The Effects of Noise on Parametrically Excited Systems with Nonlinear Damping
Abstract
Parametric oscillators are a class of resonating systems in which a parameter, such as
stiffness in a mechanical system or capacitance in an electrical system, is periodically
modulated in order to alter the system response in a desired manner. A resonance effect
occurs when the pump frequency is near twice of resonant frequency. Such systems
are said to be “parametrically pumped,” and this pump can, above a certain amplitude
threshold, destabilize the system in the absence of nonlinearities. Parametric resonance
is widely observed in nature and has been employed in a large variety of engineered
systems, most notably in micro-electro-mechanical systems (MEMS). Considering both
open and closed loop operations of a parametric oscillator, this work expands on previous
studies by embracing nonlinear damping and multiplicative noise in the modeling
and analysis and investigates their effects. Fluctuations due to noise, signal-to-noise
ratio (SNR), and power spectral density (PSD) for an open loop system are computed
and are compared with stochastic simulations. Phase diffusion for a phase-locked loop
(PLL) is also analyzed, which plays a pivotal role in time-keeping devices. The main
conclusions are relevant to SNR aspects of sensors and to the frequency stability of
time-keeping systems. It is shown that multiplicative noise serves as an ultimate limiting
factor in the resolution of sensors and precision of clocks.