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gibson:teaching:fall-2014:math445:lab7

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====== Math 445 lab 7: differential equations ====== **Problem 1:** In class we developed a linear system of differential equations of the form $dx/dt = f(x) = Ax$ for the plane pendulum, where <latex> x = \left(\begin{array}{l} x_1\\ x_2\end{array} \right) = \left(\begin{array}{l} \theta \\ v_{\theta} \end{array} \right) </latex> We used the approximation $\sin \theta \approx \theta$ for small $\theta$, i.e. small amplitude oscillations. A chief result of this derivation was that the frequency of oscillation of the pendulum is given by $\omega = \sqrt{g/l}$, where $g$ is the acceleration due to gravity and $l$ is the length of the pendulum. However if $\theta$ gets large the approximation $\sin \theta \approx \theta$ is no longer valid. This lab problem is an investigation of how the frequency of oscillation of the pendulum changes when the amplitude of the oscillation is large. **(a)** Revise the derivation from class to develop a nonlinear system of differential equations $dx/dt= f(x)$ that is valid for large $\theta$. **(b)** Set up Matlab code to integrate this system of equations numerically, using Matlab's ''ode45'' function. **%%(c)%%** Determine the frequency of oscillation of the nonlinear pendulum for for the constants $g=9.8 \; m/s^2$ and $l=1.0 \; m$ and a variety of oscillation amplitudes. The oscillation amplitude is given by the angle $\theta_0$ at which the pendulum is released at time $t=0$, with no initial velocity. Determine the frequency $\omega$ for $\theta_0 = 0.05, 0.10, ..., 0.30$ and plot $\omega$ versus $\theta_0$. **(d)** For moderate amplitudes, the frequency $\omega$ of the nonlinear pendulum should vary with $\theta_0$ as <latex> \omega = \sqrt{g/l} + c \; \theta_0^2 </latex> for some value of $c$. Determine what the value of $c$ is from your graph in %%(c)%%. **Problem 2:** For the linear pendulum, we produced in lecture //time series// plots of position $\theta$ and (angular) velocity $d\theta/dt$ as a function of time, and a //phase portrait// of $\theta$ versus $d\theta/dt$. With no damping, the equations of motion of the linear pendulum are \begin{eqnarray*} \frac{d}{dt} \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right) = \left[ \begin{array}{cc} 0 & 1 \\ -g/l & 0 \end{array} \right] \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right) \end{eqnarray*} {{:gibson:teaching:fall-2014:math445:timeseries_linear_nodamp.png?nolink&400}} {{:gibson:teaching:fall-2014:math445:phaseportrait_linear_nodamp.png?nolink&400}}

gibson/teaching/fall-2014/math445/lab7.1417470516.txt.gz · Last modified: 2014/12/01 13:48 by gibson