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An engine needs two pistons, in theory, to move like this.

one cycle

But this thing must be applied in experiment, by conversion of the motion of a rotating shaft to a motion like this.

With those sharp turn-arounds the mechanism can't be practically made.

So I need an approximation which is possible to be made using pistons, crankshaft, cam, etc. But this part isn't related to math.

For example the function $\sin x$ is easily made using a rotating wheel and two rods joined to each other, one of them to the wheel, with hinges, and the outer one being constrained only to move in one dimension.

My first try was these functions:

$$2 \left|\cos(\pi t)\right| - \frac12\ ,\ 2 \left|\sin(\pi t)\right| - \frac12$$

Which are like this:

d

But the sharp turn-around problem isn't still fixed.

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    And $\sin x$ is what engines actually use, constructed exactly as you say. It is the simplest mechanism that converts rotation to linear motion. I don't see why you want the sharp corners at the bottom of the stroke. – 2017-01-19
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    @RossMillikan I wanted the best approximation. If it is $\sin x$, then I have to use that and revise my theoretical calculations. I want to reduce after-experiment mathematical corrections as much as possible. And: I want the straight sides at the top more than I want sharp corners in the bottom. – 2017-01-19

2 Answers 2

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You can Fourier analyze your desired pattern. All the harmonics will be odd due to the symmetry of your trace. You get a more accurate trace of the pattern at the price of complexity. The fundamental is just the usual crankshaft/connecting rod mechanism. Now at the point the connecting rod joins the crankshaft insert another circle turning at three times the frequency. Connect the existing crankshaft to the circumference of the circle and center the new circle on the crankshaft. Make the diameter of the new circle the amplitude of the third order Fourier term. You can now insert another circle at five times the frequency. Keep going as long as you want.

Added: see the diagram below. The rectangle is the piston, which moves vertically in the cylinder. The dark line is the connecting rod, which is connected to the outside of the green circle. The green circle rotates at three times the engine speed. The center of the green circle is connected by the thin line to the normal spot on the crankshaft. The offset of the crankshaft is the amplitude of the fundamental Fourier component. The radius of the green circle is the amplitude of the third harmonic. You can stack up the green circles to get more components. enter image description here

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    very clever. Since I don't know much about Fourier series application, would you draw the mechanism for the first three steps tor example? – 2017-01-19
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    Epicycles return! – 2017-01-19
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    @amd: Exactly... – 2017-01-19
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    @RossMillikan I didn't get how the mechanism works. If you know Mathematica, Please make an animation and post the code. That way I can study the notebook and get what you mean. sorry. – 2017-01-20
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    If the axis of the crankshaft is at $(0,0)$ the point the connecting rod attaches to is $(r \cos \omega t, r \sin \omega t)$ The small circle offsets this by $(r' \cos 3\omega t, r' \sin 3\omega t)$ That should let you make the animation you want. You choose the ratio of the radii to get the best approximation to your desired curve. – 2017-01-20
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If I understand well your question, you are looking for an equation which is the closer possible to the graphical representation given

Here is such a function: $y=abs(\max(\min(\sin(p*x),2*a),-2*a)-a).$

where, say $0.3

Here is Matlab implementation and the resulting curve:

clear all;close all;hold on;axis equal

a=0.4;p=2;

t=0:0.01:2*pi;L=length(t);f=a*ones(1,L);

plot(t,abs(max(min(sin(b*t),2*f),-2*f))-f)

enter image description here

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    Not actually. I want to practically make that engine (or a model of it). So the resulting function must be apply-able by mechanical objects, pistons, shafts, gears, etc. This is more important that finding a curve that fits. – 2017-01-20