Operational Theory

Piston Displacement Velocity, and Acceleration (page 3)

 
 

 

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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) + √{l2 - r2sin2(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)

 

 

crank angle

rpm

time

w rad/sec

wt

crank radius (r)

con rod length (l)

Mean Piston Speed

Velocity

0

120

0

12.56637

0

1

2.5

8

0

5

120

0.006944

12.56637

0.087266

1

2.5

8

1.531656836

10

120

0.013889

12.56637

0.174533

1

2.5

8

3.041717733

15

120

0.020833

12.56637

0.261799

1

2.5

8

4.509053104

20

120

0.027778

12.56637

0.349066

1

2.5

8

5.913453345

25

120

0.034722

12.56637

0.436332

1

2.5

8

7.236057381

30

120

0.041667

12.56637

0.523599

1

2.5

8

8.459744544

35

120

0.048611

12.56637

0.610865

1

2.5

8

9.569479222

40

120

0.055556

12.56637

0.698132

1

2.5

8

10.55259917

45

120

0.0625

12.56637

0.785398

1

2.5

8

11.39904

50

120

0.069444

12.56637

0.872665

1

2.5

8

12.10149022

55

120

0.076389

12.56637

0.959931

1

2.5

8

12.65547333

60

120

0.083333

12.56637

1.047198

1

2.5

8

13.05935542

65

120

0.090278

12.56637

1.134464

1

2.5

8

13.31427922

70

120

0.097222

12.56637

1.22173

1

2.5

8

13.4240272

75

120

0.104167

12.56637

1.308997

1

2.5

8

13.39481898

80

120

0.111111

12.56637

1.396263

1

2.5

8

13.23504958

85

120

0.118056

12.56637

1.48353

1

2.5

8

12.95497725

90

120

0.125

12.56637

1.570796

1

2.5

8

12.56637061

95

120

0.131944

12.56637

1.658063

1

2.5

8

12.08212631

100

120

0.138889

12.56637

1.745329

1

2.5

8

11.51586883

105

120

0.145833

12.56637

1.832596

1

2.5

8

10.88154486

110

120

0.152778

12.56637

1.919862

1

2.5

8

10.19302427

115

120

0.159722

12.56637

2.007129

1

2.5

8

9.463719867

120

120

0.166667

12.56637

2.094395

1

2.5

8

8.706236948

125

120

0.173611

12.56637

2.181662

1

2.5

8

7.932063031

130

120

0.180556

12.56637

2.268928

1

2.5

8

7.151306538

135

120

0.1875

12.56637

2.356194

1

2.5

8

6.372491753

140

120

0.194444

12.56637

2.443461

1

2.5

8

5.602415488

145

120

0.201389

12.56637

2.530727

1

2.5

8

4.846068928

150

120

0.208333

12.56637

2.617994

1

2.5

8

4.10662607

155

120

0.215278

12.56637

2.70526

1

2.5

8

3.38549803

160

120

0.222222

12.56637

2.792527

1

2.5

8

2.682450413

165

120

0.229167

12.56637

2.879793

1

2.5

8

1.995778981

170

120

0.236111

12.56637

2.96706

1

2.5

8

1.322536981

175

120

0.243056

12.56637

3.054326

1

2.5

8

0.658805893

180

120

0.25

12.56637

3.141593

1

2.5

8

0

 

 

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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