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.

Crankshaft geometry

Definitions

"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

Angular velocity

The crankshaft angular velocity is related to the engine revolutions per minute (RPM)::$omega=\; frac\{2picdot\; rpm\}\{60\}$

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:
:$l^2\; =\; r^2\; +\; x^2\; -\; 2cdot\; rcdot\; xcdotcos\; A$

Equations wrt 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 wrt crank angle (by rearranging the triangle relation): :$l^2\; -\; r^2\; =\; x^2\; -\; 2cdot\; rcdot\; xcdotcos\; A$:$l^2\; -\; r^2\; =\; x^2\; -\; 2cdot\; rcdot\; xcdotcos\; A\; +\; r^2\; [(cos^2\; A\; +\; sin^2\; A)\; -\; 1]$:$l^2\; -\; r^2\; +\; r^2\; -\; r^2sin^2\; A\; =\; x^2\; -\; 2cdot\; rcdot\; xcdotcos\; A\; +\; r^2\; cos^2\; A$:$l^2\; -\; r^2sin^2\; A\; =\; (x\; -\; r\; cdot\; cos\; A)^2$:$x\; -\; r\; cdot\; cos\; A\; =\; sqrt\{l^2\; -\; r^2sin^2\; A\}$:$x\; =\; rcos\; A\; +\; sqrt\{l^2\; -\; r^2sin^2\; A\}$

Velocity

Velocity wrt crank angle (take first derivative, using the chain rule)::

Acceleration

Acceleration wrt crank angle (take second derivative, using the chain rule and the quotient rule)::

Equations wrt time (Time Domain)

Angular velocity derivatives

If angular velocity is constant, then:$A\; =\; omega\; t\; ,$ and the following relations apply:

:$frac\{dA\}\{dt\}\; =\; omega$

:$frac\{d^2\; A\}\{dt^2\}\; =\; 0$

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 "ω"t in the equations, and then scale for angular velocity as follows:

Position

Position wrt time is simply::$x\; ,$

Velocity

Velocity wrt time (using the chain rule): :

Acceleration

Acceleration wrt time (using the chain rule and product rule, and the angular velocity derivatives)::

caling for angular velocity

You can see that x is unscaled, x' is scaled by "ω", and x" is scaled by "ω"².
To convert x' from velocity vs angle [inch/rad] to velocity vs time [inch/s] multiply x' by "ω" [rad/s] .
To convert x" from acceleration vs angle [inch/rad²] to acceleration vs time [inch/s²] multiply x" by "ω"² [rad²/s²] .
"Note that dimensional analysis shows that the units are consistent."

Velocity maxima

The velocity maxima and minima do not occur at crank angles "(A)" of plus or minus 90°.
The velocity maxima and minima occur at crank angles that depend on rod length "(l)" and half stroke "(r)".

Example graph

The graph shows x, x', x" wrt to crank angle for various half strokes, where L = rod length "(l)" and R = half stroke "(r)": for position, [inches/rad] for velocity, [inches/rad²] for acceleration.
The horizontal axis units are crank angle degrees.]

See also

* Reciprocating engine
* Stroke
* Piston
* Connecting rod
* Crankshaft
* Scotch yoke