In class we derived four different mathematical models of the plane pendulum, listed here in order of decreasing mathematical complexity.
Nonlinear damped pendulum. This is the most physically realistic model. It includes a linear model of air resistance the term and is accurate for large displacement angles .
Nonlinear undamped pendulum. This model neglects air resistance. It is derived from the previous model by setting the air resistance constant to zero.
Linear damped pendulum. This is valid only for small oscillations. You can derive it from the nonlinear damped pendulum model by substituting the small angle approximation for small .
Linear undamped pendulum. The simplest pendulum model. It neglects air resistance and employs the small-angle approximation.
In each of these, the variables are
In class we showed that the linear undamped pendulum has a solution of the form where and is an arbitrary initial angular displacement. (Note that must be small for the small-angle approximation to be valid!)
A 2nd-order ordinary differential equation in the scalar variable can be transformed into a 1st order equation in the vector variable , using the substitution
For example, to transform the linear undamped pendulum equation , let and . Differentiate those two equations to get and . Now note that, according to the linear undamped pendulum equation, . Putting all this together, we can write
This equation is now of the form , so it can be solved numerically with Matlab's