient sample data. The probabilistic theory cannot directly measure the reliability of structures with epistemic uncertainty, ie, subjective randomness and fuzziness. Hence, a hybrid reliability analysis (HRA) problem will be caused when the aleatory and epistemic uncertainties coexist in a structure. In this paper, by combining the probability theory and the uncertainty theory into a chance theory, a probability‐uncertainty hybrid model is established, and a new quantification method based on the uncertain random variables for the structural reliability is presented in order to simultaneously satisfy the duality of random variables and the subadditivity of uncertain variables; then, a reliability index is explored based on the chance expected value and variance. Besides, the formulas of the chance theory‐based reliability and reliability index are derived to uniformly assess the reliability of structures under the hybrid aleatory and epistemic uncertainties. The numerical experiments illustrate the validity of the proposed method, and the results of the proposed method can provide a more accurate assessment of the structural system under the mixed uncertainties than the ones obtained separately from the probability theory and the uncertainty theory.     

质量偏心Timoshenko梁的振动波特性研究

在研究船舶、潜艇等工程结构的低频振动时,通常可以将其简化为质量在截面内分布非均匀的梁结构,此质量偏心会引起弯-纵耦合。针对弯-纵耦合的质量偏心Timoshenko梁,推导了其截止频率的解析表达式;探讨了质量偏心对其纵振波、传播弯曲波及衰减弯曲波波数的影响规律;研究了三组波数下纵向/弯曲位移比随频率及质量偏心的变化。分析结果表明,质量偏心会降低梁的截止频率,偏心率越大,降低越明显;弯曲衰减波会在截止频率处转变为弯曲传播波;质量偏心使得非频散的纵向振动波转变为频散波;纵向振动与弯曲振动的耦合在质量偏心率或频率增大时,会进一步加强。     

Modeling of fatigue crack propagation using dual boundary element method and Gaussian Monte Carlo method

This paper studies the modeling of fatigue crack propagation on a multiple crack site of a finite plate using deterministic and probabilistic methods. Stress intensity factor has been calculated by the combined deterministic approach of the dual boundary element method (DBEM) and the probabilistic approach of the Gaussian Monte Carlo method. The Gaussian Monte Carlo method has been incorporated to simulate the random process of the fatigue crack propagation. A finite plate of aluminum alloy 2024-T3 with a thickness of 1.6 mm and 14 holes is analyzed and the fatigue life of the plate is predicted by following a linear elastic law of fracture mechanics. The results of fatigue life predicted by DBEM-Monte Carlo method are in good agreement with experimental ones. The same approach is also applied to two other engineering applications of a gear tooth and a bracket.     

基于频响函数综合的浮筏隔振系统误差传递分析

采用基于频响函数综合的子结构方法对浮筏隔振系统进行建模分析和误差传递分析。针对测试频响函数的一些常见误差,如测试噪声对频响函数幅值和相位的污染,同一子结构的频响函数不满足互易性,原点频响函数的虚部小于零等,分析其在综合时由于误差传递和放大效应对综合结果的影响。并采用矩方法对误差传递进行定量分析,将矩方法的结果和采用Monte Carlo方法的结果进行对比,结果表明矩方法具有较高的效率和精度。     

Probabilistic learning for modeling and quantifying model-form uncertainties in nonlinear computational mechanics

Recently, a novel nonparametric probabilistic method for modeling and quantifying model-form uncertainties in nonlinear computational mechanics was proposed. Its potential was demonstrated through several uncertainty quantification (UQ) applications in vibration analysis and nonlinear computational structural dynamics. This method, which relies on projection-based model order reduction to achieve computational feasibility, exhibits a vector-valued hyperparameter in the probability model of the random reduced-order basis and associated stochastic projection-based reduced-order model. It identifies this hyperparameter by formulating a statistical inverse problem, grounded in target quantities of interest, and solving the corresponding nonconvex optimization problem. For many practical applications, however, this identification approach is computationally intensive. For this reason, this paper presents a faster predictor-corrector approach for determining the appropriate value of the vector-valued hyperparameter that is based on a probabilistic learning on manifolds. It also demonstrates the computational advantages of this alternative identification approach through the UQ of two three-dimensional nonlinear structural dynamics problems associated with two different configurations of a microelectromechanical systems device.     

Low-velocity heavy-mass impact response of slender metal foam core sandwich beam

The objective of this work is to investigate the dynamic large deflection response of fully clamped metal foam core sandwich beam struck by a low-velocity heavy mass. Analytical solution and ‘bounds’ of dynamic solutions are derived, respectively. Also, finite element analysis is carried out to obtain the numerical solution of the problem. Comparisons of the dynamic, the quasi-static and numerical solutions for the non-dimensional maximum deflection of the sandwich beam with non-dimensional initial kinetic energy of the striker are presented for different cases of mass ratio, impact velocity and location. It is seen that the dynamic solution approaches the quasi-static one as the mass ratio of the striker to the beam is large enough, the quasi-static solution is in good agreement with the numerical results and both solutions lie in the ‘bounds’ of dynamic solutions. The quasi-static and numerical results for the impact force against the maximum deflection of the sandwich beam are obtained. It shows that the quasi-static solution can offer adequate accuracy to predict the low-velocity heavy-mass impact response of fully clamped sandwich beam.     

基于动力吸振器的结构声传递抑制

研究了无限长梁上动力吸振器抑制弯曲振动波传递的机理,分别推导了一个和两个动力吸振器下无限长梁上透射系数的计算公式,讨论了动力吸振器质量、阻尼及个数对结构声传递抑制的影响.数值计算结果表明,动力吸振器的阻抑效果具有较强的频率选择性.其质量的增加对阻抑效果有利;阻尼增加虽降低了最佳的阻抑效果,但抑制的频带宽了,改善了阻抑效果的频率选择性;在总质量相同的条件下,多个动力吸振器的阻抑效果较单个的阻抑效果好.     

High-Fidelity Computational Optimization for 3-D Flexible Wings: Part II—Effect of Random Geometric Uncertainty on Design

The effect of geometric uncertainty due to statistically independent, random, normally distributed shape parameters is demonstrated in the computational design of a 3-D flexible wing. A first-order second-moment statistical approximation method is used to propagate the assumed input uncertainty through coupled Euler CFD aerodynamic/finite element structural codes for both analysis and sensitivity analysis. First-order sensitivity derivatives obtained by automatic differentiation are used in the input uncertainty propagation. These propagated uncertainties are then used to perform a robust design of a simple 3-D flexible wing at supercritical flow conditions. The effect of the random input uncertainties is shown by comparison with conventional deterministic design results. Sample results are shown for wing planform, airfoil section, and structural sizing variables.     

Full-field sensitivity analysis through dimension reduction and probabilistic surrogate models

Computational mechanics models often are compromised by uncertainty in their governing parameters, especially when the operating environment is incompletely known. Computational sensitivity analysis of a spatially distributed process to its governing parameters therefore is an essential, but often costly, step in uncertainty quantification. A sensitivity analysis method is described which features probabilistic surrogate models developed through equitable sampling of the parameter space, proper orthogonal decomposition (POD) for compact representations of the process’ variability from an ensemble of realizations, and cluster-weighted models of the joint probability density function of each POD coefficient and the governing parameters. Full-field sensitivities, i.e. sensitivities at every point in the computational grid, are computed by analytically differentiating the conditional expected value function of each POD coefficient and projecting the sensitivities onto the POD basis. Statistics of the full-field sensitivities are estimated by sampling the surrogate model throughout the parameter space. Major benefits of this method are: (1) the sensitivities are computed analytically and efficiently from regularized surrogate models, and (2) the conditional variances of the surrogate models may be used to estimate the statistical uncertainty in the sensitivities, which provides a basis for pursuing more data to improve the model. Synthetic examples and a physical example involving near-ground sound propagation through a refracting atmosphere are presented to illustrate the properties of the surrogate models and how full-field sensitivities and their uncertainties are computed.     

A critical reliability evaluation of fibre reinforced composite materials based on probabilistic micro and macro-mechanical analysis

In probabilistic composite mechanics, uncertainty modelling may be introduced at a constituent (micro-scale), ply (meso-scale) or component (macro-scale) level. Each of these approaches has particular advantages/limitations and appropriate fusing and benchmarking is desirable in order to improve confidence in probabilistic performance estimates of composite structures. In the present study, random variable based micro and macro-scale reliability analyses are critically compared through a limit state formulation based on the analytical stress tensor components of a rectangular simply supported orthotropic FRP composite plate and the Tsai–Hill failure criterion. The study aims to promote cross-fertilisation of alternative uncertainty modelling approaches in a multi-scale analysis framework. Propagation of uncertainty from micro to macro-scale, and the corresponding influence of changes in random variability on the reliability estimates is quantified. The importance of benchmarking experimentally-based probability distributions of mechanical properties through micro-scale modelling is illustrated, and the confidence that can be placed on reliability estimates is quantified through a series of numerical examples.     

Stochastic nonlinear free vibration analysis of elastically supported functionally graded materials plate with system randomness in thermal environment

This paper presents the stochastic nonlinear free vibration response of elastically supported functionally graded materials (FGMs) plate resting on two parameter Pasternak foundation having Winkler cubic nonlinearity with random system properties subjected to uniform and nonuniform temperature changes with temperature independent (TID) and dependent (TD) material properties. System properties such as material properties of each constituent’s material, volume fraction index and foundation parameters are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strains using modified C⁰ continuity. A direct iterative based nonlinear finite element method in conjunction with first order perturbation technique (FOPT) developed by last two authors for the composite plate is extended for FGM plate to compute the second order statistics (mean and coefficient of variation) of the nonlinear fundamental frequency. The present outlined approach has been validated with those results available in the literature and independent Monte Carlo simulation (MCS).     

Quantification of uncertainty modelling in stochastic analysis of FRP composites

The extensive use of FRP composite materials in a wide range of industries, and their inherent variability, has prompted many researchers to assess their performance from a probabilistic perspective. This paper attempts to quantify the uncertainty in FRP composites and to summarise the different stochastic modelling approaches suggested in the literature. Researchers have considered uncertainties starting at a constituent (fibre/matrix) level, at the ply level or at a coupon or component level. The constituent based approach could be further classified as a random variable based stochastic computational mechanics approach (whose usage is comparatively limited due to complex test data requirements and possible uncertainty propagation errors) and the more widely used morphology based random composite modelling which has been recommended for exploring local damage and failure characteristics. The ply level analysis using either stiffness/strength or fracture mechanics based models is suggested when the ply characteristics influence the composite properties significantly, or as a way to check the propagation of uncertainties across length scales. On the other hand, a coupon or component level based uncertainty modelling is suggested when global response characteristics govern the design objectives. Though relatively unexplored, appropriate cross-fertilisation between these approaches in a multi-scale modelling framework seems to be a promising avenue for stochastic analysis of composite structures. It is hoped that this review paper could facilitate and strengthen this process.     

楔形阻尼层复合梁式吸振器设计及实验研究

针对弹性薄板结构在固有模态频率处振动过大的情况,对其进行了吸振控制的理论及实验研究。以典型简支板为控制对象,运用有限元方法设计了变阻尼层复合梁式吸振器并分析了其吸振效果,进一步通过实验对所获结论进行了验证,数值分析和实验测试结果一致表明：随着长度增加,用于吸振模态个数增加,吸振频带变宽,但每个频率上的吸振效果下降;对于板振动系统,具有较好的吸振效果的安装区域为激振点附近区域;复合梁用于吸振的模态阶数越低,吸振量越大。     

Data-driven uncertainty quantification and propagation in structural dynamics through a hierarchical Bayesian framework

In the presence of modeling errors, the mainstream Bayesian methods seldom give a realistic account of uncertainties as they commonly underestimate the inherent variability of parameters. This problem is not due to any misconceptions in the Bayesian framework since it is robust with respect to the modeling assumptions and the observed data. Rather, this issue has deep roots in users’ inability to develop an appropriate class of probabilistic models. This paper bridges this significant gap, introducing a novel Bayesian hierarchical setting, which breaks time-history vibration responses into several segments so as to capture and identify the variability of inferred parameters over the segments. Since the computation of the posterior distributions in hierarchical models is expensive and cumbersome, novel marginalization strategies, asymptotic approximations, and maximum a posteriori estimations are proposed and outlined in a computational algorithm aiming to handle both uncertainty quantification and propagation. For the first time, the connection between the ensemble covariance matrix and hyper distribution parameters is characterized through approximate estimations. Experimental and numerical examples are employed to illustrate the efficacy and efficiency of the proposed method. It is observed that, when the segments correspond to various system operating conditions and input characteristics, the proposed method delivers robust parametric uncertainties with respect to unknown phenomena such as ambient conditions, input characteristics, and environmental factors.     

A study on importance measures and a quantification algorithm in a fire PRA model

In the current quantification of fire probabilistic risk assessment (PRA), when components are damaged by a fire, the basic event values of the components become ‘true’ or one (1), which removes the basic events related to the components from the minimal cut sets, and which makes it difficult to calculate accurate component importance measures. A new method to accurately calculate an importance measure such as Fussell-Vesely in fire PRA is introduced in this paper. Also, a new quantification algorithm in the fire PRA model is proposed to support the new calculation method of the importance measures. The effectiveness of the new method in finding the importance measures is illustrated with an example of evaluating cables’ importance.