Math 260 Lab #7 Overdamped pendulum

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Email your completed file to tleise@amherst.edu by Monday 4pm.

A simple mechanical example of a nonlinear oscillator is a pendulum. Let L be the length of the pendulum, m the mass, and g the acceleration due to gravity. Suppose a constant torque Γ is applied at the pendulum’s pivot point to keep it in motion.

We will assume lots of damping is present, for example, the pendulum could be immersed in a viscous fluid that hinders the swinging motion of the “bob”, so that we can neglect the second derivative term (the acceleration), just as we did with the overdamped bead on a rotating hoop. In this situation, the faster we try to move the bob, the more the fluid resists the motion. We let b be the damping coefficient.

 The DE describing this system is

(mL^2)(Overscript[θ, ¨]) + b(OverDot[θ]) + mgLsinθ = Γ

Assume all of the parameters are positive, including the torque (so it moves the pendulum counterclockwise).

Note: Energy is not conserved in this system, as the damping drains energy and the applied torque adds energy.

(Image from Wikipedia entry on the pendulum.)

Exercise 1. Apply a change of variables τ=t/T to the DE. Determine what T should be to put the DE into the form

ϵ(((d^2)θ)/(d(τ^2)))+((dθ)/(dτ))+sinθ=γ

What are ϵ and γ in terms of the original parameters?

Exercise 2. What must be true about the original parameters m, L, b, and g for ϵ << 1? (So that we can neglect the second derivative term.)

Exercise 3. 
(a) What are the fixed points of the reduced system ((dθ)/(dτ))=γ-sinθ? (We are working on the circle here.)

(b) For what range of values of γ do they exist? What is the stability of each fixed point?

(c) A saddle-node bifurcation occurs here at what value of γ? What is the value of theta at this bifurcation?

(d) What happens qualitatively as γ>1 decreases toward 1? You should observe a bottleneck occurring -- at what angle does the motion slow?

The app below show solutions to the reduced system for different values of γ, so you can see what happens. Press the play button (triangle) for tt to get a movie of the pendulum’s motion (the bob is the red dot).

Manipulate[sol=NDSolve[{D[x[t],t]==γ-Sin[x[t]],x[0]==4},x[t],{t,0,50}];
Show[ParametricPlot[{s Cos[x[t]-Pi/2],s Sin[x[t]-Pi/2]}/.sol/.t->tt,{s,0,1}],ListPlot[{Cos[x[t]-Pi/2],Sin[x[t]-Pi/2]}/.sol/.t->tt,PlotStyle->{Red,PointSize[Large]}],PlotRange->{{-1.05,1.05},{-1.05,1.05}}],{γ,0,2},{tt,0,50}]

Please note that in this movie, theta = 0 is at the bottom, to imitate where the pendulum bob would be when theta = 0.