内共振关系对弹簧摆动力学行为的影响

INFLUENCE OF INTERNAL RESONANCE ON DYNAMICS OF SPRING PENDULUMS

基于数值方法,以弹簧摆为对象,讨论了不同的内共振关系对一类平方、立方非线性系统动力学行为的影响.结果表明,对1:1内共振的情况,两个模态的振动均可能发生在偏离原来平衡位置的新的平衡位置附近,即出现平衡位置飘移的现象.能量可以从低阶(摆动)模态传递到高阶(呼吸)模态,但不能从高阶(呼吸)模态传递到低阶(摆动)模态.然而对1:3内共振的情况,这种能量在两个模态之间的传递却非常弱.从仿真结果来看,对1:1和1:3内共振的情况,等幅的周期解是稳定的;但对1:2内共振的情况,出现的是调幅的周期运动即拍振,且拍频与初始条件有关.

On the assumption that both quadratic and cubic nonlinear terms are considered,the influence of internal resonance on the dynamics of spring-pendulum systems was investigated by numerical simulations.Based on the numerical results,it is observed that vibrations in both modes can take place away from its static equilibrium position for the case of 1：1 internal resonance.For any kind of internal resonance,it is found that transfer of energy from pendulum mode into spring mode is possible,whereas the reverse transfer of energy is impossible.Moreover,transfer of energy between the two modes is very weak for the case of 1：3 internal resonance.Periodic motions with constant amplitude are stable for both 1：1 and 1：3 internal resonance.However the periodic motions for the case of 1：2 internal resonance exhibit a phenomenon known as beats,and the beat frequency is associated with initial conditions.