Date of Award


Document Type


Degree Name

Master of Science (MS)


Mechanical and Civil Engineering

First Advisor

Steven W. Shaw

Second Advisor

Brian A. Lail

Third Advisor

Hector Gutierrez

Fourth Advisor

Ashok Pandit


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.