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<h1 id="computational-mechanics-03--initial-value-problems">Computational Mechanics 03- Initial Value Problems</h1>
<h2 id="learning-to-frame-engineering-equations-as-numerical-methods">Learning to frame engineering equations as numerical methods</h2>
<p>Welcome to Computational Mechanics Module #3! In this module we will explore some more data analysis, find better ways to solve differential equations, and learn how to solve engineering problems with Python.</p>
<p><a href="./notebooks/01_Catch_Motion.ipynb">01_Catch_Motion</a></p>
<ul>
<li>Work with images and videos in Python using <code>imageio</code>.</li>
<li>Get interactive figures using the <code>%matplotlib notebook</code> command.</li>
<li>Capture mouse clicks with Matplotlib’s <code>mpl_connect()</code>.</li>
<li>Observed acceleration of falling bodies is less than <span class="math inline">\(9.8\rm{m/s}^2\)</span>.</li>
<li>Capture mouse clicks on several video frames using widgets!</li>
<li>Projectile motion is like falling under gravity, plus a horizontal velocity.</li>
<li>Save our hard work as a numpy .npz file <strong>Check the Problems for loading it back into your session</strong></li>
<li>Compute numerical derivatives using differences via array slicing.</li>
<li>Real data shows free-fall acceleration decreases in magnitude from <span class="math inline">\(9.8\rm{m/s}^2\)</span>.</li>
</ul>
<p><a href="./notebooks/02_Step_Future.ipynb">02_Step_Future</a></p>
<ul>
<li>Integrating an equation of motion numerically.</li>
<li>Drawing multiple plots in one figure,</li>
<li>Solving initial-value problems numerically</li>
<li>Using Euler’s method.</li>
<li>Euler’s method is a first-order method.</li>
<li>Freefall with air resistance is a more realistic model.</li>
</ul>
<p><a href="./notebooks/03_Get_Oscillations.ipynb">03_Get_Oscillations</a></p>
<ul>
<li>vector form of the spring-mass differential equation</li>
<li>Euler’s method produces unphysical amplitude growth in oscillatory systems</li>
<li>the Euler-Cromer method fixes the amplitude growth (while still being first</li>
<li>order)</li>
<li>Euler-Cromer does show a phase lag after a long simulation</li>
<li>a convergence plot confirms the first-order accuracy of Euler’s method</li>
<li>a convergence plot shows that modified Euler’s method, using the derivatives</li>
<li>evaluated at the midpoint of the time interval, is a second-order method</li>
<li>How to create an implicit integration method</li>
<li>The difference between <em>implicit</em> and <em>explicit</em> integration</li>
<li>The difference between stable and unstable methods</li>
</ul>
<p><a href="./notebooks/04_Getting_to_the_root.ipynb">04_Getting_to_the_root</a></p>
<ul>
<li>How to find the 0 of a function, aka root-finding</li>
<li>The difference between a bracketing and an open methods for finding roots</li>
<li>Two bracketing methods: incremental search and bisection methods</li>
<li>Two open methods: Newton-Raphson and modified secant methods</li>
<li>How to measure relative error</li>
<li>How to compare root-finding methods</li>
<li>How to frame an engineering problem as a root-finding problem</li>
<li>Solve an initial value problem with missing initial conditions (the shooting</li>
<li>method)</li>
<li><em>Bonus: In the Problems you’ll consider stability of bracketing and open methods.</em></li>
</ul>
</body>
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	<h1 id="computational-mechanics-03--initial-value-problems">Computational Mechanics 03- Initial Value Problems</h1>
	<h2 id="learning-to-frame-engineering-equations-as-numerical-methods">Learning to frame engineering equations as numerical methods</h2>
	<p>Welcome to Computational Mechanics Module #3! In this module we will explore some more data analysis, find better ways to solve differential equations, and learn how to solve engineering problems with Python.</p>
	<p><a href="./notebooks/01_Catch_Motion.ipynb">01_Catch_Motion</a></p>
	<ul>
	<li>Work with images and videos in Python using <code>imageio</code>.</li>
	<li>Get interactive figures using the <code>%matplotlib notebook</code> command.</li>
	<li>Capture mouse clicks with Matplotlib’s <code>mpl_connect()</code>.</li>
	<li>Observed acceleration of falling bodies is less than <span class="math inline">\(9.8\rm{m/s}^2\)</span>.</li>
	<li>Capture mouse clicks on several video frames using widgets!</li>
	<li>Projectile motion is like falling under gravity, plus a horizontal velocity.</li>
	<li>Save our hard work as a numpy .npz file <strong>Check the Problems for loading it back into your session</strong></li>
	<li>Compute numerical derivatives using differences via array slicing.</li>
	<li>Real data shows free-fall acceleration decreases in magnitude from <span class="math inline">\(9.8\rm{m/s}^2\)</span>.</li>
	</ul>
	<p><a href="./notebooks/02_Step_Future.ipynb">02_Step_Future</a></p>
	<ul>
	<li>Integrating an equation of motion numerically.</li>
	<li>Drawing multiple plots in one figure,</li>
	<li>Solving initial-value problems numerically</li>
	<li>Using Euler’s method.</li>
	<li>Euler’s method is a first-order method.</li>
	<li>Freefall with air resistance is a more realistic model.</li>
	</ul>
	<p><a href="./notebooks/03_Get_Oscillations.ipynb">03_Get_Oscillations</a></p>
	<ul>
	<li>vector form of the spring-mass differential equation</li>
	<li>Euler’s method produces unphysical amplitude growth in oscillatory systems</li>
	<li>the Euler-Cromer method fixes the amplitude growth (while still being first</li>
	<li>order)</li>
	<li>Euler-Cromer does show a phase lag after a long simulation</li>
	<li>a convergence plot confirms the first-order accuracy of Euler’s method</li>
	<li>a convergence plot shows that modified Euler’s method, using the derivatives</li>
	<li>evaluated at the midpoint of the time interval, is a second-order method</li>
	<li>How to create an implicit integration method</li>
	<li>The difference between <em>implicit</em> and <em>explicit</em> integration</li>
	<li>The difference between stable and unstable methods</li>
	</ul>
	<p><a href="./notebooks/04_Getting_to_the_root.ipynb">04_Getting_to_the_root</a></p>
	<ul>
	<li>How to find the 0 of a function, aka root-finding</li>
	<li>The difference between a bracketing and an open methods for finding roots</li>
	<li>Two bracketing methods: incremental search and bisection methods</li>
	<li>Two open methods: Newton-Raphson and modified secant methods</li>
	<li>How to measure relative error</li>
	<li>How to compare root-finding methods</li>
	<li>How to frame an engineering problem as a root-finding problem</li>
	<li>Solve an initial value problem with missing initial conditions (the shooting</li>
	<li>method)</li>
	<li><em>Bonus: In the Problems you’ll consider stability of bracketing and open methods.</em></li>
	</ul>
	</body>
	</html>