Piston motion equations


The motion of a non-offset piston connected to a crank through a connecting rod, can be expressed through several mathematical equations. This article shows how these motion equations are derived, and shows an example graph.

Crankshaft geometry

Definitions

″Angular velocity″

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

Triangle relation

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

Equations with respect to angular position (Angle Domain)

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

Position

Position with respect to crank angle :

Velocity

Velocity with respect to crank angle :

Acceleration

Acceleration with respect to crank angle :

Equations with respect to time (time domain)

Angular velocity derivatives

If angular velocity is constant, then
and the following relations apply:

Converting from angle domain to time domain

The equations that follow describe the reciprocating motion of the piston with respect to time. If time domain is required instead of angle domain, first replace A with ωt in the equations, and then scale for angular velocity as follows:

Position

Position with respect to time is simply:

Velocity

with respect to time :

Acceleration

with respect to time :

Scaling for angular velocity

You can see that x is unscaled, x' is scaled by ω, and x" is scaled by ω². To convert x' from velocity vs angle to velocity vs time multiply x' by ω . To convert x" from acceleration vs angle to acceleration vs time multiply x" by ω² . Note that dimensional analysis shows that the units are consistent.

Velocity maxima/minima

Acceleration zero crossings

The velocity maxima and minima do not occur at crank angles ' of plus or minus 90°. The velocity maxima and minima occur at crank angles that depend on rod length ' and half stroke , and correspond to the crank angles where the acceleration is zero.

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 idea that velocity maxima/minima occur when crank-rod angle is right angled.

Example

For rod length 6" and crank radius 2", numerically solving the acceleration zero-crossings finds the velocity maxima/minima to be at crank angles of ±73.17615°. Then, using the triangle sine law, it is found that the crank-rod angle is 88.21738° and the rod-vertical angle is 18.60647°. 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 graph of piston motion

The graph shows x, x', x" with respect to crank angle for various half strokes, where L = rod length ' and R = half stroke ':
for position, for velocity, for acceleration.
The horizontal axis units are crank angle degrees.
Pistons motion animation with same rod length and crank radius values in graph above :