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