Stochastic Dynamics And Applications For A Kind Of Non-Smooth Time Delay System

Non-smooth factors exist widely in both natural phenomena and engineering fields,which makes the study of non-smooth systems essential for a better understanding of the world.Vibro-impact systems,as a typical class of non-smooth systems,have attracted extensive attention in the field of dynamics for their special nonlinear structures and complex dynamical behaviors.The mechanism of non-smooth factors in nonlinear systems can be further explored through an in-depth study of the properties and behaviors of vibro-impact systems.Based on the problem of unavoidable collisions among the components of mechanical devices during operation,this thesis develops a theory based on stochastic time-delayed dynamical systems,considering the existence of random factors and time delay effects in the real world,and then explores the stochastic dynamical behavior of the system.Firstly,a one-sided vibro-impact system under two uncorrelated Gaussian white noise excitations with time delay and the fractional-order derivative term is investigated.The fractional-order derivative terms are expressed as a sum of linear damping and restoring forces using generalized harmonic functions.Then,the original system is simplified based on non-smooth transformations to obtain an approximate equivalent stochastic system.The corresponding one-dimensional equations and the corresponding Fokker-Planck-Kolmogorov(FPK)equations are obtained using the stochastic averaging method.The stationary probability density function of the equivalent system is obtained by solving the FPK equation.The effectiveness of the approximate method and the stochastic dynamical behavior of the system with different parameters are verified by numerical simulations.Secondly,a bilateral constrained stochastic time-delayed vibro-impact energy harvesting system is investigated.The original system is transformed into an approximate equivalent system without the piezoelectric differential equations using generalized harmonic functions and integral electrical equations.Based on the difference in initial energy,the undisturbed system motion is classified into two forms:collision-free vibration and repeated bilateral collisions.Subsequently,the FPK equations corresponding to the two forms of motion are obtained by using the stochastic averaging method,and the stationary probability density function of the approximately equivalent system is obtained by solving those FPK equations.The proposed method is verified to be highly accurate with the help of numerical methods.The effects of different parameters on the system response and the variation of the system mean square voltage with parameters are discussed.Finally,the BP neural network is used to break through the limitations of single-parameter tuning.Given the expected values of the system mean square voltage and proton-to-baffle distance,the corresponding adjustable parameters:electromechanical time constant ratio,displacement time delay feedback term coefficient and velocity time delay feedback term coefficient are predicted accordingly by constructing an appropriate BP neural network to achieve the predetermined expected values.