Stability And Hopf Bifurcation Analysis Of Stochastic Mechanical Systems

The stability of a stochastic mechanical system is an important part of the study of dynamic characteristics.It can be used to analyze the conditions for maintaining its stability.In this paper,the mechanical system under random excitation is taken as the main research object,and the stability of such systems is analyzed.The research on the optimization of the damping of the mechanical system with parameters,the stability of the rotor operation and the bifurcation of the gear transmission system is studied.Due to the complexity of the system connection structure studied,the material itself is uneven.Qualitative,as well as being influenced by other random factors,eventually leads to system uncertainties.Therefore,the nonlinear stochastic dynamics theory is used to conduct detailed research on the dynamic behavior of mechanical systems with random components.The main contents are as follows:1.The research progress of random mechanical systems at home and abroad is described in detail,as well as the objectives of this paper.Then the basic contents of nonlinear stochastic dynamics theory are described,including the stochastic averaging principle,the largest Lyapunov exponent method and the specific concepts of boundary theory and main content.2.The stability and Hopf bifurcation of a mechanical system with parameters in the damping optimization problem is studied.By using the stochastic nonlinear dynamical quasi-nonintegrable Hamilton system theory,the system is gradually converged to a one-dimensional differential equation.By calculating the maximum Lyapunov exponent and analyzing the local stability according to its characteristics,the global stability of the system is analyzed by the singular boundary theory of the stochastic differential equation.Finally,the stationary probability density function and the joint probability density function are obtained.Through numerical simulation,the process from stable to Hopf bifurcation is obtained.3.The stochastic stability and Hopf bifurcation of a four-dimensional rotor system are studied.The four-dimensional stochastic nonlinear system is analyzed by applying the quasi non-integrable Hamiltonian system theory.It is weakly convergent to a one-dimensional stochastic diffusion process with probability 1 based on the principle of stochastic averaging.But in the calculation of the drift diffusion index,in order to reduce the computational complexity and avoid the multiple integral,the polar coordinate transformation is introduced to obtain the stochastic differential equation.Then analyze the local stability and global stability.Through the numerical simulation,the process from stable to Hopf bifurcation is obtained.4.The stability and Hopf bifurcation of a transmission system with random perturbation are studied.The Gaussian white noise is used to replace the internal and external randomeffects of the system.By means of the stochastic averaging principle,the non-integrable Hamiltonian system is converged to a one-dimensional random diffusion process.The maximum Lyapunov exponent is calculated,and the condition of local stability of the system is obtained by using the relation between maximum Lyapunov and zeros.Then with the solution of FPK equation,the probability of Hopf bifurcation is simulated by using the image of stationary probability density function.