  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.
