UNIVERSITY OF GLASGOW

Degrees of M.Eng., B.Eng., M.Sc. and B.Sc. in Engineering

Flight Mechanics 3 (ENG3060)

4th May 2023

09:30 - 11:30

ATTEMPT ALL QUESTIONS

TOTAL MARKS AVAILABLE: 60

The numbers in square brackets in the right-hand margin indicate the marks allotted to the part of the
question against which the mark is shown. These marks are for guidance only.
DATA SHEETS ARE INCLUDED AT THE END OF THE EXAM PAPER
An electronic calculator may be used provided that it does not have a facility for either
textual storage or display, or for graphical display.

PLEASE SEE BELOW FOR INSTRUCTIONS ON HOW TO ANSWER:

• Script books will NOT be used in this exam.

• You will be issued with three A3 double sided answer sheets.

• Answer only one question on each A3 answer sheet.

• Ensure you add CLEARLY your GUID and date of birth on each answer sheet. GUID goes on front
and back.

• Continuation answer sheets (pink) are available on request. Please ensure you add CLEARLY your
GUID, date of birth and question number to all continuation sheets. GUID goes on front and back.

Page 1 of 6 Continued overleaf

Q 1)
a) An aircraft air data sensor provides velocity information (u, v, w) = (43, −5, 7)ms−1 . Calculate
the angle of attack α and sideslip angle β.

[4]

b) Wind tunnel measurements of an aircraft model indicate measured aircraft body axis loads
(−6.4, −3.0, −93.0) N at α = 26.1o , β = −12.8o . What are the drag, side force and lift (D, S, L)?

[4]

c) An aircraft is at angle of attack α and zero sideslip. The aircraft rolls by small angle ∆ϕ. Show
that this leads to sideslip angle β = α∆ϕ.

[4]

d) Consider a conventionally mounted, unswept wing. The installed wing dihedral angle is Γ. Show
that in the event of a roll angle disturbance ∆ϕ, the change of angle of attack on the port and
starboard wings are

∆αp = −αΓ∆ϕ,
∆αs = αΓ∆ϕ

respectively, and explain why this will lead to a restoring rolling moment in the event of a roll
disturbance.

[8]

Page 2 of 6 Continued overleaf

Q 2)
a) Consider the aircraft fin depicted in the sketch in figure Q2. The fin aerodynamic centre is
at position (x, 0, z) in body axes. For aircraft body rates (p, q, r), show that that there is an
additional v velocity component ∆v at the fin due to the body rates given by ∆v = rx − pz.
Then show that the effective aerodynamic sideslip angle at the fin is βf in = β + rx−pz
V .
f

[5]

b) The fin in a) above has aerodynamic lift curve slope a and wetted area Af in . The dynamic
pressure is q. Derive an expression for the fin side force Sf in to include the effects of sideslip
angle β and body rates (p, q, r), and show that the fin rolling and yawing moments Lroll and N
due to Sf in are

Lroll = zSf in
N = −xSf in ,

where small angle assumptions have been made.

[5]

c) The aircraft in figure Q2 has two propellers mounted on each wing as shown. Both propeller
rotational axes are in the x direction and rotate at speed Ω, and each propeller moment of inertia
is Iprop . Aircraft moments of inertia are Ixx , Iyy and Izz , and only the Ixz product of inertia is
non-zero. Derive expressions for the rotational equations of motion, and describe what happens
if the rudder on the fin is used to yaw the aircraft?.

[10]

x
β flight speed vector Vf
propeller speed Ω

z y
starboard wing

aircraft fin

Figure Q2: Aircraft configuration for Q2

Page 3 of 6 Continued overleaf

Q 3)
a) Aircraft mass distribution data are given in table Q3, where the origin of (x, y, z) is at the aircraft
aft end.
i) Show that the centre of mass is at ordinate (6.69, 0, 1.35) m.
ii) Calculate the moment of inertia Ixx and the product of inertia Ixz about the centre of mass.

[10]

b) Calculate the principal moments of inertia for an aircraft with moments of inertia Ixx = 90 kgm2 ,
Iyy = 6000 kgm2 , Izz = 6000 kgm2 , Ixz = 200 kgm2 , Iyz = Ixy =0, and show that the principal
axes are oriented by about 2 degrees in pitch.

[10]

Table Q3: Aircraft discrete masses and positions for Q3.

Component mass [kg] (x, y, z) position [m]

Airframe 350 (6,0,1.5)

Engine and propeller 75 (12,0,1.2)
Fuel 100 (7.5,0,0.8)
Wings 50 (6.5,0,2)
Pilot 75 (11,0,1.8)
Cargo 150 (3,0,1)

Page 4 of 6 Continued overleaf

Data sheet

1. For any vector A within an axis system rotating at speed ω

dA
= Ȧxyz + ω × A,
dt
where the usual notation applies.

2. Euler angle transformation matrices for yaw ψ, pitch θ, roll ϕ are

     
cos ψ sin ψ 0 cos θ 0 − sin θ 1 0 0
Tψ = − sin ψ cos ψ 0 , Tθ =  0 1 0  , Tϕ = 0 cos ϕ sin ϕ  .
0 0 1 sin θ 0 cos θ 0 − sin ϕ cos ϕ

3. Velocity at a position P relative to inertial axis system at origin O and local axes at position O′

v OP = v OO′ + v O′ P + ω × rO′ P ,

where the symbols have their usual meaning.

4. The acceleration at a position P relative to inertial axis system at origin O and local axes at position
O′ is

aOP = v̇ OO′ + ω × v OO′ + ω̇ × rO′ P + ω × ω × rO′ P + aO′ P + 2 ω × v O′ P ,

xyz xyz

where the symbols have their usual meaning.

5. Inertia matrix is given by

 
Ixx −Ixy −Ixz
I = −Ixy Iyy −Iyz 
−Ixz −Iyz Izz
where the moments of inertia are
Z Z Z
Ixx = ′2 ′2
(y + z ) dm, Iyy = (x′2 + z ′2 ) dm, Izz = (x′2 + y ′2 ) dm,
body body body

and the products of inertia are

Z Z Z
Ixy = ′ ′
x y dm, Ixz = x′ z ′ dm, Iyz = y ′ z ′ dm.
body body body

Page 5 of 6 Continued overleaf

6. Aerodynamic forces relative to aircraft axes

Page 6 of 6 End of question paper