Study Of Dynamic Behavior Of A Class Of Two Degree Of Freedom Nonlinear Vibration System

As a classical model of coupled nonlinear vibration,the spring pendulum is being noted and studied by more scholars.In this paper,the free vibration and the forced vibration of a spring pendulum are studied through theoretical analysis and numerical simulation.The dynamic behaviors of the breathing mode and the pendulum-like mode of the spring pendulum system under the conditions of different frequency ratios,initial conditions,external excitation frequencies and amplitudes are obtained.(1)For the free vibration of spring pendulum,the vibration states under different frequency ratios and different initial conditions are analyzed.It is obtained that when there is only an initial extension displacement,the spring pendulum only has the vibration of the breathing mode,and the pendulum-like mode is static and has nothing to do with the natural frequency ratios of the system.When there is only an initial pendulum angle,the spring pendulum system,with the frequency ratio being 2:1,has a relatively obvious phenomenon of internal resonance,which is shown as the beat vibration.When the frequency ratio of the spring is 1:1 and 3:1,the internal resonance phenomenon is not obvious.When the initial displacements applied to the small ball are all exists,in the case of the frequency ratio being 2:1,the spring pendulum will have the internal resonance phenomenon,which is shown as the beat vibration.In other cases,the period of the breathing mode is constant,and the period of the pendulum-like mode is its multiple times.(2)The vibration states under different frequency ratios,external excitation frequencies and amplitudes are analyzed,for the spring pendulum system driven by a horizontal direction.The response amplitude of the spring pendulum has the maximum value when the coordination parameter is zero,and the natural frequency ratio is 2:1,and it is the internal resonance state.With the increasing or decreasing of the coordination parameters,the response amplitude decreases rapidly and tends to the level.This is the non-internal resonance state at this time.The response amplitude of the spring pendulum will increase with the increase of the amplitude of the external excitation.The amplitude of the pendulum-like mode first increases linearly with the increase of the excitation frequency,then slows down,and decreases after the maximum value,and the amplitude of the breathing mode increases exponentially.(3)The vibration state under different frequency ratios and external excitation amplitudes are analyzed,for the spring pendulum system driven by a excitation of vertical direction.The response amplitude of the system increases with the increase of the excitation amplitude,but the growth rates of the two mode are different.On the whole,the increase of the pendulum-like mode is larger than that of the breathing mode.The amplitude of breathing mode first decreases and then increases with the increase of coordination parameter.On the contrary,the amplitude of pendulum-like mode first increases,then decreases and returns to zero.