Piston motion equations

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Piston motion in engines is driven by a rotating crank connected to a piston through a rod. The piston’s position can be described in two ways: as a function of the crank angle A, and as a function of time t.

Geometry and variables
- The crank radius is r and the connecting rod length is l (constants).
- The piston position is x, measured along the cylinder.
- The crank angle A is the angle turned by the crank.
- The crank center, crank pin, and piston pin form a triangle; from this geometry, x(A) is determined.

Angle-domain equations
- The position x is written as a function of the crank angle x(A).
- The corresponding velocity and acceleration with respect to angle are x′(A) = dx/dA and x″(A) = d^2x/dA^2.
- These relations are obtained by differentiating the geometry relations (triangle/Pythagorean relations) with respect to A.
- Because the rod links the piston to the rotating crank, the motion is not simple harmonic. The rod’s swing modifies the motion from a pure SHM (unlike a Scotch Yoke, which gives SHM).

Time-domain equations
- The crank rotates at angular velocity ω. If ω is constant, ω = 2πN/60, where N is the engine speed in revolutions per minute.
- The angle and time relate by A = ω t. Then the time-domain motion follows from the angle-domain equations:
- Position: x(t) = x(A) with A = ω t
- Velocity: ẋ(t) = ω x′(A)
- Acceleration: ẍ(t) = ω^2 x″(A)
- In other words, the time-domain equations are simply scaled versions of the angle-domain ones: x is unchanged in shape, but velocities scale with ω and accelerations with ω^2.

Key points about maxima and minima
- The speed extrema (velocity maxima/minima) occur where the acceleration crosses zero, not necessarily when the crank is at a right angle to the rod.
- For example, with l = 6 inches and r = 2 inches, the acceleration zeros (where velocity is extremal) occur at crank angles A ≈ ±73.18 degrees. At those instants, the corresponding angles in the linkage are not right angles: the rod-vertical angle is about 18.61 degrees and the crank-rod angle about 88.22 degrees. The sum of these angles is 180 degrees, showing the crank and rod are not at a right angle when velocity extrema occur.

Graphs and typical values
- The angle-domain curves x(A), x′(A), and x″(A) can be plotted for a fixed rod length and different half-stroke values r (the crank radius).
- In hobbyist auto/hotrod work, inches are a convenient unit. Typical dimensions are a 6-inch rod length and a 2-inch crank radius. The same equations can be plotted in these units (L for rod length, R for half-stroke).

Summary
- Piston motion is described both in angle domain and time domain.
- The angle-domain relations come from the crank-rod-piston geometry; the time-domain relations come from relating angle to time through ω.
- The motion is not pure SHM; the rod’s angle influences the motion.
- Velocity extrema depend on l and r and do not require a right-angle crank-rod configuration.