# torsional vibration of the wing of an aircraft

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I.D. Number __________________________   Desk No ________________
Faculty of Science, Engineering and Computing
April/May Examinations 2018/2019
Level 6
MODULE:             AE6022:  Further Aerospace Structures, Materials and Dynamics   DURATION:       Three Hours
Instructions to Candidates
This paper contains TEN questions in TWO sections: Section A and Section B
Section A:  Answer ALL SEVEN short questions
Section B:  Answer TWO long questions only
Short questions carry 8 marks each. Long questions carry 22 marks each.
THIS PAPER MUST BE HANDED IN AT THE END OF THIS EXAMINATION
CANDIDATES  ARE  PERMITTED  TO  BRING  ONE  APPROVED  CALCULATOR INTO THIS EXAMINATION: from either Casio FX83 or Casio FX85 series (with any suffix), FX115MS, FX570ES or FX991ES
Invigilators are under instruction to remove any other calculators
Candidates are reminded that the major steps in all arithmetical calculations are to be set out clearly
Stationery
Attached
Tables and Formulae (6 Pages)
Number of Pages: 1 – 9
+Tables and Formulae: 6 Pages
SECTION A
1.         The torsional vibration of the wing of an aircraft may be modelled as two shafts with torsional stiffnesses k1and k2 and two discs representing the two engines with polar moments of inertia of J1 and J2 as shown in the figure Q1. The damping provided by the shafts are assumed to be proportional to the mass and stiffness distribution. The table Q1 shows the undamped modal frequencies, modal damping ratio and mass normalised modal vectors (mode shapes).
The two engines generate sinusoidal moment of M = 200sin150t at node J1 .
Write the equations of motion in terms of Principal coordinates
(4 marks)
(b)       Using the modal analysis method, find expressions for both angular displacements θ1(t) and θ2(t).
(4 marks)
Mode
1
2
Natural frequency (Hz)
5.20
9.16
Damping ratio
0.02
0.04
Modal vector:
0.58 1.00
-0.45 1.00
Table Q1
Figure Q1
Continued…
2.
In a two degree of freedom structure, the following dynamic properties are given:
Modal damping ratio:                    = 0.01
= 0.02
Mass-normalised modal matrix  is given by:
Express one Frequency Response Function (either Direct or Transfer) in Receptance form for the above structure. Calculate the response levels at resonance, sketch the responses.
(8 marks)
3.      A jet aircraft is performing an accelerated climb at the airspeed of 250m/s at the altitude of 10,000 m. The jet aircraft weighs 170,000N, having a wing area of 50m2.The thrust available T = 110,000N at 10,000m. Given the drag polar equation:
Calculate the specific energy height He and specific excess power Ps. Assume the climb angle is small.
(4 marks)
Find the acceleration if the rate of climb is 12m/s.
(4 marks)
Continued…
The lateral stability derivatives for an aircraft under the cruise conditions are given by:
,  and .
where Y, N, and  are the side force, yawing moment, rate of sideslip and  rate of yaw, respectively. If the airspeed is 90 m/s, calculate the natural frequency, damping ratio, period and time-to-half amplitude using the Dutch roll approximation given in the datasheet. Sketch the yaw rate and roll rate versus time.
(8 marks)
5. Two holes have been drilled through a long steel bar as shown in Figure Q5(a). The stress concentration factor for a hole in a plate is shown in Figure Q5(b)
Determine the maximum stress at A and B when P=36 kN.
(4 marks)
If allowable stress in the bar is 120 MPa, how much is allowable load P? In this case what are the stresses at A and B?
(4 marks)
Figure Q5(a)

Question 5 continued on page 5…
Continuation of Question 5:

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Figure Q5(b). Stress concentration around a hole.
6.           For the state of the stress shown in Figure Q6:
(a) Construct the Mohr’s circle, and determine the magnitude and the direction of principal stresses.
(5 marks)
(b) Determine the magnitude and the direction of maximum shear stress and the corresponding normal stress.
(3 marks)

Continued…
7.           The stage II fatigue crack growth of a steel plate used in an aircraft part can be estimated from:
where the constants are appropriate when stress intensity values are expressed in  and crack length in m. The steel fracture toughness is 110.
A centred cracked tensile (CCT) specimen from this steel with width W=100 mm and an initial crack length of 10mm is cyclically loaded between 40 MPa and 180 MPa.
Assume geometry factor  during crack propagation.
Calculate the critical crack length when crack start to grow.
marks)
Estimate the life of the plate by advancing the crack from mm to a final crack length of 10mm. Assume Y is approximately constant when the crack is between 5mm to 7mm and when it is between 7mm to 10mm. Calculate Y at the average crack length in these ranges.
(6 marks)
Continued…
SECTION B
A half-wing of an aircraft may be idealised as a three degree of freedom cantilever beam with the fuselage assumed as the rigid support. The mass of the half-wing is lumped into three equally spaced locations with values of 520 kg, 310 kg and 200 kg as shown in Figure Q8. The length of the half-wing is 9 m, with modulus of elasticity E = 72 GN/m2 and equivalent uniform second moment of area I = 4 x 10-4 m4.
(a)       Complete the following Flexibility matrix of the model and obtain the Dynamical matrix of the system.
(6 marks)
(b)       The mode shapes for the first two modes are found to be:
where T  is the Transpose
Calculate the two associated natural frequencies.
(7 marks)
Using the orthogonality properties of modal vectors or otherwise, calculate the third mode shape and the associated natural frequency.  Sketch the three mode shapes showing the wing deflections for each mode.
State all the necessary assumptions clearly.
(9 marks)
Question 8 continued on page 8…
Continuation of Question 8:
You may use the following static deflection relationships for part (a):
Continued…
9.      A disaster relief transport aircraft takes off from airport A, carrying 5000kg of medical supply to an area affected by natural disaster. It climbs to an altitude of 7000m and starts to perform cruise climb. The corresponding cruise distance is 3000km. When it reaches the target drop zone, it descends to a lower altitude dropping the medical supply by parachute. Immediately, it climbs to a higher altitude and carries out a second cruise climb to the airport B. The fuel used during the descent, airdrop and climb to second cruise altitude is 400kg. The weight and performance characteristics of the transport aircraft are given as follows:
Operating empty mass = 18,700kg
Trip fuel mass =   8000kg
Medical supply mass = 5000kg
Contingency and diversion fuel = 900kg
Weight fractions for different flight phases:
Taxi, take-off: Wf/Wi = 0.97
Climb: Wf/Wi = 0.98
Descent: Wf/Wi = 0.995
Landing: Wf/Wi = 0.997
Wing area S = 110m2
Profile drag coefficient CDo=0.018
Aspect ratio = 8
Wing efficiency factor e = 0.85
Specific fuel consumption c = 0.5 N/N-hr
Sketch and label the complete mission profile from take-off at airport A to landing at airport B.
(2 marks)
Find the maximum possible distance between the drop zone and airport B with the given amount of trip fuel (excluding contingency and diversion fuel). Assume it adopts the cruise climb method flying 14% above minimum drag speed for both first and second cruise phases. The cruise speed and lift-to-drag ratio remain the same.
(18 marks)
Question 9 continued on page 10…
Continuation of Question 9:
Find the average mass flow rate of fuel for the second cruise climb.
(2 marks)
An aluminium alloy symmetric three-cell thin-walled wing section with shear modulus of G = 26 GPa is shown in Figure Q14. The section is made up of a triangular, a rectangular and an elliptical cell. Determine:
The shear stress distribution in the section due to the application of a torque of 40 kNm.
(19 marks)
Find also the twist per unit length of the section.
(3 marks)
Note for an ellipse, the area , and the perimeter = p(a+b)
END OF EXAMINATION PAPER
ISA Table
H
T
P

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a

km
K
N.m-2
kg.m-3
m.s-1
m2.s-1
N.s.m-2

288.2
1.0133E+05
1.2252
340.3
1.4634E-05
1.7930E-05
1
281.7
8.9866E+04
1.1117
336.4
1.5837E-05
1.7607E-05
2
275.2
7.9480E+04
1.0065
332.5
1.7169E-05
1.7280E-05
3
268.7
7.0088E+04
0.9090
328.5
1.8647E-05
1.6950E-05
4
262.2
6.1616E+04
0.8190
324.5
2.0290E-05
1.6617E-05
5
255.7
5.3993E+04
0.7359
320.5
2.2122E-05
1.6280E-05
6
249.2
4.7153E+04
0.6594
316.4
2.4170E-05
1.5939E-05
7
242.7
4.1032E+04
0.5892
312.2
2.6466E-05
1.5593E-05
8
236.2
3.5571E+04
0.5248
308.0
2.9046E-05
1.5244E-05
9
229.7
3.0714E+04
0.4660
303.8
3.1955E-05
1.4891E-05
10
223.2
2.6408E+04
0.4123
299.4
3.5246E-05
1.4534E-05
11
216.7
2.2605E+04
0.3636
295.0
3.8980E-05
1.4172E-05
12
216.7
1.9308E+04
0.3105
295.0
4.5638E-05
1.4172E-05
13
216.7
1.6491E+04
0.2652
295.0
5.3434E-05
1.4172E-05
14
216.7
1.4085E+04
0.2265
295.0
6.2560E-05
1.4172E-05
15
216.7
1.2030E+04
0.1935
295.0
7.3246E-05
1.4172E-05
16
216.7
1.0275E+04
0.1653
295.0
8.5756E-05
1.4172E-05
17
216.7
8.7763E+03
0.1411
295.0
1.0040E-04
1.4172E-05
18
216.7
7.4959E+03
0.1206
295.0
1.1755E-04
1.4172E-05
19
216.7
6.4024E+03
0.1030
295.0
1.3763E-04
1.4172E-05
20
216.7
5.4684E+03
0.0879
295.0
1.6114E-04
1.4172E-05

DATA SHEET FOR SECTION A AND B
(All symbols have their usual meanings)

ii
Continued…

Speed of sound:
where  and R = 287J kg-1K-1.
Continued…
Drag polar equation:
where
19.   Brequet range equation for cruise-climb:
20.    The condition for minimum drag is:
21.    Lift coefficient at cruise speed expressed in term of lift coefficient at minimum drag speed:
22.  Condition for maximum rate of steady climb:
Angle of climb:
Rate of climb:
Specific excess power:
Continued…
Energy height:
Specific excess power, +
28.   Time to climb from energy level He1 to He2:
29.
30.
31.
32.
33. Dutch roll approximation:

Continued…
34.
Torque:
The angle of twist:
for constant G
for variable G
LEFM stress intensity factor:
Mohr’s circle
Stress transformation equations
End of data sheets for Section A and B

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