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< a accesskey = "h" href = "index.html" > UP < / a >
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< / div > < div id = "content" >
< h1 class = "title" > Determination of the optimal nano-hexapod’ s stiffness for reducing the effect of disturbances< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#org9e9f810" > 1. Disturbances< / a > < / li >
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< li > < a href = "#orgc44cf7e" > 2. Effect of disturbances on the position error< / a >
< ul >
< li > < a href = "#org524df41" > 2.1. Initialization< / a > < / li >
< li > < a href = "#orgaf88c9f" > 2.2. Identification< / a > < / li >
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< li > < a href = "#org78dd34d" > 2.3. Sensitivity to Stages vibration (Filtering)< / a > < / li >
< li > < a href = "#orgd4ea2f4" > 2.4. Effect of Ground motion (Transmissibility).< / a > < / li >
< li > < a href = "#org0448746" > 2.5. Direct Forces (Compliance).< / a > < / li >
< li > < a href = "#orgea74617" > 2.6. Conclusion< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#org6527e58" > 3. Effect of granite stiffness< / a >
< ul >
< li > < a href = "#orgd3e5fe1" > 3.1. Analytical Analysis< / a > < / li >
< li > < a href = "#org9215f81" > 3.2. Soft Granite< / a > < / li >
< li > < a href = "#org8878556" > 3.3. Effect of the Granite transfer function< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org8a88fb0" > 4. Open Loop Budget Error< / a >
< ul >
< li > < a href = "#org2ee8120" > 4.1. Load of the identified disturbances and transfer functions< / a > < / li >
< li > < a href = "#orgb1a3177" > 4.2. Equations< / a > < / li >
< li > < a href = "#org63eade9" > 4.3. Results< / a > < / li >
< li > < a href = "#orgef96b89" > 4.4. Cumulative Amplitude Spectrum< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org34c0f38" > 5. Closed Loop Budget Error< / a >
< ul >
< li > < a href = "#orgf2d36a1" > 5.1. Reduction thanks to feedback - Required bandwidth< / a > < / li >
< / ul >
< / li >
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< li > < a href = "#org0953c03" > 6. Conclusion< / a > < / li >
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< / ul >
< / div >
< / div >
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< p >
In this document is studied how the stiffness of the nano-hexapod will impact the effect of disturbances on the position error of the sample.
< / p >
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< p >
It is divided in the following sections:
< / p >
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< ul class = "org-ul" >
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< li > Section < a href = "#org17d3d6a" > 1< / a > : the disturbances are listed and their Power Spectral Densities (PSD) are shown< / li >
< li > Section < a href = "#orgf9e4300" > 2< / a > : the transfer functions from disturbances to the position error of the sample are computed for a wide range of nano-hexapod stiffnesses< / li >
< li > Section < a href = "#orgd4105b6" > 3< / a > :< / li >
< li > Section < a href = "#org5d05990" > 4< / a > : from both the PSD of the disturbances and the transfer function from disturbances to sample’ s position errors, we compute the resulting PSD and Cumulative Amplitude Spectrum (CAS)< / li >
< li > Section < a href = "#orgd3503fb" > 5< / a > : from a simplistic model is computed the required control bandwidth to reduce the position error to acceptable values< / li >
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< / ul >
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< div id = "outline-container-org9e9f810" class = "outline-2" >
< h2 id = "org9e9f810" > < span class = "section-number-2" > 1< / span > Disturbances< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< p >
< a id = "org17d3d6a" > < / a >
< / p >
< p >
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The main disturbances considered here are:
< / p >
< ul class = "org-ul" >
< li > \(D_w\): Ground displacement in the \(x\), \(y\) and \(z\) directions< / li >
< li > \(F_{ty}\): Forces applied by the Translation stage in the \(x\) and \(z\) directions< / li >
< li > \(F_{rz}\): Forces applied by the Spindle in the \(z\) direction< / li >
< li > \(F_d\): Direct forces applied at the center of mass of the Payload< / li >
< / ul >
< p >
The level of these disturbances has been identified form experiments which are detailed in < a href = "disturbances.html" > this< / a > document.
< / p >
< p >
The measured Amplitude Spectral Densities (ASD) of these forces are shown in Figures < a href = "#org6b4e47c" > 1< / a > and < a href = "#orgb7b8e77" > 2< / a > .
< / p >
< p >
In this study, the expected frequency content of the direct forces applied to the payload is not considered.
< / p >
< div id = "org6b4e47c" class = "figure" >
< p > < img src = "figs/opt_stiff_dist_gm.png" alt = "opt_stiff_dist_gm.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 1: < / span > Amplitude Spectral Density of the Ground Displacement (< a href = "./figs/opt_stiff_dist_gm.png" > png< / a > , < a href = "./figs/opt_stiff_dist_gm.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "orgb7b8e77" class = "figure" >
< p > < img src = "figs/opt_stiff_dist_fty_frz.png" alt = "opt_stiff_dist_fty_frz.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Amplitude Spectral Density of the “ parasitic” forces comming from the Translation stage and the spindle (< a href = "./figs/opt_stiff_dist_fty_frz.png" > png< / a > , < a href = "./figs/opt_stiff_dist_fty_frz.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
< div id = "outline-container-orgc44cf7e" class = "outline-2" >
< h2 id = "orgc44cf7e" > < span class = "section-number-2" > 2< / span > Effect of disturbances on the position error< / h2 >
< div class = "outline-text-2" id = "text-2" >
< p >
< a id = "orgf9e4300" > < / a >
< / p >
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< p >
In this section, we use the Simscape model to identify the transfer function from disturbances to the position error of the sample.
We do that for a wide range of nano-hexapod stiffnesses and we compare the obtained results.
< / p >
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< / div >
< div id = "outline-container-org524df41" class = "outline-3" >
< h3 id = "org524df41" > < span class = "section-number-3" > 2.1< / span > Initialization< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
< p >
We initialize all the stages with the default parameters.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
< / pre >
< / div >
< p >
We use a sample mass of 10kg.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeSample(< span class = "org-string" > 'mass'< / span > , 10);
< / pre >
< / div >
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< p >
We include gravity, and we use no controller.
< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeSimscapeConfiguration(< span class = "org-string" > 'gravity'< / span > , < span class = "org-constant" > true< / span > );
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initializeController();
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initializeDisturbances(< span class = "org-string" > 'enable'< / span > , < span class = "org-constant" > false< / span > );
initializeLoggingConfiguration(< span class = "org-string" > 'log'< / span > , < span class = "org-string" > 'none'< / span > );
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgaf88c9f" class = "outline-3" >
< h3 id = "orgaf88c9f" > < span class = "section-number-3" > 2.2< / span > Identification< / h3 >
< div class = "outline-text-3" id = "text-2-2" >
< p >
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The considered inputs are:
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< / p >
< ul class = "org-ul" >
< li > < code > Dwx< / code > : Ground displacement in the \(x\) direction< / li >
< li > < code > Dwy< / code > : Ground displacement in the \(y\) direction< / li >
< li > < code > Dwz< / code > : Ground displacement in the \(z\) direction< / li >
< li > < code > Fty_x< / code > : Forces applied by the Translation stage in the \(x\) direction< / li >
< li > < code > Fty_z< / code > : Forces applied by the Translation stage in the \(z\) direction< / li >
< li > < code > Frz_z< / code > : Forces applied by the Spindle in the \(z\) direction< / li >
< li > < code > Fd< / code > : Direct forces applied at the center of mass of the Payload< / li >
< / ul >
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< p >
The outputs are < code > Ex< / code > , < code > Ey< / code > , < code > Ez< / code > , < code > Erx< / code > , < code > Ery< / code > , < code > Erz< / code > which are the 3 positions and 3 orientations errors of the sample.
< / p >
< p >
We initialize the set of the nano-hexapod stiffnesses, and for each of them, we identify the dynamics from defined inputs to defined outputs.
< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > Ks = logspace(3,9,7); < span class = "org-comment" > % [N/m]< / span >
< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org78dd34d" class = "outline-3" >
< h3 id = "org78dd34d" > < span class = "section-number-3" > 2.3< / span > Sensitivity to Stages vibration (Filtering)< / h3 >
< div class = "outline-text-3" id = "text-2-3" >
< p >
The sensitivity the stage vibrations are displayed:
< / p >
< ul class = "org-ul" >
< li > Figure < a href = "#orgf55ba1b" > 3< / a > : sensitivity to vertical spindle vibrations< / li >
< li > Figure < a href = "#orgce1ac2c" > 4< / a > : sensitivity to vertical translation stage vibrations< / li >
< li > Figure < a href = "#org1a24ee2" > 5< / a > : sensitivity to horizontal (x) translation stage vibrations< / li >
< / ul >
< div id = "orgf55ba1b" class = "figure" >
< p > < img src = "figs/opt_stiff_sensitivity_Frz.png" alt = "opt_stiff_sensitivity_Frz.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Sensitivity to Spindle vertical motion error (\(F_{rz}\)) to the vertical error position of the sample (\(E_z\)) (< a href = "./figs/opt_stiff_sensitivity_Frz.png" > png< / a > , < a href = "./figs/opt_stiff_sensitivity_Frz.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "orgce1ac2c" class = "figure" >
< p > < img src = "figs/opt_stiff_sensitivity_Fty_z.png" alt = "opt_stiff_sensitivity_Fty_z.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > Sensitivity to Translation stage vertical motion error (\(F_{ty,z}\)) to the vertical error position of the sample (\(E_z\)) (< a href = "./figs/opt_stiff_sensitivity_Fty_z.png" > png< / a > , < a href = "./figs/opt_stiff_sensitivity_Fty_z.pdf" > pdf< / a > )< / p >
< / div >
< div id = "org1a24ee2" class = "figure" >
< p > < img src = "figs/opt_stiff_sensitivity_Fty_x.png" alt = "opt_stiff_sensitivity_Fty_x.png" / >
< / p >
< p > < span class = "figure-number" > Figure 5: < / span > Sensitivity to Translation stage \(x\) motion error (\(F_{ty,x}\)) to the error position of the sample in the \(x\) direction (\(E_x\)) (< a href = "./figs/opt_stiff_sensitivity_Fty_x.png" > png< / a > , < a href = "./figs/opt_stiff_sensitivity_Fty_x.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgd4ea2f4" class = "outline-3" >
< h3 id = "orgd4ea2f4" > < span class = "section-number-3" > 2.4< / span > Effect of Ground motion (Transmissibility).< / h3 >
< div class = "outline-text-3" id = "text-2-4" >
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< p >
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The effect of Ground motion on the position error of the sample is shown in Figure < a href = "#org212587b" > 6< / a > .
< / p >
< div id = "org212587b" class = "figure" >
< p > < img src = "figs/opt_stiff_sensitivity_Dw.png" alt = "opt_stiff_sensitivity_Dw.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Sensitivity to Ground motion (\(D_{w}\)) to the position error of the sample (\(E_y\) and \(E_z\)) (< a href = "./figs/opt_stiff_sensitivity_Dw.png" > png< / a > , < a href = "./figs/opt_stiff_sensitivity_Dw.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
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< / div >
< div id = "outline-container-org0448746" class = "outline-3" >
< h3 id = "org0448746" > < span class = "section-number-3" > 2.5< / span > Direct Forces (Compliance).< / h3 >
< div class = "outline-text-3" id = "text-2-5" >
< p >
The effect of direct forces/torques applied on the sample (cable forces for instance) on the position error of the sample is shown in Figure < a href = "#orga33395f" > 7< / a > .
< / p >
< div id = "orga33395f" class = "figure" >
< p > < img src = "figs/opt_stiff_sensitivity_Fd.png" alt = "opt_stiff_sensitivity_Fd.png" / >
< / p >
< p > < span class = "figure-number" > Figure 7: < / span > Sensitivity to Direct forces and torques applied to the sample (\(F_d\), \(M_d\)) to the position error of the sample (< a href = "./figs/opt_stiff_sensitivity_Fd.png" > png< / a > , < a href = "./figs/opt_stiff_sensitivity_Fd.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-orgea74617" class = "outline-3" >
< h3 id = "orgea74617" > < span class = "section-number-3" > 2.6< / span > Conclusion< / h3 >
< div class = "outline-text-3" id = "text-2-6" >
< div class = "important" >
< p >
< / p >
2020-04-07 15:26:29 +02:00
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org6527e58" class = "outline-2" >
< h2 id = "org6527e58" > < span class = "section-number-2" > 3< / span > Effect of granite stiffness< / h2 >
< div class = "outline-text-2" id = "text-3" >
< p >
< a id = "orgd4105b6" > < / a >
< / p >
< / div >
< div id = "outline-container-orgd3e5fe1" class = "outline-3" >
< h3 id = "orgd3e5fe1" > < span class = "section-number-3" > 3.1< / span > Analytical Analysis< / h3 >
< div class = "outline-text-3" id = "text-3-1" >
< div id = "org8fb9606" class = "figure" >
< p > < img src = "figs/2dof_system_granite_stiffness.png" alt = "2dof_system_granite_stiffness.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > Figure caption< / p >
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< / div >
< p >
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If we write the equation of motion of the system in Figure < a href = "#org8fb9606" > 8< / a > , we obtain:
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< / p >
\begin{align}
m^\prime s^2 x^\prime & = (c^\prime s + k^\prime) (x - x^\prime) \\
ms^2 x & = (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (x_w - x)
\end{align}
< p >
If we note \(d = x^\prime - x\), we obtain:
< / p >
\begin{equation}
\label{org4396920}
\frac{d}{x_w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
\end{equation}
< / div >
< / div >
< div id = "outline-container-org9215f81" class = "outline-3" >
< h3 id = "org9215f81" > < span class = "section-number-3" > 3.2< / span > Soft Granite< / h3 >
< div class = "outline-text-3" id = "text-3-2" >
< p >
Let’ s initialize a soft granite that will act as an isolation stage from ground motion.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeGranite(< span class = "org-string" > 'K'< / span > , 5e5< span class = "org-type" > *< / span > ones(3,1), < span class = "org-string" > 'C'< / span > , 5e3< span class = "org-type" > *< / span > ones(3,1));
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Ks = logspace(3,9,7); < span class = "org-comment" > % [N/m]< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(Ks)< / span >
initializeNanoHexapod(< span class = "org-string" > 'k'< / span > , Ks(< span class = "org-constant" > i< / span > ));
G = linearize(mdl, io);
G.InputName = {< span class = "org-string" > 'Dwx'< / span > , < span class = "org-string" > 'Dwy'< / span > , < span class = "org-string" > 'Dwz'< / span > , < span class = "org-string" > 'Fty_x'< / span > , < span class = "org-string" > 'Fty_z'< / span > , < span class = "org-string" > 'Frz_z'< / span > , < span class = "org-string" > 'Fdx'< / span > , < span class = "org-string" > 'Fdy'< / span > , < span class = "org-string" > 'Fdz'< / span > , < span class = "org-string" > 'Mdx'< / span > , < span class = "org-string" > 'Mdy'< / span > , < span class = "org-string" > 'Mdz'< / span > };
G.OutputName = {< span class = "org-string" > 'Ex'< / span > , < span class = "org-string" > 'Ey'< / span > , < span class = "org-string" > 'Ez'< / span > , < span class = "org-string" > 'Erx'< / span > , < span class = "org-string" > 'Ery'< / span > , < span class = "org-string" > 'Erz'< / span > };
Gdr(< span class = "org-constant" > i< / span > ) = {minreal(G)};
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org8878556" class = "outline-3" >
< h3 id = "org8878556" > < span class = "section-number-3" > 3.3< / span > Effect of the Granite transfer function< / h3 >
< / div >
< / div >
< div id = "outline-container-org8a88fb0" class = "outline-2" >
< h2 id = "org8a88fb0" > < span class = "section-number-2" > 4< / span > Open Loop Budget Error< / h2 >
< div class = "outline-text-2" id = "text-4" >
< p >
< a id = "org5d05990" > < / a >
< / p >
< / div >
< div id = "outline-container-org2ee8120" class = "outline-3" >
< h3 id = "org2ee8120" > < span class = "section-number-3" > 4.1< / span > Load of the identified disturbances and transfer functions< / h3 >
< div class = "outline-text-3" id = "text-4-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > './mat/dist_psd.mat'< / span > , < span class = "org-string" > 'dist_f'< / span > );
load(< span class = "org-string" > './mat/opt_stiffness_disturbances.mat'< / span > , < span class = "org-string" > 'Gd'< / span > )
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgb1a3177" class = "outline-3" >
< h3 id = "orgb1a3177" > < span class = "section-number-3" > 4.2< / span > Equations< / h3 >
< / div >
< div id = "outline-container-org63eade9" class = "outline-3" >
< h3 id = "org63eade9" > < span class = "section-number-3" > 4.3< / span > Results< / h3 >
< div class = "outline-text-3" id = "text-4-3" >
< p >
Effect of all disturbances
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > freqs = dist_f.f;
< span class = "org-type" > figure< / span > ;
hold on;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(Ks)< / span >
plot(freqs, sqrt(dist_f.psd_rz)< span class = "org-type" > .*< / span > abs(squeeze(freqresp(Gd{< span class = "org-constant" > i< / span > }(< span class = "org-string" > 'Ez'< / span > , < span class = "org-string" > 'Frz_z'< / span > ), freqs, < span class = "org-string" > 'Hz'< / span > ))));
< span class = "org-keyword" > end< / span >
hold off;
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'yscale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'ASD $\left[\frac{m}{\sqrt{Hz}}\right]$'< / span > )
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'southwest'< / span > );
xlim([2, 500]);
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgef96b89" class = "outline-3" >
< h3 id = "orgef96b89" > < span class = "section-number-3" > 4.4< / span > Cumulative Amplitude Spectrum< / h3 >
< div class = "outline-text-3" id = "text-4-4" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > freqs = dist_f.f;
< span class = "org-type" > figure< / span > ;
hold on;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(Ks)< / span >
plot(freqs, sqrt(flip(< span class = "org-type" > -< / span > cumtrapz(flip(freqs), flip(dist_f.psd_ty< span class = "org-type" > .*< / span > abs(squeeze(freqresp(Gd{< span class = "org-constant" > i< / span > }(< span class = "org-string" > 'Ez'< / span > , < span class = "org-string" > 'Fty_z'< / span > ), freqs, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2)))), < span class = "org-string" > '-'< / span > , ...
< span class = "org-string" > 'DisplayName'< / span > , sprintf(< span class = "org-string" > '$k = %.0g$ [N/m]'< / span > , Ks(< span class = "org-constant" > i< / span > )));
< span class = "org-keyword" > end< / span >
plot([freqs(1) freqs(end)], [10e< span class = "org-type" > -< / span > 9 10e< span class = "org-type" > -< / span > 9], < span class = "org-string" > 'k--'< / span > , < span class = "org-string" > 'HandleVisibility'< / span > , < span class = "org-string" > 'off'< / span > );
hold off;
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'yscale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'CAS $[m]$'< / span > )
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'southwest'< / span > );
xlim([2, 500]); ylim([1e< span class = "org-type" > -< / span > 10 1e< span class = "org-type" > -< / span > 6]);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > freqs = dist_f.f;
< span class = "org-type" > figure< / span > ;
hold on;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(Ks)< / span >
plot(freqs, sqrt(flip(< span class = "org-type" > -< / span > cumtrapz(flip(freqs), flip(dist_f.psd_rz< span class = "org-type" > .*< / span > abs(squeeze(freqresp(Gd{< span class = "org-constant" > i< / span > }(< span class = "org-string" > 'Ez'< / span > , < span class = "org-string" > 'Frz_z'< / span > ), freqs, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2)))), < span class = "org-string" > '-'< / span > , ...
< span class = "org-string" > 'DisplayName'< / span > , sprintf(< span class = "org-string" > '$k = %.0g$ [N/m]'< / span > , Ks(< span class = "org-constant" > i< / span > )));
< span class = "org-keyword" > end< / span >
plot([freqs(1) freqs(end)], [10e< span class = "org-type" > -< / span > 9 10e< span class = "org-type" > -< / span > 9], < span class = "org-string" > 'k--'< / span > , < span class = "org-string" > 'HandleVisibility'< / span > , < span class = "org-string" > 'off'< / span > );
hold off;
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'yscale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'CAS $[m]$'< / span > )
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'southwest'< / span > );
xlim([2, 500]); ylim([1e< span class = "org-type" > -< / span > 10 1e< span class = "org-type" > -< / span > 6]);
< / pre >
< / div >
< p >
Ground motion
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > freqs = dist_f.f;
< span class = "org-type" > figure< / span > ;
hold on;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(Ks)< / span >
plot(freqs, sqrt(flip(< span class = "org-type" > -< / span > cumtrapz(flip(freqs), flip(dist_f.psd_gm< span class = "org-type" > .*< / span > abs(squeeze(freqresp(Gd{< span class = "org-constant" > i< / span > }(< span class = "org-string" > 'Ez'< / span > , < span class = "org-string" > 'Dwz'< / span > ), freqs, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2)))), < span class = "org-string" > '-'< / span > , ...
< span class = "org-string" > 'DisplayName'< / span > , sprintf(< span class = "org-string" > '$k = %.0g$ [N/m]'< / span > , Ks(< span class = "org-constant" > i< / span > )));
< span class = "org-keyword" > end< / span >
plot([freqs(1) freqs(end)], [10e< span class = "org-type" > -< / span > 9 10e< span class = "org-type" > -< / span > 9], < span class = "org-string" > 'k--'< / span > , < span class = "org-string" > 'HandleVisibility'< / span > , < span class = "org-string" > 'off'< / span > );
hold off;
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'yscale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'CAS $E_y$ $[m]$'< / span > )
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'northeast'< / span > );
xlim([2, 500]); ylim([1e< span class = "org-type" > -< / span > 10 1e< span class = "org-type" > -< / span > 6]);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > freqs = dist_f.f;
< span class = "org-type" > figure< / span > ;
hold on;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(Ks)< / span >
plot(freqs, sqrt(flip(< span class = "org-type" > -< / span > cumtrapz(flip(freqs), flip(dist_f.psd_gm< span class = "org-type" > .*< / span > abs(squeeze(freqresp(Gd{< span class = "org-constant" > i< / span > }(< span class = "org-string" > 'Ex'< / span > , < span class = "org-string" > 'Dwx'< / span > ), freqs, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2)))), < span class = "org-string" > '-'< / span > , ...
< span class = "org-string" > 'DisplayName'< / span > , sprintf(< span class = "org-string" > '$k = %.0g$ [N/m]'< / span > , Ks(< span class = "org-constant" > i< / span > )));
< span class = "org-keyword" > end< / span >
plot([freqs(1) freqs(end)], [10e< span class = "org-type" > -< / span > 9 10e< span class = "org-type" > -< / span > 9], < span class = "org-string" > 'k--'< / span > , < span class = "org-string" > 'HandleVisibility'< / span > , < span class = "org-string" > 'off'< / span > );
hold off;
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'yscale'< / span > , < span class = "org-string" > 'lin'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'CAS $E_y$ $[m]$'< / span > )
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'northeast'< / span > );
xlim([2, 500]);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > freqs = dist_f.f;
< span class = "org-type" > figure< / span > ;
hold on;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(Ks)< / span >
plot(freqs, sqrt(flip(< span class = "org-type" > -< / span > cumtrapz(flip(freqs), flip(dist_f.psd_gm< span class = "org-type" > .*< / span > abs(squeeze(freqresp(Gd{< span class = "org-constant" > i< / span > }(< span class = "org-string" > 'Ey'< / span > , < span class = "org-string" > 'Dwy'< / span > ), freqs, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2)))), < span class = "org-string" > '-'< / span > , ...
< span class = "org-string" > 'DisplayName'< / span > , sprintf(< span class = "org-string" > '$k = %.0g$ [N/m]'< / span > , Ks(< span class = "org-constant" > i< / span > )));
< span class = "org-keyword" > end< / span >
plot([freqs(1) freqs(end)], [10e< span class = "org-type" > -< / span > 9 10e< span class = "org-type" > -< / span > 9], < span class = "org-string" > 'k--'< / span > , < span class = "org-string" > 'HandleVisibility'< / span > , < span class = "org-string" > 'off'< / span > );
hold off;
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'yscale'< / span > , < span class = "org-string" > 'lin'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'CAS $E_y$ $[m]$'< / span > )
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'northeast'< / span > );
xlim([2, 500]);
< / pre >
< / div >
< p >
Sum of all perturbations
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > psd_tot = zeros(length(freqs), length(Ks));
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(Ks)< / span >
psd_tot(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = dist_f.psd_gm< span class = "org-type" > .*< / span > abs(squeeze(freqresp(Gd{< span class = "org-constant" > i< / span > }(< span class = "org-string" > 'Ez'< / span > , < span class = "org-string" > 'Dwz'< / span > ), freqs, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2 < span class = "org-type" > +< / span > ...
dist_f.psd_ty< span class = "org-type" > .*< / span > abs(squeeze(freqresp(Gd{< span class = "org-constant" > i< / span > }(< span class = "org-string" > 'Ez'< / span > , < span class = "org-string" > 'Fty_z'< / span > ), freqs, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2 < span class = "org-type" > +< / span > ...
dist_f.psd_rz< span class = "org-type" > .*< / span > abs(squeeze(freqresp(Gd{< span class = "org-constant" > i< / span > }(< span class = "org-string" > 'Ez'< / span > , < span class = "org-string" > 'Frz_z'< / span > ), freqs, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2;
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > freqs = dist_f.f;
< span class = "org-type" > figure< / span > ;
hold on;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(Ks)< / span >
plot(freqs, sqrt(flip(< span class = "org-type" > -< / span > cumtrapz(flip(freqs), flip(psd_tot(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ))))), < span class = "org-string" > '-'< / span > , ...
< span class = "org-string" > 'DisplayName'< / span > , sprintf(< span class = "org-string" > '$k = %.0g$ [N/m]'< / span > , Ks(< span class = "org-constant" > i< / span > )));
< span class = "org-keyword" > end< / span >
plot([freqs(1) freqs(end)], [10e< span class = "org-type" > -< / span > 9 10e< span class = "org-type" > -< / span > 9], < span class = "org-string" > 'k--'< / span > , < span class = "org-string" > 'HandleVisibility'< / span > , < span class = "org-string" > 'off'< / span > );
hold off;
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'yscale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'CAS $E_z$ $[m]$'< / span > )
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'northeast'< / span > );
xlim([1, 500]); ylim([1e< span class = "org-type" > -< / span > 10 1e< span class = "org-type" > -< / span > 6]);
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org34c0f38" class = "outline-2" >
< h2 id = "org34c0f38" > < span class = "section-number-2" > 5< / span > Closed Loop Budget Error< / h2 >
< div class = "outline-text-2" id = "text-5" >
< p >
< a id = "orgd3503fb" > < / a >
< / p >
< / div >
< div id = "outline-container-orgf2d36a1" class = "outline-3" >
< h3 id = "orgf2d36a1" > < span class = "section-number-3" > 5.1< / span > Reduction thanks to feedback - Required bandwidth< / h3 >
< div class = "outline-text-3" id = "text-5-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > wc = 1< span class = "org-type" > *< / span > 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ; < span class = "org-comment" > % [rad/s]< / span >
xic = 0.5;
S = (s< span class = "org-type" > /< / span > wc)< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wc);
bodeFig({S}, logspace(< span class = "org-type" > -< / span > 1,2,1000))
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > wc = [1, 5, 10, 20, 50, 100, 200];
S1 = {zeros(length(wc), 1)};
S2 = {zeros(length(wc), 1)};
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > j< / span > < / span > = < span class = "org-constant" > 1:length(wc)< / span >
L = (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > wc(< span class = "org-constant" > j< / span > ))< span class = "org-type" > /< / span > s; < span class = "org-comment" > % Simple integrator< / span >
S1{< span class = "org-constant" > j< / span > } = 1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > L);
L = ((2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > wc(< span class = "org-constant" > j< / span > ))< span class = "org-type" > /< / span > s)< span class = "org-type" > ^< / span > 2< span class = "org-type" > *< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > wc(< span class = "org-constant" > j< / span > )< span class = "org-type" > /< / span > 2))< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > wc(< span class = "org-constant" > j< / span > )< span class = "org-type" > *< / span > 2));
S2{< span class = "org-constant" > j< / span > } = 1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > L);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > freqs = dist_f.f;
< span class = "org-type" > figure< / span > ;
hold on;
< span class = "org-constant" > i< / span > = 6;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > j< / span > < / span > = < span class = "org-constant" > 1:length(wc)< / span >
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > ,< span class = "org-string" > 'ColorOrderIndex'< / span > ,< span class = "org-constant" > j< / span > );
plot(freqs, sqrt(flip(< span class = "org-type" > -< / span > cumtrapz(flip(freqs), flip(abs(squeeze(freqresp(S1{< span class = "org-constant" > j< / span > }, freqs, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2< span class = "org-type" > .*< / span > psd_tot(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ))))), < span class = "org-string" > '-'< / span > , ...
< span class = "org-string" > 'DisplayName'< / span > , sprintf(< span class = "org-string" > '$\\omega_c = %.0f$ [Hz]'< / span > , wc(< span class = "org-constant" > j< / span > )));
< span class = "org-keyword" > end< / span >
plot(freqs, sqrt(flip(< span class = "org-type" > -< / span > cumtrapz(flip(freqs), flip(psd_tot(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ))))), < span class = "org-string" > 'k-'< / span > , ...
< span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Open-Loop'< / span > );
plot([freqs(1) freqs(end)], [10e< span class = "org-type" > -< / span > 9 10e< span class = "org-type" > -< / span > 9], < span class = "org-string" > 'k--'< / span > , < span class = "org-string" > 'HandleVisibility'< / span > , < span class = "org-string" > 'off'< / span > );
hold off;
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'yscale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'CAS $E_y$ $[m]$'< / span > )
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'northeast'< / span > );
xlim([0.5, 500]); ylim([1e< span class = "org-type" > -< / span > 10 1e< span class = "org-type" > -< / span > 6]);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > wc = logspace(0, 3, 100);
Dz1_rms = zeros(length(Ks), length(wc));
Dz2_rms = zeros(length(Ks), length(wc));
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(Ks)< / span >
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > j< / span > < / span > = < span class = "org-constant" > 1:length(wc)< / span >
L = (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > wc(< span class = "org-constant" > j< / span > ))< span class = "org-type" > /< / span > s;
Dz1_rms(< span class = "org-constant" > i< / span > , < span class = "org-constant" > j< / span > ) = sqrt(trapz(freqs, abs(squeeze(freqresp(1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > L), freqs, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2< span class = "org-type" > .*< / span > psd_tot(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )));
L = ((2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > wc(< span class = "org-constant" > j< / span > ))< span class = "org-type" > /< / span > s)< span class = "org-type" > ^< / span > 2< span class = "org-type" > *< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > wc(< span class = "org-constant" > j< / span > )< span class = "org-type" > /< / span > 2))< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > wc(< span class = "org-constant" > j< / span > )< span class = "org-type" > *< / span > 2));
Dz2_rms(< span class = "org-constant" > i< / span > , < span class = "org-constant" > j< / span > ) = sqrt(trapz(freqs, abs(squeeze(freqresp(1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > L), freqs, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2< span class = "org-type" > .*< / span > psd_tot(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )));
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > freqs = dist_f.f;
< span class = "org-type" > figure< / span > ;
hold on;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(Ks)< / span >
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > ,< span class = "org-string" > 'ColorOrderIndex'< / span > ,< span class = "org-constant" > i< / span > );
plot(wc, Dz1_rms(< span class = "org-constant" > i< / span > , < span class = "org-type" > :< / span > ), < span class = "org-string" > '-'< / span > , ...
< span class = "org-string" > 'DisplayName'< / span > , sprintf(< span class = "org-string" > '$k = %.0g$ [N/m]'< / span > , Ks(< span class = "org-constant" > i< / span > )))
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > ,< span class = "org-string" > 'ColorOrderIndex'< / span > ,< span class = "org-constant" > i< / span > );
plot(wc, Dz2_rms(< span class = "org-constant" > i< / span > , < span class = "org-type" > :< / span > ), < span class = "org-string" > '--'< / span > , ...
< span class = "org-string" > 'HandleVisibility'< / span > , < span class = "org-string" > 'off'< / span > )
< span class = "org-keyword" > end< / span >
hold off;
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'yscale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Control Bandwidth [Hz]'< / span > ); ylabel(< span class = "org-string" > '$E_z\ [m, rms]$'< / span > )
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'southwest'< / span > );
xlim([1, 500]);
< / pre >
< / div >
< / div >
< / div >
< / div >
2020-04-07 15:58:26 +02:00
< div id = "outline-container-org0953c03" class = "outline-2" >
< h2 id = "org0953c03" > < span class = "section-number-2" > 6< / span > Conclusion< / h2 >
2020-04-07 15:26:29 +02:00
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2020-04-07 15:58:26 +02:00
< p class = "date" > Created: 2020-04-07 mar. 15:57< / p >
2020-04-07 15:26:29 +02:00
< / div >
< / body >
< / html >
