Stochastic Bifurcation Analysis Of Typical Nonlinear Oscillators With Delayed Fractional Order Controller

In the nonlinear vibration of structure the e istence of random interference ill cause the system to generate stochastic bifurcation hich ill often have an adverse effect on the system Fractional order controller has become a hotspot in the field of structural control because of its infinite memory function and genetic characteristics E isting research results sho that fractional order controllers can achieve better control effect than that of traditional integral-order controllers Time delay is very common in the control system and cannot be ignored The reason is that time delay is often the main factor that leads to the bad performance or even the instability of the control system ased on the analysis and summary of the research status of stochastic bifurcation fractional order control and time delay this paper studies the stochastic bifurcation of typical nonlinear oscillators ith delayed fractional order PD controllerFirstly the generalized harmonic function is used to develop the stochastic averaging method of quasi-integrable Hamiltonian systems ith delayed fractional order PD controller under Gaussian hite noise or ideband noise The average It? stochastic differential equation and its drift and diffusion coefficients are givenFor the stochastic P-bifurcation of single-degree-of-freedom(SDOF)strongly nonlinear system ith delayed fractional order PD controller under Gaussian hite noise e citation the stochastic average method in Chapter 2 is used to obtain the average It? stochastic differential equation and the corresponding Fokker-Planck-Kolmogorov(FPK)equation is solved to obtain the stationary probability density function of the system Taking van del Pol oscillator and Rayleigh-Duffing oscillator as e amples the effect of fractional order and time delay on the response of the system are discussed The results sho that the change of time delay and fractional order may cause thestochastic P-bifurcation hich implies that the delayed fractional order PD controller could provide as an effective tool to anti-control of stochastic bifurcation Furthermore the time delay could deteriorate the effectiveness of the fractional order PD controllerFor the stochastic Hopf bifurcation of multi-degree-of-freedom(MDOF)quasi-integrable Hamiltonian system ith delayed fractional order PD controller under ideband noise e citation the stochastic average method is used to obtain the average It? stochastic differential equation y analyzing the boundary properties of the equations H=0 and H?? the e pression of the average bifurcation parameter of the average system is obtained the criterion of stochastic Hopf bifurcation caused by the delayed fractional order PD control force in the original system is given As an e ample t o-degree-of-freedom coupled Rayleigh oscillators are studied the effect of time delay and fractional order on stochastic Hopf bifurcation and the stability of the system are analyzed The research results sho that the time delay ill cause stochastic Hopf bifurcation in addition as the fractional order increases the system ill become more stable.