The Effects of Noise on Parametrically Excited Systems with Nonlinear Damping
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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.