Dynamics Of Fractional-Order Mathieu Oscillator

With many scholars having carried out detailed discussions on the definition,properties and calculation methods of fractional calculus,it has gradually turned from mathematical theory to engineering application.The fractional-order derivative is introduced into the Mathieu oscillator,which not only enriches the application of fractional calculus in dynamics,but also reveals the functions of fractional differential term in this kind of system.Therefore,three kinds of Mathieu oscillators with fractional-order derivative are taken as objects to study the action mode of fractional differential term on the system in this thesis,and it mainly includes the following aspects.(1)The primary resonance of fractional Mathieu oscillator with quadratic damping under forced excitation is studied,and the method of multiple scales is used to calculate the approximate solution of the primary resonance of the system and its accuracy is verified by numerical method.Then,Lyapunov’s first method is used to quantitatively calculate the stability condition of the steady-state response,and the amplitudefrequency responses of the system are analyzed.Furthermore,the effects of fractional differential term on the amplitude-frequency curve of the system are investigated by numerical simulation.It is found that the fractional differential term has some functions which are applicable to optimize the system.(2)The periodic responses and quasi-periodic motions of a van der Pol-Mathieu oscillator with fractional-order derivative under a harmonic external excitation is studied.Firstly,the averaging method is used to obtain the approximate solution of periodic responses and the method of multiple scales is used to obtain the approximate solutions of quasi-periodic motions.Then the effects of various components of the system on the amplitude-frequency curve are analyzed,and the stability of the steadystate response is determined by using the stability condition.Finally,the effects of fractional differential term on the vibration amplitude of the system are analyzed in a wide frequency band.(3)The influences of fractional parameters on the stability boundary of the quasiperiodic Mathieu oscillator are studied.Firstly,the conditions of the periodic solutions are obtained by the perturbation method,and the approximate expressions of the transition curves are also gotten.The accuracy of the approximate analytical solution is verified by comparing with the numerical solution,and they are in good agreement with each other.Moreover,the approximate expressions of transition curves under different conditions are summarized,and the general forms of equivalent linear damping and equivalent linear stiffness are obtained by analyzing their formal characteristics.Finally,the effects of fractional-order parameters on the size of stability region and the position of transition curves are analyzed intuitively through numerical method.At the end of this thesis,the research results are summarized and several problems encountered in the research process are put forward.