Non-random Vibration Analysis Based On Interval Process

Time-varying or dynamic uncertain parameters are widely encountered in practical engineering,such as dynamic loads applied on structures and material properties with deterioration.Traditionally,the stochastic process model is employed to quantify these uncertain parameters,and structural uncertainty analysis is performed using probability theory methods.The premise of using the stochastic process model is that a large number of testing samples are required to construct precise probability distribution information or other statistic characteristics of time-varying parameters at arbitrary time point.However,in many real-world engineering problems,it is often difficult or even impossible to obtain sufficient testing samples due to limitations in test conditions or cost.As a novel mathematical tool for quantifying time-varying uncertain parameters,interval process model uses upper and lower bounds to describe uncertainty of the time-varying parameters,hence it has good engineering applicability and low dependence on testing samples.Structural non-random vibration analysis method is a non-probabilistic analysis method developed by combining the interval process model with classical vibration theory.In the non-random vibration analysis,both the input excitation and the output response of a system are interval processes,so it has some distinct advantages over the random vibration analysis method,such as convenient for engineers to understand and easy to use.However,the theoretical researches on the interval process model and the non-random vibration analysis method are still in their preliminary stage,and there are a series of problems to be solved in theory and application.Therefore,this thesis makes some contributions on interval process and non-random vibration analysis,aiming to improve and complement the interval process theory and enhance the applicability and application range of non-random vibration analysis methods in practical engineering.In the aspect of interval process theory,the concepts of differential and integral of interval process are proposed and its characteristic parameters are deduced.The researches on non-random vibration analysis mainly include,deriving the analytical formulations of the dynamic response boundary functions of discrete linear structures and complex continuum structures subjected to uncertain excitations;investigating non-random vibration analysis of the nonlinear single-degree-of-freedom vibration system;and developing a sensitivity analysis method for the dynamic displacement response function of the discrete linear structures.The main work of this thesis is as follows:(1)The concepts of differentiation and integration of interval process are proposed,which complements and improves the interval process theory.Firstly,the interval limit operator is defined,and it can be proved that the interval limit operator is a linear operator,namely,it can exchange operation order with other linear operators.Based on the interval limit operator,the definition of continuity of interval process is given.Secondly,based on the limit and continuity of interval process,the definitions of the differential and integral of the interval process are given,and the characteristic functions of the differential and integral of the interval process are derived respectively.Finally,it is concluded that differentiating an interval process,one can obtain an interval process while integrating an interval process,one can obtain an interval variable.(2)A non-random vibration analysis method for discrete linear vibration systems is proposed,through which analytical formulations the dynamic response boundary functions of a system under uncertain excitations can be obtained.Based on the Duhamel's integral and the real modal superposition method,the relationship between the structural dynamic displacement response functions and the dynamic external excitations can be obtained.Introducing the interval process uncertainty into the external excitations,the dynamic displacement responses of the structure are also uncertain and can be described by interval process model.According to the interval process theory,the middle point function and the radius function of the dynamic displacement responses of the structure are deduced respectively,based on which the analytical formulations the upper and lower boundary functions of the dynamic displacement responses of the structure can be obtained.Based on the differential of interval process and the Leibniz's formula,the analytical formulations of dynamic boundary functions for the velocity response and acceleration response of the structure are derived.(3)Aiming at a large number of complex continuum structures existing in practical engineering,a non-random vibration analysis method based on Green's kernel function is developed to obtain the upper and lower boundary functions of the dynamic response of the structures.According to the amount of loads applied on a structure,the continuum structure problems can be divided into the single load problem and the multiple loads problem.Through the knowledge of structural dynamics,the relationship between structural dynamic responses and external dynamic loads is established.The Green's kernel functions from the load points to the response measurement points can be calculated for the structures using the finite element method.Based on the Green's kernel functions,the formulations of the upper and lower boundary functions of the dynamic responses of the structure are derived for the single load problem and the multiple loads problem,respectively.If the numerical calculation error introduced by using the finite element method is not considered,the proposed method can be regarded as an analytical solution method.For the convenience of use in practical engineering,the numerical solution methods of the upper and lower bounds for the dynamic response at any time point are also given.(4)Based on the interval K-L(Karhunen-Loeve)expansion,a non-random vibration analysis method for solving the dynamic displacement response boundary of the nonlinear single-degree-of-freedom system is proposed.Firstly,the displacement of the system at a certain time point can be expressed as a function of a series of standard uncorrelated interval variables through using the interval K-L expansion.Then,computing the lower and upper bounds of the displacement response of the system at the given time point can be converted to solving two optimization models.Finally,EGO(Efficient Global Optimization)method is used to solve the above optimization models for obtaining the lower and upper bounds of displacement response of the system at the given time point,and further the dynamic displacement response boundary curves of the nonlinear single-degree-of-freedom system can be obtained.(5)A sensitivity analysis method for the dynamic response of structures in non-random vibration analysis.Firstly,a new formulation for the middle point function and radius function of dynamic displacement response of a structure is introduced.Then,using the chain derivation rule,the sensitivities of the middle point function and the radius function of the structural response with respect to the design variables are converted into the partial derivatives of the mass matrix and the stiffness matrix with respect to the design variables.By using the numerical difference method,the partial derivatives of the mass matrix and stiffness matrix with respect to the design variables at the nominal value is approximated.Since the partial derivatives of the mass matrix and stiffness matrix with respect to the design variables are obtained by the numerical method,the proposed sensitivity analysis method is a semi-analytic analysis method.