118 lines
4.1 KiB
Org Mode
118 lines
4.1 KiB
Org Mode
#+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]]
|