gibson:teaching:spring-2016:math445:lab11
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gibson:teaching:spring-2016:math445:lab11 [2016/04/21 09:42] gibson [Problem 2: the linear damped pendulum] |
gibson:teaching:spring-2016:math445:lab11 [2016/04/26 12:42] (current) vining |
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| Compare the analytic solution $\theta(t) = \theta_0 \cos \omega t$ of the linear undamped pendulum to a numerical solution computed with Matlab's ''ode45'' function. Use constants | Compare the analytic solution $\theta(t) = \theta_0 \cos \omega t$ of the linear undamped pendulum to a numerical solution computed with Matlab's ''ode45'' function. Use constants | ||
| - | * $g = 9./8$ (meters per second^2) | + | * $g = 9.8$ (meters per second^2) |
| * $\ell = 1.0$ (meters) | * $\ell = 1.0$ (meters) | ||
| * $\theta_0 = 0.1$ (radians) | * $\theta_0 = 0.1$ (radians) | ||
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| for the constants $g=9.8$ and $\ell=1.0$ and a variety of oscillation amplitudes. | for the constants $g=9.8$ and $\ell=1.0$ 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 a variety of values in the range | 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 a variety of values in the range | ||
| - | $0 < \theta_0 < \pi/2$ and plot $\omega$ versus $\theta_0$. | + | $0 < \theta_0 < 2 \approx 115^{\circ}$ and plot $\omega$ versus $\theta_0$. |
| - | **(d)** For moderate amplitudes, the frequency $\omega$ of the nonlinear pendulum should vary with $\theta_0$ as | + | **(d)** For this range of amplitudes, the frequency $\omega$ of the nonlinear pendulum should vary with $\theta_0$ as |
| <latex> | <latex> | ||
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| </latex> | </latex> | ||
| - | for some value of $c$. Determine what the value of $c$ is from your graph in %%(c)%%. Do this by adding a curve of the above form to your plot from %%(c)%%, and adjusting the constant $c$ until this curve is close to the numerical data computed in %%(c)%%, for moderately small $\theta_0$. What is the value of $c$? What does this mean for the accuracy of a pendulum clock? Would a real pendulum clock run fast or slow compared to the idealized linear clock if the amplitude of oscillation is too big? | + | for some value of $c$. Determine what the value of $c$ is from your graph in %%(c)%%. Do this by adding a curve of the above form to your plot from %%(c)%%, and adjusting the constant $c$ until this curve passes through the data points computed in %%(c)%%. What is the value of $c$? What does this mean for the accuracy of a pendulum clock? Would a real pendulum clock run fast or slow compared to the idealized linear clock if the amplitude of oscillation is too big? |
| ---- | ---- | ||
| - | ====Problem 3: nonlinear damped pendulum==== | + | ====Problem 4: nonlinear damped pendulum==== |
| - | For this lab problem, you are to recreate the previous four plots for the nonlinear damped pendulum, whose equations of motion are | + | For this lab problem, you are to recreate the time series and phase portrait for the nonlinear damped pendulum, whose equations of motion are |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| Line 125: | Line 125: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | Use parameter values $g=9.8, \ell=1$, $m=1$, and $\alpha = 1$. 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. | + | Use parameter values $g=9.8, \ell=1$, $m=1$, and $\alpha = 1$. 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. |
| + | ---- | ||
| Turn in your code, your plots, and answer the following questions | Turn in your code, your plots, and answer the following questions | ||
| + | ====Questions (to be answered at the end of your lab)==== | ||
| **(a)** Describe the differences that you see in the phase portraits of the nonlinear pendulum compared to the linear pendulum. | **(a)** Describe the differences that you see in the phase portraits of the nonlinear pendulum compared to the linear pendulum. | ||
gibson/teaching/spring-2016/math445/lab11.1461256953.txt.gz ยท Last modified: 2016/04/21 09:42 by gibson
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