Bouncing Bob Pendulum

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This Demonstration simulates the motion of a double pendulum with the upper bob bouncing back from an obstruction along the vertical axis.

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Mathematica's built-in function WhenEvent, triggered whenever the upper bob crosses the vertical, reverses and reduces the angular speed of the upper pendulum and changes the behavior of the lower pendulum accordingly.

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Contributed by: Erik Mahieu (November 2013)
With additional contributions by: Franz Brandhuber

Open content licensed under CC BY-NC-SA


Snapshots


Details

In this Demonstration, the upper arm has both mass and length set to 1. The lower arm has mass and length . The angular positions of the arms are and .

Lagrangian mechanics can be used to derive the equations of motion of the double pendulum. The potential energy and kinetic energy of the pendulum are

,

.

Adding the kinetic energy and subtracting the potential energy gives the Lagrangian, .

Substituting this into the Euler–Lagrange equations results in the equations of motion:

,

xx .

During the solution of these differential equations, the event θ[t]⩵0 triggers WhenEvent, resulting in an action for each of the angular positions of the bobs: is replaced by and is replaced by , where is the rebound ratio.

The formula for the rebound actions was taken from [1].

Snapshot 3 shows that for a 100% elastic impact (), the total energy stays constant during the animation.

Reference

[1] S. Timoshenko and D. H. Young, "Lagrangian Equations for Impulsive Forces," Advanced Dynamics, New York: McGraw-Hill, 1948 pp. 225–230.



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