Operational Theory

Piston Displacement Velocity, and Acceleration (page 2)

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Piston Displacement. (distance moved by piston from TDC) Using the combination of the Sine and Cosine rules, the piston displacement (s) at any crank angle (q )can be found.

x = √(l2 + r2 - 2lrcosa)

To find angle a, use the sine rule to find angle b, and then use  the 180 rule.

Then s = (l + r) - x

This however is rather long winded and will not help later to find the velocity of the piston at any point.

If  the cosine rule is written out for x using angle q,  we get:

x = rcosq  + √(l2 - r2 + 2xrcosq )  which simplifies to   x = rcosq  + √(l2 - r2sin2q )

To see how this is done using the quadratic equation formula CLICK HERE

Instead of using angles in degrees, so that  the velocity at any crank angle can be found, the crank angle in degrees is converted into radians with respect to time (t) (in seconds).

For example, if an engine is rotating at 120 rpm it is rotating at 2 revs /sec. There are 2p radians in 360°, so in one second, the crank will have rotated 4p radians. This rotational speed in radians per second is given the notation w, so in this particular case the engine speed w = 4p rad/sec. Taking TDC as a starting point, to find the crank angle at any time after TDC, multiply w by t.

For example at 0.125seconds after TDC crank angle = 4p × 0.125 = 0.5p radians (about 1.570796 radians)

so the formula becomes x = rcos(wt) + √{l2 - r2sin2(wt)}

and the distance (s) moved by the piston = (l + r) - [rcos(wt) + √{l2 - r2sin2(wt)}]

The information can be entered on a spreadsheet and the piston position for any crank angle, conrod length: crank radius ratio calculated. Below is an extract from a spreadsheet for the example previously discussed.

 crank angle rpm time t w rad/sec wt crank radius (r) con rod length (l) s 0 120 0 12.56637 0 1 2.5 0 5 120 0.006944 12.56637 0.087266 1 2.5 0.005324988 10 120 0.013889 12.56637 0.174533 1 2.5 0.021230276 15 120 0.020833 12.56637 0.261799 1 2.5 0.047507725 20 120 0.027778 12.56637 0.349066 1 2.5 0.083813442 25 120 0.034722 12.56637 0.436332 1 2.5 0.129672366 30 120 0.041667 12.56637 0.523599 1 2.5 0.184484853 35 120 0.048611 12.56637 0.610865 1 2.5 0.247535384 40 120 0.055556 12.56637 0.698132 1 2.5 0.318003552 45 120 0.0625 12.56637 0.785398 1 2.5 0.394977457 50 120 0.069444 12.56637 0.872665 1 2.5 0.477469566 55 120 0.076389 12.56637 0.959931 1 2.5 0.56443501 60 120 0.083333 12.56637 1.047198 1 2.5 0.65479212 65 120 0.090278 12.56637 1.134464 1 2.5 0.747444787 70 120 0.097222 12.56637 1.22173 1 2.5 0.841305996 75 120 0.104167 12.56637 1.308997 1 2.5 0.935321614 80 120 0.111111 12.56637 1.396263 1 2.5 1.028493322 85 120 0.118056 12.56637 1.48353 1 2.5 1.119899399 90 120 0.125 12.56637 1.570796 1 2.5 1.208712153 95 120 0.131944 12.56637 1.658063 1 2.5 1.294210884 100 120 0.138889 12.56637 1.745329 1 2.5 1.375789677 105 120 0.145833 12.56637 1.832596 1 2.5 1.452959705 110 120 0.152778 12.56637 1.919862 1 2.5 1.525346282 115 120 0.159722 12.56637 2.007129 1 2.5 1.592681311 120 120 0.166667 12.56637 2.094395 1 2.5 1.65479212 125 120 0.173611 12.56637 2.181662 1 2.5 1.711587883 130 120 0.180556 12.56637 2.268928 1 2.5 1.763044785 135 120 0.1875 12.56637 2.356194 1 2.5 1.80919102 140 120 0.194444 12.56637 2.443461 1 2.5 1.850092438 145 120 0.201389 12.56637 2.530727 1 2.5 1.885839473 150 120 0.208333 12.56637 2.617994 1 2.5 1.916535661 155 120 0.215278 12.56637 2.70526 1 2.5 1.94228794 160 120 0.222222 12.56637 2.792527 1 2.5 1.963198683 165 120 0.229167 12.56637 2.879793 1 2.5 1.979359378 170 120 0.236111 12.56637 2.96706 1 2.5 1.990845782 175 120 0.243056 12.56637 3.054326 1 2.5 1.997714385 180 120 0.25 12.56637 3.141593 1 2.5 2

The graph of crank angle against piston displacement can also be plotted: From both graph and table it can be seen that the piston has moved half the stroke at 79° and has moved about 1.21m at 90°

If the Con Rod : Crank radius ratio is increased (i.e. making the con rod longer, then mid stroke moves towards 90° crank angle)

The instantaneous velocity at any crank angle can also be calculated.

This is shown on the NEXT PAGE.

If you are a subscriber to marinediesels, then the spreadsheet can be downloaded HERE and different values entered for engine speed, conrod length and crank radius, and the effects noted.

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