Investigation of 3:1 Internal Resonance of Electrostatically Actuated Microbeams With Flexible Supports

Interaction between modes due to internal resonance has many applications in MEMS devices. In this paper, we investigate the modal interaction through 3 : 1 internal resonance of an electrostatically actuated microbeam with flexible supports in the form of rotational and transversal springs. The static displacement and the first three modal frequencies are obtained at the applied DC voltage by a reduced order model for a specified ratio of electrode gap and thickness. We then obtain the value of applied voltage for which 3 : 1 internal resonance exists for four different combinations of unequal end support stiffnesses. We calculate the coefficients of the coupled dynamical equations of first two modes for all the four cases and solve them by using numerical time integration and the method of multiple scales. We observe the interaction between the first and the second mode when each of the modes is independently excited by an external source. When the second mode is externally excited, interestingly, we also find that the undriven mode response amplitude is twice that of the driven mode.