Instability of a Large Coupled Microbeam Array Initialized at Its Two Ends

Abstract

A simple approximate method is suggested to determine the critical value for instability of a large parallel array of mutually attracting microbeams, based on instability analysis of a small array of only a few microbeams at the ends of the original large array. First, it is verified by a simplified spring system that equilibrium deflections of all intermediate microbeams (except those at the two ends of the parallel array) are negligibly small, and instability of the large microbeam array is initialized at the two ends of the array. Therefore, the critical value for instability of the original large array is determined by the critical value for instability of a small array of only a few microbeams at the two ends with its innermost microbeam fixed. The results obtained for the spring system show that the relative errors in the critical value between the original large array and the substitute small array are less than 2% when only three or four springs at each end are considered. In particular, the relative errors quickly converge to zero when the number of springs considered in the substitute small array further increases. This simple substitution method is used to approximately determine the critical value for instability of a large array of mutually attracting microbeams, and the results are compared with those obtained by other methods based on the instability analysis of the original large array, which contains a large number of microbeams. The present work offers a simple method to reduce the instability analysis of a large array of microbeams to a much simpler problem of a small array of only a few microbeams.