From 31ce8a417e2a6dec2ed324b1a79d7833f8e02b82 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Mon, 4 May 2020 15:14:35 +0200 Subject: [PATCH] Add study on the centrifugal forces --- docs/figs/centrifugal.png | Bin 0 -> 16653 bytes docs/figs/centrifugal_forces_rpm.pdf | Bin 0 -> 15621 bytes docs/figs/centrifugal_forces_rpm.png | Bin 0 -> 35617 bytes docs/figs/max_force_rpm.pdf | Bin 0 -> 12901 bytes docs/figs/max_force_rpm.png | Bin 0 -> 27822 bytes org/centrifugal_forces.org | 117 +++++++++++++++++++++++++++ 6 files changed, 117 insertions(+) create mode 100644 docs/figs/centrifugal.png create mode 100644 docs/figs/centrifugal_forces_rpm.pdf create mode 100644 docs/figs/centrifugal_forces_rpm.png create mode 100644 docs/figs/max_force_rpm.pdf create mode 100644 docs/figs/max_force_rpm.png create mode 100644 org/centrifugal_forces.org diff --git a/docs/figs/centrifugal.png b/docs/figs/centrifugal.png new file mode 100644 index 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zWng8iuul^3ZY5kg9j=7r!STMV9QFc9A-e%uKahzA^s2ZZva?nkRXKdc@#-A5{C=gt z4U^9D_UE}szOTT7+uKA> zX8NG^t1fnTgtXciJJgb1B9pk{iS`|e4=XMtnyRK?|2zx<*9yBi4s^I}HVsE3j!*nQ yEcH1QJvefZgSAchp(HI&SUn|#leEeS$@c+gD literal 0 HcmV?d00001 diff --git a/org/centrifugal_forces.org b/org/centrifugal_forces.org new file mode 100644 index 0000000..475b892 --- /dev/null +++ b/org/centrifugal_forces.org @@ -0,0 +1,117 @@ +#+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) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+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]]