In this paper, we investigated the realization of all possible internal resonance conditions between the first three modes of an electrostatically actuated, straight, clamped-hinged microbeam. We first obtained the static displacement and the first three natural frequencies around the deflected shape of the beam by using Galerkin based reduced order model and the finite element method. We then found all six possible commensurable relations between these frequencies as a function of two non-dimensional parameters (\(\alpha _1\) and \(\alpha _2 V_{\mathrm{dc}}^2\)) that depend on beam dimensions, material properties, and external forcing. Furthermore, we also examined the governing equations to check the feasibility of dynamic modal interaction. We found that although dynamical modal coupling was theoretically possible for all resonance conditions upon external excitation, the 3:1 internal resonance between the first two modes seemed to be the most experimentally feasible. Hence, we carried out a detailed forced vibrations analysis corresponding to this condition by solving the governing nonlinear equations using numerical time integration and the method of multiple scales. The system exhibited dynamically rich internal resonance behavior that can be controlled with the two non-dimensional parameters (\(\alpha _1\) and \(\alpha _2 V_{\mathrm{dc}}^2\)) and external damping. Specifically, we observed saddle node bifurcations, jump conditions, and internal loops in the amplitude response curves as a function of excitation frequency and amplitude. Overall, our work provides a systematic methodology and simple rules for in-depth exploration of internal resonance in microbeams. The findings of this paper could also assist in the development of sensors based on internal resonance.

在本文中，我们研究了静电驱动的直夹铰链微梁的前三种模式之间所有可能的内部共振条件的实现。我们首先使用基于伽辽金的降阶模型和有限元方法获得了梁偏转形状周围的静态位移和前三个固有频率。然后，我们发现这些频率之间所有六种可能的可公度关系是两个无量纲参数（\(\alpha _1\)和\(\alpha _2 V_{\mathrm{dc}}^2\)）的函数) 取决于梁尺寸、材料属性和外力。此外，我们还检查了控制方程以检查动态模态相互作用的可行性。我们发现，尽管理论上动态模态耦合对于外部激励下的所有共振条件都是可行的，但前两种模式之间的 3:1 内部共振似乎在实验上是最可行的。因此，我们通过使用数值时间积分和多尺度方法求解控制非线性方程，对这种情况进行了详细的受迫振动分析。该系统表现出动态丰富的内部共振行为，可以通过两个无量纲参数（\(\alpha _1\)和\(\alpha _2 V_{\mathrm{dc}}^2\) 进行控制）) 和外部阻尼。具体而言，我们观察到作为激励频率和幅度函数的幅度响应曲线中的鞍状节点分叉、跳跃条件和内部回路。总的来说，我们的工作为深入探索微梁内部共振提供了系统的方法和简单的规则。本文的发现还有助于开发基于内部共振的传感器。