A cantilever micro-resonator electrostatically actuated near half of the natural frequency is investigated. Hamilton’s principle is used to derive the partial-differential equation of motion for a general non-uniform sensor. Nonlinearities arise due to the electrostatic and Casimir forces. The electrostatic actuation introduces parametric coefficients in both linear and nonlinear parts of the governing equation. A direct approach is taken using the method of multiple scales resulting in a phase-amplitude relationship for the system. Numerical results for a uniform capacitive resonator micro-sensor are provided and tested numerically using a reduced-order model of the governing equation of motion.

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