主共振与内共振下纵横耦合轴系动力学分析

Dynamic analysis of the coupled longitudinal-transverse shafts under primary and internal resonances

考虑Von Karman非线性位移-应变关系,利用Hamilton原理建立了轴系纵横耦合下的动力学模型。利用Galerkin法对偏微分方程进行离散,采用多尺度法求解了离散方程。研究了纵向主共振并伴随内共振(由纵向第一阶固有频率近似等于横向第一阶正进动与反进动频率之和而产生)联合激励时轴系的动力学响应。研究表明随着系统参数以及激励载荷的不同,轴系出现不同的动力学特性。当激励载荷小于一临界值时,纵向激励力只能激起纵向振动,系统响应与线性系统一样;当载荷超过临界值时,纵向激励力同时激起了轴系的横向正进动与反进动频率,此时纵向振动出现能量饱和现象,能量从纵向渗透到横向。能量在正反进动模态间的分配与其正反进动频率成反比,从而使反进动幅值大于正进动幅值。同时响应中也出现跳跃现象。数值分析结果与摄动分析结果一致。

A nonlinear dynamic model for coupled longitudinal-transverse shafts is established by Hamilton′s principle with consideration of the Von Karman′s nonlinear strain-displacement relationship.The ordinary equations are obtained by Galerkin method and are solved by the method of multiple scales.The longitudinal primary resonance is investigated with special consideration of the case of internal resonance(because the first longitudinal natural frequency approximately equals to the sum of the first transverse forward and backward natural frequency).Research shows that the shaft has different dynamic characters with different excitation loads and parameters.Only the longitudinal natural frequency could be excited and the response is linear if the longitudinal excitation load is smaller than a critical value.Otherwise,the first transverse forward and backward natural frequency could be excited.There is saturation phenomenon in the longitudinal motion and extra energy is transferred to the transverse mode.The energy distribution between the forward and backward mode is inversely proportional to their frequency,which result in that the backward amplitude is bigger than the forward amplitude.Meanwhile,the response has jump phenomenon.Results of perturbation method are validated with numerical simulations.