The Dynamics of the Pendulum By Tori Akin and Hank Schwartz.

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The Dynamics of the Pendulum By Tori Akin and Hank Schwartz
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Transcript of The Dynamics of the Pendulum By Tori Akin and Hank Schwartz.

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  • The Dynamics of the Pendulum By Tori Akin and Hank Schwartz
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  • 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
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  • Parameters To Consider m-mass (or lack thereof) L-length g-gravity -damping term I -applied torque Result: v=-g*sin()/L =v
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  • Methods Nondimensionalization Linearization XPP/Phase Plane analysis Bifurcation Analysis Theoretical Analysis
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  • Nondimensionalization Let =sqrt(g/L) and d /dt= =vv v=-g*sin()/L -sin()
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  • Systems and Equations Simple Pendulum =v v=-sin() Simple Pendulum with Damping = v v=-sin()- v Simple Pendulum with constant Torque = v v=-sin()+I
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  • Hopf Bifurcation Simple Pendulum with Damping = v v=-sin()- v Jacobian: Trace=- Determinant=cos( ) Vary from positive to zero to negative
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  • The Simple Pendulum with Constant Torque and No Damping The theta null cline: v = 0 The v null cline: =arcsin(I) Saddle Node Bifurcation I=1 Jacobian: = v v=-sin()+I
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  • Driven Pendulum with Damping = v v = -sin() v + I Limit Cycle The theta null cline: v = 0 The v null cline: v = [ I sin()] / I = sin() and as cos 2 () = 1 sin 2 () we are left with cos() = (1-I 2 ) Characteristic polynomial- 2 + + (1-I 2 ) = 0 which implies = { [ 2 - 4(1-I 2 ) ] } / 2 Jacobian:
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  • Homoclinic Bifurcation
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  • Infinite Period Bifurcation
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  • Bifurcation Diagram
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  • Non-uniform Torque and Damped Pendulum = 1 = v v = -sin() v + Icos()
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  • Double Pendulum
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  • 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!