Operational Theory

Piston Displacement Velocity, and Acceleration (page 2)

 
 

 

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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.

 

 

 

crank angle

rpm

time t

w rad/sec

wt

crank radius (r)

con rod length (l)

s

0

120

0

12.56637

0

1

2.5

0

5

120

0.006944

12.56637

0.087266

1

2.5

0.005324988

10

120

0.013889

12.56637

0.174533

1

2.5

0.021230276

15

120

0.020833

12.56637

0.261799

1

2.5

0.047507725

20

120

0.027778

12.56637

0.349066

1

2.5

0.083813442

25

120

0.034722

12.56637

0.436332

1

2.5

0.129672366

30

120

0.041667

12.56637

0.523599

1

2.5

0.184484853

35

120

0.048611

12.56637

0.610865

1

2.5

0.247535384

40

120

0.055556

12.56637

0.698132

1

2.5

0.318003552

45

120

0.0625

12.56637

0.785398

1

2.5

0.394977457

50

120

0.069444

12.56637

0.872665

1

2.5

0.477469566

55

120

0.076389

12.56637

0.959931

1

2.5

0.56443501

60

120

0.083333

12.56637

1.047198

1

2.5

0.65479212

65

120

0.090278

12.56637

1.134464

1

2.5

0.747444787

70

120

0.097222

12.56637

1.22173

1

2.5

0.841305996

75

120

0.104167

12.56637

1.308997

1

2.5

0.935321614

80

120

0.111111

12.56637

1.396263

1

2.5

1.028493322

85

120

0.118056

12.56637

1.48353

1

2.5

1.119899399

90

120

0.125

12.56637

1.570796

1

2.5

1.208712153

95

120

0.131944

12.56637

1.658063

1

2.5

1.294210884

100

120

0.138889

12.56637

1.745329

1

2.5

1.375789677

105

120

0.145833

12.56637

1.832596

1

2.5

1.452959705

110

120

0.152778

12.56637

1.919862

1

2.5

1.525346282

115

120

0.159722

12.56637

2.007129

1

2.5

1.592681311

120

120

0.166667

12.56637

2.094395

1

2.5

1.65479212

125

120

0.173611

12.56637

2.181662

1

2.5

1.711587883

130

120

0.180556

12.56637

2.268928

1

2.5

1.763044785

135

120

0.1875

12.56637

2.356194

1

2.5

1.80919102

140

120

0.194444

12.56637

2.443461

1

2.5

1.850092438

145

120

0.201389

12.56637

2.530727

1

2.5

1.885839473

150

120

0.208333

12.56637

2.617994

1

2.5

1.916535661

155

120

0.215278

12.56637

2.70526

1

2.5

1.94228794

160

120

0.222222

12.56637

2.792527

1

2.5

1.963198683

165

120

0.229167

12.56637

2.879793

1

2.5

1.979359378

170

120

0.236111

12.56637

2.96706

1

2.5

1.990845782

175

120

0.243056

12.56637

3.054326

1

2.5

1.997714385

180

120

0.25

12.56637

3.141593

1

2.5

2

 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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