Random Response Analysis Of Duffing System With Pendulum Under Narrow-band Random Excitation

The Duffing system with pendulum is one of the most popular objects in physics research.In daily life,we can often contact with these systems,such as the large pendulum of amusement park,the friction pendulum support of bridge,etc.Therefore,it is of great theoretical and practical significance to study the response of the Duffing system with pendulum which has a very broad research background and constantly develop new theoretical methods.In this paper,we study the dynamic response of a Duffing system with pendulum under narrow-band random excitation and the governing equations of the system are given.Firstly,the response of the linear Duffing system with pendulum is studied.The amplitude phase coupling equation of the system is obtained by using the method of mutiple scales.The analytical expression of the first-order uniform expansion of the solution is derived.We qualitatively analyze the stability of the system and obtain the analytical expression of the amplitude in the steady state motion.By combining the linear method with the moment method,the analytical expressions of the first-order and second-order steady state moments of the linear Duffing system with pendulum are obtained,and its stability is analyzed,and the existence of bifurcation behavior is obtained by analyzing its dynamic behavior.In the following chapter,the stochastic response analysis of nonlinear Duffing system with pendulum under narrow-band random excitation is discussed.In the study of nonlinear system,the analytical expression of the firstorder uniform expansion of the solution is also derived by using the multiple scale method.By combining the linear method with the moment method,the first-order and second-order steady-state moments of the system are obtained in the case of random disturbance.Then,we analyze the stability of the system and give the necessary and sufficient conditions of the stability of the Duffing system with pendulum.By setting the system parameters for numerical simulation,the analysis results of the system response are verified.By comparison,it is found that the dynamic behavior of the time history diagram and phase diagram of the system changes essentially under different initial values and different parameters.The response of the system is very sensitive to the initial value.With the increase of the random noise intensity,the limit cycle dissipates and becomes a diffusive limit cycle.When the excitation amplitude increases,the width of the limit cycle also increases.