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

(wiki-TODO: {{{1}}})

Symbolic math

(wiki-TODO: {{{1}}})

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

Lag of center of mass motion to driving motion gives a phase shift.
Non-recuperation of phase shifted center of mass motion due to finite speed of sound. (wiki-TODO: {{{1}}})

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: {{{1}}})

Scaling & limits of model

(wiki-TODO: {{{1}}})

Accidental heatpump

See page: Accidental heatpump

Symbolic math

(wiki-TODO: {{{1}}})

Qunatitative example numbers

(wiki-TODO: {{{1}}})

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.

Related

• Friction in gem-gum technology
• Friction
• Accidental heatpump
• Accidentally suggestive