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<h2 class="hd hd-2 unit-title">Introduction to Damped, Driven Oscillation</h2>
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Up until this point, we have analyzed the behavior of undriven systems by doing the following:<br/> [mathjaxinline]\bullet[/mathjaxinline] deriving the equations of motion of a system <br/> [mathjaxinline]\bullet[/mathjaxinline] solving for the most general solution to these equations of motion <br/> [mathjaxinline]\bullet[/mathjaxinline] solving for the time evolution of a system from a set of initial conditions <br/></p><p>
We have done this for ideal systems, and systems with damping. To use a metaphor, it is as if we have been practicing setting up dominos and watching them fall. </p><p>
Now we will introduce a time-varying force that continuously interacts with the system—the force will be a simple sinusoidal function. Again, we seek to analyze the behavior of driven systems by completing the steps above. We will learn how to accomplish these steps when driving is considered. </p>
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<h3 class="hd hd-2">L4v1: Introduce a Driving Force</h3>
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<h2 class="hd hd-2 unit-title">L4v2: Complex Solution to Driven Harmonic Motion</h2>
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Solving Driven Harmonic Motion
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<b class="bfseries">NOTE: We will work through this ungraded example in the lesson video below, but you are encouraged to try it on your own first!</b>
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Consider the equation of motion for the damped harmonic oscillator WITH driving, where [mathjaxinline]\omega _{d}[/mathjaxinline] is the driving frequency. </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\ddot{\theta }(t) + \Gamma \dot{\theta }(t) + \omega _0^2 \theta (t) = f_{0}\cos (\omega _{d}t)[/mathjax]</td>
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We have previously used [mathjaxinline]\theta (t)=\operatorname {Re}(z(t))[/mathjaxinline] to write equations of motion in complex notation, thereby making them more straightforward to solve. We will again use this strategy. </p>
<p>
Using the ansatz [mathjaxinline]z(t)=A e^{i(\omega _{d} t - \delta )}[/mathjaxinline], which of the following correctly represents the relation between the variables [mathjaxinline]A[/mathjaxinline], [mathjaxinline]\omega _ d[/mathjaxinline], [mathjaxinline]\omega _0[/mathjaxinline], [mathjaxinline]\Gamma[/mathjaxinline], [mathjaxinline]\delta[/mathjaxinline], and [mathjaxinline]f_{0}[/mathjaxinline]? </p>
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<text> a) [mathjaxinline]\left[ -\omega _{d}^{2} + i\omega _{d}\Gamma + \omega _{0}^{2} \right] A = f_{0} e^{-i\delta }[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\left[ -\omega _{d}^{2} + i\omega _{d}\Gamma + \omega _{0}^{2} \right] A = f_{0} e^{i\delta }[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\left[ -\omega _{d}^{2} + i\omega _{d}\Gamma + \omega _{0}^{2} \right] A = f_{0} e^{i(\omega _{d}t - \delta )}[/mathjaxinline]</text>
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<text> d) [mathjaxinline]\left[ -\omega _{d}^{2} + i\omega _{d}\Gamma + \omega _{0}^{2} \right] A = f_{0} e^{-i(\omega _{d}t + \delta })[/mathjaxinline]</text>
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<h3 class="hd hd-2">L4v2: Complex Solution to Driven Harmonic Motion</h3>
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<h2 class="hd hd-2 unit-title">L4Q1: Definition of omegas</h2>
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Definition of Omegas
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We have introduced several different "omegas" related to the damped, driven oscillator system: [mathjaxinline]\omega _{d}[/mathjaxinline], [mathjaxinline]\omega _{0}[/mathjaxinline], and [mathjaxinline]\omega[/mathjaxinline]. In the following problem, identify the appropriate definition for each. </p>
<p><b class="bfseries">(Part a)</b> Definition of [mathjaxinline]\omega _{d}[/mathjaxinline]: </p>
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<text> a) the natural frequency of the system</text>
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<text> b) the frequency of the transient response</text>
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<text> c) the frequency of the driving force</text>
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<text> d) the damping frequency</text>
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<p><b class="bfseries">(Part b)</b> Definition of [mathjaxinline]\omega _{0}[/mathjaxinline]: </p>
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<text> a) the natural frequency of the system</text>
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<text> b) the frequency of the transient response</text>
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<text> c) the frequency of the driving force</text>
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<text> d) the damping frequency</text>
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<p><b class="bfseries">(Part c)</b> Definition of [mathjaxinline]\omega[/mathjaxinline]: </p>
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<text> a) the natural frequency of the system</text>
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<text> b) the frequency of the transient response</text>
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<text> c) the frequency of the driving force</text>
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<text> d) the damping frequency</text>
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Relating Omegas
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Write the expression for [mathjaxinline]\omega[/mathjaxinline] in terms of <code>omega_d</code> for [mathjaxinline]\omega _{d}[/mathjaxinline], <code>omega_0</code> for [mathjaxinline]\omega _{0}[/mathjaxinline], and <code>Gamma</code> for [mathjaxinline]\Gamma[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega =[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
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</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L4Q2: Particular and Homogeneous Solution [WITH SIMULATION]</h2>
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Graphing Damped Oscillator Solution
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The complete solution ("particular" solution plus "homogeneous" solution) to the equation of motion of a damped, driven pendulum is: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\theta (t) = A(\omega _{d})\cos (\omega _{d}t - \delta (\omega _{d})) + B e^{-\frac{\Gamma }{2}t} \cos (\omega t + \alpha )[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
where </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]A(\omega _{d}) = \frac{f_{0}}{\sqrt {\left(\omega _{0}^{2} - \omega _{d}^{2}\right)^{2} + \omega _{d}^{2}\Gamma ^{2}}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
and </p>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\delta (\omega _{d}) = \arctan {\frac{\omega _{d}\Gamma }{\left(\omega _{0}^{2} - \omega _{d}^{2}\right)}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Below are three different plots, which depict the complete solution, [mathjaxinline]\theta (t)[/mathjaxinline], of a damped-driven system. Each is driven at a different frequency, but has the same resonance frequency. Consider the following questions about the driving regimes that are shown in the plots. </p>
<p>
<b class="bfseries">Note: We expect that each of the plots represents one of the possible choices to the questions, and that each choice is represented by only one plot.</b>
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<p><b class="bfseries">(Part a)</b> Which of the plots depicts a driven system for which [mathjaxinline]\omega _{d} \lt \omega _{0}[/mathjaxinline]: <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input inline">
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<p><b class="bfseries">(Part b)</b> Which of the plots depicts a driven system for which [mathjaxinline]\omega _{d} \approx \omega _{0}[/mathjaxinline]: <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div class="inputtype option-input inline">
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<p><b class="bfseries">(Part c)</b> Which of the plots depicts a driven system for which [mathjaxinline]\omega _{d} \gt \omega _{0}[/mathjaxinline]: <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div class="inputtype option-input inline">
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<h2>Run the Interactive Python Visualization that Generated the Plots Above!</h2><p>The widget is run in a Jupyter notebook, accessible through the button below. <b>NOTE: The notebook may take up to 3 mintues to load! Please be patient!</b></p><p><div align="center"><a href="https://mybinder.org/v2/gh/mitx-803/vis/master?filepath=damped_driven_evolultion_1.ipynb" class="btn btn-primary" target="_blank" style="color:#FFFFFF;">ACCESS JUPYTER NOTEBOOK HERE</a></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">How to Run Jupyter Notebooks (expand this section if you need a reminder!)<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Running Notebooks on an External Server</h3></p><p>To access a simulation, click the "ACCESS JUPYTER NOTEBOOK HERE" button. This will bring you to a loading page, hosted by <i class="itshape">mybinder.org</i> (the loading time is anywhere from 20 seconds to 3 minutes). The Jupyter notebooks are run externally to the course, on a server which runs an instance of Python. There is no need to install Python or related dependencies!</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_06.html" width="720" height="590" scrolling="no" frameborder="0"/></div><p><h3>Initializing the Program</h3></p><p>Once loaded, you will see a Jupyter notebook in your browser! You will have to click a button to initialize the program. The button is indicated in the instructions within the notebook, and also shown below.</p><div align="center"><img width="700" src="/assets/courseware/v1/3f6c044fc06f79d82bb2e8a97f7dd11a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_binder_initialize_button.png"/></div><p/><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_07.html" width="720" height="602" scrolling="no" frameborder="0"/></div><p><h3>Instructions and Source Code</h3></p><p>Each notebook has self-contained instructions on how to use the Python simulation. Additionally, you may toggle the button at the bottom of the notebook to view/augment the source code.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_08.html" width="720" height="608" scrolling="no" frameborder="0"/></div><p><h3>Saving/Running Notebooks Locally</h3></p><p>Finally, you can dowload each notebook to run locally. Additionally, you can visit the git repository to download all notebooks in the course. In order to run notebooks locally, you must install Python and its dependencies. We cannot help with this process, but we encourage you to look at the resources below, if you are interested.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_09.html" width="720" height="609" scrolling="no" frameborder="0"/></div><p><h3>External Links</h3><br/> [mathjaxinline]\bullet[/mathjaxinline] git repository: <a href="https://github.com/mitx-803/vis" target="blank">github.com/mitx-803/vis</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on Jupyter notebooks: <a href="https://jupyter.org/" target="blank">Jupyter Notebooks</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on installing Python through Anaconda: <a href="https://www.anaconda.com/distribution/" target="blank">Anaconda</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on the Binder community: <a href="https://mybinder.readthedocs.io/en/latest/" target="blank">Binder</a><br/></p><p><h3>Dependencies</h3></p><p>
The visualizations run on Python 3. Dependencies for running Python code locally (not through Binder) are stated in the git repository, and include (but are not limited to) the following:
<br/> [mathjaxinline]\bullet[/mathjaxinline] scipy
<br/> [mathjaxinline]\bullet[/mathjaxinline] numpy
<br/> [mathjaxinline]\bullet[/mathjaxinline] ipywidgets
<br/> [mathjaxinline]\bullet[/mathjaxinline] nbinteract
<br/> [mathjaxinline]\bullet[/mathjaxinline] matplotlib
<br/> [mathjaxinline]\bullet[/mathjaxinline] pandas
<br/> [mathjaxinline]\bullet[/mathjaxinline] IPython
<br/> [mathjaxinline]\bullet[/mathjaxinline] ffmpeg
<br/> [mathjaxinline]\bullet[/mathjaxinline] jupyter-contrib-nbextensions
<br/>
</p><p>
You will have to find resources that explain how to install these appropriately for your system, if they are not already installed with your Python package.
</p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/><h2>What You Should See</h2><p>When the notebook is initialized, you will see the following visualization. Follow question prompts within the notebook.</p><div align="center"><img width="800" src="/assets/courseware/v1/3c72e675fd473de40f16e7db2aef8a92/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_damped_driven_evolultion_1.png"/></div><p/>
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<p><h2>DAMPED, DRIVEN OSCILLATORY MOTION</h2></p><p>
In this lesson, we have learned that the complete solution to the equation of motion of a damped, driven oscillator is: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\theta (t) = \theta _{\mathrm{particular}}(t) + \theta _{\mathrm{homogeneous}}(t)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>1</span>)</span></td></tr></table><p>
where the particular solution reflects the behavior of the system with driving </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\theta _{\mathrm{particular}}(t) = A(\omega _{d})\cos (\omega _{d}t - \delta (\omega _{d}))[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>2</span>)</span></td></tr></table><p>
where </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]A(\omega _{d}) = \frac{f_{0}}{\sqrt {\left(\omega _{0}^{2} - \omega _{d}^{2}\right)^{2} + \omega _{d}^{2}\Gamma ^{2}}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>3</span>)</span></td></tr></table><p>
and </p><table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\delta (\omega _{d}) = \arctan {\frac{\omega _{d}\Gamma }{\left(\omega _{0}^{2} - \omega _{d}^{2}\right)}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>4</span>)</span></td></tr></table><p>
The homogeneous solution, also known as the transient solution, reflects the behavior of the system in the absence of driving. The form of the homogeneous solution depends on the damping regime of the system. Below, we briefly review each of the damping regimes, as discussed in the previous lesson. </p><p><h3>Underdamped Motion</h3></p><p>
In the case of underdamped motion, [mathjaxinline]\omega _{0} \gt \frac{\Gamma }{2}[/mathjaxinline], and [mathjaxinline]\omega = \sqrt {\omega _{0}^2 - \left(\frac{\Gamma }{2}\right)^2}[/mathjaxinline] is purely real. The solution is: </p><table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \theta _{\mathrm{homogeneous}}(t) = B e^{-\frac{\Gamma }{2}t} \cos (\omega t + \alpha )[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>5</span>)</span></td></tr></table><p><h3>Critically Damped Motion</h3></p><p>
In the case of critically damped motion, [mathjaxinline]\omega _{0} = \frac{\Gamma }{2}[/mathjaxinline], and [mathjaxinline]\omega = 0[/mathjaxinline]. The solution is: </p><table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \theta _{\mathrm{homogeneous}}(t) = (B + Ct) e^{-\frac{\Gamma }{2}t}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>6</span>)</span></td></tr></table><p><h3>Overdamped Motion</h3></p><p>
In the case of overdamped motion, [mathjaxinline]\omega _{0} \lt \frac{\Gamma }{2}[/mathjaxinline], and [mathjaxinline]\omega = i\sqrt {\left(\frac{\Gamma }{2}\right)^2 - \omega _{0}^2}[/mathjaxinline] is imaginary. The solution is: </p><table id="a0000000008" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \theta _{\mathrm{homogeneous}}(t) = B e^{-(\frac{\Gamma }{2} + |\omega |)t} + C e^{-(\frac{\Gamma }{2} - |\omega |)t}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>7</span>)</span></td></tr></table>
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