9711

Bouncing Bob Pendulum

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

SNAPSHOTS

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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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