The angle domain equations above show that the motion of the piston (connected to rod and crank) is notsimple harmonic motion, but is modified by the motion of the rod as it swings with the rotation of the crank. This is in contrast to the
Scotch Yoke which directly produces simple harmonic motion.
Example graphs
Example graphs of the angle domain equations are shown below.
From the foregoing, you can see that the time domain equations are simply scaled forms of the angle domain equations: is unscaled, is scaled by ω, and is scaled by ω².
To convert the angle domain equations to time domain, first replace A with ωt, and then
scale for angular velocity as follows: multiply by ω, and multiply by ω².
Velocity maxima and minima
By definition, the velocity
maxima and minima
occur at the acceleration zeros (crossings of the horizontal axis).
Crank angle not right-angled
The velocity maxima and minima (see the acceleration zero crossings in the graphs below) depend on rod length and half stroke and do not occur when the crank angle is right angled.
Crank-rod angle not right angled
The velocity maxima and minima do not necessarily occur when the crank makes a right angle with the rod. Counter-examples exist to disprove the statement "velocity maxima and minima only occur when the crank-rod angle is right angled".
Example
For rod length 6" and crank radius 2" (as shown in the example graph below), numerically solving the acceleration zero-crossings finds the velocity maxima/minima to be at crank angles of ±73.17615°. Then, using the triangle
law of sines, it is found that the rod-vertical angle is 18.60647° and the crank-rod angle is 88.21738°. Clearly, in this example, the angle between the crank and the rod is not a right angle. Summing the angles of the triangle 88.21738° + 18.60647° + 73.17615° gives 180.00000°. A single counter-example is sufficient to disprove the statement "velocity maxima/minima occur when crank makes a right angle with rod".
Example graphs of piston motion
Angle Domain Graphs
The graphs below show the angle domain equations for a constant rod length (6.0") and various values of half stroke (1.8", 2.0", 2.2").
Note in the graphs that L is rod length and R is half stroke..
The vertical axis units are
inches for position, [inches/rad] for velocity, [inches/rad²] for acceleration. The horizontal axis units are crank angle
degrees.
Animation
Below is an animation of the piston motion equations with the same values of rod length and crank radius as in the graphs above
Piston motion animation with the various half strokes from the graph above (using the same color code)
Units of Convenience
Note that for the
automotive/
hotrod use-case the most convenient (used by enthusiasts) unit of length for the piston-rod-crank geometry is the
inch, with typical dimensions being 6" (inch) rod length and 2" (inch) crank radius. This article uses units of inch (") for position, velocity and acceleration, as shown in the graphs above.