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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