Rolling Moment due to rate or roll Stability and Control of aircraft
11th January 2013
Abstract The main objective of this experiment is to t o investigate the dimensionless rolling-momentdue-to-rate-of-roll derivative L derivative L p. During the course of study of a straight tapered wing which was modelled in open circuit wind tunnel at range of air velocity, Experimental value of L of L p was obtained. Theories such as Strip theory, Modified strip Theory and Lifting line theory will be used to obtain theoretical predictions for both elliptical wing models and straight tapered wing models, which will provide necessary meaning into comparing the difference or similarities between experimental and theoretical prediction of L of L p. Higher air velocity led to greater L greater L p value as expected due to need for wing to be greater resistance against the t he roll. There is a significant percentage error in the results obtained due to several factors which has been explained later in the report.
Table of Contents Abstract............................................................................................................. ................................................................................................................................................... ...................................... 1 Introduction .................................................................... .......................................................................................................................................... ........................................................................ .. 2 Background Theory ................................................................................................................................. ................................................................................................................................. 2 Theoretical estimates for L p .............................................................. ................................................................................................................ .................................................. 2 Derivation for Theoretical estimate for L p of Straight-tapered Straight-tapered wing .................................................. 4 Apparatus and Instrumentation Instrumentation ........................................................... ............................................................................................................. .................................................. 7 Experimental Procedure ......................................................................................................... ......................................................................................................................... ................ 7 Results.......................................................................................................................... ..................................................................................................................................................... ........................... 8 Raw data ..................................................................... ........................................................................................................................................... ........................................................................ .. 8 Calculation of Experimental value of L p ................................................................... ............................................................................................ ......................... 10 Calculation of angle of attack of the wings relative to the wind ...................................................... ...................................................... 15 Calculation of Reynolds Number based on mean wing chord .......................................................... 16 Calculating Theoretical values of L p .............................................................. .................................................................................................. .................................... 17 Discussion................................................................................... Discussion.............................................................................................................................................. ........................................................... 20 Errors..................................................................................................................................................... ..................................................................................................................................................... 21 Conclusion......................................................................................................... ............................................................................................................................................. .................................... 21 References ........................................................... .............................................................................................................................. ................................................................................. .............. 22
Page 1
Introduction The dimensionless rolling-moment-due-to-rate-of-roll derivative, L p is one of the most important lateral aerodynamic derivatives. L p is equal to the roll rate of the aircraft, p when an aircraft is rolling. For conventional aircraft the major contribution to L p comes from the wings which provide great resistance to rolling causing such motion to be heavily damped. The downward-going wing has its incidence increased so that the lift on it will also normally increase. The converse is true for the upward-going wing. The net effect is a rolling moment opposing the motion. The moments acting on the plane balance each other and thus the aircraft resists any rolling that occurs and remains stable.
Background Theory Theoretical estimates for L p Elliptical wings
Consider the chord wise strip of an elliptical wing of span dy at a distance y from the axis of rotation shown in Figure 1 below.
Figure 1: An elliptical wing
Simple Strip Theory
When the elliptical wing is rolling at an angular rate p, the element will have an incidence change given by tan-1 for moderate rate of roll. The lift curve slope of the profile is a∞ which is given by: a
dC L d
[1]
Where dC L is the change in lift coefficient and dα is the change in incidence. Change in incidence is or can be written as:-
[2]
Page 2
Change in lift coefficient is given by:dl
dC L
1 2
[3]
2
U S w
Where dl is the change in lift, ρ is the density of air and S w is surface area of the wing. Now substituting equation [2] and [3] into equation [1] and then rearranging will give following equation. dl =
Where
1 2
a py U c. y
[4]
Area of the element, c = c(y) is the local chord.
The rolling moment of the element can be found by multiplying equation 4 with y distance from the axis of rotations as shown below: dL = y × dl dL
1 2
a p U cy 2 y
[5]
The negative sign in equation [5] indicates the fact that the rolling moment opposes the motion of the wing. But equation [5] is only for rolling moment of chordwise strip shown in Figure 1. To find the total rolling moment for the whole elliptic al wing, equation [5] will have to be integrated along its whole span. The integration is shown below: s
L
1
s 2
a p U cy 2 dy
s
1
L U pa cy 2 dy 2 s
[6]
If the lift-curve slope, a∞ is assumed uniform across the span of an elliptical wing. Then C L
L 1 2
Where p
U 2 S w 2 s
a p 16
[7]
p 2 s U
Hence, the rolling moment due to rate of roll for an elliptical wing is given by:
Page 3
L P
( L / p) p 0 1 2
U S w ( 2 s)
2
a
[8]
16
Expression of L p given in equation [8] only works for an elliptical wing configuration for a two dimensional case. Modified Strip Theory The simple strip theory does not allow for the downwash associated with the trailing vortex. But the modified strip theory makes the allowance for the downwash by correcting the liftcurve slope for the effects of finite aspect ratio, AR and using the wing angle of attack, aw instead of lift curve slope, a∞. Thus the expression for the modified strip theory for an untwisted elliptical wing is given below:
L P
a a 161 AR
[9]
Lifting Line Theory
The modification of lift-curve slope, a∞ used above is strictly based on symmetrical spanwise loading of the wing. But while rolling, the incremental loading is anti-s ymmetrical. The resulting expression for the lifting line theory for an untwisted elliptical wing is shown below:-
L P
a
2a 161 AR
[10]
This makes the wing effectively acting as two separate wings, each of half the aspect ratio.
Derivation for Theoretical estimate for L p of Straight-tapered wing Consider the straight-tapered wing planform shown in figure 2. The three theories can be derived based on geometry of wing.
Figure 2: A straight tapered wing
Simple Strip Theory
Page 4
The dimensionless rolling moment due to ate of roll derivative, L p defined by:
∞
∞
[11]
Rearranging equation [11] gives:
∞
[12]
Equating [11] and [12] gives:
∫ ∫ ∞
∞
∞
∞
∞
∞
Now, let
∫
[13]
Calculating a specific value for ‘ x’ allows the modification of the elliptical wing formulas. Firstly solving the integral in the denominator of equation [13]
Substituting
* + . / . / . / 0 1 0 1 in R.H.S of above equation gives:
Page 5
∫
[14]
In this experiment, Geometries of wing were measured to be :-
Substituting these geometries of wing in equation [14] gives :-
∫ ∫
[15]
Recalling equation[13]. Here,
and
2
=0.0486 m
Now x can be calculated by substituting those values mentioned above and [15] into equation [13].
14.30
Recalling the original L p equation:
∫ ∞
Hence by r eplacing ‘x’ back into the equation, L p estimates using the strip theory for a straight-tapered wing is found which is:-
∞
Although the value for x was calculated for the two dimensional case, the value can still applied to 3D cases. Hence we can adjust the modified strip theory and lifting line theories for straight tapered wings: Modified Strip Theory Modified strip theory can be adjusted to:
, ∞
∞
[17]
Lifting Line Theory Lifting line theory can be adjusted to:
Page 6
, ∞
∞
[18]
Apparatus and Instrumentation The Straight tapered wing model of moderate aspect ratio is placed in the working section of open circuit wind tunnel. The model wing is mounted on a freely rotating shaft and is aligned with the direction of the flow. The wing swiftly attains a steady roll-rate p, since the applied moment is balanced by the counteracting aerodynamic moment.
Figure I: Straight tapered wing model
An external rolling moment is applied by adding weights to a weight pan suspended by a cord wound around a bobbin connected on the shaft. The rolling moment is applied by adding weights of upto 2.5 kg in an increment of 0.5 kg to the weight pan.
Figure II: Weight pan suspended by a cord
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Electric timer was used to record time taken for shaft to complete 10 revolutions. The speed of rotation is estimated from the frequency of pulse generated photo-electrically by a beam of light, which shines through two drilled holes at right angles to each other and to the axis of rotation, through a boss on the shaft. Figure III: Electric timer
Experimental Procedure 1) The Atmospheric Pressure and room temperature of the room (where experiment is going to be conducted) were noted. 2) The dimensions of the straight tapered wing were measured using measuring tape. 3) The distance travelled by weight pan for five revolutions was measured in order to estimate the effective radius of bobbin on which the cord was wound. 4) The lever handle was turned in the clockwise direct ion to rewind the cord and gear was engaged to make sure cord stayed in place and full y wounded before starting the motion. 5) The motion was started with tunnel reference pressure of 10 mmH 2O by disengaging the gear and releasing weight from the rest. 6) The time displayed for ten revolutions of shaft was noted and time was reset to zero afterwards. 7) Steps 4 to 6 were repeated for two more tunnel reference pressure of 12 mmH 2O and 14 mmH 2O, with series of mass upto 2.5 kg for positive rate of roll and for negative rate of roll incrementing the mass by 0.5 kg in each case.
Results Raw data Atmospheric pressure, P atm = 745 mmHg Atmospheric temperature, T atm = 21°C
Page 8
Chord Tip
Span Figure 1: Geometry of Straight tapered wing
Wing span = 51.5 cm =0.515 m Wing tip = 6.4 cm =0.064 m Wing chord = 12.5 cm =0.125 m Length before 5 revolution, l 1= 92 cm =0.92 m Length after 5 revolution, l 2=58.5 cm =0.585 m Δl= l1- l2 =0.92-0.585 m
=0.335 m Table 1: Raw data from experiment Mass[kg]
Pressure[mmH2O]
10
12
14
0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3
Time in seconds for 10 revolutions Clockwise Anticlockwise 32.21 31.19 15.15 15.03 9.05 9.17 6.38 6.84 4.94 4.96 3.35 3.09 33.11 31.18 16.02 16.10 10.06 10.11 7.03 7.12 6.01 5.99 4.35 4.13 34.05 31.84 16.38 16.17 10.93 11.07 7.90 7.90 6.03 6.03 5.34 5.33
Page 9
Calculation of Experimental value of Lp From equation [11] ,
∞
Rearranging the equation gives:-
∞
Experimental value of L p can be obtained by dividing the gradient of linear portion of “ L
need to be deduced . against p” graph by
. Before doing that, all the parameters given in above equation
Calculating density of air , according to ideal gas law. Rearranging the equation gives
√ √
Calculating air velocity inside the wind tunnel for each Press ure. Calculation is demonstrated for lowest tunnel reference Pressure10 mmH 2O. ∞
Where k is tunnel calibration constant which is equals to 1.03.
∞
Following the steps, air velocity for two other tunnel reference pressure is calculated which is shown in Table 2 below. Table 2: Tunnel reference pressure and corresponding air velocity Tunnel reference pressure [mmH2O] 10 12 14
Tunnel reference Pressure [Pa] 98.1 117.72 137.34
Air velocity [m/s] 13.09 14.34 15.48
Calculating area of wing, S w by assuming wing as two trapeziums.
Where in this case
Page 10
Hence
Calculating rate of roll, p.
Where T is time period and can be obtained by the time t aken for wing to complete 10 revolutions divided by 10. Calculation is demonstrated for 0.5 kg loading which took 32.21 s to revolve 10 times clockwise at lowest speed.
Following the steps, rate of roll is obtained using corresponding data and shown in Table 4, Table 5 and Table 6
Calculating applied rolling moment, L:
Where
is mass,
and is radius of the bobbin
needs to be calculated using formula
Where is number of revolutions the wing experiences. For the purpose of measuring the cord length 5 revolutions were observed. ΔL is the change of the length of the chord before and af ter the 5 revolutions. Rearranging the equation gives:-
Demonstrating to find L for 0.5kg mass.
Following the steps, applied rolling moment, L is obtained using corresponding data and shown in Table 4, Table 5 and Table 6
Page 11
Table 4: Results for 10 mmH 2O Time period, T [s]
Roll rate, p [Rad s- ]
Mass [kg]
Applied Rolling Moment, L [Nm]
Clockwise
Anticlockwise
Clockwise
Anticlockwise
0
0.00
3.22
3.12
1.95
-2.01
0.5
0.05
1.52
1.50
4.15
-4.18
1
0.10
0.91
0.92
6.94
-6.85
1.5
0.16
0.64
0.68
9.85
-9.19
2
0.21
0.49
0.50
12.72
-12.67
2.5
0.26
0.34
0.31
18.76
-20.33
Table 5: Results for 12 mmH 2O Time period, T [s]
Roll rate, p [Rad s- ]
Mass [kg]
Applied Rolling Moment, L [Nm]
Clockwise
Anticlockwise
Clockwise
Anticlockwise
0
0.00
3.31
3.12
1.90
-2.02
0.5
0.05
1.60
1.61
3.92
-3.90
1
0.10
1.01
1.01
6.25
-6.21
1.5
0.16
0.70
0.71
8.94
-8.82
2
0.21
0.60
0.60
10.45
-10.49
2.5
0.26
0.44
0.41
14.44
-15.21
Table 6: Results for 14 mmH 2O Time period, T [s]
Roll rate, p [Rad s- ]
Mass [kg]
Applied Rolling Moment, L [Nm]
Clockwise
Anticlockwise
Clockwise
Anticlockwise
0
0.00
3.41
3.18
1.85
-1.97
0.5
0.05
1.64
1.62
3.84
-3.89
1
0.10
1.09
1.11
5.75
-5.68
1.5
0.16
0.79
0.79
7.95
-7.95
2
0.21
0.60
0.60
10.42
-10.42
2.5
0.26
0.53
0.53
11.77
-11.79
Page 12
APPLIED ROLLING MOMENT AGAINST ROLL RATE FOR U =13.09m/s ∞
0.30 0.26
0.26
0.25 0.21
m N / L
0.21
0.20
0.16
0.16
0.15
ANTICLOCKWISE MOMENT 0.10
0.05 0.05
-30.00
-20.00
CLOCKWISE MOMENT
0.10
0.10
0.05
0.00 0.00 -10.00 0.00
0.00 10.00
20.00
30.00
Roll Rate (p)/rad s-1
Figur2: Graph of Applied rolling moment against roll rate for U ∞=13.09m/s
APPLIED ROLLING MOMENT AGAINST ROLL RATE FOR U =14.34m/s ∞
0.30
0.26
0.21
m N / L
0.16
0.15
0.10
-10.00
ANTICLOCKWISE MOMENT
0.05
0.05
0.000.00 -5.00 0.00
Roll Rate
CLOCKWISE MOMENT
0.10
0.10
0.05
-15.00
0.21
0.20
0.16
-20.00
0.26
0.25
0.00 5.00 (p)/rad s-1
10.00
15.00
20.00
Figure 3: Graph of Applied rolling moment against roll rate for U ∞=14.34m/s
Page 13
APPLIED ROLLING MOMENT AGAINST ROLL RATE FOR U =15.48m/s ∞
0.30
0.26
0.26 0.25
0.21
m N / L
0.21
0.20
0.16
0.16
0.15
ANTICLOCKWISE MOMENT CLOCKWISE MOMENT
0.10
0.05
-15.00
-10.00
0.10
0.10
0.05
0.05
0.00 0.00 -5.00 0.00
0.00 5.00
Roll Rate (p)/rad s-1
10.00
15.00
Figure 4: Graph of Applied rolling moment against roll rate for U ∞=15.48m/s
Table7 : Obtained experimental Air velocity [m s-1]
L p values
Gradient of linear portion of L against p graph Clockwise
Anticlockwise
13.09
0.01930
0.02228
14.34
0.02529
15.48
0.02623
for each air velocity according to roll direction 1/2ρU∞Sw(2s) 2
Experimental L p Clockwise
Anticlockwise
0.09955
0.19387
0.22381
0.02353
0.10906
0.23190
0.21576
0.02676
0.11773
0.22280
0.22731
Note that L p value is a negative quantity since it is acting in the opposite direction to actual roll. Now Standard Deviation can be used to refine the results obtained for experimental value of L p
̅ ∑ √ Page 14
Table 9:Standard deviation method outlined
x
0.19387
0.219241667
x-
2
(x- )
-0.025371667
0.000644
0.2319
0.012658333
0.000160
0.2228
0.003558333
0.000013
0.22381
0.004568333
0.000021
0.21576
-0.003481667
0.000012
0.22731
0.008068333
0.000065
0.000152
0.01235
Experimental value of L p= -0.19387±0.01235 which should lie between -0.20622 and 0.18152.
Calculation of angle of attack of the wings relative to the wind Angle of attack,
∞
Where y is the distance from the axis Ox of rotation = 0.2575 m p is rate of roll at stall which can be found using the value of p where the graph diverges from a straight line in Figure 2 to Figure 4 and diversion of linear trend on those graphs signify stalling. U ∞ is velocity of air in tunnel
Calculation of
for u∞ =13.09 m/s with clockwise roll is shown below:-
( )
Table 10: Obtained angle of attack of wing tips Air velocity -1 [ms ]
-1
Roll rate, p [Rad s ]
Angle of attack at stall, AoA[degree]
Clockwise
Anticlockwise
Clockwise
Anticlockwise
13.09
9.85
9.19
11.00
10.25
14.34
6.25
8.82
6.40
9.00
15.48
7.95
7.95
7.53
7.53
Page 15
Calculation of Reynolds Number based on mean wing chord
̅
The Reynolds number based on the mean chord for each air velocity can be obtained by ∞
equation:
Where =1.18kg/m3 , Mean chord
̅
∞ is
air velocity,
̅
is mean chord of the wing and
needs to be obtained using the formula:
̅ ̅ ̅
Hence, For the slowest speed of 13.09 m/s.
Following the steps, Reynolds numbers are obtained for each tunnel reference pressure and tabulated in Table 11.
Table 11: Reynolds number and L/ U ∞ for each air velocity which is used to plot graph in Figure 5 -1
[ ms ]
Reynolds Number
Applied Rolling Moment, L [Nm]
L /U
13.09
80085.12
0.00
U
∞
-1
14.34
87732.64
Roll rate, p [Rad s ]
∞
Clockwise
Anticlockwise
0.000
1.95
-2.01
0.05
0.004
4.15
-4.18
0.10
0.008
6.94
-6.85
0.16
0.012
9.85
-9.19
0.21
0.016
12.72
-12.67
0.26
0.020
18.76
-20.33
0.00
0.000
1.90
-2.02
0.05
0.003
3.92
-3.90
0.10
0.007
6.25
-6.21
0.16
0.011
8.94
-8.82
0.21
0.015
10.45
-10.49
0.26
0.018
14.44
-15.21
Page 16
15.48
94707.2
L/U
∞
0.00
0.000
1.85
-1.97
0.05
0.003
3.84
-3.89
0.10
0.006
5.75
-5.68
0.16
0.010
7.95
-7.95
0.21
0.014
10.42
-10.42
0.26
0.017
11.77
-11.79
against Roll rate for each air velocity 0.025
Re=80085.10 Clockwise
0.020
Re=80085.10 Anticlockwise
0.015
∞
Re=87732.64 Clockwise
U / L
Re=87732.64 Anticlockwise
0.010
Re=94707.2 Clockwise
0.005
Re=94707.2 Anticlockwise 0.000 -30.00
-20.00
-10.00
0.00
10.00
20.00
30.00
Roll Rate (p)/rad s -1 Figure 5: Graph of L/U against roll rate for each air velocity and roll direction
Calculating Theoretical values of L p Theoretical values of L p are obtained using the theories deduced in Background and Theory section, in order to compare our Experimental values with. For Elliptical planform,
Strip Theory
∞
∞
The strip theory is to be applied for two values of a∞ , 5.7 rad-1 and 2 rad-1: For a∞= 5.7 rad-1
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For a∞= 2 rad-1
, , , ,
Modified Strip Theory
∞
∞
Aspect ratio, AR of tested wing needs to be calculated using formula:
5.457 Now, Modified strip theory is to be applied for two values of a∞ , 5.7 rad-1 and 2 rad-1 For a∞= 5.7 rad-1
For a∞= 2 rad-1
Lifting line theory
∞
∞
Lifting line theory is to be applied for two values of a∞ , 5.7 rad-1 and 2 rad-1 For a∞= 5.7 rad-1
For a∞= 2 rad-1
, ,
For straight tapered planform
Strip Theory
Page 18
, , , , , , ∞
For a∞= 5.7 rad-1
For a∞= 2 rad-1
Modified Strip Theory
∞
∞
For a∞= 5.7 rad
-1
For a∞= 2 rad-1
Lifting Line Theory
∞
∞
For a∞= 5.7 rad-1
For a∞= 2 rad-1
All theoretical values of L p for elliptical wing model and straight tapered wing model are tabulated in Table 12 and Table 13 respectively.
Table 12: Theoretical L p values for elliptical wing model -1
a∞ [rad ]
Strip Theory
Modified Strip Theory
Lifting Line Theory
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-0.35625 -0.39270
5.7 2π
-0.26736 -0.28738
-0.21397 -0.22660
Table 13: Theoretical L p values for straight tapered wing model -1
a∞ [rad ]
Strip Theory
Modified Strip Theory
Lifting Line Theory
5.7
-0.39860 -0.43938
-0.29914 -0.32154
-0.23940 -0.25354
2π
Discussion Comparing the graphs from Figure 2- Figure 4 , 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. This suggests that the effect of roll damping has lessened. Also, Higher air velocity led to greater L p value as expected due to need for wing to be greater resistance against the roll. 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, 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 is affecting the roll rate. From Table 12 and Table 13 , it can be observed that differ ent theories had significant differences in L p values. In order to assess the accuracy of the experimental results against the theoretical values, the percentage error between the theoretical values and the average experimental value have been tabulated in Table 14 and Table 15. The average experimental value of L p = -0.19387 Table 14: Percentage error for elliptical wing a∞ [rad-1]
Strip Theory
Modified Strip Theory
Lifting Line Theory
5.7
45.58% 50.63%
27.49% 32.54%
9.39% 14.44%
2π
Table 15: Percentage error for straight tapered wing -1
a∞ [rad ]
Strip Theory
Modified Strip Theory
Lifting Line Theory
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5.7 2π
51.36% 55.88%
35.19% 39.71%
19.02% 23.53%
From Table 14 and Table 15, it can be observed that the results of all theories for elliptical wing models are closer to the experimental values than the results of all theories for the straight-tapered wing. The lifting line theory result for elliptical wing is closer to the experimental value than the lifting line theory results for the straight tapered wing and explains that the lift distribution of experimentally used straight-tapered wing is similar t o that of considered elliptical wings . In table 10, the angle of attack at stall decreases as the air velocity is increased for anticlockwise roll direction while in the case clockwise roll direction, the result shows anomalous behaviour. The proportionality between angle of attack and air velocity is expected to be indirect since a higher velocity equates to a greater amount of energy which can be used to overcome stall. For straight tapered wings, the lifting line theor y was the closest match to the experimental value of L p . This is because it is based on the same planform as the wing model used in the experiment. Oout of all Theoretical values of L p only -0.21397 barely matches the experimental value of L p which was found to lie in between -0.20622 and -0.18152.
Errors There were many factors that would have affecte d the results. The main source of error in this experiment is human error as most of the results deducted are mainly based on recording time for the shaft to complete ten revolutions. The person’s ability of disengaging the gear and releasing the weight at the same time might have affected time. The person might have delayed in releasing the weight while the gear had been disengaged already. Another cause that would have affected the time to complete 10 revolutions is motion of the shaft. In reality there was presence of some lateral movement whereas in the calculations steady rotations were assumed. The way to eliminate error is improving use of timing mechanism by using accurate light gates rather than small beam light pulses. Also fluctuation in readings of tunnel reference pres sure could have contributed towards the error. It had to be made sure the test was carried out in one constant reading of tunnel reference pressure by resetting back often. Another way to improve the experimental results and to reduce the percentage error crept in the calculations is to repeat the measurements and obtain the values for each tunnel reference pressure and roll direction.
Conclusion Overall it could be said that the experiment was reasonably successful. The experimental results were fairly accurate as they confirmed closely to the theoretical values predicted by
Page 21
most of the theories. However, there were minor discrepancies in the results due to errors in time recording for shaft to revolve 10 times. If this experiment is to be conducted again, the sources of human errors could be eliminated by making obvious improvements. This would have given a small percentage error and a better approximation of L p value.
References
Bandu, N.P.(2004) Performance, Stability, Dynamics, And Control of Airplanes. 2nd edition. Virginia: American Institute of Aeronautics and Astronautics. DEN 303, Rolling moment due to rate of roll Laboratory Experiment handout . Queen Mary University of London, 2012-1013. DEN 303, Stability and control of aircraft Supplementary notes. Queen Mary University of London, 2012-2013. Dole, C.E., Lewis, J.E (2000). Flight Theory and Aerodynamics: A practical guide for operational safety. 2nd ed. Canada: John Wiley & Sons. p268-271 Phillips, W.F (2004). Mechanics of Flight . New Jersey: John Wiley & Sons. p489-493 Swatton, P.J (2011). Principles of Flight for Pilots. West Sussex: Wiley & Sons. P161-162.
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