Dynamics And Control Of Fractional-order Parametrically Excited System

With the rapid development of computer technology,the fractional-order derivative and its applications of the nonlinear dynamics in different fields have recently been focused by many scholars.The fractional-order derivative is a new mathematical tool and the parametrically excited system is a classical model in the nonlinear dynamics.The combination of the fractional-order derivative and the parametrically excited system can not only deepen the theoretical study of the fractional-order derivative,but also promote the engineering application of the fractional-order parametrically excited system model and accelerate the development of the dynamics and control of the high speed vehicle system.So it has a wide application prospect.The main content of this paper is organized as follows.(1)The research status and methods of the dynamical response and control of fractional-order parametrically excited system are elaborated.Besides,the main innovations are also described.(2)Air spring of the traffic vehicle system is modeled by the fractional-order derivative.The influence rule of the parameters of the air spring system on the air spring dynamical characteristics is obtained.In the end,the model parameters are identified and analyzed.(3)Firstly,the model of the Mathieu equation with fractional-order derivative is presented to model the pantograph-catenary system.Then the dynamical characteristics of Mathieu equation with fractional-order derivative are analytically studied by the Lindstedt-Poincare method and the multiple-scale method.The stability boundaries and the corresponding periodic solutions on these boundaries are analytically obtained.The effects of the fractional-order parameters on the stability boundaries are studied.In the last section,the response of Mathieu equation with two kinds of van der Pol(VDP)fractional-order terms is investigated.The approximately analytical solution is obtained by the averaging method.Thesteady-state solution,existence conditions and stability condition for the steady-state solution are presented.The influences of the two kinds of VDP fractional coefficients and fractional orders are analyzed.(4)The incremental harmonic balance(IHB)method is extended to analyze the dynamical properties of fractional-order nonlinear oscillator.The general forms of the fractional-order derivative are founded based on IHB method.Then the dynamical analysis of strongly nonlinear fractional-order Mathieu–Duffing equation with the constant excitation and the forced excitation are studied by IHB method respectively.(5)The dynamical characteristics of an autonomous Duffing oscillator and a parametrically excited system under fractional-order feedback coupling with time delay are investigated respectively.The first-order approximate analytical solution is obtained by the averaging method.The equivalent stiffness and equivalent damping coefficients are defined by the feedback coefficient,fractional order and time delay.It is found that the fractional-order feedback coupling with time delay has the functions of both delayed velocity feedback and delayed displacement feedback simultaneously.Finally,the main results and innovators are summarized in the last Section,and the existing problems and the research in the future are also pointed out in this section.