HW2, MAE 200A. Fall 2005. UCI

Nasser Abbasi

Problem 1

Given this simple pendulum, plot the constant energy curves in MATH


sys.png

Answer

The constant energy curves are curves in the Y-X plane where energy is constant. The Y-axis represents $\dot{\theta}$, and the X-axis represents $\theta $

We are told to assume there is no damping.

We assume the pendulum is given an initial force when in the initial position ($\theta =0^{0}$) that will cause it to swing anticlock wise. The pendulum will from this point obtain an energy which will remain constant. The higher the energy the pendulum posses, the larger the angle $\theta $ it will swing by will be. If the energy is large enough to cause the pendulum to reach $\theta =\pi $ (i.e. upright position) with an non zero angular velocity, then the pendulum will swing continue to rotate in the same direction and will not swing back and forth.

To be able to solve this, I therefore need to find the range of angle the pendulum will swing for a given energy level. And for this range of angle determine the angular speed, this will give me the required phase plot.

I will first derive the expression for the energy $E$ for the pendulum above, which will be a function of MATH

MATH

Hence MATH

For $\dot{\theta}$ to be real, MATH, hence MATH

$\ $

So, for $\theta =0$, $E\geq 0$

For MATH

For MATH

For MATH

For MATH

The above means that for energy level $mLg$ for example, the angle range to plot the constant energy curve for will be between MATH

So, For each energy level, I will generate a plot for all the angles up to the angle limit allowed by that energy level. For each angle, I will find $\dot{\theta}$ from equation (1)

The following is the resulting plot and below that is the program I wrote to generate this plot. This was done for MATH

I show different plots for different granuality in the energy levels. Different quantum is used for different plots.

By the quantum I mean that the energy levels for the escape energy will change from its lower level by this quantum factor. So, for a quantum of 1/2 for example, this means the energy is being incremented for each level by an amount which 1/2 the previuse energy level.


sys4.png

This is a zoom to a smaller area to better show the energy lines


sys5.png
This below is a plot using different quantum


sys2.png


sys3.png

If we need to calculate the equilibrium points, we can do find the rate of change of the energy, and set that to zero as follows

MATH

substitue into the above the following system dynaics equation MATH

We are told to set $c=0$, hence the equation becomes

MATH so we get

MATH

For $\dot{E}=0$ means $\theta =\pm n\pi $ or MATH


code.png