弹性耦合可伸缩双梁系统的稳态
Steady states of elastically-coupled extensible double-beam systems
摘要：
给定$β\in\mathbb{R}$和$\varrho,k>0$，我们分析了一个抽象版本的非线性平稳模型的无量纲形式$$\begin{cases} u"" - \Big(β+ \varrho\int_0^1 |u'(s)|^2\,{\rm d} s\Big)u" +k(u-v) = 0 v"" - \Big(β+ \varrho\int_0^1 |v'(s)|^2\,{\rm d} s\Big)v" -k(u-v) = 0 \end{cases} $$描述平衡的弹性耦合可扩展的双梁系统均匀压缩轴向载荷。建立了方程存在非平凡解的充要条件，并给出了其显式的封闭表达式，特别地，证明了方程的解最多存在三个非零Fourier模.尽管该系统的对称性，非对称的解决方案出现，以及解决方案的弹性能量未能均匀分布。这种特征在长期动力学分析中具有一定的相关性，因为它可能导致两个梁之间的非对称能量交换，模仿从垂直振荡到扭转振荡的过渡。
英文摘要：
Given $β\in\mathbb{R}$ and $\varrho,k>0$, we analyze an abstract version of the nonlinear stationary model in dimensionless form $$\begin{cases} u"" - \Big(β+ \varrho\int_0^1 |u'(s)|^2\,{\rm d} s\Big)u" +k(u-v) = 0 v"" - \Big(β+ \varrho\int_0^1 |v'(s)|^2\,{\rm d} s\Big)v" -k(u-v) = 0 \end{cases} $$ describing the equilibria of an elastically-coupled extensible double-beam system subject to evenly compressive axial loads. Necessary and sufficient conditions in order to have nontrivial solutions are established, and their explicit closed-form expressions are found. In particular, the solutions are shown to exhibit at most three nonvanishing Fourier modes. In spite of the symmetry of the system, nonsymmetric solutions appear, as well as solutions for which the elastic energy fails to be evenly distributed. Such a feature turns out to be of some relevance in the analysis of the longterm dynamics, for it may lead up to nonsymmetric energy exchanges between the two beams, mimicking the transition from vertical to torsional oscillations.
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