Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation
Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation
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DOI：
10.1016/j.chaos.2005.04.045
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发表时间：
2006-02
影响因子：
7.8
通讯作者：
Liqun Chen;Xiaodong Yang
中科院分区：
文献类型：
--
作者：
Liqun Chen;Xiaodong Yang
本文研究了轴向加速粘弹性梁横向运动的分叉和混沌。开尔文模型用于描述梁材料的粘弹性特性，拉格朗日应变用于解释由于梁的小但有限的拉伸而产生的几何非线性。横向运动由非线性偏微分方程控制。应用伽辽金方法将偏微分方程截断为一组常微分方程。当伽辽金截断基于受到相同边界约束的线性非平移梁的本征函数时，提出了一种通过重新组合非线性项的计算技术。该方案在实际计算中很容易实现。当假设传输速度为具有较小谐波变化的恒定平均速度时，基于 4 项伽辽金截断对庞加莱图进行数值计算，以识别动力学行为。给出了改变以下参数之一的分岔图：轴向速度波动幅度、平均轴向速度和梁粘度系数，而其他参数不变。
This paper investigates bifurcation and chaos in transverse motion of axially accelerating viscoelastic beams. The Kelvin model is used to describe the viscoelastic property of the beam material, and the Lagrangian strain is used to account for geometric nonlinearity due to small but finite stretching of the beam. The transverse motion is governed by a nonlinear partial-differential equation. The Galerkin method is applied to truncate the partial-differential equation into a set of ordinary differential equations. When the Galerkin truncation is based on the eigenfunctions of a linear non-translating beam subjected to the same boundary constraints, a computation technique is proposed by regrouping nonlinear terms. The scheme can be easily implemented in practical computations. When the transport speed is assumed to be a constant mean speed with small harmonic variations, the Poincaré map is numerically calculated based on 4-term Galerkin truncation to identify dynamical behaviors. The bifurcation diagrams are present for varying one of the following parameter: the axial speed fluctuation amplitude, the mean axial speed and the beam viscosity coefficient, while other parameters are unchanged.