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首页> 外文期刊>Applied Mathematical Modelling >Nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading in the subcritical buckling regime
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Nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading in the subcritical buckling regime

机译：亚临界弯曲状态下谐波轴向载荷下高速梁的非线性横向振动

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摘要

Equations of motion of a hyperelastic beam under time-varying axial loading are derived via the extended Hamilton's principle in this work, where the transverse vibration is coupled with the longitudinal vibration, and nonlinear vibrations of the beam in the subcritical buckling regime are investigated. Complex nonlinear boundary conditions of the beam are determined under some geometric constraints. The critical buckling load is first determined through linear bifurcation analysis. Effects of material and geometric parameters on the forced longitudinal vibration of the beam are numerically investigated. Steady harmonic shapes of the beam at different times under harmonic axial loading are determined. The beam is in the barreling deformation state even when the axial load is not in excess of the critical buckling load. The governing equation for the nonlinear transverse vibration of the beam is obtained by decoupling its equations of motion. Natural frequencies of the free linearized transverse vibration of the beam are studied. By applying the eigenfunction expansion method, the governing equation for the nonlinear transverse vibration of the beam transforms to a series of strongly nonlinear ordinary differential equations (ODEs). Two-to-one internal resonance of the beam is studied by the numerical integration method and its phase-plane portraits are obtained. The harmonic balance method and pseudo arc-length method are used to determine steady-state periodic solutions of the beam from the strongly nonlinear ODEs, and amplitude-frequency responses of the beam are determined. Effects of the external mean axial load, excitation amplitude, and damping coefficient on the amplitude-frequency response of the beam are numerically investigated. Combined effects of the external excitation amplitude and frequency on response amplitudes are also investigated.
机译：在该工作中通过延长的汉密尔顿原理推导出时变轴向负载下的超弹性光束的运动方程，其中横向振动与纵向振动联接，并且研究了亚临界屈曲方案中的梁的非线性振动。梁的复杂非线性边界条件在一些几何约束下确定。首先通过线性分叉分析确定关键屈曲负载。物质和几何参数对梁的强制纵向振动的影响在数值上进行了数量研究。确定在谐波轴向载荷下不同时间的梁的稳定谐波形状。即使当轴向载荷不超过关键屈曲负荷时，光束也处于枪管变形状态。通过去耦其运动方程来获得用于光束的非线性横向振动的控制方程。研究了梁的自由线性化横向振动的自然频率。通过应用特征函数扩展方法，光束非线性横向振动的控制方程变换为一系列强非线性常微分方程（ODES）。通过数值积分法研究了光束的双向内部共振，并获得了其相平面肖像。谐波平衡法和伪弧长方法用于确定光束的稳态周期解，从强不动性的杂波确定光束的幅度频率响应。外部平均轴向载荷，激发幅度和阻尼系数对光束的幅度频率响应的影响。还研究了外部激励幅度和频率对响应幅度的综合影响。

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