M.Sc Thesis

M.Sc StudentMusnikov Lior
SubjectNonlinear Dynamics of Cantilevers
DepartmentDepartment of Mechanical Engineering
Supervisor PROF. David Elata
Full Thesis textFull thesis text - English Version


Microelectromechanical (MEMS) resonators are a building block in communication devices and in various sensors. Resonators are commonly used as filters, oscillators, and sensors. MEMS resonators are becoming increasingly prevalent due to their superior performance and high quality-factor. Seemingly, the simplest structure of a MEMS resonator is a cantilever beam that is electrostatically actuated and sensed. In this research we investigate the dynamic response of a cantilever beam, and specifically focus on the dependence of its natural frequency on the amplitude of its vibration. Additionally, we examine the effect of geometric parameters and material moduli on the natural frequency of cantilevers. To do so, we begin by analyzing a one-dimensional (1-D) model of the vibrating beam, and then we use insights gained from this analysis to expand our modeling by considering 2-D and 3‑D models of the beam. We study in each step how the beam inertia and stiffness are affected, and how they affect the natural frequency.

The natural frequency of vibrating cantilevers is hardly affected by the amplitude of the tip vibration. However, for resonators and oscillators even a small change in natural frequency may be relevant. For example, a 10-6 relative change in frequency suggests an error of 1 second per 12 days, where a clocking-oscillator is considered.

We perform our investigation using analytic derivation of exact solutions, and approximate numerical solutions implemented in finite differences and in finite elements codes. Specifically, we present an algorithm for calculating the natural frequency of the cantilever vibrations based on an analysis of an equivalent static state. We validate our analysis using a full dynamic time-dependent simulation and present the advantages of our suggested algorithm over the time-dependent simulation.