gibson:teaching:fall-2014:math445:lab7
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gibson:teaching:fall-2014:math445:lab7 [2014/12/01 14:05] gibson |
gibson:teaching:fall-2014:math445:lab7 [2014/12/02 09:19] (current) gibson |
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| - | ====== Math 445 lab 7: differential equations ====== | + | ====== Math 445 lab 7: the pendulum (numerical integration of differential eqns) ====== |
| **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 | **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 | ||
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| \end{eqnarray*} | \end{eqnarray*} | ||
| - | and the time series and phase portrait (for $g=9.8$ and $l=1$) look like | + | where $(x_1, x_2) = (\theta, d\theta/dt$). 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}} | {{:gibson:teaching:fall-2014:math445:timeseries_linear_nodamp.png?nolink&400}} {{:gibson:teaching:fall-2014:math445:phaseportrait_linear_nodamp.png?nolink&400}} | ||
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| \begin{eqnarray*} | \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) | + | \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*} | \end{eqnarray*} | ||
| + | |||
| + | Use parameter values $g=9.8, l=1$ and $\alpha = 0$ or $1$ for no damping or damping, respectively. For the time series plots, initiate the pendulum with $x_1 = \theta=0$ and $x_2 = d\theta/dt = 2$. For the phase portraits, show the range $8 \leq \theta \leq 8$ on the horizontal axis and $-10 \leq d\theta/dt \leq 10$ on the vertical. On top of the quiver plots, show trajectories with initial conditions $\theta=0$ and a variety of $d\theta/dt$ ranging from -10 to 10 in steps of 1. | ||
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| + | Turn in your code, your plots, and answer the following questions | ||
| + | |||
| + | **(a)** Describe the differences that you see in the phase portraits of the nonlinear pendulum compared to the linear pendulum. | ||
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| + | **(b)** For the undamped nonlinear pendulum, you should see trajectories with three distinctly different types of behavior. Give a physical interpretation of each of those three types of trajectories in terms of the motion of the pendulum. | ||
| + | |||
| + | **%%(c)%%** You should also see three distinct type of trajectories for damped nonlinear pendulum. Give a physical interpretation of each of those types of trajectories. | ||
| + | |||
| + | **(d)** The damped nonlinear pendulum has quite a few more distinct groups of trajectories than the ones shown in shown in your phase portrait. Based on your physical interpretation, what would those be? Revise your nonlinear damped phase portrait to show some of these other trajectories. | ||
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