The Trigonometric Zigzag theory is utilized in this research for analytically evaluating the forced vibration responses of smart multilayered laminated composite plates with piezoelectric actuators and sensors. This theory, as the name suggests, incorporates a trigonometric function, namely the secant function for describing the nonlinear behavior of transverse shear stresses through the thickness of the smart composite plates. The kinematics for the in-plane displacement components are obtained by superposing a globally varying nonlinear field through the thickness of the plate structure on a piecewise linearly varying zigzag field with slope discontinuities at the layer interfaces. The model also satisfies the inter-laminar continuity conditions of tractions at the interfaces of the multilayered plate. The equations of motion are derived using Hamilton’s principle, and the separation of the variables technique is extended to assume the solutions for the primary variables in space and time and solved analytically using Navier’s solution technique along with Newmark’s time integration scheme. A detailed analytical investigation of the dynamic behavior of the smart laminated plate coupled with piezoelectric materials like PVDF and piezoelectric fiber-reinforced composite (PFRC) is carried out by considering several forms of the time-dependent electromechanical excitations and also covering different geometrical and material features of the smart plate structure. The responses are found to be in close agreement with the elasticity solutions and some new results are also presented to show the dynamic controlling capacity of the piezoelectric layers.

中文翻译：

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使用三角锯齿理论的智能复合板的受迫振动响应

本研究利用三角之字形理论分析评估具有压电致动器和传感器的智能多层层压复合板的受迫振动响应。顾名思义，该理论包含一个三角函数，即用于描述通过智能复合板厚度的横向剪切应力的非线性行为的正割函数。平面内位移分量的运动学是通过将通过板结构厚度的全局变化非线性场叠加在层界面处具有斜率不连续的分段线性变化之字形场上来获得的。该模型还满足多层板界面处牵引的层间连续性条件。运动方程是利用汉密尔顿原理推导出来的，变量分离技术被扩展为假设空间和时间上的主要变量的解，并使用纳维尔解技术和纽马克时间积分方案进行解析求解。通过考虑多种形式的时间相关机电激励并涵盖不同的几何和材料特征，对智能层压板与压电材料（如 PVDF 和压电纤维增强复合材料（PFRC））的动态行为进行了详细的分析研究的智能板结构。发现响应与弹性解非常一致，并且还提出了一些新结果以显示压电层的动态控制能力。