Vibrational modes in MEMS resonators

Advances in microfabrication technology have enabled micromechanical systems (MEMS) to become a core component in a manifold of applications. For many of these applications the vibrational eigenmodes of a MEMS resonator play a crucial role. However, despite this wide-spread use of the notion of vibrational modes, the general properties of vibrational modes are not well known. In this review, we aim to provide a general overview of various aspects of vibrational modes in MEMS resonators. Vibrational eigenmodes are a type of solution to deformation problems in linear elasticity which give rise to a spatial and a temporal eigenvalue problem coupled by a common eigenvalue. The spatial eigenvalue problem determines a mode shape while the temporal eigenvalue problem defines an eigenfrequency at which the structure vibrates. To reduce the modelling complexity, we introduce different simplified elastic models for different eigenmodes like flexural modes in beams and strings in one dimension or plates and membranes in two dimensions. Even though very different eigenmode solutions can be found in MEMS resonators, the dynamics of vibrational modes are governed by generic laws. In the linear regime these laws connect vibrational eigenmodes to the phenomenon of resonance. In the nonlinear regime also other phenomena like bifurcations, multi-stability or parametric amplification are observed. Besides general properties of vibrational eigenmodes, we discuss how different aspect of vibrational modes are utilized in different applications. MEMS timing devices can be optimized by exploring the damping in different eigenmodes and resonant amplification is utilized in the measurement of acceleration, mass or fluid properties. While measurements with MEMS resonators are often performed in the linear regime, the modal dynamics in atomic force microscopy are inherently nonlinear. Beyond classical mechanics, we discuss how vibrational modes in MEMS resonators recently entered the quantum world.