The Dynamics of the Pendulum

By Tori Akin and Hank Schwartz

An Introduction

• What is the behavior of idealized pendulums?• What types of pendulums will we discuss?– Simple– Damped vs. Undamped– Uniform Torque– Non-uniform Torque

Parameters To Consider

m-mass (or lack thereof)L-lengthg-gravityα-damping termI-applied torqueResult: v’=-g*sin(θ)/L

θ‘=v

Methods

• Nondimensionalization• Linearization• XPP/Phase Plane analysis• Bifurcation Analysis• Theoretical Analysis

Nondimensionalization

• Let ω=sqrt(g/L) and dτ/dt= ω• θ‘=v→v• v’=-g*sin(θ)/L →-sin(θ)

Systems and Equations

• Simple Pendulum– θ‘=v– v‘=-sin(θ)

• Simple Pendulum with Damping– θ‘= v– v‘=-sin(θ)- αv

• Simple Pendulum with constant Torque– θ‘= v– v‘=-sin(θ)+I

Hopf Bifurcation

• Simple Pendulum with Damping– θ‘= v– v‘=-sin(θ)- αv

• Jacobian: • Trace=- α• Determinant=cos(θ)• Vary α from positive to zero to negative

The Simple Pendulum with Constant Torque and No Damping

• The theta null cline: v = 0• The v null cline: θ=arcsin(I) • Saddle Node Bifurcation I=1• Jacobian:

• θ‘= v• v‘=-sin(θ)+I

Driven Pendulum with Damping• θ’ = v• v’ = -sin(θ) –αv + I• Limit Cycle• The theta null cline: v = 0• The v null cline: v = [ I – sin(θ)] / α• I = sin(θ) and as• cos2(θ) = 1 – sin2(θ) we are left with• cos(θ) = ±√(1-I2)• Characteristic polynomial- λ2 + α λ + √(1-I2) = 0 which

implies λ = { ‒α±√ [α2- 4√(1-I2) ] } / 2• Jacobian:

Homoclinic Bifurcation

Infinite Period Bifurcation

Bifurcation Diagram

Non-uniform Torque and Damped Pendulum

• τ’ = 1• θ’ = v• v’ = -sin(θ) –αv + Icos(τ)

Double Pendulum

Results

• Basic Workings Various Oscillating Systems• Hopf Bifurcation-Simple Pendulum• Homoclinic Global Bifurcation-Uniform Torque• Chaotic Behavior• Saddle Node Bifurcation• Infinite Period Bifurcation• Applications to the real world

Thank You!