Rolling moment due to rate of role Introduction When an aircraft to roll, a resistance to the rolling is generated by the wing, the dimensionless

rolling moment is an important parameter of the lateral stability of an aircraft and it is equal to

the roll rate of the wing. This rolling moment is important since it contribute to the aircraft stability.

The can be estimated theoretically (simple strip theory, modified strip theory and lifting line

theory) for elliptical and tapered wings.

Derivation for theoretical estimate for of straight tapered wing

Simple strip theory

The rolling moment is defined by

=1

2

2 2 2

=

1

2

2 =1

2 (2)

2

=

(2)2/ 2

Modified strip theory

It can be adjusted to:

=

14.30{1+

}

Lifting line theory

=

14.30{1+2

}

Apparatus

A straight tapered wing of moderate AR mounted on a freely rotating shaft in the working section of

open wind tunnel.

A pan to carry the weight attached to a bobbin by a rope.

A three kilograms

Procedure 1 The atmospheric pressure and the temperature were recorded.

2 The span of the tapered wing was measured.

3 The distance travelled by the weight pan after ten revolutions was measured in order to estimate

the effective radius of the bobbin, in which the chord is wounded

4 The chord rewind by turning the lever handle clockwise

5 The weight pan was released from the rest and the wind tunnel reference pressure indicated

10.5mmH2O for the first set of the experience.

6 An increasing mass by 0.5 Kg increment up to 3 Kg clockwise and anticlockwise.

7 The rolling moment were recorded for each mass increment.

8 the steps 5, 6 and 7 were repeated for higher values of tunnel reference pressure 12.6 and

15mmH2O.

Results

Atmospheric pressure, P= 762.4 mmHg.

Atmospheric temperature, T = 21C.

Span=0.502m

Tip chord=0.06m

Root chord=0.125m

Wing surface=0.04642

Bobbin radius=0.0102m

For 10.5mmH2O

Mass (kg) Time period Clockwise(s)

Time period Anticlockwise(s)

Impulse, L/U (Ns)

Roll rate , P (rad/s) clockwise

Roll rate, P (rad/s) anticlockwise

Freestream Velocity

V(

)

0 31.01 42.38 0 2.026180363

-1.482582659

14.54488273 0.5 15.36 17.86 0.0037669

23 4.0906154

34 -

3.518020889

1 9.02 10.31 0.007533846

6.965837369

-6.09426315

1.5 7.03 7.21 0.011300769

8.93767469

-8.714542728

2 5.38 5.39 0.015067692

11.6787831

-11.6571156

2.5 3.83 3.9 0.018834615

16.40518357

- 16.11073156

For 12.5mmH2OO

Mass (kg) Time period Clockwise

(s)

Time period Anticlockwise

(s)

Impulse,

L/U (Ns)

Roll rate , P (rad/s) clockwise

Roll rate, P (rad/s) anticlockwise

Freestream Velocity

V(

)

13.27760061

0 31.28 44.29 0 2.0086909

55

-

1.4186464

91

0.5 15.92 18.19 0.0034387

15

3.9467244

39

-

3.4541975

3

1 10.2 11.32 0.0068774

29

6.1599855

95

-

5.5505170

56

1.5 7.19 8.18 0.0103161

44

8.7387834

59

-

7.6811556

32

2 6.13 6.01 0.0137548

58

10.249894

47

-

10.454551

26

2.5 4.95 5.19 0.0171935

73

12.693303

65

-

12.106330

07

Table 2

For 15mmH2O

Mass (kg) Time period Clockwise(s)

Time period Anticlockwise(s)

Impulse,

L/U (Ns)

Roll rate , P (rad/s) clockwise

- Roll rate, P (rad/s)

anticlockwise

Freestream Velocity

V(

)

15.86976812

0 32.27 46.38 0 1.94706703 -

1.354718695

0.5 16.11 18.97 0.003151634 3.900177099 -

3.312169376

1 11.08 12.36 0.006303268 5.670744862 -

5.083483258

1.5 8.09 8.97 0.009454902 7.766607302 -

7.004665894

2 6.2 7.17 0.012606536 10.13416985 -

8.763159424

2.5 5.16 5.93 0.01575817 12.17671571 -

10.59559074

Table 3

Gradient of the linear portion L vs p

graph

Experimental

Speed Clockwise Anticlockwise

clockwise

anticlockwise mmH2O

10 0.0015 -0.0014 0.00704006 0.213066356 -

0.198861932

12 0.0014 -0.0016 0.00704006 0.198861932 -0.22727078

14 0.0014 -0.0017 0.00704006 0.198861932 -

0.241475203

=

0.3787846

Table 4

Figure 1 shows the rolling moment vs roll rate clockwise and anticklocwise

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

-20 -15 -10 -5 0 5 10 15 20

L/ N

m

ROLL RATE (P)/RAD S-1

Applied rolling moment Vs roll rate 10.5mmH2O

clockwise

anticlockwise

Figure 2 shows the rolling moment vs roll rate clockwise and anticklocwise

Figure 3shows the rolling moment vs roll rate clockwise and anticklocwise

Calculation of chordial Reynolds number Reynolds number is given by the following equation

=

=2

=0.046435

0.502=0.0925

For 10.5mmH2O correspend for 13.27m/s

=1.20313.27760.0925

1.813105 =81490.9

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

-15 -10 -5 0 5 10

L/ N

m

ROLL RATE (P)/RAD S-1

Applied rolling moment Vs roll rate 12.5mmH2O

clockwise

anticlockwise

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

-15 -10 -5 0 5 10 15

L/ N

m

ROLL RATE (P)/RAD S-1

Applied rolling moment Vs roll rate 15mmH2O

clockwise

anticlockwise

For 12.5mmH2O corrspend for 14.54488273

=1.20314.544880.0925

1.813105 =89272.9

For 15mmH2O corrspend for 15.869768

=1.20315.8697680.0925

1.813105 =97404.332

Calculation angle of attack

= tan1

Where is the rate of roll at stalling and can be approximated from the graph of Applied rolling

moment against roll rate at the point where the curve diverges from the linear portion and y is the

distance from OX axis.

For 13.277m/s p=0.213

= 17.22 For clockwise and = 17.44 for anticlockwise

For 14.544m/s p=0.19886

= 12.355 For clockwise and = 11.97 for anticlockwise.

For 15.869m/s p=0.19886

= 10.9 for clockwise and = 9.6 for anticlockwise.

Calculating theoretical values of Lp

Strip theory

=(

L

p)p0

1

2(2)2

= 16

The strip theory is applied for two values of, 5.7 1 and 21

For 5.7 1

= 0.356

For 21

= 0.3927

Modified strip theory

=

16{1+

} And applied for two values, 5.7 1 and 21

=(2)2

=

0.252

0.0464=4.74

For 5.7 1 Lp=-0.267

21 = 0.287

Lifting line theory

=

16{1+2

} And applied for two values, 5.7 1 and 21

5.7 1

= 0.201

= 0.215

For straight tapered wing

Strip theory

For 5.7 1 = 0.3986

21 = 0.439

Modified strip theory

For 5.7 1 = 0.299

21 = 0.3215

Lifting line theory

For 5.7 1 = 0.2394

21 = 0.25354 .

Error calculations

The percentage error can be calculated as follow:

100

The average experimental values of = 0.3787846

The percentage error for the elliptical wing and the experimental results

1 Strip theory Modified strip theory Lifting line theory

5.7 20.6% 24.16% 27.702%

2 19.4% 23.18% 26.71%

Table 5

The percentage error for the tapered wings and the experimental results

1 Strip theory Modified strip theory Lifting line theory

5.7 19.50% 22.66% 25.82%

2 18.62% 21.78% 24.93%

Table 6

Discussion:

The figure 1,2 and 3 show that when the air speed increase the rolling moment increase and this

could be obvious because the wing has larger rolling moment to resist and also it was notice that as

the air speed increased, the linear portion of the curve became larger, which means that the angle of

attack, at which the wing stall, increased. The stall, which occurs at lower angle of attack for lower

speed suggest that the drag affect the roll rate.

The figure 1,2and 3 show also that the linear part of the clockwise part is larger than the one for

anticlockwise, which suggest less effect for the rolling moment at the anticlockwise to resist the roll.

The experimental values of and the theoretical calculated one are largely different.

Comparing the results shows that the theoretical elliptical results are closer to the experimental

than the tapered wing one

Errors

There are many source of error in the experimental.

The human error, which occurs from recording the time. The pan weight may release before the

timing starts either the opposite is true. Recording the time more than one and allows more than

ten revolutions if there is a space can reduce this error.

The pressure difference seems to be not fixed because the fluctuations again running more than one

time the experiment can reduce the error.

Conclusion

In this report the theoretical values for the rolling moment were calculated, for the elliptical and

tapered wings and the values from the experiment were calculated.

There were significant difference between the theoretical and experimental results. Nevertheless,

the experiment shows the effect of increasing speed on the angle of attack and the rolling moment,

which is very realistic.

The experiment can be more successful if considering to reduce the error to minimum.

References R.Vepa, lecture notes, Queen Mary university of London, 2014.

Lab hand-out note, Queen Mary university of London,2014

Appendix

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

-20 -15 -10 -5 0 5 10 15 20

L/ N

m

ROLL RATE (P)/RAD S-1

Applied rolling moment Vs roll rate 15mmH2O

10.5clockwise

10.5anticclowise

12.5clockwise

12.5anticklockwise

15clockwise

15anti

y = 0.0014x - 0.0021

y = -0.0016x - 0.0022

0

0.002

0.004

0.006

0.008

0.01

0.012

-10 -5 0 5 10

L/ N

M

ROLL RATE (P)/RAD S-1

+ 12.6mmH2O

- 12.6mmH2O

Linear (+ 12.6mmH2O)

Linear (- 12.6mmH2O)

x x bar x-xbar (x-xbar)squared

(x-xbar)squared/6 SIGMA

0.21306636 0.21306636 2.22045E-16 4.93038E-32 0.000269021 0.01640186

0.19886193 -0.01420442 0.000201766

0.19886193 -0.01420442 0.000201766

0.19886193 -0.01420442 0.000201766

0.22727078 0.014204424 0.000201766

0.2414752 0.028408847 0.000807063

y = 0.0014x - 0.0018

y = -0.0017x - 0.0025

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

-20 -10 0 10 20

L/ N

m

ROLL RATE (P)/RAD S-1

+ 15mmH2O

- 15mmH2O

Linear (+ 15mmH2O)

Linear (- 15mmH2O)

y = 0.0015x - 0.0026

y = -0.0014x - 0.001

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

-20 -10 0 10 20

L/ N

m

ROLL RATE (P)/RAD S-1

+ 10.5mmH2O

- 10.5mmH2O

Linear (+ 10.5mmH2O)

Linear (- 10.5mmH2O)