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<h2 class="hd hd-2 unit-title">Introduction to Driven Coupled Oscillators</h2>
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We have previously solved for the motion of a system of coupled oscillators by first determining the eigenvalues and eigenvectors of the system (i.e. the characteristics of the normal modes), and then by writing a linear combination of the normal modes to produce a general solution. This is called the homogeneous solution. </p><p>
Now we turn to systems where one mass is driven by an external force. Our analysis is exactly analogous to the case of one driven oscillator, which was previously explored. The complete solution for a system of driven coupled oscillators is, again, the sum of particular and homogeneous solutions. </p><p>
We will explore the properties of the particular solution for driven systems with two or more oscillators. </p>
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<h2 class="hd hd-2 unit-title">L9v1: Review of Normal Modes and Coupled Oscillators</h2>
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<h3 class="hd hd-2">L9v1: Review of normal modes and coupled oscillators</h3>
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<h2 class="hd hd-2 unit-title">L9Q1: Driven Springs with a String</h2>
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Driven Springs with a String
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<p>
Consider the following system, which we explored in previous lessons. Here we have two coupled masses [mathjaxinline]m_{1}[/mathjaxinline] and [mathjaxinline]m_{2}[/mathjaxinline], attached to springs with spring constants [mathjaxinline]k_{1}[/mathjaxinline] and [mathjaxinline]k_{2}[/mathjaxinline], constrained to move vertically like pistons. The masses are attached to each other by a massless string, which provides approximately constant tension with magnitude [mathjaxinline]T=\kappa L[/mathjaxinline] in the direction of the string. </p>
<p>
Additionally, there is now a driving force [mathjaxinline]\vec{F}_{d}=-F_{0}\cos (\omega _{d} t)\hat{y}[/mathjaxinline], that pushes [mathjaxinline]m_1[/mathjaxinline] up and down. </p>
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<p><b class="bfseries">(Part a)</b> What are the equations of motion for each mass in the system? Define [mathjaxinline]+\hat{y}[/mathjaxinline] to point downward, [mathjaxinline]y_1[/mathjaxinline] and [mathjaxinline]y_2[/mathjaxinline] to be the displacements from the equilibrium position with no driving force, and recall that the component of the string tension in the vertical direction is [mathjaxinline]T_{y} = \pm \kappa (y_{2} - y_{1})[/mathjaxinline]. </p>
<p>
Express your answers in terms of <code>t</code>, <code>F_0</code> for [mathjaxinline]F_{0}[/mathjaxinline], <code>omega_d</code> for [mathjaxinline]\omega _{d}[/mathjaxinline], <code>y_1</code> for [mathjaxinline]y_{1}[/mathjaxinline], <code>y_2</code> for [mathjaxinline]y_{2}[/mathjaxinline], <code>m_1</code> for [mathjaxinline]m_{1}[/mathjaxinline], <code>m_2</code> for [mathjaxinline]m_{2}[/mathjaxinline], <code>k_1</code> for [mathjaxinline]k_{1}[/mathjaxinline], <code>k_2</code> for [mathjaxinline]k_{2}[/mathjaxinline], and <code>kappa</code> for [mathjaxinline]\kappa[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]m_{1}\ddot{y}_1=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]m_{2}\ddot{y}_2=[/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>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<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>
<tr class="formulainput">
<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]
</td>
</tr>
<tr class="formulainput">
<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]
</td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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>
<tr class="formulainput">
<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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Driven Springs with a String - Matrices I
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<p><b class="bfseries">(Part b)</b> Identify the matrices [mathjaxinline]\textbf{M}[/mathjaxinline], [mathjaxinline]\textbf{K}[/mathjaxinline], and [mathjaxinline]\textbf{F}[/mathjaxinline] based on the equations of motion that you derived above where [mathjaxinline]\textbf{F}[/mathjaxinline] is defined as a <i class="itshape">time-independent</i> column vector, such that [mathjaxinline]\textbf{F}\cos (\omega _{d}t)[/mathjaxinline] represents the time-dependent driving force (the reason for this definition will be seen in the next video)&#8212;do not include [mathjaxinline]\cos (\omega _{d}t)[/mathjaxinline] in this expression for [mathjaxinline]\textbf{F}[/mathjaxinline]. </p>
<p>
Express your answers in terms of <code>F_0</code> for [mathjaxinline]F_{0}[/mathjaxinline], <code>m_1</code> for [mathjaxinline]m_{1}[/mathjaxinline], <code>m_2</code> for [mathjaxinline]m_{2}[/mathjaxinline], <code>k_1</code> for [mathjaxinline]k_{1}[/mathjaxinline], <code>k_2</code> for [mathjaxinline]k_{2}[/mathjaxinline], and <code>kappa</code> for [mathjaxinline]\kappa[/mathjaxinline]. </p>
<p style="display:inline">[mathjaxinline]\textbf{M}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\textbf{K}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\textbf{F}=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<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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<tr class="formulainput">
<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>
<tr class="formulainput">
<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]
</td>
</tr>
<tr class="formulainput">
<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]
</td>
</tr>
<tr class="formulainput">
<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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<tr class="formulainput">
<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>
<tr class="formulainput">
<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">L9Q2: Expression in Complex Notation</h2>
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Driven Springs with a String - Possible Solution
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<p>
Again consider the system of two coupled masses, attached to springs, constrained to move vertically like pistons. The masses are attached to each other by a massless string, which provides approximately constant tension with magnitude [mathjaxinline]T=\kappa L[/mathjaxinline] in the direction of the string, and there is a driving force [mathjaxinline]\vec{F}_{d}=-F_{0}\cos (\omega _{d} t)\hat{y}[/mathjaxinline], that pushes [mathjaxinline]m_1[/mathjaxinline] up and down. </p>
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<p>
The equation of motion that we wish to solve is: </p>
<table id="a0000000014" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\ddot{\textbf{X}} = -\textbf{M}^{-1}\textbf{K}\textbf{X} + \textbf{M}^{-1}\textbf{F}\cos {\omega _{d}t}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
As we have done before, this is readily solved by first transforming to complex notation: </p>
<table id="a0000000015" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\ddot{\textbf{Z}} = -\textbf{M}^{-1}\textbf{K}\textbf{Z} + \textbf{M}^{-1}\textbf{F}e^{i\omega _{d}t}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Then we use the ansatz [mathjaxinline]\textbf{Z}=\textbf{B}e^{i\omega _{d}t}[/mathjaxinline] to concisely rewrite the equation of motion, where the vector [mathjaxinline]\textbf{B}=\begin{pmatrix} B_{1}\\ B_{2}\end{pmatrix}[/mathjaxinline] represents the amplitudes of the oscillation of [mathjaxinline]m_1[/mathjaxinline] and [mathjaxinline]m_2[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> After plugging in the ansatz above, what is the correct expression for the equation of motion? </p>
<p>
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<text> a) [mathjaxinline]i\omega _{d}\textbf{B} = -\textbf{M}^{-1}\textbf{K}\textbf{B} + \textbf{M}^{-1}\textbf{F}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]-i\omega _{d}\textbf{B} = \textbf{M}^{-1}\textbf{K}\textbf{B} + \textbf{M}^{-1}\textbf{F}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\omega _{d}^{2}\textbf{B} = \textbf{M}^{-1}\textbf{K}\textbf{B} + \textbf{M}^{-1}\textbf{F}[/mathjaxinline]</text>
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<text> d) [mathjaxinline]-\omega _{d}^{2}\textbf{B} = -\textbf{M}^{-1}\textbf{K}\textbf{B} + \textbf{M}^{-1}\textbf{F}[/mathjaxinline]</text>
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Driven Springs with a String - Matrices II
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<p><b class="bfseries">(Part b)</b> Identify the [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix and [mathjaxinline]\textbf{M}^{-1}\textbf{F}[/mathjaxinline] matrix. Express your answers in terms of <code>F_0</code> for [mathjaxinline]F_{0}[/mathjaxinline], <code>m_1</code> for [mathjaxinline]m_{1}[/mathjaxinline], <code>m_2</code> for [mathjaxinline]m_{2}[/mathjaxinline], <code>k_1</code> for [mathjaxinline]k_{1}[/mathjaxinline], <code>k_2</code> for [mathjaxinline]k_{2}[/mathjaxinline], and <code>kappa</code> for [mathjaxinline]\kappa[/mathjaxinline]. </p>
<p style="display:inline">[mathjaxinline]\textbf{M}^{-1}\textbf{K}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\textbf{M}^{-1}\textbf{F}=[/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>
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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>
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<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<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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<tr class="formulainput">
<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>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<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>
<tr class="formulainput">
<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>
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<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]
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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>
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<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>
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<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>
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<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>
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<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<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">L9v4: Cramer's Rule</h2>
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<h2 class="hd hd-2 unit-title">Cramer's Rule Resource</h2>
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<p><h2>Note on Cramer's Rule and Other Methods</h2></p><p>
As discussed in the previous video, Cramer's rule is a method for quickly solving systems of equations. However, we would like to note that you may not find it feasible or necessary to solve systems of equations in this way. For instance, it is perfectly acceptable to use an equation solver like Mathematica, or some other resource. </p><p>
For completeness, we summarize Cramer's method here. </p><p><h2>Cramer's Rule</h2></p><p>
Consider a system of equations, represented in matrix notation as: </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]\textbf{A}\textbf{X}=\textbf{B}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>1</span>)</span></td></tr></table><p>
Here, [mathjaxinline]\textbf{A}[/mathjaxinline] is some matrix of the following form: </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]\textbf{A} = \begin{pmatrix} a_{11} & & a_{12} & & a_{13} & & \cdots \\ a_{21} & & a_{22} & & a_{23} & & \cdots \\ a_{31} & & a_{32} & & a_{33} & & \cdots \\ \vdots & & \vdots & & \ddots \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>2</span>)</span></td></tr></table><p>
[mathjaxinline]\textbf{X}[/mathjaxinline] is a vector containing the variables [mathjaxinline]x_{1}[/mathjaxinline], [mathjaxinline]x_{2}[/mathjaxinline], [mathjaxinline]\cdots[/mathjaxinline], </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]\textbf{X}= \begin{pmatrix} x_{1}\\ x_{2}\\ x_{3}\\ \vdots \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>3</span>)</span></td></tr></table><p>
and [mathjaxinline]\textbf{B}[/mathjaxinline] is also a vector containing the variables [mathjaxinline]b_{1}[/mathjaxinline], [mathjaxinline]b_{2}[/mathjaxinline], [mathjaxinline]\cdots[/mathjaxinline], </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]\textbf{B} = \begin{pmatrix} b_{1}\\ b_{2}\\ b_{3}\\ \vdots \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>4</span>)</span></td></tr></table><p>
We wish to solve for each of the variables [mathjaxinline]x_{1}[/mathjaxinline], [mathjaxinline]x_{2}[/mathjaxinline], [mathjaxinline]\cdots[/mathjaxinline]. Let's say we first want to solve for the first element of [mathjaxinline]\textbf{X}[/mathjaxinline], which is [mathjaxinline]x_{1}[/mathjaxinline]. We simply replace the first column of [mathjaxinline]\textbf{A}[/mathjaxinline] with the vector [mathjaxinline]\textbf{B}[/mathjaxinline] and call this a new matrix [mathjaxinline]\textbf{A}_{1}[/mathjaxinline]: </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]\textbf{A}_{1} = \begin{pmatrix} b_{1} & & a_{12} & & a_{13} & & \cdots \\ b_{2} & & a_{22} & & a_{23} & & \cdots \\ b_{3} & & a_{32} & & a_{33} & & \cdots \\ \vdots & & \vdots & & \ddots \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>5</span>)</span></td></tr></table><p>
Then the solution to [mathjaxinline]x_{1}[/mathjaxinline] is given by the determinant of [mathjaxinline]\textbf{A}_{1}[/mathjaxinline] divided by the determinant of [mathjaxinline]\textbf{A}[/mathjaxinline]: </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]x_{1}=\frac{|\textbf{A}_{1}|}{|\textbf{A}|}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>6</span>)</span></td></tr></table><p>
Similarly, if we want to solve for the second element of [mathjaxinline]\textbf{X}[/mathjaxinline], which is [mathjaxinline]x_{2}[/mathjaxinline], we simply replace the second column of [mathjaxinline]\textbf{A}[/mathjaxinline] with the vector [mathjaxinline]\textbf{B}[/mathjaxinline] and call this a new matrix [mathjaxinline]\textbf{A}_{2}[/mathjaxinline]: </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]\textbf{A}_{2} = \begin{pmatrix} a_{11} & & b_{1} & & a_{13} & & \cdots \\ a_{21} & & b_{2} & & a_{23} & & \cdots \\ a_{31} & & b_{3} & & a_{33} & & \cdots \\ \vdots & & \vdots & & \ddots \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>7</span>)</span></td></tr></table><p>
Then the solution to [mathjaxinline]x_{2}[/mathjaxinline] is given by: </p><table id="a0000000009" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]x_{2}=\frac{|\textbf{A}_{2}|}{|\textbf{A}|}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>8</span>)</span></td></tr></table><p>
and similarly for the other elements of [mathjaxinline]\textbf{X}[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">L9Q3: Solving for the Amplitudes</h2>
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Driven Springs with a String - Cramer&#39;s Rule
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Again consider the system we have been working with, consisting of two coupled masses, attached to springs and a string, constrained to move vertically like pistons, and with a driving force acting on one mass. Now, to simplify things, let [mathjaxinline]m_{1} = m_{2} = m[/mathjaxinline] and [mathjaxinline]k_{1} = k_{2} = k[/mathjaxinline]. </p>
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Use Cramer's rule (or another resource) to solve the complex equation of motion for the amplitudes [mathjaxinline]B_{1}[/mathjaxinline] and [mathjaxinline]B_{2}[/mathjaxinline]. Express your answers in terms of <code>F_0</code> for [mathjaxinline]F_{0}[/mathjaxinline], <code>m</code>, <code>k</code>, <code>kappa</code> for [mathjaxinline]\kappa[/mathjaxinline], and <code>omega_d</code> for [mathjaxinline]\omega _{d}[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]B_{1} =[/mathjaxinline]</p>
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enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
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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]
</td>
</tr>
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<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]
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<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>
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<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>
<tr class="formulainput">
<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">L9Q4: Practice with a Driven Coupled System</h2>
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Driven Springs with a String - Eigenfrequencies
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Again consider the system we have been working with, consisting of two coupled masses, attached to springs and a string, constrained to move vertically like pistons, and with a driving force acting on one mass. Again, let [mathjaxinline]m_{1} = m_{2} = m[/mathjaxinline] and [mathjaxinline]k_{1} = k_{2} = k[/mathjaxinline] for simplification. </p>
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<p><b class="bfseries">(Part a)</b> Determine the driving frequencies, [mathjaxinline]\omega _{1}[/mathjaxinline] and [mathjaxinline]\omega _{2}[/mathjaxinline], that maximize [mathjaxinline]B_{1}[/mathjaxinline]. Report your answers by defining [mathjaxinline]\omega _{1} \lt \omega _{2}[/mathjaxinline] Express your answers in terms of <code>F_0</code> for [mathjaxinline]F_{0}[/mathjaxinline], <code>m</code>, <code>k</code>, and <code>kappa</code> for [mathjaxinline]\kappa[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\omega _{1}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\omega _{2}=[/mathjaxinline]</p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<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>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<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>
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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]
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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>
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<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>
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<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>
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<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>
<tr class="formulainput">
<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>
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<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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Driven Springs with a String - Driving Frequency
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<p><b class="bfseries">(Part b)</b> Determine the driving frequency that satisfies the following condition: [mathjaxinline]B_{1}=0[/mathjaxinline]. Express your answer in terms of <code>F_0</code> for [mathjaxinline]F_{0}[/mathjaxinline], <code>m</code>, <code>k</code>, and <code>kappa</code> for [mathjaxinline]\kappa[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\omega _{d}=[/mathjaxinline]</p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<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">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L9Q5: Driven Coupled Oscillators [WITH SIMULATION]</h2>
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Driven Coupled Oscillator Amplitudes I
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As in the video, here we show a plot of the amplitudes of two coupled, driven oscillators, as a function of driving frequency. The two colored curves (red and blue) represent the two masses of the system, and only one of the masses (red) is driven. As a side comment relating to previous discussion, note that there is one value of [mathjaxinline]\omega _ d[/mathjaxinline] for which the red curve is zero, but no such point for the blue curve. </p>
<p>
Note that different driving frequencies are shown on this plot as green lines labeled [mathjaxinline]\omega _{1}[/mathjaxinline] though [mathjaxinline]\omega _{4}[/mathjaxinline]. </p>
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Below are [mathjaxinline]4[/mathjaxinline] different plots, labeled (a) through (d), which each show the amplitudes of the coupled oscillators (red and blue) as a function of time, and the driving force (green). Each plot represents a different driving frequency. In the questions below, your task will be to match each driving frequency ([mathjaxinline]\omega _1[/mathjaxinline] to [mathjaxinline]\omega _4[/mathjaxinline] in the figure above) with one of the plots. </p>
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<p style="display:inline">Driving frequency [mathjaxinline]\omega _{1}[/mathjaxinline]:</p>
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Driven Coupled Oscillator Amplitudes II
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<p><b class="bfseries">(Part b)</b> Consider the higher frequency resonance (i.e. a driving frequency for which the two amplitudes are very large), shown as a dashed red and blue vertical line near [mathjaxinline]\omega _{d}=3.9~ \rm {Hz}[/mathjaxinline]. How will the behavior of the oscillators change when varying the driving frequency, starting from below this resonance to above? </p>
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<text> a) Both oscillators stay in phase with each other</text>
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<text> b) The oscillators are initially in phase, then are driven out of phase</text>
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<text> c) Both oscillators stay out of phase with each other</text>
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<text> d) The oscillators are initially out of phase, then are driven in phase</text>
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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=driven_coupled_pendula_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/v1/075d36898ee7864b4c08924be32ebfc6/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_driven_coupled_pendula_1.png"/></div><p/>
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<h2 class="hd hd-2 unit-title">Summary</h2>
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<p>
In this lesson, we began by reviewing our knowledge of normal modes in coupled systems. Then, we considered the case when one of the masses in a system is driven by an external force of the form [mathjaxinline]\vec{F}_{d}(t)=\vec{F}_{0}\cos (\omega _{d} t)[/mathjaxinline] </p><p>
The total solution is a sum of homogeneous (undriven) and particular (driven) solutions: </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]\textbf{X}_{\mathrm{tot}}= \textbf{X}_{\mathrm{hmg}} + \textbf{X}_{\mathrm{prt}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>1</span>)</span></td></tr></table><p>
Recall that in the last few lessons we solved for the homogeneous solution: </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]\textbf{X}_{\mathrm{hmg}} = \alpha \cos (\omega _{1}t + \phi _{1}) \textbf{A}^{(1)} + \beta \cos (\omega _{2}t + \phi _{2}) \textbf{A}^{(2)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>2</span>)</span></td></tr></table><p>
In the presence of damping, the homogeneous component will exponentially decay away (while having frequencies slightly modified from the undamped oscillator condition) while the particular solution will persist </p><p>
Next, we were able to use the matrix formulation that is now familiar to us to solve for the particular solution. Specifically, we expressed the equations of motion of the driven, coupled system in the following way: </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]\ddot{\textbf{X}} = -\textbf{M}^{-1}\textbf{K}\textbf{X} + \textbf{M}^{-1}\textbf{F}\cos {\omega _{d}t}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>3</span>)</span></td></tr></table><p>
As we saw previously, this equation of motion is readily solved by first transforming to complex notation: </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]\ddot{\textbf{Z}} = -\textbf{M}^{-1}\textbf{K}\textbf{Z} + \textbf{M}^{-1}\textbf{F}e^{i\omega _{d}t}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>4</span>)</span></td></tr></table><p>
Using the ansatz, [mathjaxinline]\textbf{Z}=\textbf{B}e^{i\omega _{d}t}[/mathjaxinline], we arrived at the following matrix equation: </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]\left(\textbf{M}^{-1}\textbf{K} - \omega _{d}^{2}\textbf{I} \right) \textbf{B} = \textbf{M}^{-1}\textbf{F}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>5</span>)</span></td></tr></table><p>
The next task was to solve for the amplitudes of the vector [mathjaxinline]\textbf{B}[/mathjaxinline]. This was done using Cramer's rule, and we refer you to that section of the lesson. </p><p>
Ultimately, the particular solution of the driven, coupled system can be expressed as: </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]\textbf{X}_{\mathrm{prt}}=\textbf{B}\cos (\omega _{d}t)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>6</span>)</span></td></tr></table>
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