Centrifugal Forces
+Table of Contents
+ ++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 \([m]\) +
+
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 = 0.1; % 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] | +
|---|---|
| light | +63.2 | +
| heavy | +0.1 | +
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. +
+ + +
+
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 \(100\,[N]\)), the maximum rotation speed as a function of the sample’s mass is shown in Figure 3. +
+ +F_max = 100; % Maximum accepted centrifugal forces [N] + +R = 0.1; + +M_sample = 0:1:100; +M_reflector = 15; ++
+
Figure 3: Maximum rotation speed as a function of the sample mass for an allowed centrifugal force of \(100\,[N]\)
+