Piston motion equations

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

Acceleration

Acceleration wrt crank angle (take second derivative, using the chain rule and the quotient rule)::egin{array}{lcl} x" & = & frac{d^2x}{dA^2} \ & = & -rcos A - frac{r^2cos^2 A}{sqrt{l^2-r^2sin^2 A-frac{-r^2sin^2 A}{sqrt{l^2-r^2sin^2 A - frac{r^2sin A cos A .(-frac{1}{2})cdot(-2).r^2sin Acos A}{left (sqrt{l^2-r^2sin^2 A} ight )^3} \ & = & -rcos A - frac{r^2(cos^2 A -sin^2 A)}{sqrt{l^2-r^2sin^2 A-frac{r^4sin^2 A cos^2 A}{left (sqrt{l^2-r^2 sin^2 A} ight )^3}end{array}

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

Acceleration

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

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


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Perpetual motion — For other uses, see Perpetual motion (disambiguation). Robert Fludd s 1618 water screw perpetual motion machine from a 1660 wood engraving. This device is widely credited as the first recorded attempt to describe such a device in order to produce …   Wikipedia

  • Crankshaft — For other uses, see Crankshaft (disambiguation). Crankshaft (red), pistons (gray) in their cylinders (blue), and flywheel (black) The crankshaft, sometimes casually abbreviated to crank, is the part of an engine which translates reciprocating… …   Wikipedia

  • Crank (mechanism) — A crank is an arm attached at right angles to a rotating shaft by which reciprocating motion is imparted to or received from the shaft. It is used to change circular into reciprocating motion, or reciprocating into circular motion. The arm may be …   Wikipedia

  • Kinematics — Classical mechanics Newton s Second Law History of classical mechanics  …   Wikipedia

  • Timeline of Islamic science and engineering — This timeline of Islamic science and engineering covers the general development of science and technology in the Islamic world during the Islamic Golden Age, usually dated from the 7th to 16th centuries.From the 17th century onwards, the advances …   Wikipedia

  • Timeline of historic inventions — The timeline of historic inventions is a chronological list of particularly important or significant technological inventions. Note: Dates for inventions are often controversial. Inventions are often invented by several inventors around the same… …   Wikipedia

  • thermodynamics — thermodynamicist, n. /therr moh duy nam iks/, n. (used with a sing. v.) the science concerned with the relations between heat and mechanical energy or work, and the conversion of one into the other: modern thermodynamics deals with the properties …   Universalium

  • Thermodynamics — Annotated color version of the original 1824 Carnot heat engine showing the hot body (boiler), working body (system, steam), and cold body (water), the letters labeled according to the stopping points in Carnot cycle …   Wikipedia

  • physical science, principles of — Introduction       the procedures and concepts employed by those who study the inorganic world.        physical science, like all the natural sciences, is concerned with describing and relating to one another those experiences of the surrounding… …   Universalium

  • Otto cycle — See also: Otto engine and Four stroke engine Thermodynamics …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”