Dynamic Analysis, Dynamic Reliability And Optimization Design Of Mechanism With Uncertain Parameters

Mechanisms with random parameters or interval parameters is considered in this paper. The system dynamic characteristic, dynamic response and dynamic reliability are investigated in the scenario that physical parameters of system, geometric dimensions of components and applied loads are all random or interval variables. The main research works can be described as follows:1. Dynamic characteristic analysis for torsional vibration of a gear-rotor system with random parameters and dynamic characteristics analysis of elastic linkage mechanism with interval parameters.Firstly, based on the generalized random factor method, the time-variable natural frequencies were analyzed for the torsional vibration of a gear-rotor system with random physical and geometrical parameters. The stiffness and mass. matrices of the system were discomposed into a sum of matrices with same random factors, respectively. Based on the Rayleigh quotient formula, the natural frequencies of this system were represented as a sum of partial ones. And then the mathematic characteristics expressions of the natural frequencies were obtained by utilizing the algebra synthesis method for solving a random function. Finally, numerical examples showed that the influence of the randomness of the physical and geometrical parameters on the natural frequencies, and verified the feasibility of the proposed method. Secondly, based on the interval factor method, the analysis of system dynamic characteristic for interval elastic linkage mechanism is presented. The stiff and mass matrices of the system are discomposed into a sum of matrices with same interval factors, respectively. By using interval factor method, all of the interval parameters can be expressed as the products of two parts corresponding to the interval factors and deterministic value. Based on the Rayleigh's quotient formula, the computational expressions for the lower, upper bonds and mean value of system dynamic characteristic are developed using the interval operations. The effects of the uncertainty of the bar length and radius, mass density, and elastic modulus on mechanism dynamic characteristic are studied through an example.2. Dynamic response of gear with random parameters and torsional vibration of Gear-Rotor with random parametersFirstly, the dynamic response of gear based on probability is investigated. Considering the randomness of the system physical parameters, and the geometric dimensions parameters, as well as the randomness of the amplitude peak of applied load and the time-varying stiffness of gear pairs, the numerical characteristics of the system dynamic response are formulated from the dynamic expression of Duhamel integral by the random factor method. The influences of the randomness of physical parameters and geometric dimensions parameters on the randomness of the mean square value of system displacement are inspected and some significant conclusions are obtained as follows:the randomness of geometric parameters have greater effected on displacement response of system and the time-varying stiffness of system have impacted on the response of system through the example. Secondly, the vibration equations of torsional vibration of Gear-Rotor with random physical parameters and geometrical parameters under random excitation were established. Dynamic equations with random parameters and time-varying stiffness were reformulated as static governing equations by using the Newmark-βstep-by-step integration method. Considering system parameters and force as inhomogeneous random parameters, the mean value and the variance of dynamic displacement response were calculated by moment method for solving the characteristic of a function with random variable. It can be shown from the numerical examples that the time-varying stiffness of system have impacted on the response of system, the influences of the randomness of physical parameters, geometrical parameters and the amplitude of loads on system dynamic response can not be ignored, and the randomness of geometric parameters have greater effected on displacement response of system.3. Nonlinear dynamic analysis of response of gear-rotor with random parameters and dynamic response optimization for gear-rotor with random parameters based on reliability.Firstly, the nonlinear dynamical model of gear-rotor is established, where the random physical and geometrical parameters, the backlash, the time-varying stiffness, and the friction between teeth and the static transmission error are all included. Nonlinear dynamic equations are transformed into static governing equations by exploiting the Newmark-βstep-by-step integration method. The mean value and the variance of dynamic displacement response are calculated by algebra synthesis method and moment method for obtaining the numerical characteristics of a function with random variables. The influences of the randomness of physical parameters, geometrical parameters, the backlash and the friction coefficient on system dynamic response are studied through an example and some useful conclusions are obtained. Secondly, the randomness of the physical parameters, geometrical parameters and the stochastic loads applied Gear-Rotor with random parameters is considered. In this case, the various parameters of the system are regarded as the design variables, the root of stochastic vibration acceleration minimum is considered as the objective function with the reliability constraints of dynamic stress between teeth of gear and bearing torque and gear static constraint. Genetic algorithm is applied, where constraints based on reliability probability are converted to equivalent constraints based on corresponding numerical characteristics. Example shows that the randomness of the parameters of system can not be neglected.4. Dynamic response of rigid-elastic coupling linkage mechanism with uncertain parameters.The vibration equations of rigid-elastic coupling linkage mechanism with random or interval physical and geometrical parameters are established by incorporating random or interval excitation into the equations. Dynamic equations with random or interval parameters are transformed into static governing equations by the Newmark-βstep-by-step integration method. The mean value and variance or the upper and lower limit of dynamic displacement response are calculated by algebra synthesis method.and moment method or interval algorithm. The influences of the uncertainty of the bar length and radius, mass density, and elastic modulus on mechanism dynamic response are studied through an example and some useful conclusions are obtained.5. Dynamic reliability analysis of gear with stochastic parameters under stationary random excitation and dynamic reliability analysis of elastic linkage mechanism with stochastic parameters under stationary random excitation.Firstly, a method for calculating dynamic reliability of gear with stochastic parameters stationary random excitation is proposed under the condition of single degree random vibration. Considering the randomness of the structural physical parameters and the geometric dimensions parameters, the mean value and the variance of the mean square value of the structural displacement and stationary random excitation are computed firstly from the expressions of system stationary response in frequency domain by utilizing the random variable's functional moment method and the algebra synthesis method. And then the expressions of the mean value and the variance of stochastic gear dynamic reliability are deduced from the Poisson formula for calculating dynamic reliability. Finally, the influence of the randomness of system physical and geometric dimensions parameters on the system dynamic reliability is analyzed through comparing with the Monte Carlo method, validating the feasibility of this method. Secondly, the stationary random vibration analysis of elastic linkage mechanism with random with random parameters was investigated. Firstly, based on the Rayleigh quotient formula, the expressions of the mean value and the variance of the time-variable natural frequencies of elastic linkage mechanism with random physical and geometrical parameters were formulated by using the generalized random factor method. From the expression of dynamic stationary random response of the frequency domain, the computational expressions of the mean value, variance and variation coefficient of the mean square value of elastic displacement and velocity responses under the stationary random excitation were developed by means of the random variable's functional moment method and the algebra synthesis method. And then the expressions of the mean value and the variance of system dynamic reliability are deduced from the Poisson formula of calculating dynamic reliability. Finally, the influence of the random of mechanism physical and geometric dimensions parameters on the system dynamic reliability is analyzed through the examples, validating the feasibility of this method.