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Few months ago I opened the boards of an old, scrap radio-cum-tape-recorder (Philips DR183); whose tuning knob (that adjusts the gang); was very peculiar.

( * I did not found its datasheet on web)

inside

The channel-indicator is a narrow rod at the tip of red plastic-thing. That indicator moves along a straight-linear segment when I turns the wheel.

what its work look like

However, it is just a rough sketch. I could not figure out, what mathematical principle/ geometric theorem being used in this mechanic. It is neither like pulley and rope; and nor-like a piston-with-crankshaft. My question is, on what-basis they have chosen the angles, distances and ratios to make such movements? Or, For say if I try to make one with cardboard etc; what angles, lengths, ratios etc I have to choose, and in which logical basis?

Here I've provided some of its allowed conformations.

Steps

Please feel free to move this question if it is more appropriate for any other SE site, such as mathematics.SE .

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Chebyshev Link Motion. (cognate form known as the Lambda Motion)

It's a classic, also sometimes seen in its original form using circular motion to provide a 6th order approximation to a straight line.

The crucial dimensions are in a 2:4:5 ratio (or 1:2:2.5 in the animation linked above)

This is an improvement on the earlier (by about a century!) James Watt Parallel Link motion, which provides a fifth order approximation - and was the invention he was most proud of - as well as sparking the whole field of four-bar linkages as field of systematic study.

Some time ago I simulated these linkages ( in a spreadsheet) and plotted the errors - the Chebyshev, designed for 1 metre of linear travel, gives an equi-ripple error reaching 0.6mm off the straight line. (For other travels, the error scales accordingly).

You can actually tune the Watt linkage's error to be even lower than this, despite it being only 5th order - if you have space for slightly longer links.

Analysis with a spreadsheet isn't particularly difficult.

Start with a fixed point at some coordinate, X1 = (0,0) and a length, say, R=5 and use the equation of rotary motion (X^2 + Y^2 = R^2) to plot multiple values of Y1 at different angles of rotation. From a second fixed point, say, X2 = (4,0) and R2 = 5, find values of Y2 such that Y2 is also a fixed distance (R3 = 2) away from each Y position. This turns out to be solving quadratics, and not all angles have valid roots. Finally plot the midpoint of the Y-Y2 bar and if you've chosen the dimensions above you have the regular Chebyshev link motion.

You'll see that cancellation of two rotary motions is at the heart of it, but the rotation of the third link introduces complications and the devil is in the details.

As the other answer says, synthesis is more "involved" - to the extent that some of the world's top mathematicians devoted decades of their lives to it, and as far as I know, synthesising any motion you might want is not yet a solved problem.

I believe this is related to the use of Chebyshev's polynomials in electronic filter design, where equi-ripple errors on a straight line passband (or stopband) are also important.

Finally note that the straight line portion is also practically linear in distance per degree rotation (with a relatively fast transit through the curved part of the path).

This leads to "Tchebyshevs plantigrade machine" which maintains a practically constant speed of perfectly linear motion. I am glad to see that animation - I "invented" it independently after seeing the above motions, and actually built a slightly simpler form in Meccano years ago, though I'm not surprised it already existed and had a name.

enter image description here

If ever there was a steampunk walking robot, to an authentic 19th century design, this is it.

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  • $\begingroup$ I'm no engineering student, however it looks to me, both the Chebyshev's lambda and Watt's linkage are using some sort of 'cancellation' of 2 opposite- curved movement giving rise to a straightline (or approx-straightline) $\endgroup$ – Always Confused Jan 12 '17 at 14:36
  • $\begingroup$ Your writing is beautiful, especially the way of conclusion. The robot is terrific. $\endgroup$ – Always Confused Jan 12 '17 at 14:50
  • $\begingroup$ Thanks. Added a bit about the analysis - your insight about cancellation is close. $\endgroup$ – Brian Drummond Jan 12 '17 at 15:18
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The mechanism shown in your picture is known as a four-bar linkage. These mechanisms are used quite often in mechanical engineering to convert rotational motion into complex trajectories, and vice-versa.

The specific mechanism you've shown, as BrianDrummond mentioned in his comment, is Chebyshev's Lambda Mechanism. This four-bar linkage is often used to produce a linear motion from the rotational motion of a crank, and can be seen in some simple walking robots. There are other ways to do produce linear motion from rotational motion, such as a Slider-crank, another type of four-bar linkage.

As for how to design such mechanisms, the methods are too involved to give an adequate answer on this site, but you can easily find resources online by searching "four-bar linkage synthesis." Keep in mind that there are two different ways of going about the design: graphical methods and analytical methods. With graphical methods you place a series of anchor points and trajectories on a piece of paper, then using some simple geometrical tricks you can easily generate a four-bar that performs that task. Analytical methods involve having a mathematical expression of the trajectory and using that to solve for your four-bar.

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