We present an alternative to averaging methods for vibrational control design of second-order systems. This method is based on direct application of the stability map of the linearization of the system at the desired operating point. The paper focuses on harmonic forcing, for which the linearization is Mathieu’s equation, but somewhat more general periodic forcing functions may be handled. When it is applicable, this method achieves significantly greater functionality than previously reported approaches. This is demonstrated on two sample systems. One is the vertically driven inverted pendulum, and the other is an input-coupled bifurcation control problem arising from electrostatic MEMS comb drives.

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