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-length
g-gravity
α-damping term
I-applied torque
Result: v’=-g*sin(θ)/L
θ‘=v
Methods
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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
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Jacobian:
Trace=- α
Determinant=cos(θ)
Vary α from positive to zero to negative
The Simple Pendulum with Constant
Torque and No Damping
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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
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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
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Thank You!
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