Add study on the centrifugal forces

org/centrifugal_forces.org

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+#+TITLE:Centrifugal Forces
+#+SETUPFILE: ./setup/org-setup-file.org
+
+* Introduction                                                        :ignore:
+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.
+
+This is the case then the sample is moved by the micro-hexapod.
+
+The centrifugal forces are defined as represented Figure [[fig:centrifugal]] where:
+- $M$ is the total mass of the rotating elements in $[kg]$
+- $\omega$ is the rotation speed in $[rad/s]$
+- $r$ is the distance to the rotation axis in $[m]$
+
+#+name: fig:centrifugal
+#+caption: Centrifugal forces
+[[file:./figs/centrifugal.png]]
+
+* Matlab Init                                                :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+  <<matlab-dir>>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+  <<matlab-init>>
+#+end_src
+
+* Parameters
+We define some parameters for the computation.
+
+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 $15\,kg$).
+
+#+begin_src matlab
+  M_light = 16; % mass of excentred parts mooving [kg]
+  M_heavy = 65; % [kg]
+#+end_src
+
+For the light mass, the rotation speed is 60rpm whereas for the heavy mass, it is equal to 1rpm.
+#+begin_src matlab
+  w_light = 2*pi; % rotational speed [rad/s]
+  w_heavy = 2*pi/60; % rotational speed [rad/s]
+#+end_src
+
+Finally, we consider a mass eccentricity of $10\,mm$.
+#+begin_src matlab
+  R = 0.1; % Excentricity [m]
+#+end_src
+
+* Centrifugal forces for light and heavy sample
+From the formula $F_c = m \omega^2 r$, we obtain the values shown below.
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data = [M_light*R*w_light^2;
+          M_heavy*R*w_heavy^2];
+  data2orgtable(data, {'light', 'heavy'}, {'Force [N]'}, ' %.1f ');
+#+end_src
+
+#+RESULTS:
+|       | Force [N] |
+|-------+-----------|
+| light |      63.2 |
+| heavy |       0.1 |
+
+* Centrifugal forces as a function of the rotation speed
+The centrifugal forces as a function of the rotation speed for light and heavy sample is shown on Figure [[fig:centrifugal_forces_rpm]].
+
+#+begin_src matlab :exports none
+  ws = 0:1:60; % [rpm]
+
+  figure;
+  hold on;
+  plot(ws, M_light*(2*pi*ws/60).^2*R, 'DisplayName', sprintf('$M = %.0f$ [kg]', M_light))
+  plot(ws, M_heavy*(2*pi*ws/60).^2*R, 'DisplayName', sprintf('$M = %.0f$ [kg]', M_heavy))
+  hold off;
+  xlabel('Rotation Speed [rpm]'); ylabel('Centrifugal Force [N]');
+  legend('Location', 'northwest');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/centrifugal_forces_rpm.pdf', 'width', 'wide', 'height', 'tall')
+#+end_src
+
+#+name: fig:centrifugal_forces_rpm
+#+CAPTION: Centrifugal forces function of the rotation speed
+#+RESULTS:
+[[file:figs/centrifugal_forces_rpm.png]]
+
+* Maximum rotation speed as a function of the mass
+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 [[fig:max_force_rpm]]).
+
+From a specified maximum allowed centrifugal force (here set to $100\,[N]$), the maximum rotation speed as a function of the sample's mass is shown in Figure [[fig:max_force_rpm]].
+
+#+begin_src matlab
+  F_max = 100; % Maximum accepted centrifugal forces [N]
+
+  R = 0.1;
+
+  M_sample = 0:1:100;
+  M_reflector = 15;
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+  hold on;
+  plot(M_sample, 60/2/pi*sqrt(F_max/R./(M_sample + M_reflector)));
+  hold off;
+  xlim([M_sample(1), M_sample(end)]); ylim([0, 100]);
+  xlabel('Mass of the Sample [kg]'); ylabel('Rotation Speed [rpm]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/max_force_rpm.pdf', 'width', 'wide', 'height', 'tall')
+#+end_src
+
+#+name: fig:max_force_rpm
+#+CAPTION: Maximum rotation speed as a function of the sample mass for an allowed centrifugal force of $100\,[N]$
+#+RESULTS:
+[[file:figs/max_force_rpm.png]]