Equations of Piston Motion
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The motion 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.
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[edit] Geometry
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) w = crank angular velocity in rad/s
[edit] Angular Velocity
Angular velocity is related to engine speed (RPM) as follows (see revolutions per minute):
w = 2.pi.RPM/60
If angular velocity is constant, the following relations apply:
A = wt
dA/dt = w
d²A/dt² = 0
[edit] Triangle Relation
Triangle NOP shown above has the following relation:
l² = r² + x² - 2.r.x.cos(A)
[edit] Equations wrt angular position
Position wrt crank angle (rearrange triangle relation):
x = r.cos(A) + sqrt(l² - r².sin²(A))
Velocity wrt crank angle (take first derivative):
x' = dx/dA
= -r.sin(A) + (1/2).(-2.r².sin(A).cos(A))/sqrt(l² - r².sin²(A))
= -r.sin(A) - r².sin(A).cos(A)/sqrt(l² - r².sin²(A))
Acceleration wrt crank angle (take second derivative):
x" = d²x/dA²
= -r.cos(A) - r².cos²(A)/sqrt(l² - r².sin²(A)) - (-1).r².sin²(A)/sqrt(l² - r².sin²(A)) - r².sin(A).cos(A).(-1/2).(-2.r².sin(A).cos(A))/sqrt(l² - r².sin²(A))³
= -r.cos(A) - r².(cos²(A) - sin²(A))/sqrt(l² - r².sin²(A)) - (r²)².sin²(A).cos²(A)/sqrt(l² - r².sin²(A))³
Example graphs of these equations are shown below.
[edit] Equations wrt time
Let w be constant, then A = wt and d²A/dt² = 0.
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:
x
velocity wrt time (using the chain rule):
v = dx/dt = dx/dA.dA/dt = dx/dA.w = x'.w
acceleration wrt time (using the chain rule and product rule):
a = d²x/dt² = d/dt(dx/dt) = d/dt(dx/dA.dA/dt) = d/dt(dx/dA).dA/dt + dx/dA.d/dt(dA/dt) = d/dA(dx/dA).(dA/dt)² + dx/dA.d²A/dt² = d²x/dA².(dA/dt)² + dx/dA.d²A/dt² = d²x/dA².w² = x".w²
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 Graphs
Graph showing x, x', x" wrt to crank angle for various half strokes (L = rod length (l), 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].
