Vibration mode frequency formulae for micromechanical scanners

A torsional scan mirror suspended with two flexure beams can be used in various display, imaging and other scanning applications. Using various mirror shapes and flexure dimensions as parameters, a set of analytical formulae is presented to predict the natural frequency of the first five vibration modes, which are torsion, in-plane and out-of-plane sliding modes and in-plane and out-of-plane rocking modes. Mode frequencies are compared with the finite element model (FEM) predictions using ANSYS™ for a wide range of flexure beam dimensions. The formulae include the effective inertia of the flexure beams and orthotropic material anisotropy effects. The analytical formulae are verified for both isotropic (e.g. steel) and orthotropic (e.g. silicon) materials. These formulae work very well when the Euler–Bernoulli beam theory assumptions and the rigid mirror assumption are satisfied. The accuracy of analytical predictions is improved by introducing an empirical correction factor to the analytical predictions using non-dimensional flexure beam ratios. The correction factor reduces the error between analytical formulae and FEM predictions to within a few per cent for all five modes for a large range of flexure dimensions. FEM predictions and analytical formulae are partly verified by experimental results.