In this study, the nonlinear vibration control of functionally graded laminated piezoelectric cylindrical shells under simultaneous parametric axial and radial external excitations is presented. The partial differential equations of shells are derived based on Hamilton’s principle, first-order shear deformation theory (FSDT), and nonlinear von Karman relations. The coupled nonlinear ordinary differential equations are obtained by Galerkin’s procedure and solved by the method of static condensation. Two piezoelectric layers are placed on the outer and inner surfaces of the cylindrical shell each as distributed sensor and actuator. Then the constant-gain negative velocity feedback strategy is employed. Regarding the nonlinear equations of motion, for the first time, the vibration analysis and active vibration control of smart FG cylindrical shells under combined parametric and external excitations are analyzed using the multiple scales approach. The effects of various parameters such as power index, external excitation’s amplitude, and control gain on the dynamic behavior of the system are investigated, using bifurcation diagrams, phase portraits, time histories, and Poincare maps. It is shown that quasi-periodic motion is the most common behavior of the system and controller gain and power index have inevitable effects on enhancing the quasi-periodic response of the system. Care should be exerted in selecting the parameters to have the desired response in the broad range of excitation frequency.

中文翻译：

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带有压电层的FG圆柱壳的非线性动力学分析与控制

在这项研究中，提出了同时参数轴向和径向外部激励下功能梯度层压压电圆柱壳的非线性振动控制。壳的偏微分方程是基于汉密尔顿原理、一阶剪切变形理论（FSDT）和非线性von Karman关系推导出来的。耦合非线性常微分方程由Galerkin程序得到，并采用静凝聚法求解。两个压电层放置在圆柱壳的外表面和内表面上，每个作为分布式传感器和致动器。然后采用恒定增益负速度反馈策略。关于非线性运动方程，第一次，采用多尺度方法分析了参数和外部联合激励下智能FG圆柱壳的振动分析和主动振动控制。使用分岔图、相图、时间历程和庞加莱图研究了功率指数、外部激励幅度和控制增益等各种参数对系统动态行为的影响。结果表明，准周期运动是系统最常见的行为，控制器增益和功率指标对增强系统的准周期响应具有不可避免的作用。在选择参数时应小心谨慎，以便在宽广的激励频率范围内获得所需的响应。使用分岔图、相图、时间历程和庞加莱图研究了功率指数、外部激励幅度和控制增益等各种参数对系统动态行为的影响。结果表明，准周期运动是系统最常见的行为，控制器增益和功率指标对增强系统的准周期响应具有不可避免的作用。在选择参数时应小心谨慎，以便在宽广的激励频率范围内获得所需的响应。使用分岔图、相图、时间历程和庞加莱图研究了功率指数、外部激励幅度和控制增益等各种参数对系统动态行为的影响。结果表明，准周期运动是系统最常见的行为，控制器增益和功率指标对增强系统的准周期响应具有不可避免的作用。在选择参数时应小心谨慎，以便在宽广的激励频率范围内获得所需的响应。结果表明，准周期运动是系统最常见的行为，控制器增益和功率指标对增强系统的准周期响应具有不可避免的作用。在选择参数时应小心谨慎，以便在宽广的激励频率范围内获得所需的响应。结果表明，准周期运动是系统最常见的行为，控制器增益和功率指标对增强系统的准周期响应具有不可避免的作用。在选择参数时应小心谨慎，以便在宽广的激励频率范围内获得所需的响应。