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.
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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