the dynamics of the pendulum by tori akin and hank schwartz
TRANSCRIPT
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!