gibson:teaching:spring-2016:math445:lab11
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gibson:teaching:spring-2016:math445:lab11 [2016/04/21 08:37] gibson |
gibson:teaching:spring-2016:math445:lab11 [2016/04/26 12:42] (current) vining |
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| In 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 $\theta$. |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| \frac{d^2\theta}{dt^2} + \frac{\alpha}{m} \frac{d\theta}{dt} + \frac{g}{\ell} \sin \theta = 0 | \frac{d^2\theta}{dt^2} + \frac{\alpha}{m} \frac{d\theta}{dt} + \frac{g}{\ell} \sin \theta = 0 | ||
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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 $d\theta/dt$ versus $\theta$ plot, plot just the numerical solution. Show a number of numerical solutions, using initial conditions $\vec{x}_0 = [\theta_0, 0]$ with $\theta_0$ varying from $0.01$ to $0.10$ in steps of $0.01$. Superimpose a quiver plot that shows the vector field $\vec{f}(\vec{x})$ which governs the time-evolution of the pendulum. | For the $d\theta/dt$ versus $\theta$ plot, plot just the numerical solution. Show a number of numerical solutions, using initial conditions $\vec{x}_0 = [\theta_0, 0]$ with $\theta_0$ varying from $0.01$ to $0.10$ in steps of $0.01$. Superimpose a quiver plot that shows the vector field $\vec{f}(\vec{x})$ which governs the time-evolution of the pendulum. | ||
| - | The plots should look something like this (but with smaller values of $\theta$): | + | The plots should look something like this: |
| + | {{:gibson:teaching:spring-2016:math445:lab11:problem1a.png?400|}} {{:gibson:teaching:spring-2016:math445:lab11:problem1b.png?400|}} | ||
| - | {{:gibson:teaching:fall-2014:math445:timeseries_linear_nodamp.png?nolink&400}} {{:gibson:teaching:fall-2014:math445:phaseportrait_linear_nodamp.png?nolink&400}} | + | ---- |
| ====Problem 2: the linear damped pendulum==== | ====Problem 2: the linear damped pendulum==== | ||
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| and the time series and phase portrait for (for $g=9.8$, $\ell=1$, $m=1$, and $\alpha=1$) look like | and the time series and phase portrait for (for $g=9.8$, $\ell=1$, $m=1$, 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}} | + | {{:gibson:teaching:spring-2016:math445:lab11:problem2a.png?direct&400|}} {{:gibson:teaching:spring-2016:math445:problem2b.png?direct&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 $d\theta/dt = 0$). | 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 $d\theta/dt = 0$). | ||
| Line 105: | Line 105: | ||
| 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> | ||
| Line 113: | Line 113: | ||
| </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/lab11.1461253023.txt.gz · Last modified: 2016/04/21 08:37 by gibson
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