A single perfectly rigid reciprocating rod transmits power only intermittently; near the moment between its movement in one direction and the movement in another, the power it transmits falls to zero, unless it experience an infinite acceleration in that movement, and thus too an infinite force, if its mass not be infinitesimal. This must happen twice per cycle.
Two such rods can transmit power continuously by virtue of being out of phase. If the resistance against which they push and pull be variable, they can even transmit constant power; this is the case for example if they are each in simple harmonic motion in quadrature and connected to equal idealized dashpots; for the power transmitted by one varies as k sin² xt, the other as k cos² xt = k(1 - sin² xt).
A more interesting case is where two pushrods push and pull a crankshaft rotating at a constant speed: a pushrod whose displacement at time t is y sin xt, whose velocity is thus xy cos xt, has a lever arm for its crank of k cos xt. If its force should vary in proportion, z cos xt, then the power it transmits will be xyz cos² xt, just as in the dashpot case, and summing with a pushrod in quadrature provides constant power transmission.
Now we can see also that the torque on the crankshaft from the first pushrod is kz cos² xt, and so the pushrod in quadrature brings us also to constant torque.
If we want to transmit power some distance, pushrods are impractical because they will buckle unless supported. If we use four cranks instead of two, we can substitute four longitudinally oscillating cables rather than two pushrods, transmitting power only through two of the cables at any given moment, and clearly getting the same constant power transmission.
This is temptingly analogous to two-phase AC power transmission, strongly suggesting the possibility of improving efficiency by using three cables oscillating at phase angles of 0°, 120°, and 240°, rather than four quadrature angles. The situation is not totally analogous; if the tension on each cable varies in proportion to its velocity as before, the power transmitted by a cable at phase angle φ is ReLU(sin xt + φ)². I think these still sum up to a constant, but if not, varying the tension according to some different curve can clearly provide constant power transmission over three cables. I’m not entirely sure that this would also provide constant torque, but actually I don’t care.
That’s because the reason I think this is interesting is for continuously transmitting power between parts of a flexure; flexagons aside, most flexures can’t manage continuous rotation, so the traditional forms of continuous mechanical power transmission such as belts, shafts, and gears are unhelpful. Cable transmission also has the generally-noted property of fitting conveniently into smaller spaces.
Such oscillating cable transmission might use radically non-sinusoidal force and displacement profiles — for example, think of four cables, of which at any given time three are under high tension transmitting power, while the fourth is under low tension but higher velocity, being returned to its starting position. This sort of thing can, I think, increase the material efficiency of such power transmission systems.
A cable tensioned at 2 GPa traveling at 1 m/s is transmitting 2 gigawatts per square meter, which is 2 kilowatts per square millimeter. If the cable can only be safely tensioned to 200 MPa, it can only transmit 200 W/mm² at this speed. In theory you could increase the cable speed arbitrarily — for example, your wimpy 200 MPa cable would transmit 4 MW/mm² at 20 km/s — but, aside from concerns about sonic booms and friction heating, the process of reversing the cable’s direction of movement involves accelerating it, which also requires force, and that force adds to the cable’s tension.
However, for cable lengths short compared to the free breaking length of the cable material, this acceleration can reach many gees before the load on a cable during the return stroke equals the load during the power delivery stroke. Indeed, the number of gees is precisely the ratio of the cable length to the free breaking length. So, for example, gel-spun UHMWPE at 3 GPa and 0.96 g/cc has a free breaking length of 319 km, so a 100-mm-long cable can be accelerated by pulling on one end at about 31 million gees before it breaks.
Return strokes faster than some 10–100 times the acoustic length of the cable will result in waves noticeably propagating back and forth in the cable, unless it’s properly acoustically terminated to prevent such reflections. This is precisely analogous to the phenomenon in RF electronics with unterminated transmission lines, but I don’t know of any electrical power transmission scheme for which it is essential to keep the mean drift velocity of the charge carriers in the cable to zero, so the analogy only goes so far.
To reduce the total displacement of the cable and thus permit higher instantaneous velocities and power densities, it might be desirable to transmit power at much higher acoustic frequencies than this, such that indeed many wavelengths of the tension wave fit within the cable. Taking this step renders the power transmission capability of the cable independent of its length.
As one point among many in this possible design space, consider four parallel cables oscillating in simple sinusoidal motion in quadrature, carved from ASTM A36 steel, each 100 μm square. According to Machine Teeth A36 yields at 250 MPa, and its Young’s modulus is 200 GPa. It weighs 7.9 g/cc, so if they’re 200 mm long, each weighs about 16 mg. Suppose they’re oscillating longitudinally at 3 kHz by a distance of 100 μm. Their peak speed is only 1.9 m/s, but their peak acceleration is 36 km/s/s, 3600 gees, which requires about 570 mN peak acceleration force, working out to 57 MPa, comfortably below A36’s yield stress. Such a stress will elongate the wire by 0.03%. If the one or two wires actively transmitting power at any given time are loaded sinusoidally up to 125 MPa, the peak power transmitted on a wire is 2.4 watts, but about 1.1 W of that is “returned” during the return stroke for the wire. I think this means that the total net power is consistently about 1.3 watts for the whole assemblage.
This is a promising but not outstanding amount of power to transmit through something the thickness of a beard hair. How can we increase it?
If we increase the distance of displacement, holding constant the frequency, then we linearly increase the velocity and thus the power, at the expense of linearly increasing the acceleration. If we increase the frequency instead, while holding the displacement constant, then we linearly increase the velocity but quadratically increase the acceleration.
If we use a (nonexistent) material of the same physical characteristics except that it had a lower density, or if we were transmitting over a shorter distance, it would reduce the force required to accelerate the wires; we could trade that for a reciprocally higher acceleration. If instead we used a material with the same physical characteristics except a higher yield stress, such as a harder steel, then we could proportionally increase both the acceleration and the load force.
The totally free way to improve the system seems to be decrease the frequency linearly while increasing the distance quadratically, thus holding the acceleration constant while increasing the velocity and thus the power reciprocally with the frequency. So maybe we could increase the total displacement to 10 mm while decreasing the frequency to 300 Hz, increasing the power to about 13 watts, a much more respectable power level. Further development in this direction would seem to be dependent on very precise motion control to avoid having to space the wires further apart.
If we drop the density by a factor of 8 while multiplying the yield stress by 10, then the first would allow us to increase the frequency and power further by a factor of 2.8 (increasing the acceleration by 8), and the second would allow us to increase the frequency by a factor of 3.2, but also the tension by a factor of 3.2, increasing the power by a factor of 10. I think this means you could transmit 360 watts of mechanical power through your new hair-thin gel-spun UHMWPE cable, at least for 20 mm.
(In practice I doubt this could be sustained continuously; the imperfectly elastic nature of any real material results in some heat dissipation from stretching and relaxing it, and UHMWPE is, I suspect, worse in this aspect than steels. Moreover it melts at a very low temperature; tens of milliwatts of such dissipation would likely be fatal.)
With the acoustic traveling wave mode of energy transmission mentioned earlier, the maximum power per unit area is the energy’s elastic energy density (half its yield stress multiplied by its yield strain) multiplied by the speed of sound in the substance.
Consider how the example above compared to electrical power transmission through an electrical cable of comparable size. If you enclosed three 40 AWG solid copper wires, each 80 microns in diameter, in 20 microns of Kynar insulation, you would have a similar-sized cable. You could run three-phase AC power over it at whatever frequency was convenient; your phase-to-phase voltage would be limited by the breakdown voltage of 40 μm of Kynar, while your RMS current would be limited by the resistance per meter and heat dissipation per meter of the cable at Kynar’s maximum service temperature of 149°. You can decrease the radius of the conductor and increase the thickness of the Kynar by an equal amount, thus increasing the voltage linearly but also increasing resistance as the square of the remaining wire.
The dielectric strength is supposedly “1700 V/mil” for a short period of time; if we figure that’s 1000 V/milli-inch in practice, that’s about 40 volts per micron, so 1600 volts peak, 1100 VAC RMS phase-to-phase. 40-gauge wire is rated for 90 milliamps over short runs. I think this ends up at a few hundred watts, too, so it’s the same ballpark.