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<h2 class="hd hd-2 unit-title">PS1.1. Playground antics</h2>
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Modeling a swing
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<p>
Jeremy is sitting on the swing shown in the sketch below. </p>
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<ol class="enumerate">
<li value="1">
<p>
Use geometry and Newton's second law to find the equation of motion for Jeremy in terms of the angle [mathjaxinline]\theta[/mathjaxinline]. We are going to make a series of approximations. </p>
<ul class="itemize">
<li>
<p>
Approximate Jeremy as a point mass [mathjaxinline]M[/mathjaxinline] situated on the seat of the swing. </p>
</li>
<li>
<p>
Other parts of the swing (cables, seat) are massless. </p>
</li>
<li>
<p>
The only force present is gravity (no friction or wind resistance). </p>
</li>
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<li value="2">
<p>
For small angles [mathjaxinline]\theta[/mathjaxinline], you can express the equation you found in the form </p>
<table id="a0000000510" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]m \ddot{\theta } + k \theta = 0\; .[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(3.95)</td>
</tr>
</table>
<p>
Find [mathjaxinline]m[/mathjaxinline] and [mathjaxinline]k[/mathjaxinline] in terms of the parameters in our model: [mathjaxinline]L[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], and [mathjaxinline]M[/mathjaxinline]. </p>
<p>
(Note that both answer boxes are graded together since the answers are ambiguous up to a constant multiple.) </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div class="multi-inputs-group" role="group"><p style="display:inline">[mathjaxinline]m=[/mathjaxinline] <div id="formulaequationinput_ps1-tab1-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<p>
Adding friction and wind resistance, you obtain a differential equation of the form </p>
<table id="a0000000511" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]m \ddot{\theta } + b \dot{\theta } + k \theta = 0\; .[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(3.96)</td>
</tr>
</table>
<p>
Check the solution to the ODE with [mathjaxinline]m=1[/mathjaxinline], [mathjaxinline]b=6[/mathjaxinline], and [mathjaxinline]k=5[/mathjaxinline] when Jeremy's friend Alice holds his swing at an angle of [mathjaxinline]\pi /4[/mathjaxinline] and releases him from rest. </p>
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Mathlet exploration
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Jeremy's friend Alice wants to set the swing into motion by periodically pushing and pulling it. Add a force to the system with input signal [mathjaxinline]f(t)= \cos (\omega t)[/mathjaxinline]. This changes the equation you found for small angles [mathjaxinline]\theta[/mathjaxinline]. From experience, Alice learned that Jeremy will gain most from her actions if she forces the swing at a phase lag of [mathjaxinline]\pi /2[/mathjaxinline] with respect to [mathjaxinline]\theta (t)[/mathjaxinline]; i.e., when Jeremy is at the left/rightmost position, she applies no force, whereas she pushes or pulls most when Jeremy passes through the lowest point [mathjaxinline]\theta = 0[/mathjaxinline]. Here we want to explore if this intuitive idea truly gives largest gain. </p>
<iframe style=" display:block; border-left-width: 0px; padding-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; border-right-width: 0px; border-top-width: 0px; border-bottom-width: 0px;" src="https://mathlets1803.surge.sh/ampPhaseSecondOrderIV.html" width="1100 px" height="640 px"/>
<ol class="enumerate">
<li value="1">
<p>
In the Mathlet, set [mathjaxinline]m=1[/mathjaxinline], [mathjaxinline]b=0.2[/mathjaxinline], [mathjaxinline]k=2.0[/mathjaxinline], and [mathjaxinline]\omega =1.5[/mathjaxinline]. By moving the mouse over the graphing window or Bode amplitude plot, measure the amplitude of the system response. What is the gain for this choice of [mathjaxinline]\omega[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps1-tab1-problem2_2_1" class=" capa_inputtype textline">
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<p>
Using the same parameters, tune [mathjaxinline]\omega[/mathjaxinline] manually until you find the resonant frequency [mathjaxinline]\omega _ r[/mathjaxinline] with highest gain. Report this value of [mathjaxinline]\omega _ r[/mathjaxinline]. Judging from the plot of input and system response, estimate the phase lag [mathjaxinline]\phi[/mathjaxinline] at resonance. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _ r=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\phi (\omega _ r)=[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_ps1-tab1-problem2_4_1">degrees (not radians)</span>
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<p>
Keeping [mathjaxinline]m=1[/mathjaxinline], find the phase lag at resonance for 2 other choices of [mathjaxinline]b[/mathjaxinline] and [mathjaxinline]k[/mathjaxinline]. Do you think Alice's intuition of [mathjaxinline]\phi = \pi /2[/mathjaxinline] when [mathjaxinline]\omega =\omega _ r[/mathjaxinline] is correct? (Hint: Look at the Nyquist plot.) <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ps1-tab1-problem2_5_1">
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We now want to check this mathematically. To this end, use ERF to compute the complex gain [mathjaxinline]G(i\omega )[/mathjaxinline] of the system where we set the leading coefficient to be [mathjaxinline]1[/mathjaxinline]. </p>
<table id="a0000000522" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\ddot x + b\dot x + kx = \cos (\omega t).[/mathjax]</td>
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Calculate the resonant frequency [mathjaxinline]w_ r[/mathjaxinline]. (Assume that [mathjaxinline]b &lt; \sqrt {2k}[/mathjaxinline].) </p>
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<p style="display:inline">[mathjaxinline]\omega _ r=[/mathjaxinline]</p>
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What is the input frequency [mathjaxinline]\omega _{\pi /2}[/mathjaxinline] needed for the phase lag to be equal to [mathjaxinline]\pi /2[/mathjaxinline]? (Enter as a math expression in terms of [mathjaxinline]k[/mathjaxinline].) </p>
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<p style="display:inline">[mathjaxinline]\omega _{\pi /2}=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">PS1.2. Operators from their response</h2>
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Identifying operators from the system response
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In many applications in engineering, properties of a system are discovered by observing the system response to a given signal. Suppose [mathjaxinline]L[/mathjaxinline] is a linear differential operator of the form [mathjaxinline]L = D^2+b D+c 1[/mathjaxinline], where [mathjaxinline]b[/mathjaxinline] and [mathjaxinline]c[/mathjaxinline] are constants. Deduce the values of [mathjaxinline]b[/mathjaxinline] and [mathjaxinline]c[/mathjaxinline] for the following observed input/output relationships: </p>
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[mathjaxinline]L(e^{-x} x^3) = e^{-x} (6x +3x^2 + x^3)[/mathjaxinline] <p style="display:inline">[mathjaxinline]b=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps1-tab2-problem1_2_1" class=" capa_inputtype inline textline">
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</div></div> <p style="display:inline">[mathjaxinline]c=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps1-tab2-problem1_3_1" class=" capa_inputtype inline textline">
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[mathjaxinline]Le^ x = a[/mathjaxinline] and [mathjaxinline]Le^{3x} = a[/mathjaxinline] (for the same [mathjaxinline]L[/mathjaxinline]). <p style="display:inline">[mathjaxinline]b=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps1-tab2-problem1_4_1" class=" capa_inputtype inline textline">
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[mathjaxinline]L \cos x = \sqrt {2} \cos (x + \pi /2)[/mathjaxinline] <p style="display:inline">[mathjaxinline]b=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_ps1-tab2-problem1_6_1" class=" capa_inputtype inline textline">
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Determining stability
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Consider the population model [mathjaxinline]\dot x-kx=-ax[/mathjaxinline], where [mathjaxinline]k[/mathjaxinline] is the growth rate of some population and [mathjaxinline]a[/mathjaxinline] is a harvest rate (measured as a fraction of the current total population). For what values of [mathjaxinline]a[/mathjaxinline] is this system stable? Is there a value of [mathjaxinline]a[/mathjaxinline] that leads to what a biologist would call a &#8220;stable" population &#8211; that is, one that neither blows up nor dies off to zero? </p>
<p>
(If no such [mathjaxinline]a[/mathjaxinline] exists, enter dne for &#8220;does not exist".) </p>
<p>
<p style="display:inline">[mathjaxinline]a=[/mathjaxinline]</p>
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Is the system stable in the signals/systems sense (in the sense used in this course) for that value of [mathjaxinline]a[/mathjaxinline] (or those values of [mathjaxinline]a[/mathjaxinline], if any such exist)? </p>
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Pole diagram
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Suppose [mathjaxinline]k=2[/mathjaxinline]. Draw the pole diagram of the transfer function [mathjaxinline]G(s)[/mathjaxinline] for this system, for [mathjaxinline]a=0[/mathjaxinline] [no harvesting], [mathjaxinline]a=2[/mathjaxinline] [harvesting at the same rate as growth] and [mathjaxinline]a=4[/mathjaxinline] [harvesting twice as fast as growth]. </p>
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Using your answers from above and the pole diagrams, comment on the dynamics of the system in each of these cases. How could these dynamics be deduced from the pole diagram? </p>
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<h2 class="hd hd-2 unit-title">PS1.4. Driving through the spring</h2>
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Spring-mass-dashpot system
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This Mathlet models the spring-mass-dashpot system discussed in class (more information can be found by clicking the &#8220;Help" tab on the Mathlet), but driven through the spring (rather than through the dashpot). </p>
<iframe style=" display:block; border-left-width: 0px; padding-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; border-right-width: 0px; border-top-width: 0px; border-bottom-width: 0px;" src="https://mathlets1803.surge.sh/ampPhaseSecondOrderI.html" width="1100 px" height="640 px"/>
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Select the [Bode plot]. You should see the graphs of the gain [mathjaxinline]g(\omega )[/mathjaxinline] and of the phase gain relative to the input signal, [mathjaxinline]-\phi (\omega )[/mathjaxinline], both as a function of [mathjaxinline]\omega[/mathjaxinline]. </p>
<p><b class="bf">(a)</b> Let [mathjaxinline]b=0.5[/mathjaxinline] and [mathjaxinline]k=4.0[/mathjaxinline]. You should see that [mathjaxinline]\omega =2.0[/mathjaxinline] is pretty close to a resonant peak. What is the exact resonant frequency [mathjaxinline]\omega _ r[/mathjaxinline] (for which [mathjaxinline]g(\omega )[/mathjaxinline] is maximal)? Calculate this by writing down [mathjaxinline]g(\omega )[/mathjaxinline] and setting the derivative to zero. </p>
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(Enter to 4 decimal places.) </p>
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<p style="display:inline">[mathjaxinline]\omega _ r=[/mathjaxinline]</p>
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<p><b class="bf">(b)</b> Determine the frequency [mathjaxinline]\omega _{\pi /2}[/mathjaxinline] at which [mathjaxinline]\phi (\omega _{\pi /2})=\pi /2[/mathjaxinline]. Hint: if [mathjaxinline]\phi =\pi /2[/mathjaxinline], what is [mathjaxinline]\mathrm{Re\, }(G(i\omega ))[/mathjaxinline]? </p>
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(Enter a positive number.) </p>
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<p style="display:inline">[mathjaxinline]\omega _{\pi /2}=[/mathjaxinline]</p>
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Transfer function
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(The following cannot be checked using this checker.) </p>
<p><b class="bf">(c)</b> The transfer or system function of this system is [mathjaxinline]G(s)=k/p(s)[/mathjaxinline], displayed on the Mathlet if you select [Nyquist plot]. Where are the poles of [mathjaxinline]G(s)[/mathjaxinline]? Sketch the pole diagram of [mathjaxinline]G(s)[/mathjaxinline]. Then sketch the graph of [mathjaxinline]|G(s)|[/mathjaxinline]; this is a surface lying over the complex plan (which we think of as the floor). </p>
<p><b class="bf">(d)</b> Consider how [mathjaxinline]|G(s)|[/mathjaxinline] looks along the imaginary axis (i.e. project your sketch of [mathjaxinline]|G(s)|[/mathjaxinline] onto the vertical plane meeting the imaginary axis). Compare this curve with the plot of amplitude versus [mathjaxinline]\omega[/mathjaxinline], which can be seen on the Bode plot. Comment on the relationship between the location of the poles of [mathjaxinline]|G(s)|[/mathjaxinline] and the near resonant peaks in the frequency response. </p>
<p><b class="bf">(e)</b> Suppose that the dashpot is made weaker, so [mathjaxinline]b[/mathjaxinline] becomes smaller. What happens to the poles? Where are they when [mathjaxinline]b=0[/mathjaxinline]? Comment on the limitations of the mathematical model in this case. </p>
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<h2 class="hd hd-2 unit-title">PS1.5. Pole diagrams</h2>
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Two spring system
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A certain two-spring system has transfer function given by </p>
<table id="a0000000532" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]H(s)=\frac{128}{(s^2+b_1s+2)(s^2+b_2s+64)}.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
Suppose first that [mathjaxinline]b_1=2[/mathjaxinline] and [mathjaxinline]b_2=20[/mathjaxinline]. Sketch the pole diagram. </p>
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Is this system with [mathjaxinline]b_1=2[/mathjaxinline] and [mathjaxinline]b_2=20[/mathjaxinline] stable? </p>
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Transfer function
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Write down a differential equation in &#8220;input/output standard form" [mathjaxinline]P(D)x=Q(D)y[/mathjaxinline] for which this [mathjaxinline]H(s)[/mathjaxinline] is the transfer function (still with [mathjaxinline]b_1=2[/mathjaxinline] and [mathjaxinline]b_2=20[/mathjaxinline]). </p>
<p>
(Enter such that [mathjaxinline]P(D)[/mathjaxinline] and [mathjaxinline]Q(D)[/mathjaxinline] have no common factors. Type <b class="bf">I</b> for the identity operator.) </p>
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Smaller b
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Now suppose that [mathjaxinline]b_1[/mathjaxinline] and [mathjaxinline]b_2[/mathjaxinline] are both small (on the order of [mathjaxinline]0.1[/mathjaxinline] say). Make a rough sketch of the new pole diagram. </p>
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Gain
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Continue to suppose that [mathjaxinline]b_1[/mathjaxinline] and [mathjaxinline]b_2[/mathjaxinline] are both small. Which of the following is a rough sketch of the gain [mathjaxinline]|H(i\omega )|[/mathjaxinline] as a function of [mathjaxinline]\omega[/mathjaxinline] in the range [mathjaxinline]\omega =0[/mathjaxinline] up to [mathjaxinline]\omega =10[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">PS1.6. Transfer functions</h2>
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In lecture <a href="/courses/course-v1:OCW+18.031+2019_Spring/courseware/week1/lec1/3" target="_blank">here</a> we determined the equation [mathjaxinline]P(D)x=Q(D)y[/mathjaxinline] controlling a series RLC circuit, choosing the applied voltage [mathjaxinline]V[/mathjaxinline] as input signal and the voltage drop across the resistor, [mathjaxinline]V_ R[/mathjaxinline], as system response. </p><p>
Now work out the corresponding equations if we continue to take [mathjaxinline]V[/mathjaxinline] as the input signal but now choose the current [mathjaxinline]I[/mathjaxinline] or the voltage drops [mathjaxinline]V_ C[/mathjaxinline] or [mathjaxinline]V_ L[/mathjaxinline] as system response. </p>
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Case 1
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Find the transfer function [mathjaxinline]H(s)[/mathjaxinline] if we continue to take [mathjaxinline]V[/mathjaxinline] as the input signal but now choose the current [mathjaxinline]I[/mathjaxinline] as the system response. </p>
<p><b class="bf">Response</b> &#8195;&#8195;<b class="bf">Transfer function</b><br/><p style="display:inline"> [mathjaxinline]I: \phantom{response_ L} H(s)=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_ps1-tab6-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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The gain [mathjaxinline]|H(i\omega )|[/mathjaxinline] behaves, for [mathjaxinline]\omega[/mathjaxinline] large, like a constant times some power of [mathjaxinline]\omega[/mathjaxinline]: [mathjaxinline]a\omega ^ k[/mathjaxinline]. Work out what [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]k[/mathjaxinline] are for system response [mathjaxinline]I[/mathjaxinline]. </p>
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<p style="display:inline"> [mathjaxinline]I: \phantom{response_ L} a=[/mathjaxinline]</p>
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<h3 class="hd hd-3 problem-header" id="ps1-tab6-problem2-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps1-tab6-problem2-problem-progress" tabindex="-1">
Case 2
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Find the transfer function [mathjaxinline]H(s)[/mathjaxinline] if we continue to take [mathjaxinline]V[/mathjaxinline] as the input signal but now choose the voltage drop [mathjaxinline]V_ L[/mathjaxinline] as the system response. </p>
<p><b class="bf">Response</b> &#8195;&#8195;<b class="bf">Transfer function</b><br/><p style="display:inline">[mathjaxinline]V_ L:\phantom{response} H(s)=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_ps1-tab6-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<p>
The gain [mathjaxinline]|H(i\omega )|[/mathjaxinline] behaves, for [mathjaxinline]\omega[/mathjaxinline] large, like a constant times some power of [mathjaxinline]\omega[/mathjaxinline]: [mathjaxinline]a\omega ^ k[/mathjaxinline]. Work out what [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]k[/mathjaxinline] are for system response [mathjaxinline]V_ L[/mathjaxinline]. </p>
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<p style="display:inline"> [mathjaxinline]V_ L: \phantom{response} a=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]k=[/mathjaxinline]</p>
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<h3 class="hd hd-3 problem-header" id="ps1-tab6-problem3-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps1-tab6-problem3-problem-progress" tabindex="-1">
Case 3
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Find the transfer function [mathjaxinline]H(s)[/mathjaxinline] if we continue to take [mathjaxinline]V[/mathjaxinline] as the input signal but now choose the voltage drop [mathjaxinline]V_ C[/mathjaxinline] as the system response. </p>
<p><b class="bf">Response</b> &#8195;&#8195;<b class="bf">Transfer function</b><br/><p style="display:inline">[mathjaxinline]V_ C:\phantom{response} H(s)=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_ps1-tab6-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<p>
The gain [mathjaxinline]|H(i\omega )|[/mathjaxinline] behaves, for [mathjaxinline]\omega[/mathjaxinline] large, like a constant times some power of [mathjaxinline]\omega[/mathjaxinline]: [mathjaxinline]a\omega ^ k[/mathjaxinline]. Work out what [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]k[/mathjaxinline] are for system response [mathjaxinline]V_ C[/mathjaxinline]. </p>
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<p style="display:inline"> [mathjaxinline]V_ C: \phantom{response} a=[/mathjaxinline]</p>
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