Dynamic Analysis Of Vibro-impact System With Fractional Order

In the mechanical systems,due to the processing factors or wear in the use of machinery,there will be a gap between the mechanical components,and the impact vibration caused by the gap presents complex motion.Mechanical or equipment with viscoelastic materials will also produce impact vibration in the process of operation,which is more complex than the classical impact vibration.Because the impact vibration has a great influence on the mechanical system,the research on the impact vibration has great value.Firstly,based on the single side impact vibration model under harmonic external excitation,the stability and bifurcation of a single degree of freedom linear vibration impact system with fractional derivative are studied.The approximate analytical solutions of the transient and steady state of the impact vibration system are obtained by the average method,and the general solution of the system is obtained by the superposition method.The approximate analytical solution and numerical solution of the system are compared,and they are in good agreement.The correctness of the approximate analytical solution is proved.Based on the approximate analytical solution and Poincaré mapping,the problem of periodic motion stability of the impact vibration system is transformed into the problem of fixed point on the Poincarémapping section.The bifurcation behavior of the system with fractional order,frequency and amplitude of external excitation is analyzed in detail.The results show that the system has saddle node bifurcation,boundary bifurcation,period doubling bifurcation and chaotic motion.Besides,based on a two-sided symmetric collision model with two rigid constraints under harmonic external excitation,the bifurcation behavior of a single degree of freedom linear vibration collision system with fractional derivative is studied.The approximate equivalent integral order system of fractional order linear system is obtained by means of the average method.On this basis,the approximate analytical solution of bilateral symmetric impact vibration system is analyzed.According to the approximate analytical solution,the existence conditions of the periodic motion of the two-sided symmetric n-1-1 are obtained,and the stability of the periodic motion of the two-sided symmetric vibration collision system is studied by Poincaré mapping.The bifurcations of the two-sided symmetric impact vibration system are studied when the external excitation frequency,fractional order and gap change.The results show that there are edge bifurcations,tuning bifurcations,period doubling bifurcations and chaotic motions in the system.Finally,based on the two degree of freedom impact vibration model excited by external harmonic,the stability and bifurcation behavior of fractional linear two degree of freedom impact vibrator are studied.The approximate analytical solution of the system is obtained by quoting the approximate equivalent integer order system of fractional order system.The stability of n-1 periodic motion is analyzed by Poincarémapping.The bifurcation behavior of two degree of freedom impact vibration system is studied by numerical solution.It is found that there are two paths to chaos when the excitation frequency changes.One is Hopf bifurcation when the excitation frequency changes.The system changes from stable periodic motion to quasi periodic motion through Hopf bifurcation,and then to chaotic motion.On the other hand,with the increase of the excitation frequency,the system undergoes period doubling bifurcation,and then tends to chaos.In addition,when the fractional order increases,period doubling bifurcation and quasi periodic motion occur.As the fractional order continues to increase,the system eventually behaves as a stable 2-1 periodic motion.