Stability Control Report

8/12/2019 Stability Control Report

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Queen Mary University Of London

School of Engineering and Materials

Science

Rolling moment due to rate of roll

DEN 303: Stability and Control of Aircraft

Kedian Lamin

100407518



2

Abstract

This report demonstrates how to experimentally determine the dimensionless rolling moment

due to roll rate derivative L p. The experiment was carried out on a straight tapered wing with

moderate aspect ratio placed on an open circuit wind tunnel. The model was tested over a

range of three air velocities. The obtained results were analysed and compared to theoretical

predictions given by the strip theory, the modified strip theory and the lifting line theory, for

both elliptical and straight tapered planforms. The experiment was fairly successful as the

obtained values were in accordance with the theoretical estimates. It has also be observed that

the predictions given for elliptical wings were the closest to the results obtained in the

experiment as opposed to straight-tapered wings. However, only the lifting line theory

provided satisfactory predictions.

Table of Contents

Abstract …………………………………………………………………………….2

I. Introduction

………………………………………………………………………………………..3

II. Theoretical estimates of Lp (straight-tapered

wing)……………………………………………………………………………3

III. Experimental procedure………………………………………………………….6

IV. Sample

calculations……………………………………………………………………………6

V. Results…………………………………………………………………………...9

VI. Discussion…………………………………………………………………………...15

VII. References…………………………………………………………………………15



3

I. Introduction

Roll is a complex but important aspect in flight dynamics. It starts with the creation of an

asymmetric lift distribution along the wingspan which causes a rolling torque. As the plane

rolls, the wing going down has an increased incidence α, thus produces more lift, while the

other wing undergoes the opposite effect. This results in a difference in lift generated by both

wings, which creates a restoring moment that is opposing the rolling motion. After a

disturbing rolling moment is created, the roll rate p increases exponentially until flight

equilibrium is restored, and a steady roll rate is established. In order to understand the effects

of roll motion it is necessary to define important lateral stability derivatives such as the

rolling moment due to roll rate L p. For conventional aircrafts the major contribution in L p

comes from the wings which provide great resistance to rolling (roll damping). This

experiment proposes a simple method to determine the rolling damping derivative of a

straight-tapered wing planform.

II. Theoretical estimates of Lp (straight-tapered wing)

This part provides a derivation of the simple strip theory for straight-tapered wings. It does

not include theoretical estimates for elliptical wings, as these have already been derived in the

laboratory handout.

The wing planform illustrated below is a straight-tapered wing. The three theories used for

elliptic planform can be adjusted based on the geometry of the model.

Simple Strip Theory

The dimensionless rolling moment due to rate of roll derivative, L p is defined by:

Rearranging equation (1) gives:

Figure 1. Local chordwise strip distribution for an elliptic planform



4

The total rolling moment about axis Ox is given as:

This may be re-written as:

Equating (2) and (4) gives:

∫

∫

Now, let ∫ (5)

Calculating a specific value for d allows the modification of the elliptical wing formulas.

Firstly solving the integral in the denominator of equation (5)

Substituting in R.H.S of the above equation yields:

[ ]



5

*

+

*

+

In the experiment, the model used had the following characteristics :-

Substituting these values into equation [14] gives:

∫ (7)

Recalling equation (5)

∫

Here, and =0.047638 m2

Now x can be calculated by substituting those values mentioned above and (7) into equation (5)

14.29

Recalling the original L p equation:

∫

Hence by replacing d back into the equation, L p is estimated for a straight-tapered wing as:

Although the value for d was calculated for the two dimensional case, it can still be applied to

3D cases. Consequently we can adjust the modified strip theory and lifting line theories for

straight tapered wings:



6

Modified Strip Theory

Modified strip theory can be adjusted to: { } (9)

Lifting Line TheoryLifting line theory can be adjusted to: { } (10)

III. Experimental procedure

The experiment conditions (atmospheric pressure and temperature) were noted. The dimensions of the straight tapered wing were measured using measuring tape.

The distance travelled by weight pan for ten revolutions was measured in order to

estimate the effective radius of bobbin on which the cord was wound.

The lever handle was turned in the clockwise direction to rewind the cord and gear

was engaged to make sure cord stayed in place and fully wounded before starting the

motion.

The motion was started with tunnel reference pressure of 11.2 mmH 2O by disengaging

the gear and releasing weight from the rest.

The time displayed for ten revolutions of shaft was noted and time was reset to zeroafterwards.

The two previous steps were repeated for two more tunnel reference pressure of 13.2

mmH 2O and 14.5 mmH 2O, with series of masses up to 2.5 kg for positive rate of roll

and for negative rate of roll incrementing the mass by 0.5 kg in each case.

IV. Sample calculations

A. Pressure drop across the Betz manometer (∆H=11.2

mmH2O)

B. Air density and local temperature



7

C. Tunnel air speed (∆H=11.2 mmH2O)

√ √

D. Reynolds number (U∞=13.795 m.s-1

)



8

( ) ⁄ ( ) ( ) ⁄

E. Angle of attack of wing tips relative to wing (y=s,

p=4.19 rad.s-1, U∞=13.795 m.s-1)

F. Rolling moment due to rate of roll (modified striptheory, elliptical wing, a∞=5.7)

{ }

{ }



9

V. Results

A. Raw data

Atmospheric pressure, P atm = 760.50 mmHg

Atmospheric temperature, T atm = 24°C

Wing span = 51.5 cm =0.515 m

Wing tip = 6.2 cm =0.062 m

Wing chord = 12.3 cm =0.123 m

Length before 10 revolutions,

Length after 10 revolutions,

Pressure

(mmH2O)

Mass

(kg)

Time for 10 revolutions (s)

Clockwise Anticlockwise

11.2

0 29.3 36.27

0.5 14.99 16.91

1 9.22 9.92

1.5 6.95 7.35

2 5.06 5.32

2.5 3.87 4.21

13.2

0 30.09 38.24

0.5 15.22 17.15

1 9.87 11.24

1.5 7.35 7.83

2 6 6.35

2.5 4.81 5.06

14.5

0 30.32 39.310.5 15.39 17.93

1 10.33 11.31

1.5 7.2 8.28

2 6.26 6.31

2.5 5.28 5.12

Table 1. Experimental data



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B. Rolling moment variation due to rate of roll

Tunnel reference pressure

(mmH2O)

Tunnel reference Pressure

(Pa)

Wind speed

(m/s)

11.2 109.84 13.795

13.2 129.45 14.976

14.5 142.20 15.696

Table 2. Tunnel reference pressure and equivalent wind speed

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10 12 14 16 18

L ( N . m

)

p (rad/s)

Rolling moment vs. roll rate

clockwise

anti-clockwise

Figure 2. Rolling moment due to rate of roll (p=11.2 mmH2O)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10 12 14

L ( N . m

)

p (rad/s)


clockwise

anti-clockwise




11

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10 12 14

L ( N . m

)

p (rad/s)


clockwise

anti-clockwise


0

0.005

0.01

0.015

0.02

0 5 10 15 20

L / U ∞ ( N . s

)

p (rad/s)

L/U∞ vs. p Re=82813.58,

clock-wise

Re=82813.58,

anti-

clockwiseRe=89903.31,

clockwise

Re=89903.31,

anti-

clockwiseRe=94225.59,

clockwise

Re=94225.59,anti-

clockwise



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C. Experimental assessment of Lp

Air speed

(m s-1)

Slope of straight-line portion

of L vs. p graph

Experimental L p

Clockwise Anticlockwise Clockwise Anticlockwise

13.795 0.0018 0.0019 -0.23965 -0.25296

14.976 0.0016 0.0017 -0.21302 -0.22633

15.696 0.0017 0.0016 -0.22633 -0.21302

The minus sign in front of the L p values accounts for the fact that L p is indeedopposing rolling motion.

The final experimental value for L p can be obtained with the following formula:

, where is the best estimation (average) for Lp and

√ ∑

√ ∑

√

x (x-)2

-0.23965 0.0001232

-0.25296 0.00059585

-0.21302 0.00024118

-0.22633 0.0000049284 -0.22633 0.0000049284

-0.21302 0.00024118

Table 3. Experimental values of L p according to roll direction

Table 4. Standard deviation method explained



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D. Comparison with theoretical estimates

a∞ (rad-

) Strip Theory Modified Strip Theory Lifting Line Theory

Elliptical

wing

5.7 -0.3563 -0.2687 -0.2157

2π -0.3927 -0.2889 -0.2286

Straight-

tapered

wing

5.7 -0.3989 -0.3009 -0.2415

2π -0.4397 -0.3317 -0.2662

a∞ (rad-1

) Strip Theory Modified Strip Theory Lifting Line Theory

Elliptical

wing

5.7 35.85% 14.94% 5.96%

2π 41.8% 20.89% 0.0219%

Straight-

taperedwing

5.7 42.70% 24.04% 5.36%

2π 48.02% 31.10% 14.14%

Table 5. Theoretical L values (elliptical and straight-tapered wing models)

Table 6. Percentage error (elliptical and straight-tapered wing models)



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E. Incidence of the wings relative to wind at stall

Air

velocity (m/s)

Roll rate, p (rad/s) Angle of attack at stall (◦)

Clockwise Anticlockwise Clockwise Anticlockwise

13.795 4.19 3.72 4.47 3.97

14.976 8.55 5.59 8.36 5.49

15.696 6.08 5.56 5.70 5.49

VI. Discussion

From Figure 2.- Figure 4 , it can be observed that as the velocity of air is increased

inside the tunnel , the relationship between the roll rate and the applied rolling

moment becomes more linear i.e. stall occurs at a higher applied rolling moment. Thissuggests that the effect of roll damping is less critical for higher speeds.

In addition, for all three tunnel reference pressures , the corresponding graphs from

Figure 2- Figure 4 were fairly symmetrical which means that the magnitude of the

values for clockwise roll rate and anticlockwise roll rate were fairly similar. However,

in each graph, it seems that the direction of roll tends to affect the linearity of the

relationship between rolling moment and rate of roll: the trend for the clockwise roll

direction is more linear than anticlockwise roll direction which indicates that the roll

damping is losing its effect quicker for anticlockwise roll. For the lower speed, the

wing is closer to the stall angle which means fluctuations in drag across the wing isaffecting the roll rate. Also it can be seen in the same graphs that the magnitude of the

disparity in the values for clockwise and anti-clockwise roll rate increases with air

velocity.

Table 6. Highlights significant disparities in L p values for the different theories. The

accuracy of the experimental results was determined by percentage error between

theoretical values and average experimental value. This showed that predictions given

for elliptical wings were much more accurate than those given for straight-tapered

wings. It also became apparent that the lifting line theory provided predictions that

came closest to the experimental results. This may be explained by the fact that the

Table 7. Angle of incidence of wing tips near stall



straight-tapered wing used in the experiment may present a lift distribution similar to

that of an elliptical wing.

Table 7. shows that the angle of attack at stall is influenced by the direction of roll:

values are higher for clockwise rate of roll than for anti-clockwise rate of roll.

Although the results can be considered as fairly reliable, it should be noted that they

have most definitely been subject to limitations that led to uncertainties of variable

sources. The main source of error is human error, as the most important part of the

experiment involved recording the time taken to achieve ten revolutions.

In figure 5. It is seen that as roll rate increases the roll damping effect is diminished.

Noticeably, roll damping is fading fast near stall. When the roll rate increases, a

change in incidence α is observed: the left (downgoing) wingtip is flying at a higher

angle of attack, which (in this regime) produces more lift, compared to the right

wingtip. At normal airspeed, if both wing tips are flying below critical angle, this

generates large forces which oppose the rolling motion: a large amount of roll

damping is observed. However, when the angle of attack is increased beyond stall

angle, the left wingtip no longer produces more lift: the aerodynamic forces do not

oppose the initial rolling motion.

The effect roll damping loss presented above may represent an extreme case where L p

is positive: the aerodynamic forces generated by the downward going wing tip tend to

amplify rolling motion instead of nullifying it. In a particular case where both wing

tips are flying above critical angle, the aircraft may unintentionally enter spin roll.

Wing tips tend to contribute more to rolling damping that wing roots because thevalue of r is greater at this location. If an aircraft is designed such that the incidence at

the tips is set to be greater than at the roots, the former will stall first when maximum

lift coefficient is reached. This design technique is known as washin. Furthermore, it

is possible to combine it with the addition of winglets which will allow a greater

amount of lift to be generated near the wing tips.

VII. References

[1] DEN 303, Rolling moment due to rate of roll Laboratory Experiment handout . Queen

Mary University of London, 2013-1014.

[2] DEN 303, Rolling moment due to rate of roll Laboratory Experiment slides. Queen Mary

University of London, 2013-1014.

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