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Correct wrong sentence about mechanical impedance

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Thomas Dehaeze 2020-04-07 16:19:15 +02:00
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docs/uncertainty_payload.html
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@ -4,7 +4,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2020-04-01 mer. 16:14 --> <!-- 2020-04-07 mar. 16:17 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Effect of Uncertainty on the payload&rsquo;s dynamics on the isolation platform dynamics</title> <title>Effect of Uncertainty on the payload&rsquo;s dynamics on the isolation platform dynamics</title>
@ -227,7 +227,9 @@
</script> </script>
<script> <script>
MathJax = { MathJax = {
tex: { macros: { tex: {
tags: 'ams',
macros: {
bm: ["\\boldsymbol{#1}",1], bm: ["\\boldsymbol{#1}",1],
} }
} }
@ -249,17 +251,17 @@
<ul> <ul>
<li><a href="#orgcc5f0ec">1. Simple Introductory Example</a> <li><a href="#orgcc5f0ec">1. Simple Introductory Example</a>
<ul> <ul>
<li><a href="#org6264842">1.1. Equations of motion</a></li> <li><a href="#org1f20d62">1.1. Equations of motion</a></li>
<li><a href="#org4efccbf">1.2. Initialization of the payload dynamics</a></li> <li><a href="#org4efccbf">1.2. Initialization of the payload dynamics</a></li>
<li><a href="#orgb400ca3">1.3. Initialization of the isolation platform</a></li> <li><a href="#orgb400ca3">1.3. Initialization of the isolation platform</a></li>
<li><a href="#orgd0dd88b">1.4. Comparison</a></li> <li><a href="#orgd0dd88b">1.4. Comparison</a></li>
<li><a href="#org1b051ce">1.5. Conclusion</a></li> <li><a href="#org1637b13">1.5. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org1f8e63e">2. Generalization to arbitrary dynamics</a> <li><a href="#org1f8e63e">2. Generalization to arbitrary dynamics</a>
<ul> <ul>
<li><a href="#orgc4fa63e">2.1. Introduction</a></li> <li><a href="#orgc4fa63e">2.1. Introduction</a></li>
<li><a href="#org35ac80d">2.2. Equations of motion</a></li> <li><a href="#orgd6da9a7">2.2. Equations of motion</a></li>
<li><a href="#orge217a33">2.3. Impedance \(G^\prime(s)\) of a mass-spring-damper payload</a></li> <li><a href="#orge217a33">2.3. Impedance \(G^\prime(s)\) of a mass-spring-damper payload</a></li>
<li><a href="#org0ee44da">2.4. First Analytical analysis</a></li> <li><a href="#org0ee44da">2.4. First Analytical analysis</a></li>
<li><a href="#orgfe81c1c">2.5. Impedance of the Payload and Dynamical Uncertainty</a></li> <li><a href="#orgfe81c1c">2.5. Impedance of the Payload and Dynamical Uncertainty</a></li>
@ -272,7 +274,7 @@
<li><a href="#org9086831">2.8.3. Effect of the platform&rsquo;s mass \(m\)</a></li> <li><a href="#org9086831">2.8.3. Effect of the platform&rsquo;s mass \(m\)</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org43f33dc">2.9. Conclusion</a></li> <li><a href="#org3a1ebf1">2.9. Conclusion</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
@ -326,8 +328,8 @@ The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the payloa
</div> </div>
</div> </div>
<div id="outline-container-org6264842" class="outline-3"> <div id="outline-container-org1f20d62" class="outline-3">
<h3 id="org6264842"><span class="section-number-3">1.1</span> Equations of motion</h3> <h3 id="org1f20d62"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
If we write the equation of motion of the system in Figure <a href="#orgaa77a57">1</a>, we obtain: If we write the equation of motion of the system in Figure <a href="#orgaa77a57">1</a>, we obtain:
@ -434,8 +436,8 @@ The obtained dynamics from \(F\) to \(x\) for the three isolation platform are s
</div> </div>
</div> </div>
<div id="outline-container-org1b051ce" class="outline-3"> <div id="outline-container-org1637b13" class="outline-3">
<h3 id="org1b051ce"><span class="section-number-3">1.5</span> Conclusion</h3> <h3 id="org1637b13"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5"> <div class="outline-text-3" id="text-1-5">
<div class="important"> <div class="important">
<p> <p>
@ -458,12 +460,12 @@ The stiff platform dynamics does not seems to depend on the dynamics of the payl
<h3 id="orgc4fa63e"><span class="section-number-3">2.1</span> Introduction</h3> <h3 id="orgc4fa63e"><span class="section-number-3">2.1</span> Introduction</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
Let&rsquo;s now consider a general payload described by its <b>impedance</b> \(G^\prime(s) = \frac{x}{F^\prime}\) as shown in Figure <a href="#orgb54b79a">4</a>. Let&rsquo;s now consider a general payload described by its <b>impedance</b> \(G^\prime(s) = \frac{F^\prime}{x}\) as shown in Figure <a href="#orgb54b79a">4</a>.
</p> </p>
<div class="note"> <div class="note">
<p> <p>
Note here that we use the term <i>impedance</i>, however, the mechanical impedance is usually defined as the ratio of the velocity over the force \(\dot{x}/F^\prime\). We should refer to <i>resistance</i> instead of <i>impedance</i>. Note here that we use the term <i>impedance</i>, however, the mechanical impedance is usually defined as the ratio of the force over the velocity \(F^\prime/\dot{x}\). We should refer to <i>resistance</i> instead of <i>impedance</i>.
</p> </p>
</div> </div>
@ -487,8 +489,8 @@ Now let&rsquo;s consider the system consisting of a mass-spring-system (the isol
</div> </div>
</div> </div>
<div id="outline-container-org35ac80d" class="outline-3"> <div id="outline-container-orgd6da9a7" class="outline-3">
<h3 id="org35ac80d"><span class="section-number-3">2.2</span> Equations of motion</h3> <h3 id="orgd6da9a7"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<p> <p>
We have to following equations of motion: We have to following equations of motion:
@ -536,12 +538,11 @@ By eliminating \(x^\prime\) of the equations, we obtain:
</p> </p>
<div class="important"> <div class="important">
\begin{equation} \begin{equation}
\label{orgae0b162} G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime} \label{eq:impedance_mass_spring_damper}
G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime}
\end{equation} \end{equation}
<p> <p>
The impedance of a 1dof mass-spring-damper system is described by Eq. \eqref{orgae0b162}. The impedance of a 1dof mass-spring-damper system is described by Eq. \eqref{eq:impedance_mass_spring_damper}.
</p> </p>
</div> </div>
@ -945,8 +946,8 @@ Let&rsquo;s fix \(k = 10^7\ [N/m]\), \(\xi = \frac{c}{2\sqrt{km}} = 0.1\) and se
</div> </div>
</div> </div>
<div id="outline-container-org43f33dc" class="outline-3"> <div id="outline-container-org3a1ebf1" class="outline-3">
<h3 id="org43f33dc"><span class="section-number-3">2.9</span> Conclusion</h3> <h3 id="org3a1ebf1"><span class="section-number-3">2.9</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-9"> <div class="outline-text-3" id="text-2-9">
<div class="important"> <div class="important">
<p> <p>
@ -970,7 +971,7 @@ In that case, maximizing the stiffness of the payload is a good idea.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-04-01 mer. 16:14</p> <p class="date">Created: 2020-04-07 mar. 16:17</p>
</div> </div>
</body> </body>
</html> </html>

9
org/uncertainty_payload.org
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@ -337,10 +337,10 @@ The obtained dynamics from $F$ to $x$ for the three isolation platform are shown
* Generalization to arbitrary dynamics * Generalization to arbitrary dynamics
<<sec:arbitrary_dynamics>> <<sec:arbitrary_dynamics>>
** Introduction ** Introduction
Let's now consider a general payload described by its *impedance* $G^\prime(s) = \frac{x}{F^\prime}$ as shown in Figure [[fig:general_payload_impedance]]. Let's now consider a general payload described by its *impedance* $G^\prime(s) = \frac{F^\prime}{x}$ as shown in Figure [[fig:general_payload_impedance]].
#+begin_note #+begin_note
Note here that we use the term /impedance/, however, the mechanical impedance is usually defined as the ratio of the velocity over the force $\dot{x}/F^\prime$. We should refer to /resistance/ instead of /impedance/. Note here that we use the term /impedance/, however, the mechanical impedance is usually defined as the ratio of the force over the velocity $F^\prime/\dot{x}$. We should refer to /resistance/ instead of /impedance/.
#+end_note #+end_note
#+begin_src latex :file general_payload_impedance.pdf #+begin_src latex :file general_payload_impedance.pdf
@ -455,12 +455,11 @@ In order to verify that the formula is correct, let's take the same mass-spring-
By eliminating $x^\prime$ of the equations, we obtain: By eliminating $x^\prime$ of the equations, we obtain:
#+begin_important #+begin_important
#+name: eq:impedance_mass_spring_damper
\begin{equation} \begin{equation}
G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime} G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime} \label{eq:impedance_mass_spring_damper}
\end{equation} \end{equation}
The impedance of a 1dof mass-spring-damper system is described by Eq. [[eq:impedance_mass_spring_damper]]. The impedance of a 1dof mass-spring-damper system is described by Eq. eqref:eq:impedance_mass_spring_damper.
#+end_important #+end_important
And we obtain And we obtain