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| {{Unreferenced|date=December 2009}}
| | Friends contact her Felicidad and her spouse doesn't like it at all. To perform croquet is the pastime I will never stop doing. My occupation is a production and distribution officer and I'm doing pretty great monetarily. Years ago we moved to Kansas.<br><br>Also visit my web site [http://bayi.gmy.Com.tr/UserProfile/tabid/57/userId/6646/Default.aspx http://bayi.gmy.Com.tr/UserProfile/tabid/57/userId/6646/Default.aspx] |
| The motion of a non-offset [[piston]] connected to a [[Crank (mechanism)|crank]] through a [[connecting rod]] (as would be found in [[internal combustion engine]]s), can be expressed through several [[mathematical equation]]s. This article shows how these motion equations are derived, and shows an example graph.
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| ==Crankshaft geometry==
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| [[Image:Piston motion geometry.png|thumb|500px|right|Diagram showing geometric layout of piston pin, crank pin and crank center]]
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| ===Definitions===
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| ''l'' = [[Connecting rod|rod]] length (distance between [[Gudgeon pin|piston pin]] and [[crank pin]])<br> | |
| ''r'' = [[Crank (mechanism)|crank]] [[radius]] (distance between [[crank pin]] and crank center, i.e. half [[Stroke (engine)|stroke]])<br>
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| ''A'' = crank angle (from [[Cylinder (engine)|cylinder]] [[Bore (engine)|bore]] centerline at [[Dead centre|TDC]])<br>
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| ''x'' = piston pin position (upward from crank center along cylinder bore centerline)<br>
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| ''v'' = piston pin velocity (upward from crank center along cylinder bore centerline)<br>
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| ''a'' = piston pin acceleration (upward from crank center along cylinder bore centerline)<br>
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| ''ω'' = crank [[angular velocity]] in [[radians per second|rad/s]]
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| ===Angular velocity===
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| The [[crankshaft]] [[angular velocity]] is related to the engine [[revolutions per minute]] (RPM):
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| :<math>\omega= \frac{2\pi\cdot \mathrm{RPM}}{60} </math>
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| ===Triangle relation===
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| As shown in the diagram, the [[crank pin]], crank center and piston pin form triangle NOP.<br>
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| By the [[cosine law]] it is seen that:<br>
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| :<math> l^2 = r^2 + x^2 - 2\cdot r\cdot x\cdot\cos A </math>
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| ==Equations with respect to angular position (Angle Domain)==
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| The equations that follow describe the [[reciprocating motion]] of the piston with respect to crank angle.<br>
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| Example graphs of these equations are shown below.
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| ===Position===
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| Position with respect to crank angle (by rearranging the triangle relation):
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| :<math> l^2 - r^2 = x^2 - 2\cdot r\cdot x\cdot\cos A </math>
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| :<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>
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| :<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>
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| :<math> l^2 - r^2\sin^2 A = (x - r \cdot \cos A)^2</math>
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| :<math> x - r \cdot \cos A = \sqrt{l^2 - r^2\sin^2 A}</math>
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| :<math> x = r\cos A + \sqrt{l^2 - r^2\sin^2 A} </math>
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| ===Velocity===
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| Velocity with respect to crank angle (take first [[derivative]], using the [[chain rule]]):
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| :<math>
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| \begin{array}{lcl}
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| x' & = & \frac{dx}{dA} \\
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| & = & -r\sin A + \frac{(\frac{1}{2}).(-2). r^2 \sin A \cos A}{\sqrt{l^2-r^2\sin^2 A}} \\
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| & = & -r\sin A - \frac{r^2\sin A \cos A}{\sqrt{l^2-r^2\sin^2 A}}
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| \end{array}
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| </math>
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| ===Acceleration===
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| Acceleration with respect to crank angle (take second [[derivative]], using the [[chain rule]] and the [[quotient rule]]):
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| :<math>
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| \begin{array}{lcl}
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| x'' & = & \frac{d^2x}{dA^2} \\
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| & = & -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} \\
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| & = & -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}
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| \end{array}
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| </math>
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| ==Equations with respect to time (Time Domain)==
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| ===Angular velocity derivatives===
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| If angular velocity is constant, then
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| :<math>A = \omega t \,</math>
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| and the following relations apply:
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| :<math> \frac{dA}{dt} = \omega </math>
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| :<math> \frac{d^2 A}{dt^2} = 0 </math>
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| ===Converting from Angle Domain to Time Domain===
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| The equations that follow describe the [[reciprocating motion]] of the piston with respect to time.
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| If [[Time domain|time]] [[Domain (mathematics)|domain]] is required instead of angle domain, first replace A with ''ω''t in the equations, and then [[Scale factor|scale]] for angular velocity as follows:
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| ===Position===
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| Position with respect to time is simply:
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| :<math>x \,</math>
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| ===Velocity===
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| [[Velocity]] with respect to time (using the [[chain rule]]):
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| :<math>
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| \begin{array}{lcl}
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| v & = & \frac{dx}{dt} \\
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| & = & \frac{dx}{dA} \cdot \frac{dA}{dt} \\
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| & = & \frac{dx}{dA} \cdot\ \omega \\
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| & = & x' \cdot \omega \\
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| \end{array}
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| </math>
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| ===Acceleration===
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| [[Acceleration]] with respect to time (using the [[chain rule]] and [[product rule]], and the angular velocity [[Formal derivative|derivative]]s):
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| :<math>
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| \begin{array}{lcl}
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| a & = & \frac{d^2x}{dt^2} \\
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| & = & \frac{d}{dt} \frac{dx}{dt} \\
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| & = & \frac{d}{dt} (\frac{dx}{dA} \cdot \frac{dA}{dt}) \\
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| & = & \frac{d}{dt} (\frac{dx}{dA}) \cdot \frac{dA}{dt} + \frac{dx}{dA} \cdot \frac{d}{dt} (\frac{dA}{dt}) \\
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| & = & \frac{d}{dA} (\frac{dx}{dA}) \cdot (\frac{dA}{dt})^2 + \frac{dx}{dA} \cdot \frac{d^2A}{dt^2} \\
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| & = & \frac{d^2x}{dA^2} \cdot (\frac{dA}{dt})^2 + \frac{dx}{dA} \cdot \frac{d^2A}{dt^2} \\
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| & = & \frac{d^2x}{dA^2} \cdot \omega^2 + \frac{dx}{dA} \cdot 0 \\
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| & = & x'' \cdot \omega^2 \\
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| \end{array}
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| </math>
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| ===Scaling for angular velocity===
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| You can see that x is unscaled, x' is scaled by ''ω'', and x" is scaled by ''ω''². <br>
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| To convert x' from velocity vs angle [inch/rad] to velocity vs time [inch/s] multiply x' by ''ω'' [rad/s]. <br>
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| To convert x" from acceleration vs angle [inch/rad²] to acceleration vs time [inch/s²] multiply x" by ''ω''² [rad²/s²].<br>
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| ''Note that [[dimensional analysis]] shows that the [[Units of measurement|units]] are consistent.''
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| ==Velocity maxima/minima==
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| ===Acceleration zero crossings===
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| The velocity [[maxima and minima]] do ''not'' occur at crank angles ''(A)'' of plus or minus 90°. <br>
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| The velocity maxima and minima occur at crank angles that depend on rod length ''(l)'' and half stroke ''(r)'',<br>
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| and correspond to the crank angles where the acceleration is zero (crossing the horizontal axis).
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| ===Crank-Rod angle not right angled===
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| The velocity maxima and minima '''do not necessarily occur''' when the crank makes a right angle with the rod.<br>
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| Counter-examples exist to '''disprove''' the ''idea'' that velocity maxima/minima occur when crank-rod angle is right angled.
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| ===Example===
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| For rod length 6" and crank radius 2", numerically solving the acceleration zero-crossings finds the velocity maxima/minima to be at crank angles of ±73.17615°.<br>
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| Then, using the triangle sine law, it is found that the crank-rod angle is 88.21738° and the rod-vertical angle is 18.60647°.<br>
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| Clearly, in this example, the angle between the crank and the rod is not a right angle.
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| (Sanity check, summing the angles of the triangle 88.21738° + 18.60647° + 73.17615° gives 180.00000°)
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| A single counter-example is sufficient to '''disprove''' the statement ''"velocity maxima/minima occur when crank makes a right angle with rod"''.
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| ==Example graph of piston motion==
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| The graph shows x, x', x" with respect to crank angle for various half strokes, where L = rod length ''(l)'' and R = half stroke ''(r)'':
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| [[Image:Graph of Piston Motion.png|thumb|800px|left|The vertical axis units are [[inches]] for position, [inches/rad] for velocity, [inches/rad²] for acceleration.<br>
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| The horizontal axis units are crank angle [[degree (angle)|degree]]s.]]
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| {{Clr}}
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| Pistons motion animation with same rod length and crank radius values in graph above :<br>
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| [[Image:TRUE piston3 ANI.gif|left|Pistons motion animation with various half strokes|frame]]
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| {{Clr}}
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| ==See also==
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| * [[Internal combustion engine]]
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| * [[Reciprocating engine]]
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| * [[Scotch yoke]]
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| == References ==
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| * http://www.epi-eng.com/piston_engine_technology/piston_motion_basics.htm
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| ==Further reading==
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| * John Benjamin Heywood, ''Internal Combustion Engine Fundamentals'', McGraw Hill, 1989.
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| * Charles Fayette Taylor, ''The Internal Combustion Engine in Theory and Practice, Vol. 1 & 2'', 2nd Edition, MIT Press 1985.
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| ==External links==
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| * [http://www.codecogs.com/reference/engineering/dynamics/velocity_and_acceleration_of_a_piston.php codecogs] Velocity and Acceleration of a Piston
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| * http://www.animatedengines.com/otto.shtml
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| * [http://www.youtube.com/watch?v=stuaK5bk_Ck youtube] Rotating chevy 350 short block.
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| * [http://www.youtube.com/watch?v=lMIxKOM6jRA youtube] 3D animation of a V8 ENGINE
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| * [http://www.youtube.com/watch?v=sLQWEnQmmyY youtube] Inside a V8 Engine at Idle Speed
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| [[Category:Pistons|Motion equations]]
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| [[Category:Engine technology]]
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| [[Category:Mechanical engineering]]
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| [[Category:Equations]]
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