ESC 250 PROJECT 2: DOUBLE PENDULUM

Problem Statement:

The equations of motion for the system shown below (assuming small angle displacements) are given

by:

(𝑚1 + 𝑚2)𝑙1 2�̈�1 + 𝑚2𝑙1𝑙2�̈�2 + (𝑚1 + 𝑚2)𝑙1𝑔𝜃1 = 0 1)

𝑚2𝑙2 2�̈�2 + 𝑚2𝑙1𝑙2�̈�1 + 𝑚2𝑙2𝑔𝜃2 = 0 2)

Solve the system for exact solutions of 𝜃1 𝑎𝑛𝑑 𝜃2 via Laplace Transform and numerically approximate

the solution for 𝜃1(3), 𝜃2(3) using your own 4 th-order RK method and a step size of 0.25 for the

following conditions:

𝑚1 = 𝑚2 = 1 𝑘𝑔

𝑙1 = 4 𝑐𝑚, 𝑙2 = 8 𝑐𝑚

𝜃1(0) = 1°, 𝜃2(0) = −1°

�̇�1(0) = �̇�2(0) = 0°/𝑠

𝑔 = 9.81 𝑚2/𝑠

Generate a report showing your solution development for both methods. Include plots of the response

for 𝜃1, 𝜃2, �̇�1, �̇�2, �̈�1, 𝑎𝑛𝑑 �̈�2 from t = 0-2 sec for both methods (plot both methods on the same graph for

each position , velocity, and acceleration). Submit your report (as a pdf) and your Matlab script to BB.

𝜃1 𝑚1

𝜃2

𝑚2