82 lines
2.1 KiB
Org Mode
82 lines
2.1 KiB
Org Mode
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#+TITLE: Motion and Force Requirements for the Nano-Hexapod
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* Soft Hexapod
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As the nano-hexapod is in series with the other stages, it must apply all the force required to move the sample.
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If the nano-hexapod is soft (voice coil), its actuator must apply all the force such that the sample has the wanted motion.
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In some sense, it does not use the fact that the other stage are participating to the displacement of the sample.
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Let's take two examples:
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- Sinus Ty translation at 1Hz with an amplitude of 5mm
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- Long stroke hexapod has an offset of 10mm in X and the spindle is rotating
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Thus the wanted motion is a circle with a radius of 10mm
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If the sample if light (30Kg) => 60rpm
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If the sample if heavy (100Kg) => 1rpm
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From the motion, we compute the required acceleration by derive the displacement two times.
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Then from the Newton's second law: $m \vec{a} = \sum \vec{F}$ we can compute the required force.
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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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** Example
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The wanted motion is:
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\begin{align*}
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x &= d \cos(\omega t) \\
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y &= d \sin(\omega t)
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\end{align*}
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The corresponding acceleration is thus:
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\begin{align*}
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\ddot{x} &= - d \omega^2 \cos(\omega t) \\
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\ddot{y} &= - d \omega^2 \sin(\omega t)
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\end{align*}
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From the Newton's second law:
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\begin{align*}
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m \ddot{x} &= F_x \\
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m \ddot{y} &= F_y
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\end{align*}
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Thus the applied forces should be:
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\begin{align*}
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F_x &= - m d \omega^2 \cos(\omega t) \\
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F_y &= - m d \omega^2 \sin(\omega t)
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\end{align*}
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And the norm of the force is:
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\[ |F| = \sqrt{F_x^2 + F_y^2} = m d \omega^2 \ [N] \]
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For a Light sample:
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#+begin_src matlab :results value replace
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m = 30;
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d = 10e-3;
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w = 2*pi;
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F = m*d*w^2;
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ans = F
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#+end_src
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#+RESULTS:
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: 11.844
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For the Heavy sample:
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#+begin_src matlab :results value replace
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m = 80;
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d = 10e-3;
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w = 2*pi/60;
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F = m*d*w^2
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ans = F
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#+end_src
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#+RESULTS:
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: 0.008773
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