stability control report
TRANSCRIPT
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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
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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
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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
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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:
[ ]
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*
+
*
+
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:
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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
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C. Tunnel air speed (∆H=11.2 mmH2O)
√ √
D. Reynolds number (U∞=13.795 m.s-1
)
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( ) ⁄ ( ) ( ) ⁄
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)
{ }
{ }
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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)
Rolling moment vs. roll rate
clockwise
anti-clockwise
Figure 3. Rolling moment due to rate of roll (p=13.2 mmH2O)
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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)
Rolling moment vs. roll rate
clockwise
anti-clockwise
Figure 4. Rolling moment due to rate of roll (p=14.5 mmH2O)
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
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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.