Piston motion equations
From Wikipedia, the free encyclopedia
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
[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)
w = crank angular velocity in rad/s
[edit] Angular velocity
Angular velocity is related to the engine revolutions per minute (RPM):
- <math>w=\left ( \frac{2\pi\cdot rpm}{60} \right ) </math>
If angular velocity is constant, the following relations apply:
- <math>A = wt</math> <math> \frac{dA}{dt} = w </math> <math> \frac{d^2 A}{dt^2} = 0 </math>
[edit] Triangle relation
The sides of the triangle formed by the crank pin, crank center and piston pin (NOP) have the following relation with each other:
- <math> l^2 = r^2 + x^2 - 2\cdot r\cdot x\cdot\cos A </math>
[edit] Equations wrt angular position
Position wrt crank angle (rearrange triangle relation):
- <math> l^2 - r^2 = x^2 - 2\cdot r\cdot x\cdot\cos A </math>
- <math> l^2 - r^2 = x^2 - 2\cdot r\cdot x\cdot\cos A + r^2[(\cos^2 A + \sin^2 A) - 1]</math>
- <math> 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</math>
- <math> l^2 - r^2\sin^2 A = (x - r \cdot \cos A)^2</math>
- <math> \sqrt{l^2 - r^2\sin^2 A} = x - r \cdot \cos A</math>
- <math> x = r\cos A + \sqrt{l^2 - r^2\sin^2 A} </math>
Velocity wrt crank angle (take first derivative):
- <math>
\begin{array}{lcl}
x' & = & \frac{dx}{dA} \\
& = & -r\sin A + \frac{\frac{1}{2}r^2 \sin 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} </math>
Acceleration wrt crank angle (take second derivative):
- <math>
\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-2r^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^2)^2\sin^2 A \cos^2 A}{\left (\sqrt{l^2-r^2 \sin^2 A}\right )^3}
\end{array} </math>
Example graphs of these equations are shown below.
[edit] Equations wrt time
If the angular velocity w is constant then:
- <math>A = wt</math> and <math> \frac{d^2 A}{dt^2} = 0 </math>
If time domain is required instead of angle domain, first replace A with wt in the equations; and then scale for angular velocity as follows:
Position wrt time is simply:
- <math>x</math>
Velocity wrt time (using the chain rule):
- <math>
\begin{array}{lcl}
v & = & \frac{dx}{dt} \\
& = & \frac{dx}{dA}\cdot\frac{dA}{dt}\\
& = & \frac{dx}{dA}\cdot\ w\\
& = & x'\cdot w\\
\end{array} </math>
Acceleration wrt time (using the chain rule and product rule):
- <math>
\begin{array}{lcl}
v & = & \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} \frac{dA}{dt} + \frac{dx}{dA} \frac{d}{dt} \frac{dA}{dt}\\
& = & \frac{d}{dA} \frac{dx}{dA} (\frac{dA}{dt})^2 + \frac{dx}{dA} \frac{d^2A}{dt^2} \\
& = & \frac{d^2x}{dA^2} (\frac{dA}{dt})^2 + \frac{dx}{dA} \frac{d^2A}{dt^2} \\
& = & \frac{d^2x}{dA^2} \cdot w^2 \\
& = & x \cdot w^2 \\
\end{array} </math>
You can see that x is unscaled, x' is scaled by w, and x" is scaled by w². To convert x' from velocity vs angle [in/rad] to velocity vs time [in/s] multiply x' by w [rad/s]. To convert x" from acceleration vs angle [in/rad²] to acceleration vs time [in/s²] multiply x" by w² [rad²/s²].
[edit] Velocity maxima
The velocity maxima (positive and negative) do not occur at +/-90°, they occur at the acceleration zero crossings which are not at +/-90°. The 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 (L = rod length (l), R = half stroke (r)):
