Dynamic Response Analysis Of Time-varying Uncertain Structure Based On Orthogonal Polynomial Expansion

Time-varying uncertain structures exist widely in vehicles,ships,submarines,aircraft and aerospace and other delivery vehicles.Due to manufacturing and assembly errors,material degradation,unpredictable external excitation and other factors,the dynamic response of the structure has time-varying and dynamic uncertainties.Traditional dynamic structural analysis and optimization methods are generally based on determined system parameters and solved by the CAE method.However,in many practical engineering problems,due to coupling effects among several small uncertain parameters,these small uncertainties may result in a large deviation of the dynamic response of the structure.Therefore,when the dynamic response analysis of the structure is performed,the influence of time-varying uncertain parameters needs to be considered.The commonly used time-varying uncertain models include the random process model,the interval process model,and the random and interval hybrid process uncertainty model.Each of the three time-varying uncertain models has its own engineering application value.Therefore,this paper starts the interval process model and the random process model and gradually deepens into the random and interval hybrid process model.Based on these nondeterministic models,this dissertation conducted a systematical research for the polynomial expansion algorithm for time-varying uncertain structures.The main research work is as follows:(1)Chebyshev analysis method for structural response under interval process modelBased on dynamic equations of motion,a finite element model of a dynamic structural system was established.The interval process model was introduced to describe the independent uncertain parameters in the system.The Karhunen-Loeve(KL)was used to describe the correlation of the time process.Two numerical methods,namely the Interval Perturbation Method based on Karhunen-Loeve Expansion was proposed(IPM-KLE)and the Interval Chebyshev polynomial expansion based on the Karhunen-Loeve Expansion(ICM-KLE),were proposed.For the multi-degree-of-freedom linear vibration system and the shell structure system,IPM-KLE and ICM-KLE are used to calculate their upper and lower bounds of the dynamic response.Comparing with the reference solution yielded by the Monte-Carlo method,the accuracy of ICM-KLE is higher than IPM-KLE.(2)Gegenbauer analysis method for structural response of bounded random process modelFirstly,a bounded random process model was established.The correlation of time-varying uncertain parameters in the time process is described by KL expansion.Then Bounded Random Gegenbauer Polynomial Expansion Method based on KL Expansion is proposed(BRGM-KLE).For the multi-degree-of-freedom linear vibration system,its expectation and variance of dynamic response are calculated by BRGM-KLE.The numerical results show that the numerical results produced by BRGM-KLE match well with the results obtained by the Monte-Carlo method.Thus,BRGM-KLE can effectively and efficiently predict the expectation and variance of dynamic response of the time-varying uncertain structure.(3)The Gegenbauer analysis method for structural response of bounded random and interval hybrid process modelA bounded random and interval hybrid process model was established.The KL expansion was used to describe the correlation of time-varying uncertain parameters in the time process.Bounded Random and Interval Gegenbauer Polynomial Expansion Method based on Karhunen-Loeve Expansion(BRAIGM-KLE)was proposed.For the multi-degree-of-freedom linear vibration system,its upper and lower bounds of the expectation and variance of the dynamic response are estimated by BRAIGM-KLE.The numerical results show that the results yielded by BRAIGM-KLE match well with the reference results produced by the Monte-Carlo method.Thus,BRAIGM-KLE can effectively and efficiently predict the variational range of the expectation and variance of the dynamic response of the the time-varying uncertain structure.This paper discusses three uncertainties models,namely the interval process model,the bounded random process model and the bounded random and interval hybrid process model.By combining with the uncertainty analysis method,the finite element method,the KL expansion theory,the first-order perturbation expansion theory,the Chebyshev Polynomial expansion method and the Gegenbauer polynomial expansion analysis method,an interval perturbation method based on Karhunen-Loeve expansion(IPM-KLE),an interval Chebyshev polynomial expansion based on the Karhunen-Loeve expansion(ICM-KLE),a bounded random Gegenbauer polynomial expansion method based on Karhunen-Loeve expansion(BRGM-KLE),and a bounded random and interval Gegenbauer polynomial expansion method based on Karhunen-Loeve expansion(BRAIGM-KLE)were proposed.The numerical examples of the shell structure and the multi-degree-of-freedom linear vibration system are used to verify the efficiency and effectiveness of the proposed numerical analysis methods.