Difference between revisions of "Lower stiffness of smaller machinery"
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| + | With scaling down machinery to smaller the stiffness of this machinery falls. <br> | ||
| + | Also accelerations from machine motions go up. <br> | ||
| + | These two effects both increasing deformation magnitudes. But ... <br> | ||
| + | With scaling down machinery to smaller the mass of that machinery is going down. <br> | ||
| + | This is decreasing deformation magnitudes. <br> | ||
| + | |||
| + | As it turns out the overall '''the relative deflections/strains from acceleration stay scale invariant'''. <br> | ||
| + | |||
| + | This result is very important for [[applicability of macro 3D printing for nanomachine prototyping]]. <br> | ||
| + | As it means 3D printed macroscale prototypes will typically vastly underperform nanoscale target systems of equal geometry. <br> | ||
| + | Heck, if anything this may lead to massive over-engineering that, while surely working at the nanoscale, is very far from optimal. | ||
| + | |||
| + | = Macroscale style machinery: Works better rather than worse at the nanoscale = | ||
| + | |||
| + | What we absolutely do not want is to accidentally build a prototype macroscale systems with performance so high that | ||
| + | target nanoscale systems of equal geometry will not be able to replicate that performance. | ||
| + | |||
| + | The result here implies that we do not have to fear that. | ||
| + | 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 ... | ||
| + | * with a tensile modulus as high as diamond | ||
| + | * with a density as low as diamond | ||
| + | * with a maximal strain (bendability) in the double digit percentual range. | ||
| + | Such materials simply do not exist today. Ceramics come closest to stiffness but they're totally not elastic. <br> | ||
| + | (Future [[gem-gum metamaterial]]s might come close.) | ||
| + | |||
| + | That's the 🤯 degree of how much [[Macroscale style machinery at the nanoscale]] <br> | ||
| + | works better than our good old macroscale machinery. <br> | ||
| + | And that does not even factor in that we can easily afford to go a 1000x slower with speeds | ||
| + | by compensating with more nanomachinery (as is possible due to [[higher throughput of smaller machinery]]). | ||
| + | This is not an option for in comparison extremely voluminous macromachinery. | ||
| + | |||
| + | '''In summary:''' | ||
| + | * 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: [[Condervative design]] [[Exploratory engineering]] | ||
| + | |||
| + | = Analyzing scaling behavior = | ||
| + | |||
| + | 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) == | ||
| + | |||
| + | 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.''' | ||
| + | |||
| + | = Misc = | ||
{{wikitodo|explain the following}} | {{wikitodo|explain the following}} | ||
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* The consequences on design constraints based no this falling stiffness | * The consequences on design constraints based no this falling stiffness | ||
| − | + | = Related = | |
* '''[[Scaling law]]''' | * '''[[Scaling law]]''' | ||
Revision as of 16:02, 27 September 2022
With scaling down machinery to smaller the stiffness of this machinery falls.
Also accelerations from machine motions go up.
These two effects both increasing deformation magnitudes. But ...
With scaling down machinery to smaller the mass of that machinery is going down.
This is decreasing deformation magnitudes.
As it turns out the overall the relative deflections/strains from acceleration stay scale invariant.
This result is very important for applicability of macro 3D printing for nanomachine prototyping.
As it means 3D printed macroscale prototypes will typically vastly underperform nanoscale target systems of equal geometry.
Heck, if anything this may lead to massive over-engineering that, while surely working at the nanoscale, is very far from optimal.
Contents
Macroscale style machinery: Works better rather than worse at the nanoscale
What we absolutely do not want is to accidentally build a prototype macroscale systems with performance so high that target nanoscale systems of equal geometry will not be able to replicate that performance.
The result here implies that we do not have to fear that. 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 ...
- with a tensile modulus as high as diamond
- with a density as low as diamond
- with a maximal strain (bendability) in the double digit percentual range.
Such materials simply do not exist today. Ceramics come closest to stiffness but they're totally not elastic.
(Future gem-gum metamaterials might come close.)
That's the 🤯 degree of how much Macroscale style machinery at the nanoscale
works better than our good old macroscale machinery.
And that does not even factor in that we can easily afford to go a 1000x slower with speeds
by compensating with more nanomachinery (as is possible due to higher throughput of smaller machinery).
This is not an option for in comparison extremely voluminous macromachinery.
In summary:
- 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: Condervative design Exploratory engineering
Analyzing scaling behavior
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)
The critical quantity we want to be preserved across scales is relative strain:
[math]\epsilon = \Delta l / l[/math]
Strain epsilon is proportional to stress sigma.
[math]\epsilon = \sigma / E[/math]
Stress is given by applied force F.
[math]\sigma = F / A [/math]
A natural level of force for a scale is zentrifugal force from rotating a mass at that scale.
[math] F = m \omega^2 r = m (2\pi f)^2 r[/math]
Putting it all together we get:
[math]\epsilon = \sigma / E = F / (E A) = m \omega^2 r / (E A)[/math]
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:
Both stress & strain from machine motions is scale invariant.
Scaling of deformations from gravity
[math] F_{grav} = \rho V g \propto L^3[/math]
[math]\epsilon_{grav} = \sigma_{grav} / E = F_{grav} / (E A) = \rho V g / (E A) \propto L^1 [/math]
Both stress & strain from gravity go down linearly with smaller scales.
Misc
(wiki-TODO: explain the following)
- Why stiffness falls when shrinking the size of machinery (while keeping the same material)
- The consequences on design constraints based no this falling stiffness
