Hi and thanks; i did see your posts on OU.com (Fletcher here gave a link to the thread) and was likewise impressed with your work - in fact i thought i might as well leave you guys to it, as i'd hardly made any inroads at that point.
It was only after reading the patent application that i began to grasp what the rig was supposed to do, and so had to try it for myself - it's not too far astray from the topic here and might even have helped..
And i like your line of thinking. The efficiency of a crank is an interesting dynamic, precisely because of the effective force variation as a function of angle. Some of the most important Bessler images also depict crank mechanisms - the con-rod is mis-labelled as a "piston" due variations between modern and historical terminologies, i think.. but there's one aspect of a crank's operation i find particularly interesting, which i'll come back to after addressing your points...
There must be some kind of image editing software that can measure irregular areas for you, but i dunno - to do this myself i'd have to go cell-by-cell. Just from crude estimation though, i count roughly 5.5 under the first half and about 4.5 over the second half, so there does seem to be an asymmetry of about 5% between positive and negative torques, with the former being greater as you suggest.
Let's ignore for now that this is 95% smaller than the asymmetry claimed in the application, and ask whether this could be an actual energy/work asymmetry.
We have more positive torque than negative, but while superficially encouraging (it's certainly the right trend) we have to consider it in terms of the total work in vs out; which ultimately is MassGravityHeight in/out, not whatever's going on on the crank.
And it all comes down to this intrinsic inefficiency of cranks per se: as you point out, the output is a function of angle, hence we must remember that some of the area outside of the curve represents energy lobbed off of the input integral..! As you allude, the sin/cos line can only have a maximum of the peak force available, but any minimum of it down to zero (ie. in the zero degrees TDC and corresponding BDC positions described in the application and reproduced here).
In short, if we connect a crank to a vertically-falling weight via pulleys and chord, the crank cannot exploit the full force available throughout the drop, because most of the time the torque is below the maximum vertical force available. Thus one full cycle of the crank in this mechanism yields a loss on the way down, and a slightly smaller loss on the way back up. It's an asymmetric balance of input vs output losses, more lossy on the way down, and more efficient on the way back up.
Hence the fact that there's more positive than negative torque isn't enough to render an energy or work asymmetry, since there simply isn't enough torque to complete a full rotation in either direction.
This is a point to consider when viewing the above traces - the axes show torque vs time and don't mention angle, so one might reasonably assume the interactions span a full 360° - i probably should've clarified this earlier - they actually only represent about 350° of rotation, since this was the most it could manage, in either of the three configurations.
Hence both the positive and negative integrals depict somewhat less than 180° of angle, and obviously, the type of asymmetry we're after would show us an uneven distribution of +/- torque across a full 360° cycle - say 185 / 175, or whatever. Either that, or a significant variation in peak force or duration, which again isn't supported by the sim here.
If we were to try and overcome this intrinsic inefficiency of cranks, we'd be trying to achieve the maximum force available for a larger angular displacement - preferably the whole cycle. So for example, at the zero degrees position, where torque is minimal, we might try to arrange a right-angled lever to rotate the initial force around, 90° orthogonally and into the 'easy' plane of the crank. But, even if we could do this for every angle around the cycle, we'd only succeed in harnessing all of the input force, ie. we'd have eliminated the inherent losses in the mechanism, but we'd still only get 100% of any input energy back out.
A perfectly efficient crank would effectively be functionally identical to a spool and pulley system, as far as converting linear to rotary work is concerned.
I'm not sure this speaks directly to your ideas, i've read your descriptions carefully and studied your diagrams.. It might be more helpful if you could try the sim yourself, if you could find somewhere to download it, cough, sputter.. ;) ;)
