In this paper, the vibrational behavior of a functionally graded (FG) micro-resonator is investigated for three different configurations based on the Euler–Bernoulli beam model and the Von Karman nonlinear strain assumption. The three configurations include: (1) an FG micro-resonator with a fixed foundation, (2) piezoelectric layers added to the first model, and (3) a second fixed foundation added to the first model. These investigations are performed under the effect of electrostatic forces, the forces caused by the deformations of piezoelectric layers due to the applied voltage, Casimir forces, and uniform temperature changes. The equations governing the vibrational behaviors are obtained using the Hamilton’s principle and the modified couple stress theory. Static deformations and the fundamental vibrational mode are calculated using the differential quadrature method in the spatial domain for all three configurations. Furthermore, the dynamic equations are obtained from the Galerkin discretization and solved with multiple time scales technique. The results show that the frequency behavior of a micro-resonator can be adjusted using the three models.

在本文中，基于欧拉-伯努利梁模型和冯卡曼非线性应变假设，针对三种不同的配置，研究了功能梯度（FG）微谐振器的振动行为。这三种配置包括：（1）具有固定基础的FG微谐振器，（2）添加到第一模型的压电层，以及（3）添加到第一模型的第二固定基础。这些研究是在静电力，压电层因施加电压，卡西米尔力和均匀温度变化而变形而产生的力的作用下进行的。利用汉密尔顿原理和改进的耦合应力理论，获得了控制振动行为的方程。对于所有三种配置，在空间域中使用微分正交方法计算静态变形和基本振动模式。此外，从Galerkin离散化获得动力学方程，并用多个时间尺度技术对其求解。结果表明，可以使用这三种模型来调整微谐振器的频率特性。