耦合能量收集压电振子的不变流形及其参数化方法
Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators
摘要：
基于振荡器的能量收集系统旨在从机械振荡中捕获能量并将其转换为电能。广泛扩展的是基于压电材料的那些，其动力学是服从于不同耗散源的哈密顿：阻尼和耦合。这些耗散使系统进入低能量状态，这在长期内是不期望的，因为它减少了吸收的能量。为了避免或尽量减少这种情况，我们建议，两个振荡器的耦合可以受益于阿诺德扩散理论。这种现象研究了哈密顿系统中的$O(1)$能量变化，因此在能量收集应用中可能非常有用。本文是实现这一目标的第一步。我们考虑两个压电梁提交一个小的强制和通过电路耦合。通过将耦合、阻尼和强迫视为扰动，证明了未扰动系统具有一个$4$维常双曲不变流形，该流形分别具有$5$和$4$维稳定流形和非稳定流形.在扰动之后，这些是局部唯一的。利用参数化方法，数值计算了扰动流形及其稳定和非稳定流形的参数化，并研究了其内部动力学。我们证明了同宿连接时的扰动是打开的证据。
英文摘要：
Energy harvesting systems based on oscillators aim to capture energy from mechanical oscillations and convert it into electrical energy. Widely extended are those based on piezoelectric materials, whose dynamics are Hamiltonian submitted to different sources of dissipation: damping and coupling. These dissipations bring the system to low energy regimes, which is not desired in long term as it diminishes the absorbed energy. To avoid or to minimize such situations, we propose that the coupling of two oscillators could benefit from theory of Arnold diffusion. Such phenomenon studies $O(1)$ energy variations in Hamiltonian systems and hence could be very useful in energy harvesting applications. This article is a first step towards this goal. We consider two piezoelectric beams submitted to a small forcing and coupled through an electric circuit. By considering the coupling, damping and forcing as perturbations, we prove that the unperturbed system possesses a $4$-dimensional Normally Hyperbolic Invariant Manifold with $5$ and $4$-dimensional stable and unstable manifolds, respectively. These are locally unique after the perturbation. By means of the parameterization method, we numerically compute parameterizations of the perturbed manifold, its stable and unstable manifolds and study its inner dynamics. We show evidence of homoclinic connections when the perturbation is switched on.
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