Random Vibration Analysis Of Structures Based On Probability Transformation Method And Probability Density Evolution Theory

Material parameters,geometric parameters of engineering structures and external excitation present randomness.Accordingly,the efficient and accurate analysis of stochastic dynamic responses of structure has important theoretical and practical significance.Probabilistic transformation method(PTM)and probability density evolution method(PDEM)are two emerging methods for random vibration analysis of structures,and their engineering applicability and algorithm performance need to be further studied.In this thesis,PTM is extended to the random vibration analysis of linear structures.Then,the algorithm performance of PDEM is scrutinized,and is applied to the analysis of dynamic responses and reliability assessment of nonlinear structures and continuous beam.The main research contents are described as follows.The present paper calculates the PDF(probability density function)and CF(characteristic function)of the stochastic seismic dynamic response of linear elastic systems,and obtains the central moments of the response by PTM.Meanwhile,the formula of PTM is re-derived and revised and its main idea is revealed in this paper.On the basis of the preservation of probability,PTM establishes the direct relationship between the excitation CF and the response CF,with using Dirac delta function and Fourier transform relationship between PDF and CF.Given that the random excitation's CF is known,the stochastic dynamic response's CF of linear systems can be solved.Then,the corresponding PDF is easily known by inverse Fourier transform.Since CF and statistical moments have simple differential relationship,the central moments of dynamic responses can be obtained directly.Finally,numerical examples illustrate that PTM has extremely high accuracy and computational efficiency for random vibration analysis of frame structures with many degrees of freedom under non-stationary seismic excitation.Then,PDEM is used for the analysis of stochastic structure dynamic responses and dynamic reliability under random excitation.By means of random event description of the principle of probability preservation and the Dirac-delta function,the generalized probability density evolution equation(GDEE),which is core formula of PDEM,can be established.The PDF of structural responses can be obtained by the solution of GDEE with finite difference method.Meanwhile,the dynamic reliability can be achieved when the absorbing boundary condition is imposed on GDEE.The differential step size,responses interval length and responses step number in TVD schemes of finite difference method to affect the reliability calculation results of GDEE are examined in detail.Numerical examples of nonlinear frame structures and continuous beam indicate that,not only their PDF and dynamic reliability are accurate sufficiently,but also PDEM can greatly reduce the amount of calculation.Finally,the PTM and PDEM are compared,and the relationship and difference between the basic ideas of two methods are demonstrated.