Piston Velocity
It was seen on page one that the AVERAGE piston speed can be obtained from the simple formula, 2nL
To obtain the instantaneous velocity at any crank angle then use must be made of differential calculus.
This will show, that contrary to common belief, the piston is not moving with maximum velocity at 90° ATDC.
From the previous page it was shown that the distance moved by the piston at any crank angle, wt, is given by
s = (l + r)  [rcos(wt) + √{l^{2}  r^{2}sin^{2}(wt)}]
If the first differential, ds/dt of the equation is found, this gives an expression for the instantaneous velocity, v.
This equation, luckily, can be simplified to:
If you are interested in finding how these equations were obtained, CLICK HERE
The information can be entered on a spreadsheet and the velocity for 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)
It can be seen that maximum velocity is reached when the crank is at 71° past TDC and again at 71° before TDC If the Con rod : Crank radius ratio is increased then the point of maximum velocity moves closer to 90° past TDC. and the curve gets closer to a pure sine wave form. The smaller the ratio (i.e. the shorter the con rod, the higher the maximum velocity of the piston)
What must be remembered is that the negative values on the graph are showing the piston is travelling back up the cylinder. The velocity at a maximum is still approximately 13.4m/s, even though the graph shows it as a negative value (indicating reverse direction). This must also be borne in mind when looking at the graph for acceleration on the next page.
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.
Velocity/time gives acceleration: So if velocity is differentiated with respect to time then the instantaneous acceleration of the piston can be calculated for any crank angle and engine speed. This is shown on the NEXT PAGE.

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