Nonlinear vibration analysis of hyperelastic and dielectric microbeams with a control parameter using nonlinear normal modes

This study focuses on applying a nonlinear normal modes (NNM) method to analyze the nonlinear vibrations and static bifurcation of hyperelastic and dielectric microbeams in the presence of a control parameter. The microbeam mathematical model is based on the Euler–Bernoulli beam theory and von Kármán nonlinearity, and the strain energy of hyperelastic material is formulated according to the Yeoh model. The applied NNM method is based on an asymptotic approach to determine an invariant manifold. To achieve a reduced-order model, a master mode is selected and the effect of other modes is included through the manifold. The research encompasses three case studies. The first case examines a non-sandwiched hyperelastic microbeam subjected to an axial compressive force as the control parameter. The second and third case studies investigate hyperelastic dielectric microbeams sandwiched between conductive electrode layers. In the second case, the electrodes are made of hydrogel based on the Mooney–Rivlin hyperelastic model, while in the third case, the electrodes are made of aluminum. In both cases, voltage serves as the control parameter. In all three cases, static bifurcation diagrams are depicted and critical values of the control parameters are identified. Vibration analysis is performed for sub-critical control parameter values, where vibrations occur around the unbuckled state of the microbeam, and the effect of the control parameter on the vibration responses and frequencies is investigated. The first case study compares the accuracy of the NNM method with three-mode and single-mode Galerkin solutions, demonstrating close alignment with the three-mode response. The third case highlights the ability of the NNM method to detect internal resonance. Overall, this study highlights the advantages of NNM-based reduced-order models for efficiently predicting the dynamic behavior of hyperelastic microbeams in MEMS applications, providing useful insights for the design and optimization of hyperelastic MEMS devices.