轴向运动黏弹性梁受力参激振动稳定性的多尺度分析

Multi-scales Analysis on Stability of Parametrically Excited Vibration for Axially Moving of Viscoelastic Beams Subjected to Axial Disturbing Tension

研究了轴向匀速运动黏弹性梁的运动稳定性。考察轴向拉力在初始拉力的基础上做微小简谐变化的参激振动。建立了受轴向拉力参数激励时轴向运动梁的控制微分方程,黏弹性本构关系引入了物质时间导数。轴向运动梁两端的边界受由带有扭转弹簧的套筒铰支约束的混杂边界条件。应用多尺度法直接求解轴向运动梁参激振动的控制方程,并导出了当扰动拉力的频率接近未扰系统任意两个固有频率之和及任一固有频率2倍时所发生的组合共振和主共振的稳定边界方程。数值例子给出了黏弹阻尼对轴向运动黏弹性梁参激振动发生组合共振和主共振的影响,结果显示:不论组合共振还是主共振发生时,失稳区域均会随轴向运动黏弹性梁的黏弹阻尼增大而减小。

The motion stability of axially moving viscoelastic beams subjected to the parametrically excited tension is presented. The parametric vibration of axially moving beams is studied in this paper. The axial tension is characterized as a simple harmonic variation about the initial tension. The material time derivative is used in the viscoelastic constitutive relation. Asymptotic analysis is proposed to investigate the governing equation of an axially accelerating viscoelastic beam via the method of multiple scales. Beams are always fastened up by elastic joints at both ends. The supporting conditions may be formulated as simple supports with torsion springs. If the axial speed variation frequency approaches the sum of arbitrary two natural frequencies or the twice arbitrary natural frequency, the combined resonance or principal parametric resonance may occur. Analytical expressions of the instability boundary are obtained for summation and principal parametric resonance. Numerical examples show the effects of the viscous damping： whenever the instability regions for either the combined or the principal parametric resonance occcur, they will both decras while the viscous damping is increasing.