Interactions between directly and parametrically driven vibration modes   in a micromechanical resonator
  The interactions between parametrically and directly driven vibration modes of a clamped-clamped beam resonator are studied. An integrated piezoelectric transducer is used for direct and parametric excitation. First, the parametric amplification and oscillation of a single mode are analyzed by the power and phase dependence below and above the threshold for parametric oscillation. Then, the motion of a parametrically driven mode is detected by the induced change in resonance frequency in another mode of the same resonator. The resonance frequency shift is the result of the nonlinear coupling between the modes by the displacement-induced tension in the beam. These nonlinear modal interactions result in the quadratic relation between the resonance frequency of one mode and the amplitude of another mode. The amplitude of a parametrically oscillating mode depends on the square root of the pump frequency. Combining these dependencies yields a linear relation between the resonance frequency of the directly driven mode and the frequency of the parametrically oscillating mode. 
H. J. R. Westra and D. M. Karabacak and S. H. Brongersma and M.   Crego-Calama and H. S. J. van der Zant and W. J. Venstra
cond-mat.mes-hall