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)
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.