一类压电智能复合材料梁的非线性建模与初步稳定性分析
Nonlinear modeling and preliminary stabilization results for a class of piezoelectric smart composite beams
摘要：
现有的智能复合材料压电梁模型大多忽略电磁相互作用，采用线弹性理论。然而，这些相互作用实质上改变了在高频下的可控性和稳定性，并且线性模型不能表示和预测支配动态，因为机械非线性在某些应用中是显著的，例如能量收集。本文首先利用非线性弹性理论，采用一致变分法导出了单层压电梁在电磁相互作用（完全动态和静电相互作用）和不存在电磁相互作用时的运动方程。通过采用两种被广泛接受的夹层梁理论，将这种建模策略扩展到三层压电智能复合材料。对于单层和三层模型，所得到的无穷维运动方程可以用状态空间形式来表示。结果表明，完全动态非线性模型是无界边界控制系统（线性理论也是）${\bf \dot y}(t)=(\mathcal A +\mathcal N) {\bf y} (t) + \mathcal B u(t)$，静电非线性模型是无界双线性控制系统${\bf \dot y}(t)=(\mathcal A +\mathcal N){\bf y} (t) + (\mathcal B_1+ \mathcal B_2 {\bf y}) u(t)$，与线性理论形成鲜明对比。最后，我们提出了$\mathcal B^*-$型反馈控制器来镇定单压电梁模型。滤波半离散有限差分近似被用来说明结果。
英文摘要：
Existing smart composite piezoelectric beam models in the literature mostly ignore the electro-magnetic interactions and adopt the linear elasticity theory. However, these interactions substantially change the controllability and stabilizability at the high frequencies, and linear models fail to represent and predict the governing dynamics since mechanical nonlinearities are pronounced in certain applications such as energy harvesting. In this paper, first, a consistent variational approach is used by considering nonlinear elasticity theory to derive equations of motion for a single-layer piezoelectric beam with and without the electromagnetic interactions (fully dynamic and electrostatic). This modeling strategy is extended for the three-layer piezoelectric smart composites by adopting the two widely-accepted sandwich beam theories. For both single-layer and three-layer models, the resulting infinite dimensional equations of motion can be formulated in the state-space form. It is observed that the fully dynamic nonlinear models are unbounded boundary control systems (same in linear theory) ${\bf \dot y}(t)=(\mathcal A +\mathcal N) {\bf y} (t) + \mathcal B u(t)$, the electrostatic nonlinear models are unbounded bilinear control systems ${\bf \dot y}(t)=(\mathcal A +\mathcal N){\bf y} (t) + (\mathcal B_1+ \mathcal B_2 {\bf y}) u(t)$ in sharp contrast to the linear theory. Finally, we propose $\mathcal B^*-$type feedback controllers to stabilize the single piezoelectric beam models. The filtered semi-discrete Finite Difference approximations is adopted to illustrate the findings.
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