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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 However if $\theta$ gets large the approximation $\sin \theta \approx \theta$ is no longer valid.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** from air resistance, 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*} and the time series and phase portrait (for $g=9.8$ and $l=1$) look like {{:gibson:teaching:fall-2014:math445:timeseries_linear_nodamp.png?nolink&400}} {{:gibson:teaching:fall-2014:math445:phaseportrait_linear_nodamp.png?nolink&400}} Note that the temporal oscillations look like perfect sines and cosines, and the phase portrait shows that trajectories circle around the origin indefinitely. However, **if we include the damping of air resistance**, 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 & -\alpha/m \end{array} \right] \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right) \end{eqnarray*} and the time series and phase portrait for (for $g=9.8$, $l=1$, and $\alpha=1$) look like {{:gibson:teaching:fall-2014:math445:timeseries_linear_damp.png?nolink&400}} {{:gibson:teaching:fall-2014:math445:phaseportrait_linear_damp.png?nolink&400}} Now the temporal oscillations get smaller and smaller as time goes on. The phase portrait shows that all initial conditions eventually spiral into the origin, i.e. the pendulum hangs straight down ($\theta = 0$) and doesn't move $d\theta/dt = 0$). For this lab problem, you are to recreate the previous four plots for the nonlinear pendulum, whose equations of motion are \begin{eqnarray*} \frac{d}{dt} \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right) = \left( \begin{array}{c} x_2 \\ -g/l sin x_1 - \alpha/m x_2 \end{array} \right) \end{eqnarray*}
