Nonlinear internal resonances of a microbeam array near the pull-in point
Stefanie Gutschmidt, Oded Gottlieb
The dynamic response of parametrically excited microbeam arrays is
governed by nonlinear effects which directly influence their
performance. To date, documented theoretical research consists of
nonlinear lumped-mass models. While a lumped-mass approach is
useful for a qualitative understanding of the system response it
does not resolve the spatio-temporal interaction of the individual
elements in the array. Thus, we employ a consistent nonlinear
continuum model to investigate the nonlinear dynamic behavior of
an array of N nonlinearly coupled microbeams near the array's
pull-in point. The region near the pull-in point is shown to be
governed by several internal 3:1 and combination resonances. The
nonlinear equations of motion for a two beam system are solved
using the asymptotic multiple-scales method for the weakly
nonlinear system. The analytically obtained periodic response of
two coupled microbeams is verified numerically and complemented by
a numerical analysis of a three beam array which exhibits
quasiperiodic response and lengthy chaotic transients.