Operational Theory

Piston Displacement Velocity, and Acceleration (page 4)

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Piston Acceleration

The acceleration can be found by differentiating piston velocity, v with respect to time, t

To be 100% accurate, the equation:

should be differentiated.

This can be achieved, but because of the rather complex denominator, the differential is fairly horrendous.

By using the simplified equation:

to find the velocity, the maximum error is less than 1%.

The differential of this equation is far simpler, being:

The information can be entered on a spreadsheet and the acceleration at any crank angle, conrod length: crank radius ratio calculated. Below is an extract from a spreadsheet for the example previously discussed. (120rpm, conrod l = 2.5m, crank radius r = 1m)

 crank angle rpm time w rad/sec wt crank radius (r) con rod length (l) Mean Piston Speed Acceleration 0 120 0 12.56637 0 1 2.5 8 221.0791386 5 120 0.006944 12.56637 0.087266 1 2.5 8 219.518604 10 120 0.013889 12.56637 0.174533 1 2.5 8 214.8707313 15 120 0.020833 12.56637 0.261799 1 2.5 8 207.2357927 20 120 0.027778 12.56637 0.349066 1 2.5 8 196.7778667 25 120 0.034722 12.56637 0.436332 1 2.5 8 183.7203695 30 120 0.041667 12.56637 0.523599 1 2.5 8 168.3399843 35 120 0.048611 12.56637 0.610865 1 2.5 8 150.9591684 40 120 0.055556 12.56637 0.698132 1 2.5 8 131.9374582 45 120 0.0625 12.56637 0.785398 1 2.5 8 111.6618272 50 120 0.069444 12.56637 0.872665 1 2.5 8 90.53638231 55 120 0.076389 12.56637 0.959931 1 2.5 8 68.97169785 60 120 0.083333 12.56637 1.047198 1 2.5 8 47.37410113 65 120 0.090278 12.56637 1.134464 1 2.5 8 26.1352206 70 120 0.097222 12.56637 1.22173 1 2.5 8 5.622100303 75 120 0.104167 12.56637 1.308997 1 2.5 8 -13.8318347 80 120 0.111111 12.56637 1.396263 1 2.5 8 -31.9347032 85 120 0.118056 12.56637 1.48353 1 2.5 8 -48.4427595 90 120 0.125 12.56637 1.570796 1 2.5 8 -63.1654682 95 120 0.131944 12.56637 1.658063 1 2.5 8 -75.968926 100 120 0.138889 12.56637 1.745329 1 2.5 8 -86.7775454 105 120 0.145833 12.56637 1.832596 1 2.5 8 -95.5739655 110 120 0.152778 12.56637 1.919862 1 2.5 8 -102.397212 115 120 0.159722 12.56637 2.007129 1 2.5 8 -107.339181 120 120 0.166667 12.56637 2.094395 1 2.5 8 -110.539569 125 120 0.173611 12.56637 2.181662 1 2.5 8 -112.179423 130 120 0.180556 12.56637 2.268928 1 2.5 8 -112.473519 135 120 0.1875 12.56637 2.356194 1 2.5 8 -111.661827 140 120 0.194444 12.56637 2.443461 1 2.5 8 -110.000321 145 120 0.201389 12.56637 2.530727 1 2.5 8 -107.751443 150 120 0.208333 12.56637 2.617994 1 2.5 8 -105.174516 155 120 0.215278 12.56637 2.70526 1 2.5 8 -102.516409 160 120 0.222222 12.56637 2.792527 1 2.5 8 -100.002755 165 120 0.229167 12.56637 2.879793 1 2.5 8 -97.8299925 170 120 0.236111 12.56637 2.96706 1 2.5 8 -96.1584826 175 120 0.243056 12.56637 3.054326 1 2.5 8 -95.1069185 180 120 0.25 12.56637 3.141593 1 2.5 8 -94.7482023

When studying the graph, bear in mind that between 180° and 360°, as the piston is moving up the cylinder, the values are reversed, i.e. negative values show acceleration, positive values show deceleration.

The piston is accelerating fastest as it comes over TDC (221m/s2). It is still accelerating, although the rate of acceleration is dropping until it reaches 0m/s2 when the velocity is at a maximum (In this case 71° past TDC). The piston then decelerates until it reaches BDC, when it stops, changes direction and accelerates back up the cylinder.

The rate of acceleration drops until it reaches 71° before TDC when it then decelerates until the rate of deceleration reaches 221m/s2 as it reaches TDC.

What may seem at first as an anomaly is the "bump" in the curve 50° either side of BDC. The acceleration/deceleration is not at a maximum value at 180° but at about 129° and 231°. This is a function of the con rod: crank radius ratio. If this is increased, the curve becomes sinusoidal. Below is an acceleration graph for a ratio of 6:1 which is not practical, but shows the theory.

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

Why is the ratio between con rod length and crank radius (and therefore piston stroke) so important to engine designers?

Engine size. The longer the conrod the taller the engine.

When the fuel burns it happens over a period of time, not an instantaneous detonation. As it burns and tries to expand, the pressure increases. The larger the con rod:crank radius ratio, the slower the piston is moving down the cylinder close to TDC, the higher the gas load on the piston, and thus loading on the bearings.

If a 2 stroke crosshead engine is considered in which for example, the exhaust valve opens and closes 60° either side of BDC (120° and 240°) and if the engine is doing 120 rpm, then the exhaust valve is open for 0.167 seconds. The smaller the con rod:crank radius ratio the further the piston will have moved down the cylinder, and therefore the more work done before the exhaust valve opens.

Similarly, the greater the compression ratio for a given exhaust valve closing crank angle.

With a four stroke engine as the con rod:crank radius ratio gets smaller, the point of maximum piston velocity moves closer to TDC, and the piston moves away from TDC faster, creating a stronger intake pulse. The location of maximum piston velocity also influences the design of the inlet cam profile.

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