Lower stiffness of smaller machinery

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A rod is the stiffer ...

  • the bigger its cross-section A (∝L²) is and
  • the shorter its length l (∝L¹) is.

The geometry dependent stiffness (aka spring constant) k [N/m] is calculated from
the geometry independent stiffness (aka elastic Young's modulus) E [N/m²] as such:
[math]k = E ~ (A/l) \propto L^1[/math]
Thus geometry dependent stiffness falls when shrinking the size of machinery (while keeping the same material).
Also covered on page about Scaling laws.

Even diamond becomes soft like jelly – Not a problem though

With scaling down machinery to smaller sizes the stiffness of this machinery falls.
One millionth the size => One millionth the stiffness. See related page: Scaling law.
This makes even diamond jelly soft.
Which poses an obvious question:
Q: Could this maybe be a serious problem?
A: Perhaps surprisingly the answer is: No.
At least for the most part. I.e. only thermal motions are of concern.
Math covered on page: Same relative deflections across scales

Important are deflection magnitudes rather than spring constants

For the material astoundingly low spring constants are not a problem because
what is relevant are relative deflections rather the geometry dependent stiffness of the material.
So how do deflections scale?

As it turns out the relative deflections / strains ...

  • from accelerations of machinery scale with L0 (scale invariant - nice!).
  • from gravity scale with L1. – (Large machines suffocating under their own weight. A well known macroscale problem.)
  • from thermal motions scale with L-1. – (Relevant for piezomechanosynthesis and unguided covalent welding)

For the math deriving these scaling laws see Page:
Same relative deflections across scales

Consequences of slowing down for smaller machinery

Even more important than same relative deflections is keeping friction levels low.
This motivates deviating from keeping speeds constant across scales.
That is: It motivates to slow down a bit (see related page: lower friction despite higher bearing area)
With this slowdown as a better choice (that modifies all speed dependent scaling laws)
relative deflections do not just stay constant across scales.
They actually fall some for smaller machinery.
This is possible because (unlike macromachinery) nanomachinery can be run slower
as there is plenty of space for more machinery to fully compensate for
the loss in throughput thanks to higher throughput of smaller machinery.

Main page: Scaling of speeds

Example numbers – Jelly indeed

Example numbers for diamond crystolecule strut:

  • A = 1 nm²
  • l = 10 nm
  • E = 1000 GPa ≈ 10^12 N/m²

This gives:
[math]k = E ~ (A/l) = (10^{12} N/m^2) · (10^{-18} m^2) / (10^-8 m) = 100N/m = 1daN/dm [/math]
Or colloquially: 1kg/dm or 100g/cm.
This is how incredibly soft diamond gets at the nanoscale.

For 10cm long macroscale strut with same aspect ratio (thus 1cm² cross section) that would be a pretty darn low spring constant.
One would need to go to materials like quite soft rubber or jelly to reproduce this low level off a stiffness.
Jelly is probably a better analogy since it tends to rupture somewhere in the low two digit percentual range.
Just like perfect flawless diamond crystolecules do. Whereas rubber often can be stretched several 100s of percents.

Related: The feel of atoms

Misc

This scaling law is also a/the reason why extremely high pressures
are so easy to generate at the nanoscale by focusing force down into small cross-sectional areas.

(wiki-TODO: explain the following) The consequences on design-constraints / design-choices based on this falling stiffness.
E.g. striving for high stiffness providing parallel robotics geometries to counter deflections from thermal motions.

Related

Thermal motion related:

Intuitive feel related:

Off-topic

Low spring constants at the macroscale:

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