Piston motion equations

The motion of a non-offset piston connected to a crank through a connecting rod (as would be found in internal combustion engines), can be expressed through several mathematical equations. This article shows how these motion equations are derived, and shows an example graph.

l = rod length (distance between piston pin and crank pin)
r = crank radius (distance between crank pin and crank center, i.e. half stroke)
A = crank angle (from cylinder bore centerline at TDC)
x = piston pin position (upward from crank center along cylinder bore centerline)
v = piston pin velocity (upward from crank center along cylinder bore centerline)
a = piston pin acceleration (upward from crank center along cylinder bore centerline)
ω = crank angular velocity in rad/s

The crankshaft angular velocity is related to the engine revolutions per minute (RPM):

As shown in the diagram, the crank pin, crank center and piston pin form triangle NOP.
By the cosine law it is seen that:

The equations that follow describe the reciprocating motion of the piston with respect to crank angle.
Example graphs of these equations are shown below.

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