#### Title

Exploiting the Principal Parametric Resonance of an RLC" Circuit for Vibratory Energy Harvesting"

#### Date of Award

12-2016

#### Document Type

Thesis

#### Degree Name

Master of Science (MS)

#### Department

Mechanical Engineering

#### Committee Member

Mohammed F Daqaq, Committee Chair

#### Committee Member

Phanindra Tallapragada

#### Committee Member

Gang Li

#### Abstract

The use of ambient energy sources to independently power small electronic devices, a process commonly known as *energy harvesting,* has recently become a focus of research due to advances in low-power electronic applications. A particular class of energy harvesting devices, known as vibratory energy harvesters (VEHs), utilizes low-level vibrations present in numerous natural and man-made environments to generate electrical energy for electronic devices.

This work investigates the use of a new technique to harvest energy from ambient vibrations by *parametrically* exciting a resonance condition of the electric current in a nonlinear oscillating circuit. To accomplish this parametric resonance phenomenon, we consider an electromechanical coupling device, an oscillating cantilever beam with a ferromagnetic tip mass, which changes the permeability of an iron-alloy cored inductor coil to produce a harmonically-varying modulation of the inductance. Such a type of harvester possesses the potential to generate large amplitude System response that is not limited by the linear damping of the system, as is the case with directly-excited systems, but rather whose behavior is governed by the nonlinearity of the system.

In order to study the ability of such an energy harvesting system to generate electricity when subject to external vibrations, we develop a second-order differential equation to model the theoretical dynamic behavior of a parametrically-driven nonlinear circuit. Due to the complexity of the nonlinear and harmonically-varying components of the governing equation, we use the Method of Multiple Scales to derive an approximate analytical solution for the steady-state current response and output power of the circuit near the principal parametric resonant frequency. We show that the relationship of parameter modulation depth and load resistance characterize the bandwidth of the response and define a critical forcing threshold, below which no energy is harvested. The harvested power is maximized when the load resistance is half of the maximum load resistance at which the critical threshold is still achieved for a given forcing level. We also demonstrate the need for nonlinear damping in the system to attenuate the growth of the response to a physically attainable level. We show the dependence of the natural frequency of the circuit on the parametric forcing parameter, which can lead detuning of the system at different forcing levels.

An experimental set up is developed to test the assertions presented by the analytical model. Numerous parameter constraints are balanced in the experimental design in order to be able to achieve the critical forcing threshold necessary for exciting the parametric resonance condition. The frequency response behavior of the electrical current and load power in the circuit is observed by varying the natural frequency of the system, which is compared against the variation of forcing frequency presented in the theoretical section. The beam is excited at its natural frequency of 85.8 *Hz* across input accelerations ranging from 1:1*g* – 1:5*g*. A maximum output power of 28.67 *mW* across an 8 Ω resistance is achieved at an input acceleration of 1:5*g*. The behavior of the experimental data is in good agreement with the findings of the theoretical model with respect to the bandwidth, nonlinear behavior, and sensitivity to forcing and damping parameters. The analytical model under predicts the peak power measured experimentally, but the general trend is well modeled. Furthermore, several key observations are noted during the experimental procedures, notably the effects of eddy current damping on the behavior of the response and the development of quasiperiodic solutions near the saddle node bifurcation point.

#### Recommended Citation

Caldwell, Nicholas B., "Exploiting the Principal Parametric Resonance of an RLC" Circuit for Vibratory Energy Harvesting"" (2016). *All Theses*. 3041.

https://tigerprints.clemson.edu/all_theses/3041