Assume two vertical pistons with their flat head facing upward.

Is it possible to couple them to one rotating shaft in a way that the position vs. time diagram gets like this? (This is one complete cycle)


Each color being representing a piston.

The equations are:

Red: $$2 \left|\cos(\pi t)\right| - \frac12$$

Blue: $$2 \left|\sin(\pi t)\right| - \frac12$$ (The first with a phase difference of $\pi \over 2$)

I think because of the Absolute values, this can not be a simple crankshaft.

Can anyone help?


Actually I want to get as close as possible to this diagram of motion, which is the ideal theoretical case. Maybe this can help you for a better approximation.

real thing

  • 5
    $\begingroup$ That instantaneous velocity direction change is a killer. would require infinite acceleration. $\endgroup$
    – joojaa
    Jan 19, 2017 at 16:59
  • $\begingroup$ @joojaa Then we could soften the edge so that it becomes seamless. Would $\sin^2(\pi t)$ be fine enough? Is sin squared motion possible to be achieved using mechanical structures? $\endgroup$
    – AHB
    Jan 19, 2017 at 17:55
  • $\begingroup$ How exact do you need to be with any of this? A little insight as to why they have to have that position profile could help. $\endgroup$
    – JMac
    Jan 19, 2017 at 18:09
  • $\begingroup$ @JMac I added what you want. $\endgroup$
    – AHB
    Jan 19, 2017 at 18:32
  • $\begingroup$ You can do a lot with 4-bar linkages. Note that the vertical movement of the feet (ignoring the horizontal movement) is very close to the profile you're looking for. $\endgroup$
    – Dave Tweed
    Jan 20, 2017 at 13:29

1 Answer 1


You can (come close to) this motion profile with two cams on one shaft. the tops being the dwell portion of the cams, and the bottom being the peak between two accelerations. As some one else mentioned since the bottom of your motion profile is an instantaneous acceleration change, you will never achieve it perfectly. this can be minimized by reducing mass, speed, and stroke length. This is going to be the closest mechanical way to do this.

  • $\begingroup$ It would be best to machine the cams with smooth lead-ins and lead-outs either side of the sharp acceleration point. This would reduce wear and tear due to shock and vibration. $\endgroup$
    – user6335
    Nov 24, 2017 at 14:49

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