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+ UP + | + HOME +
+

Centrifugal Forces

+ + +

+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 1 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 \([m]\)
  • +
+ + +
+

centrifugal.png +

+

Figure 1: Centrifugal forces

+
+ +
+

1 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 \(15\,kg\)). +

+ +
+
M_light = 16; % mass of excentred parts mooving [kg]
+M_heavy = 65; % [kg]
+
+
+ +

+For the light mass, the rotation speed is 60rpm whereas for the heavy mass, it is equal to 1rpm. +

+
+
w_light = 2*pi; % rotational speed [rad/s]
+w_heavy = 2*pi/60; % rotational speed [rad/s]
+
+
+ +

+Finally, we consider a mass eccentricity of \(10\,mm\). +

+
+
R = 0.01; % Excentricity [m]
+
+
+
+
+ +
+

2 Centrifugal forces for light and heavy sample

+
+

+From the formula \(F_c = m \omega^2 r\), we obtain the values shown below. +

+ + + + +++ ++ + + + + + + + + + + + + + + + + + +
 Force [N]
light6.32
heavy0.01
+
+
+ +
+

3 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 2. +

+ + +
+

centrifugal_forces_rpm.png +

+

Figure 2: Centrifugal forces function of the rotation speed

+
+
+
+ +
+

4 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 3). +

+ +

+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 3. +

+ +
+
F_max = 10; % Maximum accepted centrifugal forces [N]
+
+R = 0.01;
+
+M_sample = 0:1:100;
+M_reflector = 15;
+
+
+ + +
+

max_force_rpm.png +

+

Figure 3: Maximum rotation speed as a function of the sample mass for an allowed centrifugal force of \(100\,[N]\)

+
+
+
+
+
+

Author: Dehaeze Thomas

+

Created: 2020-05-06 mer. 16:02

+
+ + diff --git a/org/centrifugal_forces.org b/org/centrifugal_forces.org index 475b892..b3244dd 100644 --- a/org/centrifugal_forces.org +++ b/org/centrifugal_forces.org @@ -42,7 +42,7 @@ For the light mass, the rotation speed is 60rpm whereas for the heavy mass, it i Finally, we consider a mass eccentricity of $10\,mm$. #+begin_src matlab - R = 0.1; % Excentricity [m] + R = 0.01; % Excentricity [m] #+end_src * Centrifugal forces for light and heavy sample @@ -51,14 +51,14 @@ 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 '); + data2orgtable(data, {'light', 'heavy'}, {'Force [N]'}, ' %.2f '); #+end_src #+RESULTS: | | Force [N] | |-------+-----------| -| light | 63.2 | -| heavy | 0.1 | +| 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]]. @@ -87,12 +87,12 @@ exportFig('figs/centrifugal_forces_rpm.pdf', 'width', 'wide', 'height', 'tall') * 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]]. +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 = 100; % Maximum accepted centrifugal forces [N] + F_max = 10; % Maximum accepted centrifugal forces [N] - R = 0.1; + R = 0.01; M_sample = 0:1:100; M_reflector = 15;