Difference between revisions of "Lower stiffness of smaller machinery"

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The geometry dependent stiffness (aka spring constant) k [N/m] is calculated from <br>
 
The geometry dependent stiffness (aka spring constant) k [N/m] is calculated from <br>
−
the geometry independent stiffness (aka elastic modulus) E [N/m²] as such: <br>
+
the geometry independent stiffness (aka elastic Young's modulus) E [N/m²] as such: <br>
 
<math>k = E ~ (A/l) \propto L^1</math> <br>
 
<math>k = E ~ (A/l) \propto L^1</math> <br>
 
Thus geometry dependent stiffness falls when shrinking the size of machinery (while keeping the same material). <br>
 
Thus geometry dependent stiffness falls when shrinking the size of machinery (while keeping the same material). <br>
 
Also covered on page about [[Scaling law]]s.
 
Also covered on page about [[Scaling law]]s.
  
−
= Even diamond becomes soft like jelly - not a problem though =
+
= Even diamond becomes soft like jelly – Not a problem though =
  
 
With scaling down machinery to smaller sizes the stiffness of this machinery falls. <br>
 
With scaling down machinery to smaller sizes the stiffness of this machinery falls. <br>
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Which poses an obvious question: <br>  
 
Which poses an obvious question: <br>  
 
'''Q:''' '''Could this maybe be a serious problem?''' <br>
 
'''Q:''' '''Could this maybe be a serious problem?''' <br>
−
'''A:''' Perhaps surprisingly the answer is: '''No.''' At least for the most part.
+
'''A:''' Perhaps surprisingly the answer is: '''No.''' <br>
 +
At least for the most part. I.e. only thermal motions are of concern. <br>Math covered on page: [[Same relative deflections across scales]]
  
 
= Important are deflection magnitudes rather than spring constants =
 
= Important are deflection magnitudes rather than spring constants =
  
−
What is relevant are deflections rather the geometry dependent stiffness of the material. <br>
+
For the material astoundingly low spring constants are not a problem because <br>
 +
what is relevant are relative deflections rather the geometry dependent stiffness of the material. <br>
 
So how do deflections scale?
 
So how do deflections scale?
  
−
'''As it turns out the relative deflections / strains from ...'''
+
'''As it turns out the relative deflections / strains ...'''
−
* accelerations of machinery scale with L<sup>0</sup> (scale invariant - nice!).
+
* from accelerations of machinery scale with L<sup>0</sup> (scale invariant - nice!).
−
* gravity scale with L¹. – <small>(Large machines suffocating under their own weight. A known macroscale problem.)</small>
+
* from gravity scale with L<sup>1</sup>. – <small>(Large machines suffocating under their own weight. A well known macroscale problem.)</small>
−
* thermal motions scale with TODO. – <small>(Relevant for [[piezomechanosynthesis]] and unguided [[covalent welding]])</small>
+
* from thermal motions scale with L<sup>-1</sup>. – <small>(Relevant for [[piezomechanosynthesis]] and unguided [[covalent welding]])</small>
  
−
= Relevance for scale transposed prototyping =
+
For the math deriving these [[scaling laws]] see Page: <br>
 +
'''[[Same relative deflections across scales]]'''
  
−
Scale invariance of deflections from machine motions is a very important result for [[applicability of macro 3D printing for nanomachine prototyping]]. <br>
+
== Consequences of slowing down for smaller machinery ==
−
This is because it means that 3D printed macroscale prototypes will typically vastly underperform nanoscale target systems of equal geometry. <br>
+
−
From the stiffness aspect there is no risk to accidentally prototype a macroscale system that then cannot be ported to the nanoscale. <br>
+
−
Heck, if anything this may lead to massive over-engineering in macroscale prototypes that, will surely working at the nanoscale, but will also be be very far from optimal.
+
  
−
= How macroscale style machinery at the nanoscale outperforms its native scale =
+
Even more important than same relative deflections is keeping friction levels low. <br>
 +
This motivates deviating from keeping speeds constant across scales. <br>
 +
That is: It motivates to slow down a bit (see related page: [[lower friction despite higher bearing area]]) <br>
 +
'''With this slowdown as a better choice''' (that modifies all speed dependent scaling laws) <br>
 +
'''relative deflections do not just stay constant across scales.''' <br>
 +
'''They actually fall some for smaller machinery.''' <br>
 +
This is possible because (unlike macromachinery) nanomachinery can be run slower <br>
 +
as there is plenty of space for more machinery to fully compensate for <br>
 +
the loss in throughput thanks to [[higher throughput of smaller machinery]].
  
−
'''Deflections being scale invariant is one important reason for why [[macroscale style machinery at the nanoscale]] works better rather than worse at the nanoscale.'''
+
Main page: [[Scaling of speeds]]
  
−
As discussed in [[applicability of macro 3D printing for nanomachine prototyping]]: <br>
+
= Example numbers – Jelly indeed =
−
What we absolutely do not want is to accidentally build a prototype macroscale systems with performance so high that <br>
+
−
target nanoscale systems of equal geometry will not be able to replicate that performance.
+
  
−
The result (math further below) implies that we do not have to fear that. <br>
+
Example numbers for diamond [[crystolecule]] strut:
−
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 ...
+
* A = 1 nm²
−
* with a tensile modulus as high as diamond
+
* l = 10 nm
−
* with a density as low as diamond
+
* E = 1000 GPa ≈ 10^12 N/m²
−
* with a maximal strain (bendability) in the double digit percentual range.
+
This gives: <br>
−
Such materials simply do not exist today. Ceramics come closest to stiffness but they're totally not elastic. <br>
+
<math>k = E ~ (A/l) = (10^{12} N/m^2) · (10^{-18} m^2) / (10^-8 m) = 100N/m = 1daN/dm </math> <br>
−
(Future [[gem-gum metamaterial]]s might come close.)
+
Or colloquially: 1kg/dm or 100g/cm. <br>
 +
This is how incredibly soft diamond gets at the nanoscale. <br>
  
−
That's the 🤯 degree of how much [[macroscale style machinery at the nanoscale]] <br>
+
For 10cm long macroscale strut with same aspect ratio (thus 1cm² cross section) that would be a pretty darn low spring constant. <br>
−
works better than our good old macroscale machinery at its native macroscale. <br>
+
One would need to go to materials like quite soft rubber or jelly to reproduce this low level off a stiffness.<br>
−
And that does not even factor in that we can easily afford to go a 1000x slower with speeds <br>
+
Jelly is probably a better analogy since it tends to rupture somewhere in the low two digit percentual range. <br>
−
by compensating with more nanomachinery (as is possible due to [[higher throughput of smaller machinery]]). <br>
+
Just like perfect flawless diamond [[crystolecule]]s do. Whereas rubber often can be stretched several 100s of percents.
−
This is not an option for the (in comparison extremely voluminous) macromachinery.
+
  
−
'''In summary:'''
+
Related: [[The feel of atoms]]
−
* steel has less elastic modulus than diamond
+
−
* steel has less elasticity than nanoscale flawless diamond
+
−
* steel cannot be moves as slow as nanoscale machinery as this slowdown cannot be compensated by mountains of more machinery
+
  
−
Related: [[Conservative design]] [[Exploratory engineering]]
+
= Misc =
  
−
= Analyzing scaling behavior =
+
This [[scaling law]] is also a/the reason why extremely [[high pressure]]s <br>
−
 
+
are so easy to generate at the nanoscale by focusing force down into small cross-sectional areas.
−
Without loss of generality only the one dimensional case of tensile strain is covered.
+
−
 
+
−
== Scaling of deformations from machine motions (spoiler: they are scale invariant) ==
+
−
 
+
−
Accelerations from machine motions go up too worsening the situation. <br>
+
−
One millionth the size => One million times the accelerations.
+
−
 
+
−
But don't fret just yet. We have a quite literally massive compensating factor.
+
−
With scaling down machinery to smaller sizes the mass of that machinery is going down. <br>
+
−
One millionth the size => (One millionth)³ the mass.
+
−
 
+
−
----
+
−
 
+
−
The critical quantity we want to be preserved across scales is relative strain: <br>
+
−
<math>\epsilon = \Delta l / l</math> <br>
+
−
Strain epsilon is proportional to stress sigma. <br>
+
−
<math>\epsilon = \sigma / E</math> <br>
+
−
Stress is given by applied force F. <br>
+
−
<math>\sigma = F / A </math> <br>
+
−
A natural level of force for a scale is zentrifugal force from rotating a mass at that scale. <br>
+
−
<math> F = m \omega^2 r = m (2\pi f)^2 r</math> <br>
+
−
Putting it all together we get: <br>
+
−
<math>\epsilon = \sigma / E = F / (E A) = m \omega^2 r / (E A)</math> <br>
+
−
 
+
−
With:
+
−
* mass m scales with <math> L^3 </math>
+
−
* frequency f scales <math> L^{-1} </math> (assuming absolute speed v is kept scale invariant)
+
−
* radius r scales with <math> L^1 </math>
+
−
* Tensile modulus being scale invariant <math> L^0 </math>
+
−
* Area scaling with <math> L^2 </math>
+
−
 
+
−
We get:
+
−
* Force F scales with <math> L^2 </math>
+
−
* Both stress & strain from machine motions scale with <math> L^0 </math>
+
−
 
+
−
In other words when assuming scale invariant speed then: <br>
+
−
'''Both stress & strain from machine motions is scale invariant.'''
+
−
 
+
−
== Scaling of deformations from gravity ==
+
−
 
+
−
<math> F_{grav} = \rho V g \propto L^3</math> <br>
+
−
<math>\epsilon_{grav} = \sigma_{grav} / E = F_{grav} / (E A) = \rho V g / (E A) \propto L^1 </math> <br>
+
−
 
+
−
'''Both stress & strain from gravity go down linearly with smaller scales.'''
+
−
 
+
−
== Scaling of deformations from thermal motions ==
+
−
 
+
−
{{wikitodo|Work that out by using equipartitioning theorem on a block the size of the scale L.}}
+
−
 
+
−
= Misc =
+
  
 
{{wikitodo|explain the following}}
 
{{wikitodo|explain the following}}
−
* The consequences on design constraints based no this falling stiffness
+
The consequences on design-constraints / design-choices based on this falling stiffness. <br>
 +
E.g. striving for high stiffness providing parallel robotics geometries to counter deflections from thermal motions.
  
 
= Related =
 
= Related =
  
 +
* '''[[Same relative deflections across scales]]'''
 +
* '''[[How macroscale style machinery at the nanoscale outperforms its native scale]]'''
 +
* '''[[Applicability of macro 3D printing for nanomachine prototyping]]'''
 +
* [[Macroscale style machinery at the nanoscale]]
 +
* [[Natural scaling of absolute speeds]]
 
* '''[[Scaling law]]'''
 
* '''[[Scaling law]]'''
 +
* [[Same absolute speeds for smaller machinery]]
 
* [[Stiffness]]
 
* [[Stiffness]]
 +
----
 +
Thermal motion related:
 +
* [[Lattice scaled stiffness]]
 
* Parallel [[Robotic manipulators]]
 
* Parallel [[Robotic manipulators]]
−
* [[Applicability of macro 3D printing for nanomachine prototyping]]
+
----
−
* [[Macroscale style machinery at the nanoscale]]
+
[[Intuitive feel]] related:
−
* [[Lattice scaled stiffness]]
+
* [[The feel of atoms]] – about what "diamond getting jelly soft" intuitively means
 +
 
 +
== 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 07:07, 3 April 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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