gibson:teaching:spring-2018:math445:lecture:pendulum

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

- : the unknown function that describes the angle of the pendulum away from vertical as a function of time.
- : the mass of the pendulum bob.
- : the length of the pendulum string
- : the acceleration of gravity
- : an air resistance constant

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 `ode45`

function.

gibson/teaching/spring-2018/math445/lecture/pendulum.txt · Last modified: 2018/04/26 05:58 by gibson