Piston motion equations

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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.

Contents

[edit] Crankshaft geometry

Diagram showing geometric layout of piston pin, crank pin and crank center
Diagram showing geometric layout of piston pin, crank pin and crank center

[edit] Definitions

l = rod length (distance between piston pin and crank pin)
r = crank radius (distance between crank pin and crank center, 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

[edit] Angular velocity

Angular velocity is related to the engine revolutions per minute (RPM):

\omega= \frac{2\pi\cdot rpm}{60}

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

[edit] Triangle relation

As shown in the diagram, the crank pin, crank center and piston pin form triangle NOP. By the cosine law,

l^2 = r^2 + x^2 - 2\cdot r\cdot x\cdot\cos A

[edit] Equations wrt angular position

Position wrt crank angle (by rearranging the triangle relation):

l^2 - r^2 = x^2 - 2\cdot r\cdot x\cdot\cos A
l^2 - r^2 = x^2 - 2\cdot r\cdot x\cdot\cos A + r^2[(\cos^2 A + \sin^2 A) - 1]
l^2 - r^2 + r^2 - r^2\sin^2 A = x^2 - 2\cdot r\cdot x\cdot\cos A + r^2 \cos^2 A
l^2 - r^2\sin^2 A = (x - r \cdot \cos A)^2
\sqrt{l^2 - r^2\sin^2 A} = x - r \cdot \cos A
x = r\cos A + \sqrt{l^2 - r^2\sin^2 A}

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

\begin{array}{lcl} x' & = & \frac{dx}{dA} \\ & = & -r\sin A + \frac{(\frac{1}{2}).(-2). r^2 \sin A \cos A}{\sqrt{l^2-r^2\sin^2 A}} \\ & = & -r\sin A - \frac{r^2\sin A \cos A}{\sqrt{l^2-r^2\sin^2 A}} \end{array}

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

\begin{array}{lcl} x'' & = & \frac{d^2x}{dA^2} \\ & = & -r\cos A - \frac{r^2\cos^2 A}{\sqrt{l^2-r^2\sin^2 A}}-\frac{-r^2\sin^2 A}{\sqrt{l^2-r^2\sin^2 A}} - \frac{r^2\sin A \cos A .(-\frac{1}{2})\cdot(-2).r^2\sin A\cos A}{\left (\sqrt{l^2-r^2\sin^2 A} \right )^3} \\ & = & -r\cos A - \frac{r^2(\cos^2 A -\sin^2 A)}{\sqrt{l^2-r^2\sin^2 A}}-\frac{r^4\sin^2 A \cos^2 A}{\left (\sqrt{l^2-r^2 \sin^2 A}\right )^3} \end{array}


Example graphs of these equations are shown below.

[edit] Equations wrt time

If the angular velocity ω is constant then:

A = \omega t \,
\frac{dA}{dt} = \omega
\frac{d^2 A}{dt^2} = 0

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 wrt time is simply:

x \,

Velocity wrt time (using the chain rule):

\begin{array}{lcl} v & = & \frac{dx}{dt} \\ & = & \frac{dx}{dA} \cdot \frac{dA}{dt} \\ & = & \frac{dx}{dA} \cdot\ \omega \\ & = & x' \cdot \omega \\ \end{array}

Acceleration wrt time (using the chain rule and product rule):

\begin{array}{lcl} a & = & \frac{d^2x}{dt^2} \\ & = & \frac{d}{dt} \frac{dx}{dt} \\ & = & \frac{d}{dt} (\frac{dx}{dA} \cdot \frac{dA}{dt}) \\ & = & \frac{d}{dt} (\frac{dx}{dA}) \cdot \frac{dA}{dt} + \frac{dx}{dA} \cdot \frac{d}{dt} (\frac{dA}{dt}) \\ & = & \frac{d}{dA} (\frac{dx}{dA}) \cdot (\frac{dA}{dt})^2 + \frac{dx}{dA} \cdot \frac{d^2A}{dt^2} \\ & = & \frac{d^2x}{dA^2} \cdot (\frac{dA}{dt})^2 + \frac{dx}{dA} \cdot \frac{d^2A}{dt^2} \\ & = & \frac{d^2x}{dA^2} \cdot \omega^2 \\ & = & x'' \cdot \omega^2 \\ \end{array}

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 [in/rad] to velocity vs time [in/s] multiply x' by ω [rad/s].
To convert x" from acceleration vs angle [in/rad²] to acceleration vs time [in/s²] multiply x" by ω² [rad²/s²].

[edit] Velocity maxima

The velocity maxima (positive and negative) do not occur at a crank angle (A) of plus or minus 90°.
The crank angles at which the velocity maxima occur vary depending on rod length (l) and half stroke (r).

[edit] 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):

The vertical axis units are inches for position, [inches/rad] for velocity, [inches/rad²] for acceleration, and the horizontal axis units are degrees
The vertical axis units are inches for position, [inches/rad] for velocity, [inches/rad²] for acceleration, and the horizontal axis units are degrees


[edit] See also