Piston motion equations

The reciprocating motion of a non-offset piston connected to a rotating crank through a connecting rod (as would be found in internal combustion engines) can be expressed by equations of motion. This article shows how these equations of motion can be derived using calculus as functions of angle (angle domain) and of time (time domain). The geometry of the system consisting of the piston, rod and crank is represented as shown in the following diagram: From the geometry shown in the diagram above, the following variables are defined: rod length (distance between piston pin and crank pin) crank radius (distance between crank center and crank pin, i.e. half stroke) crank angle (from cylinder bore centerline at TDC) piston pin position (distance upward from crank center along cylinder bore centerline) The following variables are also defined: piston pin velocity (upward from crank center along cylinder bore centerline) piston pin acceleration (upward from crank center along cylinder bore centerline) crank angular velocity (in the same direction/sense as crank angle ) The frequency (Hz) of the crankshaft's rotation is related to the engine's speed (revolutions per minute) as follows: So the angular velocity (radians/s) of the crankshaft is: As shown in the diagram, the crank pin, crank center and piston pin form triangle NOP. By the cosine law it is seen that: where and are constant and varies as changes. Angle domain equations are expressed as functions of angle. The angle domain equations of the piston's reciprocating motion are derived from the system's geometry equations as follows. Position with respect to crank angle (from the triangle relation, completing the square, utilizing the Pythagorean identity, and rearranging): Velocity with respect to crank angle (take first derivative, using the chain rule): Acceleration with respect to crank angle (take second derivative, using the chain rule and the quotient rule): The angle domain equations above show that the motion of the piston (connected to rod and crank) is not simple harmonic motion, but is modified by the motion of the rod as it swings with the rotation of the crank.