gibson:teaching:spring-2016:math445:lab11
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gibson:teaching:spring-2016:math445:lab11 [2016/04/21 06:49] 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 | ||
| Line 52: | Line 52: | ||
| ---- | ---- | ||
| - | ====Problem 1: linear undamped pendulum==== | + | ====Problem 1: the linear undamped pendulum==== |
| 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) | ||
| Line 71: | Line 71: | ||
| 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==== | ||
| + | |||
| + | If we include the damping of air resistance, the equations of motion of the linear damped 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 $g=9.8$, $\ell=1$, $m=1$, and $\alpha=1$) look like | ||
| + | |||
| + | {{: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 $d\theta/dt = 0$). | ||
| + | |||
| + | Write Matlab code to simulate the linear damped pendulum and reproduce the above plots. | ||
| ---- | ---- | ||
| - | ====Problem 2: nonlinear undamped pendulum==== | + | ====Problem 3: nonlinear undamped pendulum==== |
| 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. Specifically, you'll determine if a more realistic nonlinear pendulum clock runs faster or slower than the idealized linear clock, and by how much, depending on amplitude. | 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. Specifically, you'll determine if a more realistic nonlinear pendulum clock runs faster or slower than the idealized linear clock, and by how much, depending on amplitude. | ||
| Line 87: | Line 104: | ||
| **%%(c)%%** Determine the frequency of oscillation of the nonlinear pendulum for | **%%(c)%%** Determine the frequency of oscillation of the nonlinear pendulum for | ||
| 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 | + | 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 |
| - | $\theta_0 = 0.05, 0.10, ..., 0.30$ 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 96: | 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 the curve goes through the data. 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: linear damped pendulum==== | + | ====Problem 4: nonlinear damped pendulum==== |
| - | + | For this lab problem, you are to recreate the time series and phase portrait for the nonlinear damped pendulum, whose equations of motion are | |
| - | 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 $g=9.8$, $\ell=1$, $m=1$, 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}} | + | |
| - | + | ||
| - | 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*} | \begin{eqnarray*} | ||
| Line 121: | Line 125: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | Use parameter values $g=9.8, \ell=1$, $m=1$, 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. | + | 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/lab11.1461246574.txt.gz · Last modified: 2016/04/21 06:49 by gibson
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