Steady-State Periodic Responses of a Viscoelastic Buckled Beam in Nonlinear Internal Resonance
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摘要: 研究了内共振下简支边界屈曲黏弹性梁受迫振动稳态周期幅频响应.考虑Kelvin黏弹性本构关系,并通过对非平凡平衡位形做坐标变换,建立屈曲梁横向振动的非线性偏微分-积分模型.基于对控制方程的Galerkin截断,得到多维非线性常微分方程组.在前两阶模态内共振存在的条件下,运用多尺度法分析截断后的控制方程,利用可解性条件消除长期项,获得一阶主共振下的幅值与相角方程.通过数值算例以展示系统稳态幅频响应关系以及失稳区域,从而聚焦系统共振中存在的非线性现象,如跳跃现象、滞后现象,并讨论了双跳跃现象随轴向荷载的演化.通过直接数值方法处理截断方程,数值验证近似解析解,计算结果表明多尺度法具有较高精度.Abstract: Nonlinear vibration of a hinged-hinged viscoelastic buckled beam subjected to primary resonance in the presence of internal resonance was investigated. The governing integro-partial differential equation was derived via introduction of coordinate transform for the non-trivial equilibrium configuration, with the viscoelastic constitutive relation taken into account. Based on the Galerkin method, the governing equation was truncated to a set of infinite ordinary differential equations and the condition for internal resonance was obtained. The multiple scales method was applied to derive the modulation-phase equations. Steady-state periodic solutions to the system as well as their stability were determined. The numerical examples were focused on the nonlinear phenomena, such as double-jump and hysteresis. The generation and vanishing of a double-jumping phenomenon on the amplitude-frequency curves were discussed in detail. The Runge-Kutta method was developed to verify the accuracy of results from the multiple scales method.
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Key words:
- buckled beam /
- viscoelasticity /
- internal resonance /
- multiple scales method /
- stability
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