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

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+ | ====== 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. | ||

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