Transient Response And Stability Of Constrained Viscoelastic Stochastic System

With the rapid development of industry,viscoelastic materials have been widely applied in engineering practice.The study of dynamic mechanical properties of viscoelastic materials has been an important research topic in the field of materials mechanics engineering.The study of nonlinear dynamic behavior of viscoelastic materials is of great significance for structural design,shock absorption and engineering applications.Based on the constitutive relation of viscoelastic material,the transient response and stability of viscoelastic stochastic system with impact constrained is studied in this paper.(1)The transient response of viscoelastic nonlinear system with impact constrained under Gauss white noise is studied.By introducing the Dirac delta function,the restricted domain is transformed to the periodic boundary,so that the non smooth system is transformed into a new system with continuous Hamilton function.Then,with application of the method of stochastic averaging,the FPK equation of system is obtained.Based on Mellin transform,the complex fractional moments of probability density function is constructed.The transient probability density governed by the FPK equation is derived with solving a set of differential equations yields the complex fractional moments.Two examples are used to verify the effectiveness of the complex fractional moment method.The effect of the restitution factor of viscoelastic material on the transient response of the system is discussed.The results show that when the restitution factor is determined,the error between the transient response and the steady-state response decreases with the increase of time.When the truncated term m increases,the transient probability density from the present technique converges to that from Monte Carlo simulation.(2)Based on Maxwell exponentially integrated and fractional viscoelastic material models,stochastic stabilities of nonlinear system with impact constrained subject wide-band noise are studied.Combining the stochastic averaging method and It?principle,p order mean It?differential equation is established.The p moment Lyapunov exponent expression of system is obtained.The stabilities of system under Gaussian white noise and real noise are analyzed respectively.Analytical results and the numerical simulation results from Monte Carlo are compared.For Maxwell viscoelastic material model,the restitution factor can weaken system stability;the stability of the system will be weaker with strong noise intensity;the stability region can be wider with the increasing of the viscoelastic parameters?,but narrowed with the relaxation time?increases.For frational viscoelastic material model the restitution factor can weaken system stability;the higher fractional number?contributes to the stability of the system;with the increase of the viscoelastic parameters?_?,the stable region increases;the real noise bandwidth can strengthen the stability of the system;noise intensity will reduce the stability.(3)According to above viscoelastic material constitutive models,stochastic stabilities of nonlinear system with impact constrained subject bound noise are studied.With the help of stochastic averaging method and It?principle,averaged It?differential equations of coupling are established.Applying bound non canonical transformation,the coupling It?differential equations are to be eigenvalues problem.By solving the eigenvalues,the approximate analytic solutions of the p moment Lyapunov exponent of the viscoelastic system are obtained by solving eigenvalues problem.Finally,the numerical results of Monte Carlo are used to verify the correctness of the approximate analytical results.The research shows that the greater the restitution factor of viscoelastic material can weaken the system stability.The stability of the system strengthens with the increase of viscoelastic parameter?.The system stability is stronger,as the relaxation time is longer;the strong excitation amplitude weakens the stability of the system response.It is seen that the fractional order?has effects on the almost-sure stability.