Dynamics And Stationary Reliability And Sensitivity Analysis Of Nonlinear Gear System

Multi-mesh gear systems are used widely in the industrial machinery.Traditional gear design is from the perspective of strength checking to complete.But for the actual transmission system,the dynamic characteristics during the meshing process is very important to the system,because it influence the stationary,accuracy and reliability.For the deterministic model system,the approximate analytical method,for example,harmonic balance method,is used to obtain the result,and the solutions will be verified by comparing them to those of numerical simulation.Meanwhile,some nonlinear phenomena of the system are studied,and detailed parametric studies are also performed to investigate the sensitivity of the nonlinear dynamic behavior to each key parameter.Through these ways,the structural design can be improved and the structure parameters can be optimized under different working conditions.Generally,because of the stochastic structural parameters,such as material characteristics,dimension parameters,assembly technique,and stochastic excitation,the response of the system become a stochastic process.For the stochastic model,the main studies of the structure are reliability and sensitivity.To solve these problems,the theory of the probability model in engineering and science,stochastic response analysis of nonlinear system and reliability analysis of stochastic vibration should be combined together.In this paper,the idler gear systems including three gears are studied.Based on the detail study of structural parameters,both deterministic model and stochastic model are proposed.The main content of this work include:First,the gear dynamics and stochastic dynamics theory and reliability theory under stochastic model are elaborated.The main methods for solving the dynamic characteristics of the gear transmission system are given,including numerical solution and approximate analytical solution.At the same time,the reliability and reliability sensitivity of the random vibration system based on the moment method and the reliability theory of the probability density transfer function based on the FPK equation are analyzed in detail.Second,according to the physical model of the three-gear idler,based on the experimental verification of the relevant literature,the mathematical model of the secondary transmission torsional vibration is established by the lumped mass method.The main factors causing nonlinearity in the system such as time-varying mesh stiffness,static transimission error and gear backlash are analyzed and calculated in detail.Based on the accurate modeling of the tooth profile,the time-varying meshing stiffness is deduced in detail by the material mechanics method,and compared with the ANSYS calculation results,the accuracy of the numerical solution method is verified.The transmission error is described in terms of both concepts and components.The existence of backlash causes the system to have unilateral impact and bilateral impact,which causes strong nonlinearity of the system.For the convenience of calculation,an approximate smooth curve is used instead,and the degree of approximation is calculated in detail.Finally,the established model is dimensionlessly processed,which facilitates the comparison of parameter sensitivity and reliability sensitivity,and provides a theoretical basis for structural design improvement in engineering.Third,the periodic steady state solutions for frequency response are obtained by semi-analytical harmonic balance method.Through equating the coefficients of like harmonic terms,a set of nonlinear algebraic equations are obtained.Using Newton-Raphson iteration method to solve the set of equations will fail when the Jacobian matrix singular,and this iteration is sensitive to the initial value.So arc-length continuation method is used to avoid the problem.The stability of the steady-state period-1 solutions will be determined by using the Floquet theory.The sensitivity of the design parameters,such as time-varying mesh stiffness,involute contact ratio,damping ratio,load and backlash are discussed via analytical and numerical solutions.Also,the main index of the stability of the system,the dynamic transmission error and dynamic force are studied under different speed.Fourth,consider the randomness of the structural parameter and excitation,a probability model of the idler gear system are given.Then a practical and efficient method for reliability analysis of the gear system is present by integrating the techniques of stochastic perturbation method,high order moment technique,Edgeworth series and its empirical remedy expression.when the system contain series and parallel connections,the corresponding formula should be used to get the system reliability,as the reliability is instantaneous,this paper introduce the cumulative reliability.And the Monte Carlo method is used to verify the validity of the proposed theory.The sensitivity of the reliability to the mean value and variance of orginal random parameters is obtained by the direct differential method.Finally,the influence degree of all the basic random parameters on the relioability is derived,which provides theoretical guidance for the product designing process.Finally,study the nonlinear stochastic multi-mesh gear system under Gaussian white noise excitation.For the nonlinear system,using the moment method will lose some useful probability information because of the truncation error.Through solving the FPK equation we can get the instantaneous transition probability and probability density of the stochastic process.if the system is linear and excited by Gaussian white noise,according to the denfinition of moment and the FPK equation,it is equivalent to get the first two moments.For the nonlinear stochastic gear systems,it's hard to obtain the analytical solution of the FPK equation,so using the numerical simulation is a good way.As the model is high dimension parabolic partial differential equation,the method of locally one-dimensional finite different scheme,coupled with time-splitting method is used.After obtain the instantaneous joint probability density function,we can use Rice crossing rate theory to calculate reliability.