Nonlinear coupling between modes through internal resonance has many applications in MEMS resonators. The primary objective of this work is to systematically study all possible internal resonance conditions between the first three modes and the feasibility of mode interaction in an electrostatically excited, straight, clamped-clamped beam. The static displacement and the first three natural frequencies of the beam are obtained for an applied DC voltage by using Galerkin based reduced order model and finite element method. All six possible commensurable relations between the first three frequencies of the beam are obtained by sweeping the non-dimensional parameters (\(\alpha _1\) and \(\alpha _2 V_\mathrm{dc}^2\)) which depend on beam dimensions, material properties and external forcing. It is also demonstrated by a further examination of the dynamical equations that only one resonance condition is capable of exhibiting modal coupling when externally excited. A detailed analysis is carried out for this feasible resonance condition by altering \(\alpha _1\) and \(\alpha _2 V_\mathrm{dc}^2\) and solving the relevant nonlinear coupled equations by using both numerical time integration and the method of multiple scales. From this study, we observe that the lower mode is automatically excited after driving amplitude for the higher mode reaches a critical value—a sign of mode interaction initiation. We also find that amplitude of the higher mode is saturated after interaction with the lower mode for a combination of \(\alpha _1\) and \(\alpha _2 V_\mathrm{dc}^2\). Moreover, we see that the frequency bandwidth of modal interaction increases with excitation amplitude for all combinations of \(\alpha _1\) and \(\alpha _2 V_\mathrm{dc}^2\). Finally, the role of external damping on the amplitude-frequency response curve of both modes during the mode interaction is also investigated.

通过内部谐振的模式之间的非线性耦合在MEMS谐振器中有许多应用。这项工作的主要目的是系统地研究前三种模式之间的所有可能的内部共振条件，以及在静电激发，笔直，夹紧的束中进行模式相互作用的可行性。通过使用基于Galerkin的降阶模型和有限元方法，获得了施加直流电压时梁的静态位移和梁的前三个固有频率。通过扫描无量纲参数（\（\ alpha _1 \）和\（\ alpha _2 V_ \ mathrm {dc} ^ 2 \）获得波束的前三个频率之间的所有六个可能的可衡量关系。），取决于梁的尺寸，材料特性和外力。通过对动力学方程的进一步检查还表明，在外部激励时，只有一个共振条件能够表现出模态耦合。通过更改\（\ alpha _1 \）和\（\ alpha _2 V_ \ mathrm {dc} ^ 2 \），对该可行的谐振条件进行了详细的分析。并通过数值时间积分和多尺度方法求解相关的非线性耦合方程。从这项研究中，我们观察到较高模式的驱动幅度达到临界值后，较低模式会自动激发，这是模式交互作用启动的标志。我们还发现，对于\（\ alpha _1 \）和\（\ alpha _2 V_ \ mathrm {dc} ^ 2 \）的组合，较高模式的振幅在与较低模式进行交互之后达到饱和。此外，我们看到，对于\（\ alpha _1 \）和\（\ alpha _2 V_ \ mathrm {dc} ^ 2 \）的所有组合，模态交互作用的带宽随着激励幅度的增加而增加。。最后，还研究了在模态相互作用过程中外部阻尼对两种模态的幅频响应曲线的作用。