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<title>Effect of Uncertainty on the support&rsquo;s dynamics on the isolation platform dynamics</title>
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</div><div id="content">
<h1 class="title">Effect of Uncertainty on the support&rsquo;s dynamics on the isolation platform dynamics</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgbe6e0b8">1. Simple Introductory Example</a>
<li><a href="#orgb1adbd9">1. Simple Introductory Example</a>
<ul>
<li><a href="#org3d4902a">1.1. Equations of motion</a></li>
<li><a href="#org8bd2a4a">1.2. Initialization of the support dynamics</a></li>
<li><a href="#orgefb9b71">1.3. Initialization of the isolation platform</a></li>
<li><a href="#org3bc4ad1">1.4. Comparison</a></li>
<li><a href="#org999e1c5">1.5. Conclusion</a></li>
<li><a href="#orgf20a63e">1.1. Equations of motion</a></li>
<li><a href="#org7f0c67f">1.2. Initialization of the support dynamics</a></li>
<li><a href="#org557badc">1.3. Initialization of the isolation platform</a></li>
<li><a href="#org2c66ca0">1.4. Comparison</a></li>
<li><a href="#org8e2f9eb">1.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#orge1d3484">2. Generalization to arbitrary dynamics</a>
<li><a href="#org60caf30">2. Generalization to arbitrary dynamics</a>
<ul>
<li><a href="#org3948d1f">2.1. Introduction</a></li>
<li><a href="#org18c1c3f">2.2. Equations of motion</a></li>
<li><a href="#orgc20cabb">2.3. Compliance of the Support</a></li>
<li><a href="#org67810a4">2.4. Equivalent Inverse Multiplicative Uncertainty</a></li>
<li><a href="#orge950395">2.5. Effect of the Isolation platform Stiffness</a></li>
<li><a href="#org6967854">2.6. Reduce the Uncertainty on the plant</a>
<li><a href="#orgbe4faa1">2.1. Introduction</a></li>
<li><a href="#org49bd260">2.2. Equations of motion</a></li>
<li><a href="#orgb277730">2.3. Compliance of the Support</a></li>
<li><a href="#org9880e78">2.4. Equivalent Inverse Multiplicative Uncertainty</a></li>
<li><a href="#orgcde9f95">2.5. Effect of the Isolation platform Stiffness</a></li>
<li><a href="#org71a0598">2.6. Reduce the Uncertainty on the plant</a>
<ul>
<li><a href="#orgafebadd">2.6.1. Effect of the platform&rsquo;s stiffness \(k\)</a></li>
<li><a href="#orgd9a82cb">2.6.2. Effect of the platform&rsquo;s damping \(c\)</a></li>
<li><a href="#orgd2fc303">2.6.3. Effect of the platform&rsquo;s mass \(m\)</a></li>
<li><a href="#org983db9a">2.6.1. Effect of the platform&rsquo;s stiffness \(k\)</a></li>
<li><a href="#org92a740c">2.6.2. Effect of the platform&rsquo;s damping \(c\)</a></li>
<li><a href="#org88d86c4">2.6.3. Effect of the platform&rsquo;s mass \(m\)</a></li>
</ul>
</li>
<li><a href="#orgde3616e">2.7. Conclusion</a></li>
<li><a href="#orgd68af10">2.7. Conclusion</a></li>
</ul>
</li>
</ul>
@@ -82,18 +86,18 @@ The goal is to study:
Two models are made to study these effects:
</p>
<ul class="org-ul">
<li>In section <a href="#org232d01f">1</a>, simple mass-spring-damper systems are chosen to model both the isolation platform and the flexible support</li>
<li>In section <a href="#orgb01b074">2</a>, we consider arbitrary support dynamics with multiplicative input uncertainty to study the unmodelled dynamics of the support</li>
<li>In section <a href="#org0009692">1</a>, simple mass-spring-damper systems are chosen to model both the isolation platform and the flexible support</li>
<li>In section <a href="#org407474d">2</a>, we consider arbitrary support dynamics with multiplicative input uncertainty to study the unmodelled dynamics of the support</li>
</ul>
<div id="outline-container-orgbe6e0b8" class="outline-2">
<h2 id="orgbe6e0b8"><span class="section-number-2">1</span> Simple Introductory Example</h2>
<div id="outline-container-orgb1adbd9" class="outline-2">
<h2 id="orgb1adbd9"><span class="section-number-2">1</span> Simple Introductory Example</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org232d01f"></a>
<a id="org0009692"></a>
</p>
<p>
Let&rsquo;s consider the system shown in Figure <a href="#org41bc770">1</a> consisting of:
Let&rsquo;s consider the system shown in Figure <a href="#org162a9b2">1</a> consisting of:
</p>
<ul class="org-ul">
<li>A <b>support</b> represented by a mass \(m^\prime\), a stiffness \(k^\prime\) and a dashpot \(c^\prime\)</li>
@@ -105,18 +109,18 @@ The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the suppor
</p>
<div id="org41bc770" class="figure">
<div id="org162a9b2" class="figure">
<p><img src="figs/2dof_system_stiffness_uncertainty.png" alt="2dof_system_stiffness_uncertainty.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Two degrees-of-freedom system</p>
</div>
</div>
<div id="outline-container-org3d4902a" class="outline-3">
<h3 id="org3d4902a"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div id="outline-container-orgf20a63e" class="outline-3">
<h3 id="orgf20a63e"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div class="outline-text-3" id="text-1-1">
<p>
If we write the equation of motion of the system in Figure <a href="#org41bc770">1</a>, we obtain:
If we write the equation of motion of the system in Figure <a href="#org162a9b2">1</a>, we obtain:
</p>
\begin{align}
ms^2 x &= F + (cs + k) (x^\prime - x) \\
@@ -127,14 +131,14 @@ If we write the equation of motion of the system in Figure <a href="#org41bc770"
After eliminating \(x^\prime\), we obtain:
</p>
\begin{equation}
\label{org2d73355}
\label{orgf3ed6bc}
\frac{x}{F} = \frac{m^\prime s^2 + c^\prime s + k^\prime}{ms^2(cs + k) + (ms^2 + cs + k)(m^\prime s^2 + c^\prime s + k^\prime)}
\end{equation}
</div>
</div>
<div id="outline-container-org8bd2a4a" class="outline-3">
<h3 id="org8bd2a4a"><span class="section-number-3">1.2</span> Initialization of the support dynamics</h3>
<div id="outline-container-org7f0c67f" class="outline-3">
<h3 id="org7f0c67f"><span class="section-number-3">1.2</span> Initialization of the support dynamics</h3>
<div class="outline-text-3" id="text-1-2">
<p>
Let the support have:
@@ -146,9 +150,9 @@ Let the support have:
</ul>
<div class="org-src-container">
<pre class="src src-matlab">mpi = 1e3;
cpi = 5e4;
kpi = 1e8;
<pre class="src src-matlab"> mpi = 1e3;
cpi = 5e4;
kpi = 1e8;
</pre>
</div>
@@ -156,14 +160,14 @@ kpi = 1e8;
Let&rsquo;s also consider some uncertainty in those parameters:
</p>
<div class="org-src-container">
<pre class="src src-matlab">mp = ureal('m', mpi, 'Percentage', 30);
cp = ureal('c', cpi, 'Percentage', 30);
kp = ureal('k', kpi, 'Percentage', 30);
<pre class="src src-matlab"> mp = ureal(<span class="org-string">'m'</span>, mpi, <span class="org-string">'Percentage'</span>, 30);
cp = ureal(<span class="org-string">'c'</span>, cpi, <span class="org-string">'Percentage'</span>, 30);
kp = ureal(<span class="org-string">'k'</span>, kpi, <span class="org-string">'Percentage'</span>, 30);
</pre>
</div>
<p>
The compliance of the support without the isolation platform is \(\frac{1}{m^\prime s^2 + c^\prime s + k^\prime}\) and its bode plot is shown in Figure <a href="#orgf0e5d13">2</a>.
The compliance of the support without the isolation platform is \(\frac{1}{m^\prime s^2 + c^\prime s + k^\prime}\) and its bode plot is shown in Figure <a href="#orga056e3a">2</a>.
</p>
<p>
@@ -171,7 +175,7 @@ One can see that support has a resonance frequency of \(\omega_0^\prime = 50\ Hz
</p>
<div id="orgf0e5d13" class="figure">
<div id="orga056e3a" class="figure">
<p><img src="figs/nominal_support_compliance_dynamics.png" alt="nominal_support_compliance_dynamics.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Nominal compliance of the support (<a href="./figs/nominal_support_compliance_dynamics.png">png</a>, <a href="./figs/nominal_support_compliance_dynamics.pdf">pdf</a>)</p>
@@ -179,14 +183,14 @@ One can see that support has a resonance frequency of \(\omega_0^\prime = 50\ Hz
</div>
</div>
<div id="outline-container-orgefb9b71" class="outline-3">
<h3 id="orgefb9b71"><span class="section-number-3">1.3</span> Initialization of the isolation platform</h3>
<div id="outline-container-org557badc" class="outline-3">
<h3 id="org557badc"><span class="section-number-3">1.3</span> Initialization of the isolation platform</h3>
<div class="outline-text-3" id="text-1-3">
<p>
Let&rsquo;s first fix the mass of the payload to be isolated:
</p>
<div class="org-src-container">
<pre class="src src-matlab">m = 100;
<pre class="src src-matlab"> m = 100;
</pre>
</div>
@@ -201,15 +205,15 @@ And we generate three isolation platforms:
</div>
</div>
<div id="outline-container-org3bc4ad1" class="outline-3">
<h3 id="org3bc4ad1"><span class="section-number-3">1.4</span> Comparison</h3>
<div id="outline-container-org2c66ca0" class="outline-3">
<h3 id="org2c66ca0"><span class="section-number-3">1.4</span> Comparison</h3>
<div class="outline-text-3" id="text-1-4">
<p>
The obtained dynamics from \(F\) to \(x\) for the three isolation platform are shown in Figure <a href="#org5fb09ae">3</a>.
The obtained dynamics from \(F\) to \(x\) for the three isolation platform are shown in Figure <a href="#org1927c94">3</a>.
</p>
<div id="org5fb09ae" class="figure">
<div id="org1927c94" class="figure">
<p><img src="figs/plant_dynamics_uncertainty_stiff_mid_soft.png" alt="plant_dynamics_uncertainty_stiff_mid_soft.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Obtained plant for the three isolation platforms considered (<a href="./figs/plant_dynamics_uncertainty_stiff_mid_soft.png">png</a>, <a href="./figs/plant_dynamics_uncertainty_stiff_mid_soft.pdf">pdf</a>)</p>
@@ -217,10 +221,10 @@ The obtained dynamics from \(F\) to \(x\) for the three isolation platform are s
</div>
</div>
<div id="outline-container-org999e1c5" class="outline-3">
<h3 id="org999e1c5"><span class="section-number-3">1.5</span> Conclusion</h3>
<div id="outline-container-org8e2f9eb" class="outline-3">
<h3 id="org8e2f9eb"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<div class="important" id="orgc636dcd">
<p>
The soft platform dynamics does not seems to depend on the dynamics of the support nor to be affect by the dynamic uncertainty of the support.
</p>
@@ -230,32 +234,32 @@ The soft platform dynamics does not seems to depend on the dynamics of the suppo
</div>
</div>
<div id="outline-container-orge1d3484" class="outline-2">
<h2 id="orge1d3484"><span class="section-number-2">2</span> Generalization to arbitrary dynamics</h2>
<div id="outline-container-org60caf30" class="outline-2">
<h2 id="org60caf30"><span class="section-number-2">2</span> Generalization to arbitrary dynamics</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="orgb01b074"></a>
<a id="org407474d"></a>
</p>
</div>
<div id="outline-container-org3948d1f" class="outline-3">
<h3 id="org3948d1f"><span class="section-number-3">2.1</span> Introduction</h3>
<div id="outline-container-orgbe4faa1" class="outline-3">
<h3 id="orgbe4faa1"><span class="section-number-3">2.1</span> Introduction</h3>
<div class="outline-text-3" id="text-2-1">
<p>
Let&rsquo;s now consider a general support described by its <b>compliance</b> \(G^\prime(s) = \frac{x^\prime}{F^\prime}\) as shown in Figure <a href="#orgaa4cf23">4</a>.
Let&rsquo;s now consider a general support described by its <b>compliance</b> \(G^\prime(s) = \frac{x^\prime}{F^\prime}\) as shown in Figure <a href="#org4baf379">4</a>.
</p>
<div id="orgaa4cf23" class="figure">
<div id="org4baf379" class="figure">
<p><img src="figs/general_support_compliance.png" alt="general_support_compliance.png" />
</p>
<p><span class="figure-number">Figure 4: </span>General support</p>
</div>
<p>
Now let&rsquo;s consider the system consisting of a mass-spring-system (the isolation platform) on top of a general support as shown in Figure <a href="#org524a33a">5</a>.
Now let&rsquo;s consider the system consisting of a mass-spring-system (the isolation platform) on top of a general support as shown in Figure <a href="#org36e0868">5</a>.
</p>
<div id="org524a33a" class="figure">
<div id="org36e0868" class="figure">
<p><img src="figs/general_support_with_isolator.png" alt="general_support_with_isolator.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Mass-Spring-Damper system on top of a general support</p>
@@ -263,8 +267,8 @@ Now let&rsquo;s consider the system consisting of a mass-spring-system (the isol
</div>
</div>
<div id="outline-container-org18c1c3f" class="outline-3">
<h3 id="org18c1c3f"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div id="outline-container-org49bd260" class="outline-3">
<h3 id="org49bd260"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div class="outline-text-3" id="text-2-2">
<p>
We have to following equations of motion:
@@ -279,7 +283,7 @@ We have to following equations of motion:
And by eliminating \(F^\prime\) and \(x^\prime\), we find the plant dynamics \(G(s) = \frac{x}{F}\).
</p>
<div class="important">
<div class="important" id="orgf4990df">
\begin{equation}
\frac{x}{F} = \frac{1}{ms^2 + cs + k + ms^2(cs + k)G^\prime(s)} \label{eq:plant_dynamics_general_support}
\end{equation}
@@ -287,20 +291,20 @@ And by eliminating \(F^\prime\) and \(x^\prime\), we find the plant dynamics \(G
</div>
<p>
In order to verify that the formula is correct, let&rsquo;s take the same mass-spring-damper system used in the system shown in Figure <a href="#org41bc770">1</a>:
In order to verify that the formula is correct, let&rsquo;s take the same mass-spring-damper system used in the system shown in Figure <a href="#org162a9b2">1</a>:
\[ \frac{x^\prime}{F^\prime} = \frac{1}{m^\prime s^2 + c^\prime s + k^\prime} \]
</p>
<p>
And we obtain
\[ \frac{x}{F} = \frac{m^\prime s^2 + c^\prime s + k^\prime}{(ms^2 + cs + k)(m^\prime s^2 + c^\prime s + k^\prime) + ms^2(cs + k)} \]
Which is the same transfer function that was obtained in section <a href="#org232d01f">1</a> (Eq. \eqref{eq:plant_simple_system}).
Which is the same transfer function that was obtained in section <a href="#org0009692">1</a> (Eq. \eqref{eq:plant_simple_system}).
</p>
</div>
</div>
<div id="outline-container-orgc20cabb" class="outline-3">
<h3 id="orgc20cabb"><span class="section-number-3">2.3</span> Compliance of the Support</h3>
<div id="outline-container-orgb277730" class="outline-3">
<h3 id="orgb277730"><span class="section-number-3">2.3</span> Compliance of the Support</h3>
<div class="outline-text-3" id="text-2-3">
<p>
We model the support by a mass-spring-damper model with some uncertainty.
@@ -312,16 +316,16 @@ The main resonance of the support is then \(\omega^\prime = \sqrt{\frac{m^\prime
</p>
<div class="org-src-container">
<pre class="src src-matlab">m0 = 1e3;
c0 = 5e4;
k0 = 1e8;
<pre class="src src-matlab"> m0 = 1e3;
c0 = 5e4;
k0 = 1e8;
Gp0 = 1/(m0*s^2 + c0*s + k0);
Gp0 = 1<span class="org-type">/</span>(m0<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c0<span class="org-type">*</span>s <span class="org-type">+</span> k0);
</pre>
</div>
<p>
Let&rsquo;s represent the uncertainty on the compliance of the support by a multiplicative uncertainty (Figure <a href="#orgaaa2d77">6</a>):
Let&rsquo;s represent the uncertainty on the compliance of the support by a multiplicative uncertainty (Figure <a href="#orgb0ae74c">6</a>):
\[ G^\prime(s) = G_0^\prime(s)(1 + w_I^\prime(s)\Delta_I(s)) \quad |\Delta_I(j\omega)| < 1\ \forall \omega \]
</p>
@@ -330,7 +334,7 @@ This could represent <b>unmodelled dynamics</b> or unknown parameters of the sup
</p>
<div id="orgaaa2d77" class="figure">
<div id="orgb0ae74c" class="figure">
<p><img src="figs/input_uncertainty_set.png" alt="input_uncertainty_set.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Input Multiplicative Uncertainty</p>
@@ -346,11 +350,11 @@ where \(r_0\) is the relative uncertainty at steady-state, \(1/\tau\) is the fre
The parameters are defined below.
</p>
<div class="org-src-container">
<pre class="src src-matlab">r0 = 0.5;
tau = 1/(100*2*pi);
rinf = 10;
<pre class="src src-matlab"> r0 = 0.5;
tau = 1<span class="org-type">/</span>(100<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>);
rinf = 10;
wI = (tau*s + r0)/((tau/rinf)*s + 1);
wI = (tau<span class="org-type">*</span>s <span class="org-type">+</span> r0)<span class="org-type">/</span>((tau<span class="org-type">/</span>rinf)<span class="org-type">*</span>s <span class="org-type">+</span> 1);
</pre>
</div>
@@ -358,7 +362,7 @@ wI = (tau*s + r0)/((tau/rinf)*s + 1);
We then generate a complex \(\Delta\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">DeltaI = ucomplex('A',0);
<pre class="src src-matlab"> DeltaI = ucomplex(<span class="org-string">'A'</span>,0);
</pre>
</div>
@@ -366,16 +370,16 @@ We then generate a complex \(\Delta\).
We generate the uncertain plant \(G^\prime(s)\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gp = Gp0*(1+wI*DeltaI);
<pre class="src src-matlab"> Gp = Gp0<span class="org-type">*</span>(1<span class="org-type">+</span>wI<span class="org-type">*</span>DeltaI);
</pre>
</div>
<p>
A set of uncertainty support&rsquo;s compliance transfer functions is shown in Figure <a href="#orgcac0998">7</a>.
A set of uncertainty support&rsquo;s compliance transfer functions is shown in Figure <a href="#org5d8850d">7</a>.
</p>
<div id="orgcac0998" class="figure">
<div id="org5d8850d" class="figure">
<p><img src="figs/compliance_support_uncertainty.png" alt="compliance_support_uncertainty.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Uncertainty of the support&rsquo;s compliance (<a href="./figs/compliance_support_uncertainty.png">png</a>, <a href="./figs/compliance_support_uncertainty.pdf">pdf</a>)</p>
@@ -383,8 +387,8 @@ A set of uncertainty support&rsquo;s compliance transfer functions is shown in F
</div>
</div>
<div id="outline-container-org67810a4" class="outline-3">
<h3 id="org67810a4"><span class="section-number-3">2.4</span> Equivalent Inverse Multiplicative Uncertainty</h3>
<div id="outline-container-org9880e78" class="outline-3">
<h3 id="org9880e78"><span class="section-number-3">2.4</span> Equivalent Inverse Multiplicative Uncertainty</h3>
<div class="outline-text-3" id="text-2-4">
<p>
Let&rsquo;s express the uncertainty of the plant \(x/F\) as a function of the parameters as well as of the uncertainty on the platform&rsquo;s compliance:
@@ -395,9 +399,9 @@ Let&rsquo;s express the uncertainty of the plant \(x/F\) as a function of the pa
&= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \cdot \frac{1}{1 + \frac{ms^2(cs + k)G_0^\prime(s) w_I(s)}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \Delta(s)}\\
\end{align*}
<div class="important">
<div class="important" id="org6aceb50">
<p>
We can the plant dynamics that as an inverse multiplicative uncertainty (Figure <a href="#orge738173">8</a>):
We can the plant dynamics that as an inverse multiplicative uncertainty (Figure <a href="#org85dcc28">8</a>):
</p>
\begin{equation}
\frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1}
@@ -413,7 +417,7 @@ with:
</div>
<div id="orge738173" class="figure">
<div id="org85dcc28" class="figure">
<p><img src="figs/inverse_uncertainty_set.png" alt="inverse_uncertainty_set.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Inverse Multiplicative Uncertainty</p>
@@ -421,14 +425,14 @@ with:
</div>
</div>
<div id="outline-container-orge950395" class="outline-3">
<h3 id="orge950395"><span class="section-number-3">2.5</span> Effect of the Isolation platform Stiffness</h3>
<div id="outline-container-orgcde9f95" class="outline-3">
<h3 id="orgcde9f95"><span class="section-number-3">2.5</span> Effect of the Isolation platform Stiffness</h3>
<div class="outline-text-3" id="text-2-5">
<p>
Let&rsquo;s first fix the mass of the payload to be isolated:
</p>
<div class="org-src-container">
<pre class="src src-matlab">m = 100;
<pre class="src src-matlab"> m = 100;
</pre>
</div>
@@ -445,12 +449,12 @@ And we generate three isolation platforms:
Soft Isolation Platform:
</p>
<div class="org-src-container">
<pre class="src src-matlab">k_soft = m*(2*pi*5)^2;
c_soft = 0.1*sqrt(m*k_soft);
<pre class="src src-matlab"> k_soft = m<span class="org-type">*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5)<span class="org-type">^</span>2;
c_soft = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_soft);
G_soft = 1/(m*s^2 + c_soft*s + k_soft + m*s^2*(c_soft*s + k_soft)*Gp);
G0_soft = 1/(m*s^2 + c_soft*s + k_soft + m*s^2*(c_soft*s + k_soft)*Gp0);
wiI_soft = Gp0*m*s^2*(c_soft*s + k_soft)*G0_soft*wI;
G_soft = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft)<span class="org-type">*</span>Gp);
G0_soft = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft)<span class="org-type">*</span>Gp0);
wiI_soft = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft)<span class="org-type">*</span>G0_soft<span class="org-type">*</span>wI;
</pre>
</div>
@@ -458,12 +462,12 @@ wiI_soft = Gp0*m*s^2*(c_soft*s + k_soft)*G0_soft*wI;
Mid Isolation Platform
</p>
<div class="org-src-container">
<pre class="src src-matlab">k_mid = m*(2*pi*50)^2;
c_mid = 0.1*sqrt(m*k_mid);
<pre class="src src-matlab"> k_mid = m<span class="org-type">*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50)<span class="org-type">^</span>2;
c_mid = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_mid);
G_mid = 1/(m*s^2 + c_mid*s + k_mid + m*s^2*(c_mid*s + k_mid)*Gp);
G0_mid = 1/(m*s^2 + c_mid*s + k_mid + m*s^2*(c_mid*s + k_mid)*Gp0);
wiI_mid = Gp0*m*s^2*(c_mid*s + k_mid)*G0_mid*wI;
G_mid = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid)<span class="org-type">*</span>Gp);
G0_mid = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid)<span class="org-type">*</span>Gp0);
wiI_mid = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid)<span class="org-type">*</span>G0_mid<span class="org-type">*</span>wI;
</pre>
</div>
@@ -471,35 +475,35 @@ wiI_mid = Gp0*m*s^2*(c_mid*s + k_mid)*G0_mid*wI;
Stiff Isolation Platform
</p>
<div class="org-src-container">
<pre class="src src-matlab">k_stiff = m*(2*pi*500)^2;
c_stiff = 0.1*sqrt(m*k_stiff);
<pre class="src src-matlab"> k_stiff = m<span class="org-type">*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>500)<span class="org-type">^</span>2;
c_stiff = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_stiff);
G_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + m*s^2*(c_stiff*s + k_stiff)*Gp);
G0_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + m*s^2*(c_stiff*s + k_stiff)*Gp0);
wiI_stiff = Gp0*m*s^2*(c_stiff*s + k_stiff)*G0_stiff*wI;
G_stiff = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff)<span class="org-type">*</span>Gp);
G0_stiff = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff)<span class="org-type">*</span>Gp0);
wiI_stiff = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff)<span class="org-type">*</span>G0_stiff<span class="org-type">*</span>wI;
</pre>
</div>
<p>
The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure <a href="#org89aa89f">9</a>.
The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure <a href="#org3126520">9</a>.
</p>
<div id="org89aa89f" class="figure">
<div id="org3126520" class="figure">
<p><img src="figs/plant_uncertainty_stiffness_isolator.png" alt="plant_uncertainty_stiffness_isolator.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Obtained plant for the three isolators (<a href="./figs/plant_uncertainty_stiffness_isolator.png">png</a>, <a href="./figs/plant_uncertainty_stiffness_isolator.pdf">pdf</a>)</p>
</div>
<p>
The obtain result is very similar to the one obtain in section <a href="#org232d01f">1</a>, except for the stiff isolation that experience lot&rsquo;s of uncertainty at high frequency.
The obtain result is very similar to the one obtain in section <a href="#org0009692">1</a>, except for the stiff isolation that experience lot&rsquo;s of uncertainty at high frequency.
This is due to the fact that with the current model, at high frequency, the support&rsquo;s compliance uncertainty is much higher than the previous model.
</p>
</div>
</div>
<div id="outline-container-org6967854" class="outline-3">
<h3 id="org6967854"><span class="section-number-3">2.6</span> Reduce the Uncertainty on the plant</h3>
<div id="outline-container-org71a0598" class="outline-3">
<h3 id="org71a0598"><span class="section-number-3">2.6</span> Reduce the Uncertainty on the plant</h3>
<div class="outline-text-3" id="text-2-6">
<p>
Now that we know the expression of the uncertainty on the plant, we can wonder what parameters of the isolation platform would lower the plant uncertainty, or at least bring the uncertainty to reasonable level.
@@ -515,30 +519,30 @@ Let&rsquo;s study separately the effect of the platform&rsquo;s mass, damping an
</p>
</div>
<div id="outline-container-orgafebadd" class="outline-4">
<h4 id="orgafebadd"><span class="section-number-4">2.6.1</span> Effect of the platform&rsquo;s stiffness \(k\)</h4>
<div id="outline-container-org983db9a" class="outline-4">
<h4 id="org983db9a"><span class="section-number-4">2.6.1</span> Effect of the platform&rsquo;s stiffness \(k\)</h4>
<div class="outline-text-4" id="text-2-6-1">
<p>
Let&rsquo;s fix \(\xi = \frac{c}{2\sqrt{km}} = 0.1\), \(m = 100\ [kg]\) and see the evolution of \(|w_{iI}(j\omega)|\) with \(k\).
</p>
<p>
This is first shown for few values of the stiffness \(k\) in figure <a href="#org5bc976b">10</a>
This is first shown for few values of the stiffness \(k\) in figure <a href="#org9addccd">10</a>
</p>
<div id="org5bc976b" class="figure">
<div id="org9addccd" class="figure">
<p><img src="figs/inverse_multiplicative_uncertainty_norm_few_k.png" alt="inverse_multiplicative_uncertainty_norm_few_k.png" />
</p>
<p><span class="figure-number">Figure 10: </span>caption (<a href="./figs/inverse_multiplicative_uncertainty_norm_few_k.png">png</a>, <a href="./figs/inverse_multiplicative_uncertainty_norm_few_k.pdf">pdf</a>)</p>
</div>
<p>
The norm of the uncertainty weight \(|w_iI(j\omega)|\) is displayed as a function of \(\omega\) and \(k\) in Figure <a href="#orgb283d43">11</a>.
The norm of the uncertainty weight \(|w_iI(j\omega)|\) is displayed as a function of \(\omega\) and \(k\) in Figure <a href="#orgc7a264d">11</a>.
</p>
<div id="orgb283d43" class="figure">
<div id="orgc7a264d" class="figure">
<p><img src="figs/inverse_multiplicative_uncertainty_norm_k.png" alt="inverse_multiplicative_uncertainty_norm_k.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the platform&rsquo;s stiffness \(k\) (<a href="./figs/inverse_multiplicative_uncertainty_norm_k.png">png</a>, <a href="./figs/inverse_multiplicative_uncertainty_norm_k.pdf">pdf</a>)</p>
@@ -553,12 +557,12 @@ Instead of plotting as a function of the platform&rsquo;s stiffness, we can plot
</ul>
<p>
The obtain plot is shown in Figure <a href="#org9adcd50">12</a>.
The obtain plot is shown in Figure <a href="#orgae2fb41">12</a>.
In that case, we can see that with a platform&rsquo;s resonance frequency 10 times lower than the resonance of the support, we get less than \(1\%\) uncertainty.
</p>
<div id="org9adcd50" class="figure">
<div id="orgae2fb41" class="figure">
<p><img src="figs/inverse_multiplicative_uncertainty_k_normalized_frequency.png" alt="inverse_multiplicative_uncertainty_k_normalized_frequency.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the frequency ratio \(\omega_0/\omega_0^\prime\) (<a href="./figs/inverse_multiplicative_uncertainty_k_normalized_frequency.png">png</a>, <a href="./figs/inverse_multiplicative_uncertainty_k_normalized_frequency.pdf">pdf</a>)</p>
@@ -566,15 +570,15 @@ In that case, we can see that with a platform&rsquo;s resonance frequency 10 tim
</div>
</div>
<div id="outline-container-orgd9a82cb" class="outline-4">
<h4 id="orgd9a82cb"><span class="section-number-4">2.6.2</span> Effect of the platform&rsquo;s damping \(c\)</h4>
<div id="outline-container-org92a740c" class="outline-4">
<h4 id="org92a740c"><span class="section-number-4">2.6.2</span> Effect of the platform&rsquo;s damping \(c\)</h4>
<div class="outline-text-4" id="text-2-6-2">
<p>
Let&rsquo;s fix \(k = 10^7\ [N/m]\), \(m = 100\ [kg]\) and see the evolution of \(|w_{iI}(j\omega)|\) with the isolator damping \(c\) (Figure <a href="#org983fa6b">13</a>).
Let&rsquo;s fix \(k = 10^7\ [N/m]\), \(m = 100\ [kg]\) and see the evolution of \(|w_{iI}(j\omega)|\) with the isolator damping \(c\) (Figure <a href="#org21e5135">13</a>).
</p>
<div id="org983fa6b" class="figure">
<div id="org21e5135" class="figure">
<p><img src="figs/inverse_multiplicative_uncertainty_norm_c.png" alt="inverse_multiplicative_uncertainty_norm_c.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the platform&rsquo;s damping ratio \(\xi\) (<a href="./figs/inverse_multiplicative_uncertainty_norm_c.png">png</a>, <a href="./figs/inverse_multiplicative_uncertainty_norm_c.pdf">pdf</a>)</p>
@@ -583,15 +587,15 @@ Let&rsquo;s fix \(k = 10^7\ [N/m]\), \(m = 100\ [kg]\) and see the evolution of
</div>
<div id="outline-container-orgd2fc303" class="outline-4">
<h4 id="orgd2fc303"><span class="section-number-4">2.6.3</span> Effect of the platform&rsquo;s mass \(m\)</h4>
<div id="outline-container-org88d86c4" class="outline-4">
<h4 id="org88d86c4"><span class="section-number-4">2.6.3</span> Effect of the platform&rsquo;s mass \(m\)</h4>
<div class="outline-text-4" id="text-2-6-3">
<p>
Let&rsquo;s fix \(k = 10^7\ [N/m]\), \(\xi = \frac{c}{2\sqrt{km}} = 0.1\) and see the evolution of \(|w_{iI}(j\omega)|\) with the payload mass \(m\) (Figure <a href="#orgf899c7a">14</a>).
Let&rsquo;s fix \(k = 10^7\ [N/m]\), \(\xi = \frac{c}{2\sqrt{km}} = 0.1\) and see the evolution of \(|w_{iI}(j\omega)|\) with the payload mass \(m\) (Figure <a href="#org6c5410a">14</a>).
</p>
<div id="orgf899c7a" class="figure">
<div id="org6c5410a" class="figure">
<p><img src="figs/inverse_multiplicative_uncertainty_norm_m.png" alt="inverse_multiplicative_uncertainty_norm_m.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the payload mass \(m\) (<a href="./figs/inverse_multiplicative_uncertainty_norm_m.png">png</a>, <a href="./figs/inverse_multiplicative_uncertainty_norm_m.pdf">pdf</a>)</p>
@@ -600,10 +604,10 @@ Let&rsquo;s fix \(k = 10^7\ [N/m]\), \(\xi = \frac{c}{2\sqrt{km}} = 0.1\) and se
</div>
</div>
<div id="outline-container-orgde3616e" class="outline-3">
<h3 id="orgde3616e"><span class="section-number-3">2.7</span> Conclusion</h3>
<div id="outline-container-orgd68af10" class="outline-3">
<h3 id="orgd68af10"><span class="section-number-3">2.7</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-7">
<div class="important">
<div class="important" id="orgcdbfc6b">
<p>
If the goal is to have an acceptable (\(<10\%\)) uncertainty on the plant until the highest frequency, two design choice for the isolation platform are possible:
</p>
@@ -629,7 +633,7 @@ Thus, if a stiff isolation platform is used, the recommendation is to have the l
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-05-05 mar. 10:33</p>
<p class="date">Created: 2021-02-20 sam. 23:08</p>
</div>
</body>
</html>