The plane pendulum

Mathematical models

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 $\alpha/m \; d\theta/dt$ term and is accurate for large displacement angles $\theta$ .

$\begin{eqnarray*} \frac{d^2\theta}{dt^2} + \frac{\alpha}{m} \frac{d\theta}{dt} + \frac{g}{\ell} \sin \theta = 0 \end{eqnarray*}$

Nonlinear undamped pendulum. This model neglects air resistance. It is derived from the previous model by setting the air resistance constant $\alpha$ to zero.

$\begin{eqnarray*} \frac{d^2\theta}{dt^2} + \frac{g}{\ell} \sin \theta = 0 \end{eqnarray*}$

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 $\sin \theta \approx \theta$ for small $\theta$ .

$\begin{eqnarray*} \frac{d^2\theta}{dt^2} + \frac{\alpha}{m} \frac{d\theta}{dt} + \frac{g}{\ell} \theta = 0 \end{eqnarray*}$

Linear undamped pendulum. The simplest pendulum model. It neglects air resistance and employs the small-angle approximation.

$\begin{eqnarray*} \frac{d^2\theta}{dt^2} + \frac{g}{\ell} \theta = 0 \end{eqnarray*}$

In each of these, the variables are

$\theta(t)$ : the unknown function that describes the angle of the pendulum away from vertical as a function of time.
$m$ : the mass of the pendulum bob.
$\ell$ : the length of the pendulum string
$g$ : the acceleration of gravity
$\alpha$ : an air resistance constant

In class we showed that the linear undamped pendulum has a solution of the form $\theta(t) = \theta_0 \cos \omega t$ where $\omega = \sqrt{g/\ell}$ and $\theta_0$ is an arbitrary initial angular displacement. (Note that $\theta_0$ must be small for the small-angle approximation to be valid!)

Changing a second-order ODE into a system of first-order ODEs

A 2nd-order ordinary differential equation in the scalar variable $\theta(t)$ can be transformed into a 1st order equation in the vector variable $\vec{x}(t)$ , using the substitution

$\begin{eqnarray*} \vec{x} = \left(\begin{array}{l} x_1\\ x_2 \end{array} \right) = \left(\begin{array}{l} \theta \\ d\theta/dt \end{array} \right) \end{eqnarray*}$

For example, to transform the linear undamped pendulum equation $d^2\theta/dt^2 + (g/\ell) \theta = 0$ , let $x_1 = \theta$ and $x_2 = d\theta/dt$ . Differentiate those two equations to get $d x_1/dt = d\theta/dt = x_2$ and $d x_2/dt = d^2\theta/dt^2$ . Now note that, according to the linear undamped pendulum equation, $d^2\theta/dt^2 = -(g/\ell) \theta = -(g/\ell) x_1$ . Putting all this together, we can write

$\begin{eqnarray*} \frac{d\vec{x}}{dt} = \left(\begin{array}{l} dx_1/dt\\ dx_2/dt \end{array} \right) = \left(\begin{array}{cc} 0 & 1 \\ -g/\ell & 0 \end{array} \right) \left(\begin{array}{c} x_1 \\ x_2 \end{array} \right) \end{eqnarray*}$

This equation is now of the form $d\vec{x}/dt = \vec{f}(\vec{x})$ , so it can be solved numerically with Matlab's ode45 function.

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Table of Contents

The plane pendulum

Mathematical models

Changing a second-order ODE into a system of first-order ODEs

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Table of Contents

The plane pendulum

Mathematical models

Changing a second-order ODE into a system of first-order ODEs

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