gibson:teaching:spring-2016:math445:lecture:pendulum
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gibson:teaching:spring-2016:math445:lecture:pendulum [2016/04/27 13:15] gibson created |
gibson:teaching:spring-2016:math445:lecture:pendulum [2016/04/27 13:24] (current) gibson [Changing a second-order ODE into a system of first-order ODEs] |
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| ====== The plane pendulum ====== | ====== The plane pendulum ====== | ||
| + | ==== Mathematical models==== | ||
| - | n class we derived four different mathematical models of the plane pendulum, listed here in order of decreasing mathematical complexity. | + | 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 $\theta$. | ** 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 $\theta$. | ||
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| 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!) | 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!) | ||
| - | We also showed that the 2nd-order 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 | + | |
| + | |||
| + | ---- | ||
| + | |||
| + | ==== 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*} | \begin{eqnarray*} | ||
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| \end{eqnarray*} | \end{eqnarray*} | ||
| - | If you differentiate the above equation in time and perform a few substitutions between $\theta, d\theta/dt$ and $x_1, x_2$, you get | + | 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*} | \begin{eqnarray*} | ||
gibson/teaching/spring-2016/math445/lecture/pendulum.1461788134.txt.gz · Last modified: 2016/04/27 13:15 by gibson
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