Inverted pendulum-2010

Stabilizing an inverted pendulum

Group members: Yiwei Cheng, Samuel Shapero, M. Robert Hayward

Main presentation. Papers: Yivei, Robert, Samuel

This page is dedicated to the Inverted Pendulum Project. The simulations and laboratory experiments conducted in the inverted perdulum project are part of the requirements of the class: Phys 4267A/6268 Nonlinear Dynamics & Chaos, Spring 2010.

The overarching scientific objective of our research is to provide a coherent picture on stability and bifurcations of an inverted pendulum subjected to forcing that ranges from harmonic to nonharmonic in nature through a combination of numerical studies and laboratory experiments.

Introduction

The simple pendulum has only one stable state of equilibrium, which is the vertically down orientation (a). But if the axis undergoes vertical perturbation, three different equilibrium states emerge: (a) stable down, (b) stable up, and (c) Continuous rotation in either direction. The dynamics of an inverted pendulum while subjected to periodic forcing were investigated numerically and experimentally. The following questions were to be answered.

1. What amplitudes and frequencies of periodic forcing allow the inverted pendulum to remain upright? (i.e. what are the regions of stability?) 2. Does the region of stability change when additional harmonics are added to the forcing function? 3. Can our theoretical findings be validated through laboratory experiments?

Modeling

The following equation of motion was employed and integrated with a fourth order Runge Kutta approximation in matlab to compute numerical solutions.

<math>\frac{\delta v}{\delta t}=-\frac{v}{Q \Omega}-\left[\frac{1}{\Omega^{2}}-\epsilon*cn(t,m)\right]*\sin{\theta}</math>

<math>\frac{\delta \theta}{\delta t}=v</math>, <math>Q = \frac{\omega_{0}I}{b}</math>

where $I$ is moment of inertia, $b$ is damping coefficient, $\Omega$ is forcing frequency relative to natural frequency, $\omega_{0}$, and $cn$ is the Jacobi elliptic function used for non-sinusoidal forcing. In order to find the boundary of stability numerically, we swept systematically through values of $\Omega$. For each value of Ω a binary search was used to narrow in on a value of $\epsilon$ lying on the edge of stability. In order to judge stability an initial condition $(\theta, \dot{\theta}) = (1e − 6, −1e − 6)$ was integrated for 1000 periods of the forcing function and checked for divergence.

Group Member Bios

Samuel Shapero is a second year PhD student in the Bioengineering Interdisciplinary Program at Georgia Tech. His research focuses on implementing a sparsity extraction algorithm for high dimensional inputs using nonlinear analog circuits. He has a Master’s in Electrical Engineering from Stanford, and has a strong background in analog and digital filter analysis. His research advisor is Dr. Paul Hasler.

Robert Hayward is currently working on PhD in Nuclear and Radiological Engineering (expected Fall 2010). He has strong background in modeling and simulation, numerical analysis and data analysis. He has received his Bachelor’s in Cell and Molecular Biology and Bachelor’s in Mathematics from Tulane. He has received his Master’s in Medical Physics from Georgia Tech.

Yiwei Cheng is a third year PhD student in the School of Civil and Environmental Engineering. Yiwei’s main thesis advisor is Dr Marc Stieglitz and his research focuses on understanding how eco-hydrological processes scale. Specifically, whether there are emergent properties that manifest at scale resulting from small scale non-linearity and feedbacks. His research focus areas are in the Arctic and Everglades Florida, where he conducts field experiments and simulations. Yiwei has received his Bachelor’s and Master’s in Civil Engineering from Georgia Tech.