Scale dependence in the oscillatory control of micromechanisms

This paper is concerned with the way in which nonlinear effects of oscillatory forcing depend on characteristic length scales. While vibrating beams, plates, and membranes as well as electrostatic comb motors have been incorporated in a wide variety of devices, connections with recent work in the nonlinear control theory of systems with oscillatory inputs have remained unexplored. We summarize recent developments in the control of mechanical systems with oscillatory inputs and pay particular attention to how these results depend on scale. For a number of applications, Lagrangian and Hamiltonian models capture the essential qualitative features of the dynamics of small-scale mechanical systems. The nonlinear response of such models to oscillatory forcing involves a nonintegrable relationship between inputs and configuration variables, and recent research has gone a long way toward characterizing this relationship in terms of the critical point structure of an energy-like function called the averaged potential. Part of the aim of the present paper is to show how the stability and bifurcation theory of the averaged potential depends on mechanism length scales. In addition to discussing scale dependence, we shall describe how to apply our theory to models which incorporate the effects of “stick-slip” friction at small scales