Multi-parameter Perturbation Method And Application To Nonlinear Structural Problems

Nonlinear structural problems are a kind of problems often encountered in engineering technology.However,it is difficult to obtain exact analytical solutions in most cases due to the analytical difficulties of the nonlinear differential equations.Therefore,some approximation methods are usually required to seek its approximate solution.Perturbation method is an important approximate analytical method that has been widely used in many fields of nonlinear structural analysis.The existing perturbation method is the method to selecting single perturbation parameter to participate in perturbation solution,which can effectively solve simple or weak nonlinear problems,but it is difficult to deal with complex or strong nonlinear problems.For complex nonlinear problems,some scholars try to select two parameters to participate in the perturbation solution,and achieve good results.The extension of the concept of two-parameter perturbation naturally involves the problem of needing to set more parameters.However,only the two-parameter perturbation situation is involved in the existing literature,and the three parameters and more parameters how to perturbation solution has not been determined.Therefore,the method of selecting multiple parameters to participate in the perturbation solution(i.e.,the multi-parameter perturbation method)can not be used as a general mathematical method.This paper focuses on several key issues in multi-parameter perturbation method,including in which case should be set multiple perturbation parameters,how to choose multiple parameters reasonably and how do multiple parameters expand.Then,it forms a systematic multi-parameter perturbation method,which provides an effective way for solving complex nonlinear problems.Further,the multi-parameter perturbation method is applied to some typical nonlinear structural problems,including the bending problems of circular plate with variable thickness under nonuniformly distributed load and the static and dynamic problems of piezoelectric cantilever beam.On the one hand,testing the validity and applicability of multi-parameter perturbation methods,on the other hand,provide an analytical method for studies of similar problems.The main studies of this paper are as follows:First,based on the existing parameter perturbation method,the function and significance of the perturbation parameters are analyzed,and a selection criterion for the perturbation parameters is proposed.Combined with the classification of perturbation problems,the selection problem of perturbation parameters and the parameter size problem are studied,and a set of specific selection methods for perturbation parameters are presented.Based on the classical expansion of one parameter and different combinations of multiple parameters,four expansions of multiple parameters are given.And a reasonable expansion formula for one parameter and multiple parameters is determined by comparative analysis.Based on the definition of the zero-order term and the basic properties of the asymptotic sequence,the convergence of the multi-parameter perturbation expansion is analyzed.Based on the perturbation parameter selection method and the multi-parameter perturbation expansion general formula,a systematic multi-parameter perturbation method is finally formed.Second,the application of multi-parameter perturbation method in the small and large deflection problems of circular plate with variable thickness under nonuniformly distributed load is studied.According to the plate theory of reissner and the classical elasticity theory,the small deflection equation and the large deflection equation of the circular plate with variable thickness under nonuniformly distributed load are established respectively.And the perturbation solutions of the small and large deflection equations are also solved by using multi-parameter perturbation method.The validity of the obtained perturbation solutions is verified by comparing with existing studies and numerical results.Based on the obtained perturbation solutions,the influence of variable thickness coefficient,load inequality coefficient and thickness span ratio on the displacement response of circular plate are discussed respectively,and the applicability of the classical "Chien’s method" to the small and large deflection problems is investigated.Third,the application of multi-parameter perturbation method in the static and dynamic problems of piezoelectric cantilever beam is studied.According to the stress function method,the governing equations for the bending problem of the piezoelectric cantilever beam under combined load are established.And the perturbation solutions of the governing equations are obtained by taking the piezoelectric coefficients as the perturbation parameters.According to the one-dimensional solution of the piezoelectric cantilever beem,the piezoelectric equivalent elastic modulus is obtained.And then,the vibration equation of the piezoelectric cantilever beem is derived based on the vibration equation of the general cantilever.The vibration type and frequency of the piezoelectric cantilever beem are obtained by taking the piezoelectric coefficient and damping coefficient as the perturbation parameters.The validity of the obtained perturbation solutions is verified by experimental and numerical results.Based on the obtained perturbation solutions,the coupling relationship between the force and electric functions,the influence of the piezoelectric performance on the structural response and each piezoelectric coefficient on stress,displacement and electric displacement are discussed respectively.Fourth,the application of multi-parameter perturbation method in the bending problem of functional gradient cantilever beem and functional gradient piezoelectric rectangular plate is studied.According to the classical elasticity theory and the stress function method,the governing equations of the bending problem of functional gradient cantilever beem and the functional gradient piezoelectric rectangular plate are established.And the perturbation solutions of those governing equations are obtained by the multi-parameter perturbation method.The validity of the obtained perturbation solutions is verified by numerical results.Based on the obtained perturbation solutions,the influence of functional gradient on stress and displacement response and the simplification of the perturbation expansion forms of functional gradient piezoelectric materials are discussed.