Reciprocative dissipation mechanisms in gem-gum technology

One may also want to call this more fancily "reciprocative energy dissipation".
This applies to any reciprocative motion both linear reciprocative and rotative reciprocative.
Essence is back and forth motion that requires accelerations and jerk.

Note that:
– in factory style there is much less reciprocative motion than in more general purpose kind of robotics akin to 3D printers.
– at least one more highly relevant thing …

(wiki-TODO: Add explanation of physics, math, and reasonable example values (last one hardest).)
(wiki-TODO: How does it scale and how far is that scaling reliable. Within or out of quantized regime and such.)

Mechanisms

Akhiezer damping

Physics

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Symbolic math

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Qunatitative example numbers

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Scaling & limits of model

Specifics to diamond (compared to metals or doped silicon).
It seems as if the the absence of dense electronic states in undoped diamond should reduce electron-phonon-coupling significantly. If so then by how much?
Having large amplitudes corresponds to many phonons bosonically overlapping in a few closeby modes.
Multi phonon processes may start to matter. (wiki-TODO: {{{1}}})

Non recuperated phase shift

In some but not all cases this could be partially recuperated.

Physics

A lag between center-of-mass motion and driving motion gives a phase shift.
Non-recuperation of phase shifted center of mass motion due to finite speed of sound.

★ Highly predictable assembly line motions …
– might allow for a high degrees of energy recuperation and …
– are to be expected in andvanced productive nanosystems with standard part mass productions
★ Much more inherently unpredicatble computation motions …
– will have much less chance for energy recuperation.
– (TODO: Is this fundamental or are some tricks still applicable?)

Symbolic math

Energy turnover.
Turnover power integrated over time would be zero
if not for a phase shift due to the speed of sound:
<math>E_{maxloss} = \int_0^t dt (F v)</math>
To introduce that phase shift we formulate the model
in therms of phase angle rather than time like so:
<math>v(\phi)= \hat{v} ~ \sin(\phi)</math>
<math>F(\phi)= m \hat{a} ~ \cos(\phi + \Delta \phi)</math>
For transforming the integrand:
<math>\phi = t ~2\pi/\tau_p</math> => <math>d\phi/dt = ~2\pi/\tau_p</math> => <math>dt = d\phi ~ \tau_p/(2\pi)</math>
Substituting everything in gives:
<math>E_{maxloss} = \tau_p/(2\pi) \int_0^{2\pi} d\phi (m \hat{a} ~ cos(\phi+\Delta\phi) ~ \hat{v} ~ sin(\phi))</math>
In order to factor out kinetic energy in form of <math>m \hat{v}^2/2</math>
We want to merge the period (unit: s)
with the peak-acceleration (unit: m/s²) to get peak-speed (unit: m/s):
From comparing position speed and acceleration
<math>x(t)=A⋅cos(ωt); ~~ v(t)=−A⋅ω⋅sin(ωt); ~~ a(t)=−A⋅ω^2⋅cos(ωt)</math>
we get <math>\hat{a} / \hat{v} = \omega</math> and with <math>ω⋅τ_p=2π</math>
we get <math>\hat{a} = \hat{v} ~ 2 \pi / \tau_p</math>
Substituting for <math>\hat{a}</math> gives:
<math>E_{maxloss} = m \hat{v}^2 \int_0^{2\pi} d\phi \sin(\phi)\cos(\phi + \Delta \phi)</math>
Using some trigonometric identities to make this integratable:
<math>E_{maxloss} = m \hat{v}^2 \left(\cos(\Delta \phi)\int_0^{2\pi} d\phi \sin(\phi)\cos(\phi) + \sin(\Delta \phi) \int_0^{2\pi} d\phi \sin^2(\phi)\right)</math>
First term is zero and with <math>\int_0^{2\pi} \sin^2{\phi} = \pi</math> that leaves:
Result for maximal power-loss per period:
<math>E_{maxloss} = m \hat{v}^2 \sin(\Delta \phi) ~\pi</math>
Q factor:
<math>Q_{min} = 2 \pi \frac{E_{stored}}{E_{maxloss}} = …</math>
Harmonic oscillator: <math>E_{stored} = m \hat{v}^2/2 + k x^2/2 = 2 ~ mv^2/2</math>
<math>Q_{min} = 1/(\sin(\Delta \phi) ~\pi)</math>
Effective power-loss proportional to frequency f:
<math>P_{maxloss} = E_{maxloss}/\tau_p = E_{maxloss} \cdot f = …</math>

Qunatitative example numbers

(wiki-TODO: For the nanofactory context and the compute context: What numbers are needed and what are the difficulties in getting sensible numbers for conservative estimates. Then run the numbers.)

Example context nanofactory assembly line station (~1MHz, ~5mm/s from Nanosystems)

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Example context rod logic (~1GHz from Nanosystems)

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<math>l_{rod} = …</math>; <math>v_{s} = …</math>;
<math>\tau_{shift} = l_{rod}/v_s</math>
<math>\Delta \phi = \pi ~ \tau_{shift}/\tau_p</math>

Scaling & limits of model

I seems this potential source of dissipation scales linearly with
frequency & amplitude of motions.

Discussion

Helping conservative assumptions:
★ This is a worst case scenario with zero energy recuperation.
★ Proposed frequencies for productive nanomachinery are mere MHz not GHz
… GHz is only discussed in context of mechanical logic in Nanosystems
★ Mill style manufacturing has a large fraction of non-reciprocative and predictable reciprocative motions,
… so there is a lot more potential for energy recuperation than in computation with inherently unpredictable outcomes.
… Even for rod-logic some local counter-direction motion design tricks may allow for some recuperation. But less clear.

Hurting conservative assumptions:
★ This is assuming harmonic motion across the whole motion range, sudden stops act notably worse.
=> Fourier decomposition. This is serious.
Possible remedy: Larger rod logic designs allow for designs with way softer stoppings
like e.g. using vdW force across a longer soft stopping path.

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Regarding the speed of sound in rods being potentially lower than in bulk diamond:
★ Non-bonded interactions can be even stiffer than non-bonded interactions if pressed like in a tight fit channel.
That increases speed dependent dissipation though.
And pure carbon based linear logic designs give somewhat fixed clearances.
★ Polyyne rods with axially aligned bonds (and alternating triple and single bonds)
… are not likely to be notably less stiff in this direction compared to bulk diamond. The opposite rather.
★ Larger scale rod logic from diamond slabs (possibly more viable than maximally compact polyyne rod logic)
is naturally closer to bulk diamond in stiffness properties.

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There seems to be analogy-to overlap-with …
★ wave propagation and reflection
★ reactive power (as used in electric systems but here in mechanical analogy)
Which may provide pathways to gain some better intuition here.

Derivable design principles

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Accidental heatpump

See page: Accidental heatpump

Symbolic math

(wiki-TODO: Estimate the thermal energy in a (nontrivial) degree of freedom when freely fluttering around (nontrivial), and then the thermal energy when constrained by a (steric) obstruction caused by cyclic machine motion. Then (assuming full dissipation of that energy) multiply by twice the frequency of operation. Twice since the cooling phase gets dissipated too.)

Qunatitative example numbers

(wiki-TODO: Run numbers fro the symbolic math. Then assume a plausible volume and a lower volumetric density of these mechanisms for some more estimates.)

Scaling & limits of model

It seems this should scale linearly over a very very wide range of speeds. Including proposed ~1mm/s scale.
Nontrivial things might happen near absolute zero where phonon modes freeze out.
Especially for diamond.

Derivable design principles

In case of structures for sliding interfaces: Avoid ones that have too thin walls.
Particularly avoid ones so thin that bond-bending rather than bond-stretching
becomes the main remaining source for structural rigidity.
As bond-bending is about ~20x lower stiffness than bond stretching.
(wiki-TODO: Add example images of such structures.)

Related

• Friction in gem-gum technology
• Friction
• Accidental heatpump & Equipartitioning theorem

• Accidentally suggestive