Fall 2009/JonathanS

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Problem

A simple pendulum with a length L = 0.5m is pulled back and released from an initial angle . Find an equation that gives the angle of the pendulum at any given time t.

Solution

Assuming no damping and a small angle(), the equation for the motion of a simple pendulum can be written as


Substituting values we get


Remember the identities


Now we can take the Laplace Transform to change the second order differential equation, from the t domain, into a simple linear equation, from the s domain, that's much easier to work with


Since we know that and the initial velocity we get


Now we can take the inverse Laplace Transform to convert our equation back into the time domain using the identity

We get


This will give us the angle (in degrees) of the pendulum at any given time t.


Initial Value Theorem

We can use the Initial Value Theorem as a check that our initial values for the problem are valid.

Below we will use this theorem to check the values for our problem.

This value is the initial angle we gave the pendulum so it checks out.

Final Value Theorem

We can use the Final Value Theorem as a check that our final values for the problem are valid.

Below we will use this theorem to check the values for our problem.

Bode Plot