Studies of parametrically excited non-linear MDOF systems at parametric resonances

An increasing number of MEMS devices use parametric excitation (PE) to outperform conventional designs. These systems are operated at parametric resonance. Different to standard cases of resonance, at parametric resonances the vibration's amplitudes increase faster and within a smaller interval. The amplification is much larger and only limited by non-linearities. So far the focus has mostly been on one degree of freedom (1DOF) systems. This is partly caused by a lack of methods to investigate and design multi degree of freedom (MDOF) systems time-efficiently. Restricting the systems to 1DOF ignores the opportunity to make use of PE effects only available in MDOF systems: parametric combination resonances and parametric anti-resonances, where an enhanced damping behaviour can be observed. The paper demonstrates how to approximate non-linear MDOF PE systems with 1DOF models. This leads to a generalised, dimensionless model applicable to many systems. Approaches are presented for investigating such a reduced non-linear 1DOF PE model analytically and semianalytically at parametric resonances using averaging methods. For a 2DOF system the results are validated numerically by continuation methods and time simulations. Limits of both the analytical and the semi-analytical approaches are discussed.