压电智能复合材料的动力学与静电建模及相关镇定结果
Dynamic and electrostatic modeling for a piezoelectric smart composite and related stabilization results
AI总结：
该研究针对悬臂压电智能复合梁，基于麦克斯韦方程组建立全动力学与静电型两类Rao-Nakra、Mead-Marcus摄动模型，证明解的存在唯一性，分析$B^*-$型反馈镇定效果，发现静电模型所需控制器更少且稳定性更优。
摘要：
本文研究一种由弹性层、黏弹性层和压电层完美粘结而成的悬臂式压电智能复合材料梁，其中压电层由电压源驱动。研究采用基于麦克斯韦方程组的全动力学方法与静电方法对压电层进行建模。通过假设黏弹性层的重量和刚度可忽略不计，推导得到全动力学与静电型的Rao-Nakra类模型；通过忽略面内惯性项与转动惯性项，推导得到全动力学与静电型的Mead-Marcus类模型，每类模型均为对应经典智能复合材料梁模型的摄动形式。研究将上述模型转化为状态空间形式，在合适的希尔伯特空间中证明了解的存在性与唯一性。随后，针对自然的$B^*-$型状态反馈控制器，深入分析并研究了各闭环系统的镇定问题。结果表明：含4个状态反馈控制器的全动力学Rao-Nakra模型在特定材料参数选择下不具备渐近稳定性，而静电型Rao-Nakra模型仅需3个状态反馈控制器即可实现指数稳定性（采用谱乘子法证明）。类似地，全动力学Mead-Marcus模型对部分解不具备渐近稳定性，而静电型Mead-Marcus模型仅需1个仅作用于弯矩的反馈控制器即可实现指数稳定性。
英文摘要：
A cantilevered piezoelectric smart composite beam, consisting of perfectly bonded elastic, viscoelastic and piezoelectric layers, is considered. The piezoelectric layer is actuated by a voltage source. Both fully dynamic and electrostatic approaches, based on Maxwell's equations, are used to model the piezoelectric layer. We obtain (i) fully-dynamic and electrostatic Rao-Nakra type models by assuming that the viscoelastic layer has a negligible weight and stiffness, (ii) fully-dynamic and electrostatic Mead-Marcus type models by neglecting the in-plane and rotational inertia terms. Each model is a perturbation of the corresponding classical smart composite beam model. These models are written in the state-space form, the existence and uniqueness of solutions are obtained in appropriate Hilbert spaces. Next, the stabilization problem for each closed-loop system, with a thorough analysis, is investigated for the natural $B^*-$type state feedback controllers. The fully dynamic Rao-Nakra model with four state feedback controllers is shown to be not asymptotically stable for certain choices of material parameters whereas the electrostatic model is exponentially stable with only three state feedback controllers (by the spectral multipliers method). Similarly, the fully dynamic Mead-Marcus model lacks of asymptotic stability for certain solutions whereas the electrostatic model is exponentially stable by only one state feedback controller.stable by one feedback controller applied to the bending moment only.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
