Brendan Gramp and Eadom Dessalene

Background

In this project we were tasked with the implementation of a forced damped pendulum and a damped pendulum with an oscillating pivot, along with a double pendulum. We used the Trapezoid Method, which is a second order ODE solver derived from the trapezoidal rule for computing integrals. The pendulum is suspended from a rod, and is under the effects of gravity and other effects such as magnetism and air resistance, as in the case of the forced pendulum.

Exercise 6.3.4

In this section, a forced damped pendulum is constructed. In typical physics, pendulums are all ideal with no friction or damping. However, in reality things always damp down. Damping forces will always oppose the direction of the movement of the pendulum, and can include things such as air resistance or frictional forces. Our pendulum for this part is also forced, meaning the pendulum is also dependent on time. The relevant equations are below:

$$y_1' = y_2 \\ y_2' = -\frac{g}{l} \sin y_1 - dy_2 + A \sin t $$

For this part, we chose the forcing parameter A = 10, and d = 1. We changed the step size from 0.005 to 0.1. Something important to know is that as the step size increases, the error also increases. Therefore, the movement of the pendulum at h = 0.1 is far less accurate than at h = 0.005. We found that at h = 0.005, no matter what initial conditions we set, the pendulum gets to a consistent pattern of about two and a half revolutions in both directions. At h = 0.1, the pendulum exhibits very different behavior compared to h = 0.005.
Code for this part can be seen below : https://github.com/edessa/Math_447/blob/master/Forced_Pendulum.m

Exercise 6.3.5

The purpose of this part is to change A, the forcing parameter, to 12, and to find two initial conditions such that a and b are within 0.1 radians. We found that a = pi/2.5 and b = pi/2.5 - 0.1 illustrated the first solution, and a = pi/2.5 and b = pi/2.5 - 0.1 illustrated the second solution. Included below is a video with a forcing parameter of A = 15. Code for this section is found below : https://github.com/edessa/Math_447/blob/master/Forced_Pendulum_A

Exercise 6.3.6

In this part, we are to implement a damped pendulum with an oscillating pivot, find an accurate range of A values such that the pendulum reaches a stable equilibrium. We iterated through possible A values using initial estimates to find as close as possible to an A value that resulted in equilibrium, using the following code: The relevant equations are below: $$y_1' = y_2 \\ y_2' = -(\frac{g}{l} + A\cos 2 \pi t)\sin y_1 - dy_2 $$ The final range we arrived at is : 18.37 < A < 25.65 using the above method. A video showing the iterative approach is shown below:
Code for this part is found at : https://github.com/edessa/Math_447/blob/master/Equilibrium.m

Exercise 6.3.7

The purpose of this part is to find the smallest value of A such that the pendulum reaches unstable equilibrium. The smallest value of A we found that resulted in unstable equilibrium was A = 13.8 for a step size of 0.0005, but we also found that A values varied depending on the step size chosen. For example, using a step size of 0.01 we found A to be equal to 11.8.

Exercise 6.3.8

Lastly, we were tasked with constructing a double pendulum by concatenating a rod and a bob to the end of the previous rod and bob. A video of its motion can be seen below. We did not attempt to stabilize the double pendulum. The relevant equations are below: $$y_1' = y_2 \\ y_2' = \frac{-3g \sin y_1 - g \sin y_1 - 2y_3 - 2(\sin y_1 - y_3)(y_4^2 + y_2^2\cos y_1 - y_3)}{3 - \cos 2y_1 - 2y_3} \\ y_3' = y_4 $$ Code for this part is found at : https://github.com/edessa/Math_447/blob/master/Double_Pendulum.m

Video for h = 0.005 found below :

Video for h = 0.1 found below:

Video for a = pi/2.5 shown below :

Video for b = pi/2.5 - 0.1 shown below :

Video of chaotic motion at A = 15 shown below :

Video for the iterative process starting at A = 18 can be found below:

Video for our minimum and maximum A values shown below:

A = 18.37

A = 25.65

Video for minimum A value for an unstable system with an oscillating pivot shown below:

Video for the double pendulum operation can be seen below: (Ran as pend([0 20], [pi/2 0 0 3], 400)