Matlab Primer

Today's lecture will be on MATLAB and PENDULA (plural of PENDULUM). Your next lab assignment motivated the topic.

My name is Sean Carver; I am a research scientist in Mechanical Engineering. I have been programming in MATLAB for almost 20 years and programming computers for almost 30 years.

This class is about MATLAB, not about deriving equations. So I am just going to give you the equations for the PENDULUM. There is still a lot to do to get it into MATLAB.

Pendulum on a movable support

This example comes from Wikipedia (copied legally). Consider a pendulum of mass m and length ℓ, which is attached to a support with mass M which can move along a line in the x-direction. Let x be the coordinate along the line of the support, and let us denote the position of the pendulum by the angle θ from the vertical.

Sketch of the situation with definition of the coordinates (click to enlarge)

$\ddot\theta + \frac{\ddot x}{\ell} \cos\theta + \frac{g}{\ell} \sin\theta = 0.\,$

See http://en.wikipedia.org/w/index.php?title=Lagrangian_mechanics&oldid=516894618 for a full derivation.

Your homework and class exercise for today is to implement this model.

Pendulum on a fixed support

Together we are going to implement a simpler model with support fixed and unmovable. You can see what the equations for this are easily. Just plug in

$\ddot x = \dot x = x = 0$

The second term drops out:

$\ddot\theta + \frac{g}{\ell} \sin\theta = 0.\,$

Step 1: Make the model first order

Define new variables

$\begin{align} \theta_0 & = \theta, \\ \theta_1 & = \dot\theta \end{align}$

Now we have two equations:

$\begin{align} \dot\theta_0 & = \theta_1, \\ \dot\theta_1 & = - \frac{g}{\ell} \sin\theta_0 \end{align}$