nass-simscape/org/centrifugal_forces.org

#+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 = 0.01; % 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]'}, ' %.2f ');
#+end_src

#+RESULTS:
|       | Force [N] |
|-------+-----------|
| light |      6.32 |
| heavy |      0.01 |

* 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 $10\,[N]$), 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 = 10; % Maximum accepted centrifugal forces [N]

  R = 0.01;

  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]]