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< h1 class = "title" > Centrifugal Forces< / h1 >
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< li > < a href = "#org5199302" > 1. Parameters< / a > < / li >
< li > < a href = "#orga2ea10d" > 2. Centrifugal forces for light and heavy sample< / a > < / li >
< li > < a href = "#orgf375b50" > 3. Centrifugal forces as a function of the rotation speed< / a > < / li >
< li > < a href = "#orge7fb13d" > 4. Maximum rotation speed as a function of the mass< / a > < / li >
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In this document, we wish to estimate the centrifugal forces due to the spindle’ s rotation when the sample’ s center of mass is off-centered with respect to the rotation axis.
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This is the case then the sample is moved by the micro-hexapod.
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< p >
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The centrifugal forces are defined as represented Figure < a href = "#org91ed599" > 1< / a > where:
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< li > \(M\) is the total mass of the rotating elements in \([kg]\)< / li >
< li > \(\omega\) is the rotation speed in \([rad/s]\)< / li >
< li > \(r\) is the distance to the rotation axis in \([m]\)< / li >
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< p > < img src = "./figs/centrifugal.png" alt = "centrifugal.png" / >
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< p > < span class = "figure-number" > Figure 1: < / span > Centrifugal forces< / p >
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< h2 id = "org5199302" > < span class = "section-number-2" > 1< / span > Parameters< / h2 >
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We define some parameters for the computation.
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The mass of the sample can vary from \(1\,kg\) to \(50\,kg\) to which is added to mass of the metrology reflector and the nano-hexapod’ s top platform (here set to \(15\,kg\)).
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< pre class = "src src-matlab" > M_light = 16; < span class = "org-comment" > % mass of excentred parts mooving [kg]< / span >
M_heavy = 65; < span class = "org-comment" > % [kg]< / span >
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For the light mass, the rotation speed is 60rpm whereas for the heavy mass, it is equal to 1rpm.
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< pre class = "src src-matlab" > w_light = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ; < span class = "org-comment" > % rotational speed [rad/s]< / span >
w_heavy = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 60; < span class = "org-comment" > % rotational speed [rad/s]< / span >
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Finally, we consider a mass eccentricity of \(10\,mm\).
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< pre class = "src src-matlab" > R = 0.01; < span class = "org-comment" > % Excentricity [m]< / span >
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< h2 id = "orga2ea10d" > < span class = "section-number-2" > 2< / span > Centrifugal forces for light and heavy sample< / h2 >
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From the formula \(F_c = m \omega^2 r\), we obtain the values shown below.
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< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-right" > Force [N]< / th >
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< td class = "org-left" > light< / td >
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< td class = "org-right" > 6.32< / td >
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< td class = "org-left" > heavy< / td >
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< td class = "org-right" > 0.01< / td >
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< h2 id = "orgf375b50" > < span class = "section-number-2" > 3< / span > Centrifugal forces as a function of the rotation speed< / h2 >
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The centrifugal forces as a function of the rotation speed for light and heavy sample is shown on Figure < a href = "#org87b7644" > 2< / a > .
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< / p >
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< p > < img src = "figs/centrifugal_forces_rpm.png" alt = "centrifugal_forces_rpm.png" / >
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< p > < span class = "figure-number" > Figure 2: < / span > Centrifugal forces function of the rotation speed< / p >
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< h2 id = "orge7fb13d" > < span class = "section-number-2" > 4< / span > Maximum rotation speed as a function of the mass< / h2 >
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We plot the maximum rotation speed as a function of the mass for different maximum force that we can use to counteract the centrifugal forces (Figure < a href = "#org8fe6a07" > 3< / a > ).
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< p >
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From a specified maximum allowed centrifugal force (here set to \(10\,[N]\)), the maximum rotation speed as a function of the sample’ s mass is shown in Figure < a href = "#org8fe6a07" > 3< / a > .
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< pre class = "src src-matlab" > F_max = 10; < span class = "org-comment" > % Maximum accepted centrifugal forces [N]< / span >
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R = 0.01;
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M_sample = 0< span class = "org-type" > :< / span > 1< span class = "org-type" > :< / span > 100;
M_reflector = 15;
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< div id = "org8fe6a07" class = "figure" >
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< p > < img src = "figs/max_force_rpm.png" alt = "max_force_rpm.png" / >
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< p > < span class = "figure-number" > Figure 3: < / span > Maximum rotation speed as a function of the sample mass for an allowed centrifugal force of \(100\,[N]\)< / p >
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2021-02-20 sam. 23:09< / p >
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