# Cell mechanics specifications

###### tags: `experimental setup` `SoftQC` `self-assembly`

---

Cells and holder for the beads will be designed at the mechanics workshop of LPS.
This page summarises the specifications that the solution must satisfy.

## Generalities

* The cell must be able to confine a layer of non-magnetic stainless steel beads of diameter ranging from 1.2 mm to 2.5 mm.
* The maximum size of the cavity is a square of 200 mm edge and 3 mm height.
* The top cover of the cell must be transparent to allow beads tracking.
* The top cover should not electrostatically charge in contact with the vibrated beads (plexiglas doesn't work).
* It would be nice to be able to change the cell size and shape (square, circle, possibly more). Therefore, we need a holder on which several cells could be adapted.
* The whole thing must be able to endure continuous vibration of a few hundred Hz with an amplitude of tens of micrometers for many hours.

## Shaker constraints

In the regime we are interested in (frequency of a few hundred Hz, and amplitude of a few tens of µm), it seems that the force applied by the shaker is the limiting factor.

The simulation below, provided by the constructor, is for a dispacement of 1.22 mm. Could we ask for a similar evaluation with displacements of 10 µm and slightly larger frequency ?


### Sanity check

For a frequency of $f = 300\;Hz$ and amplitude $A = 10\;\mu m$, the amplitude of the acceleration is

$$\tilde{a} = A \omega^2 = A (2 \pi f)^2 = 35.5\;m.s^{-2}$$

This is below a peak accleration of $5g = 49\;m.s^{-2}$.
The mass of the payload is limited by the maximum force applied by the shaker.

$$
m_\text{max} = \frac{F_\text{max}}{\tilde{a}}
$$

The maximum force $F_\text{max} = 0.16 \;kN$ in the picture above yields a maximum mass for the payload of $m_\text{max} = 3.55 \; kg$.

### Masses

Small beads mass: $m_S = 0.00711744\;g$
Large beads mass: $m_L = 0.0645018\;g$

The largest systems we will look at will contain 10 000 beads of each type, wheighting about $720\;g$ in total.

* The holder and cell total mass should not exceed $2.3 \;kg$.

### Geometry

* The holder must be compatible with the geometry of the shaker described by the picture below.


## Angular tolerance

* The whole setup (including the shaker) must be as horizontal as possible.

### Attempt for quantitative estimate

Consider a bead on an tilted plane, experiencing only gravity and reaction of the support.


From conservation of mechanical energy
$$
\frac{1}{2} m v^2(t) + m g z(t) = 0\\
\frac{1}{2} v^2(t) - g x(t) \sin \alpha = 0
$$
since $z = -x \sin \alpha$.

And $v^2 = \dot{x}^2 + \dot{y}^2$, so
$$
\dot{x}^2 - \frac{2 g \sin \alpha}{1 + \sin^2 \alpha} x = \dot{x}^2 - a x = 0
$$

The solution of the non-linear first order ODE is
$$
x(t) = \frac{1}{4}(2 \sqrt{a} C t + a t^2 + C^2)
$$
with C a constant.

Since $x(0) = 0$,
$$
x(t) = \frac{g \sin \alpha}{2(1 + \sin^2 \alpha)} t^2
$$

The distance travelled on the tilted plane after time t is $d(t) = \sqrt{x^2(t) + z^2(t)}$.


The model is -- hopefully -- bad, atherwise we are screwed... A crazy angular control would be necessary.

## Accelerometer

An accelerometer will be used to monitor the applied vibration.

* The accelerometer must be attached to the holder, ideally in a centered position.

There is a small screw provided with the accelerometer that can be use to fix it. See mechanical schematics below