Difference between revisions of "Lower stiffness of smaller machinery"

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'''A rod is the stiffer ...'''
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* the bigger its cross-section A (∝L²) is and
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* the shorter its length l (∝L¹) is.
  
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With scaling down machinery to smaller the stiffness of this machinery falls. <br>
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The geometry dependent stiffness (aka spring constant) k [N/m] is calculated from <br>
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Also accelerations from machine motions go up. <br>
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the geometry independent stiffness (aka elastic Young's modulus) E [N/m²] as such: <br>
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These two effects both increasing deformation magnitudes. But ... <br>
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<math>k = E ~ (A/l) \propto L^1</math> <br>
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With scaling down machinery to smaller the mass of that machinery is going down. <br>
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Thus geometry dependent stiffness falls when shrinking the size of machinery (while keeping the same material). <br>
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This is decreasing deformation magnitudes. <br>
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Also covered on page about [[Scaling law]]s.
  
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As it turns out the overall '''the relative deflections/strains from acceleration stay scale invariant'''. <br>
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= Even diamond becomes soft like jelly – Not a problem though =
  
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This result is very important for [[applicability of macro 3D printing for nanomachine prototyping]]. <br>
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With scaling down machinery to smaller sizes the stiffness of this machinery falls. <br>
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As it means 3D printed macroscale prototypes will typically vastly underperform nanoscale target systems of equal geometry. <br>
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'''One millionth the size => One millionth the stiffness.''' See related page: [[Scaling law]].<br>
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Heck, if anything this may lead to massive over-engineering that, while surely working at the nanoscale, is very far from optimal.
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'''This makes even diamond jelly soft.''' <br>
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Which poses an obvious question: <br>
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'''Q:''' '''Could this maybe be a serious problem?''' <br>
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'''A:''' Perhaps surprisingly the answer is: '''No.''' <br>
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At least for the most part. I.e. only thermal motions are of concern. <br>Math covered on page: [[Same relative deflections across scales]]
  
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= Macroscale style machinery: Works better rather than worse at the nanoscale =
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= Important are deflection magnitudes rather than spring constants =
  
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What we absolutely do not want is to accidentally build a prototype macroscale systems with performance so high that
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For the material astoundingly low spring constants are not a problem because <br>
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target nanoscale systems of equal geometry will not be able to replicate that performance.
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what is relevant are relative deflections rather the geometry dependent stiffness of the material. <br>
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So how do deflections scale?
  
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The result here implies that we do not have to fear that.
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'''As it turns out the relative deflections / strains ...'''
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In fact for us to build a macroscale system that has the same or higher performance than the target nanoscale systems we would have to use materials ...
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* from accelerations of machinery scale with L<sup>0</sup> (scale invariant - nice!).
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* with a tensile modulus as high as diamond
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* from gravity scale with L<sup>1</sup>. – <small>(Large machines suffocating under their own weight. A well known macroscale problem.)</small>
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* with a density as low as diamond
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* from thermal motions scale with L<sup>-1</sup>. – <small>(Relevant for [[piezomechanosynthesis]] and unguided [[covalent welding]])</small>
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* with a maximal strain (bendability) in the double digit percentual range.
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Such materials simply do not exist today. Ceramics come closest to stiffness but they're totally not elastic. <br>
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(Future [[gem-gum metamaterial]]s might come close.)
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That's the 🤯 degree of how much [[Macroscale style machinery at the nanoscale]] <br>
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For the math deriving these [[scaling laws]] see Page: <br>
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works better than our good old macroscale machinery. <br>
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'''[[Same relative deflections across scales]]'''
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And that does not even factor in that we can easily afford to go a 1000x slower with speeds
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by compensating with more nanomachinery (as is possible due to [[higher throughput of smaller machinery]]).
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This is not an option for in comparison extremely voluminous macromachinery.
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'''In summary:'''
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== Consequences of slowing down for smaller machinery ==
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* steel has less elastic modulus than diamond
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* steel has less elasticity than nanoscale flawless diamond
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* steel cannot be moves as slow as nanoscale machinery as this slowdown cannot be compensated by mountains of more machinery
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Related: [[Condervative design]] [[Exploratory engineering]]
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Even more important than same relative deflections is keeping friction levels low. <br>
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This motivates deviating from keeping speeds constant across scales. <br>
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That is: It motivates to slow down a bit (see related page: [[lower friction despite higher bearing area]]) <br>
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'''With this slowdown as a better choice''' (that modifies all speed dependent scaling laws) <br>
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'''relative deflections do not just stay constant across scales.''' <br>
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'''They actually fall some for smaller machinery.''' <br>
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This is possible because (unlike macromachinery) nanomachinery can be run slower <br>
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as there is plenty of space for more machinery to fully compensate for <br>
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the loss in throughput thanks to [[higher throughput of smaller machinery]].
  
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= Analyzing scaling behavior =
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Main page: [[Scaling of speeds]]
  
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Without loss of generality only the one dimensional case of tensile strain is covered.
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= Example numbers – Jelly indeed =
  
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== Scaling of deformations from machine motions (spoiler: they are scale invariant) ==
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Example numbers for diamond [[crystolecule]] strut:
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* A = 1 nm²
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* l = 10 nm
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* E = 1000 GPa ≈ 10^12 N/m²
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This gives: <br>
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<math>k = E ~ (A/l) = (10^{12} N/m^2) · (10^{-18} m^2) / (10^-8 m) = 100N/m = 1daN/dm </math> <br>
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Or colloquially: 1kg/dm or 100g/cm. <br>
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This is how incredibly soft diamond gets at the nanoscale. <br>
  
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The critical quantity we want to be preserved across scales is relative strain: <br>
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For 10cm long macroscale strut with same aspect ratio (thus 1cm² cross section) that would be a pretty darn low spring constant. <br>
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<math>\epsilon = \Delta l / l</math> <br>
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One would need to go to materials like quite soft rubber or jelly to reproduce this low level off a stiffness.<br>
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Strain epsilon is proportional to stress sigma. <br>
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Jelly is probably a better analogy since it tends to rupture somewhere in the low two digit percentual range. <br>
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<math>\epsilon = \sigma / E</math> <br>
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Just like perfect flawless diamond [[crystolecule]]s do. Whereas rubber often can be stretched several 100s of percents.
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Stress is given by applied force F. <br>
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<math>\sigma = F / A </math> <br>
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A natural level of force for a scale is zentrifugal force from rotating a mass at that scale. <br>
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<math> F = m \omega^2 r = m (2\pi f)^2 r</math> <br>
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Putting it all together we get: <br>
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<math>\epsilon = \sigma / E = F / (E A) = m \omega^2 r / (E A)</math> <br>
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With:
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Related: [[The feel of atoms]]
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* mass m scales with <math> L^3 </math>
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* frequency f scales <math> L^{-1} </math> (assuming absolute speed v is kept scale invariant)
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* radius r scales with <math> L^1 </math>
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* Tensile modulus being scale invariant <math> L^0 </math>
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* Area scaling with <math> L^2 </math>
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We get:
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= Misc =
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* Force F scales with <math> L^2 </math>
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* Both stress & strain from machine motions scale with <math> L^0 </math>
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In other words when assuming scale invariant speed then: <br>
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This [[scaling law]] is also a/the reason why extremely [[high pressure]]s <br>
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'''Both stress & strain from machine motions is scale invariant.'''
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are so easy to generate at the nanoscale by focusing force down into small cross-sectional areas.
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== Scaling of deformations from gravity ==
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<math> F_{grav} = \rho V g \propto L^3</math> <br>
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<math>\epsilon_{grav} = \sigma_{grav} / E = F_{grav} / (E A) = \rho V g / (E A) \propto L^1 </math> <br>
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'''Both stress & strain from gravity go down linearly with smaller scales.'''
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= Misc =
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{{wikitodo|explain the following}}
 
{{wikitodo|explain the following}}
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* Why stiffness falls when shrinking the size of machinery (while keeping the same material)
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The consequences on design-constraints / design-choices based on this falling stiffness. <br>
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* The consequences on design constraints based no this falling stiffness
+
E.g. striving for high stiffness providing parallel robotics geometries to counter deflections from thermal motions.
  
 
= Related =
 
= Related =
  
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* '''[[Same relative deflections across scales]]'''
 +
----
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* '''[[How macroscale style machinery at the nanoscale outperforms its native scale]]'''
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* '''[[Applicability of macro 3D printing for nanomachine prototyping]]'''
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* [[Macroscale style machinery at the nanoscale]]
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* [[Natural scaling of absolute speeds]]
 
* '''[[Scaling law]]'''
 
* '''[[Scaling law]]'''
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* [[Same absolute speeds for smaller machinery]]
 
* [[Stiffness]]
 
* [[Stiffness]]
 +
----
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Thermal motion related:
 +
* [[Lattice scaled stiffness]]
 
* Parallel [[Robotic manipulators]]
 
* Parallel [[Robotic manipulators]]
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* [[Applicability of macro 3D printing for nanomachine prototyping]]
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----
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* [[Macroscale style machinery at the nanoscale]]
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[[Intuitive feel]] related:
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* [[Lattice scaled stiffness]]
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* [[The feel of atoms]] – about what "diamond getting jelly soft" intuitively means
 +
 
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== Off-topic ==
 +
 
 +
Low spring constants at the macroscale:
 +
* [[Emulated elasticity]]
 +
* [[gem-gum]], [[Gemstone based metamaterial]]
 +
 
 +
[[Category:Pages with math]]
 +
[[Category:Scaling law]]

Latest revision as of 20:15, 18 October 2024

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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