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1 Challenge the future
Introduction to Aerospace Engineering
Lecture slides
15-12-2012
Challenge the future
Delft University of Technology
Introduction Aerospace Engineering
Flight and Mechanics
Dr. ir. Mark Voskuijl
2 Flight Mechanics
5.&6 Climbing and descending flight
3 Flight Mechanics
Contents
1. Summary horizontal flight
2. Introduction climbing and descending flight
3. Equations of motion straight and symmetric flight
4. Gliding flight
5. Climbing flight
6. Example calculations
4 Flight Mechanics
Contents
1. Summary horizontal flight
2. Introduction climbing and descending flight
3. Equations of motion straight and symmetric flight
4. Gliding flight
5. Climbing flight
6. Example calculations
5 Flight Mechanics
Equations of motion General equations of motion in symmetric flight (1)
Free Body Diagram Kinetic Diagram
L
D
W
T V
T
Zb
Ze
Xe
V
Zb
Ze
Xe
dmV
dt
dVm
dt
Xb Xb
6 Flight Mechanics
Equations of motion
: cos sin
: cos sin
T
T
F ma
dVV T D W m
dt
dV L W T mV
dt
Horizontal, steady, symmetric flight
= 1 = 0 = 0
= 1 = 0 = 0
:
:
V T D
V L W
Note: Clearly this is the most simple form of the equations of motion
7 Flight Mechanics
Summary - Jet
Force
Drag
Airspeed
Max thrust
Max. Endurance (CL / CD)max
Max. Range (CL/CD
2)max
Max. Speed (depends on Tmax)
Min. speed CL,max
8 Flight Mechanics
Summary – Propeller Aircraft
Power
Power required
Airspeed
Power available
Max. Endurance (CL
3 / CD2 )max
Max. Range (CL/CD)max
Max. Speed (depends on Pamax)
Min. speed CL,max
9 Flight Mechanics
Summary – Horizontal flight
,maxLC min
,max
2 1
L
WV
S C
max
,max
(Jet)
(Prop)a r
T D
P P
Minimum airspeed
Maximum airspeed
Maximum range 2
max
L
D
C
C
L W
T D
...LC max
2 1
L
WV
S C
max
V
F
max
L
D
C
C
0
13L DC C Ae
0L DC C Ae
2 1
L
WV
S C
prop
jet
10 Flight Mechanics
Summary – Horizontal flight
Maximum endurance
max
L
D
C
C
L W
T D
min
F
3
2
max
L
D
C
C
0L DC C Ae
03L DC C Ae
2 1
L
WV
S C
prop
jet
11 Flight Mechanics
Contents
1. Summary horizontal flight
2. Introduction climbing and descending flight
3. Equations of motion straight and symmetric flight
4. Gliding flight
5. Climbing flight
6. Example calculations
12 Flight Mechanics
Contents
1. Summary horizontal flight
2. Introduction climbing and descending flight
3. Equations of motion straight and symmetric flight
4. Gliding flight
5. Climbing flight
6. Example calculations
13 Flight Mechanics
Introduction
• How fast can an aircraft climb?
• How steep can an aircraft climb?
• How far can an aircraft glide following an engine failure?
14 Flight Mechanics
Rate of climb versus flight path angle
V
Vsin = RC
V1
RC1
V2
RC2
Example case 1 Example case 2
1 2
1 > 2 however, RC1 < RC2 ! So do not confuse the flight path angle with the rate of climb
15 Flight Mechanics
Contents
1. Summary horizontal flight
2. Introduction climbing and descending flight
3. Equations of motion straight and symmetric flight
4. Gliding flight
5. Climbing flight
6. Example calculations
16 Flight Mechanics
Equations of motion General equations of motion in symmetric flight (1)
Free Body Diagram Kinetic Diagram
L
D
W
T V
T
Zb
Ze
Xe
V
Zb
Ze
Xe
dmV
dt
dVm
dt
Xb Xb
17 Flight Mechanics
Equations of motion
: cos sin
: cos sin
T
T
F ma
dVV T D W m
dt
dV L W T mV
dt
Simplification: The aircraft is performing a straight and symmetric flight. Assumption: The thrust is assumed to be in the direction of flight (T = 0).
General equations of motion in symmetric flight (2)
18 Flight Mechanics
Equations of motion
• Straight flight: flight in which the centre of gravity of the
aircraft travels along a straight line (d/dt = 0)
• Steady flight: Flight in which the forces and moments acting on
the aircraft do not vary in time, neither in magnitude, nor in
direction (dV/dt = 0)
• Horizontal flight: The aircraft remains at a constant altitude
(= 0)
• Symmetric flight: flight in which both the angle of sideslip is
zero and the plane of symmetry of the aircraft is perpendicular to
the earth ( = 0 and the aircraft is not turning)
definitions
19 Flight Mechanics
Equations of motion
: cos sin
: cos sin
T
T
F ma
dVV T D W m
dt
dV L W T mV
dt
Straight, symmetric flight
= 1
1 = 0 = 0
: sin
:
W dV
V T D Wg dt
V L W
20 Flight Mechanics
Power equation Straight, symmetric flight
: sin
:
W dV
V T D Wg dt
V L W
Multiply first equation with airspeed
21sin potential kinetic energy increase
2a rP P dV
VW g dt
L W
So, excess power (Pa – Pr) can be used to climb or to increase airspeed
21 Flight Mechanics
Contents
1. Summary horizontal flight – (Vmin, Vmax, Range)
2. Endurance
3. Speed instability
4. Introduction climbing and descending flight
5. Equations of motion straight and symmetric flight
6. Gliding flight
7. Climbing flight
8. Example calculations
22 Flight Mechanics
Gliding flight
0 aT P
Question: How far can a Boeing 747 glide from 10km altitude?
Answer: Approximately 180 km!
23 Flight Mechanics
24 Flight Mechanics
Gliding flight
2
0
2
a
r
T P
dH W dVP W
dt g dt
2
aircraft decelerates2
r
W dVP
g dt
What if the pilot tries to fly a horizontal path?
Clearly this is not an option… glide at constant airspeed:
sinr
dHP W WV
dt
Power equation
25 Flight Mechanics
Best gliding performance
min ,min
sinr
r
P WV
RC P
Conclusion: 1. To glide as far as possible, one must glide at the
condition for minimum drag (e.g. useful for engine failure)
2. To glide as long (time wise) as possible, one must glide at the condition for minimum power required (e.g. useful for gliders)
Note: This is not treated in ‘introduction to flight’
1. RCmin (long time)
2. min (long distance)
Option 1
min min
sinDV
WV
D
Option 2
26 Flight Mechanics
Glide as far as possible
arcsin arcsinD DW L
arcsin D
L
CC
0
max
LL D
D
CC C Ae
C
Question: Does the aircraft weight influence the minimum glide angle a. Yes b. No c. ?
2 1
L
WV
S C
27 Flight Mechanics
Glide as far as possible
arcsin arcsinD DW L
arcsin D
L
CC
0
max
LL D
D
CC C Ae
C
Question: Does the aircraft weight influence the minimum glide angle a. Yes b. No c. ?
2 1
L
WV
S C
28 Flight Mechanics
Glide as far as possible
Strangely enough, the glide angle is independent of the aircraft weight!
V1, W1
V2,W2
W1 > W2 1 = 2 and V1 < V2
29 Flight Mechanics
Story: We have lost both engines
30 Flight Mechanics
Story: We have lost both engines
Question: Could the aircraft have made it to Dobbins Air Force Base
if the pilots had decided to glide there?
W = 90,000 lb (= 400500 N)
S = 1001 sq ft (=93 m2)
b = 93.3 ft (=28.4 m)
CD0 = 0.02
e = 0.85
Distance to Dobbins AFB: 20 nautical miles (=36 km)
Altitude: 7000 ft (=2134 m)
31 Flight Mechanics
Story: We have lost both engines
Answer: Yes!
Maximum distance when CL/CD is max. So:
Now the glide angle and distance can be calculated:
0
0
2 2
2 2
28.4( 8.7)
93
0.02 8.7 0.85 0.68
0.680.02 0.04
8.7 0.85
L D
LD D
bA
S
C C Ae
CC C
Ae
0.04arcsin arcsin 3.4 [deg]
0.68
distance = / tan 2134 / tan 3.4 36 km
D
L
C
C
h
32 Flight Mechanics
Glide as long as possible
0
3
,min 2
max
3Lr L D
D
CP C C Ae
C
sinrP WV
Minimum rate of descent
rP DV
D
L
CD W
C
2 1
L
WV
S C
2
3
2 Dr
L
CWP W
S C
33 Flight Mechanics
Summary gliding flight
Minimum glide angle
V
D
Minimum rate of descent
V
Pr
Minimum glide angle
34 Flight Mechanics
Contents
1. Summary horizontal flight – (Vmin, Vmax, Range)
2. Endurance
3. Speed instability
4. Introduction climbing and descending flight
5. Equations of motion straight and symmetric flight
6. Gliding flight
7. Climbing flight
8. Example calculations
35 Flight Mechanics
Climbing performance
P
V
Pr
Pa (at a certain setting of )
Vmin Vmax, (hor,steady)
Excess power
21sin potential kinetic energy increase
2a rP P dV
VW g dt
L W
36 Flight Mechanics
Climbing performance
• Maximum rate of climb (propeller)
• Steepest climb (jet)
Steady flight
37 Flight Mechanics
Climbing performance
• Maximum rate of climb (propeller)
• Steepest climb (jet)
Steady flight
38 Flight Mechanics
Maximum rate of climb
Maximum rate of climb maximum
excess power (Pa – Pr)max
sina rP PV RC
W
L W
Steady flight
39 Flight Mechanics
Solutions Maximum rate of climb propeller aircraft
P
V
Maximum excess power Pr
Pa
RCmax at Pr,min This corresponds to the following condition:
0
3
2
max
3
L
D
L D
C
C
C C Ae
Assumption: Pa is independent of airspeed
40 Flight Mechanics
Climbing performance
• Maximum rate of climb (propeller)
• Steepest climb (jet)
Steady flight
41 Flight Mechanics
Steepest climb Jet aircraft
T D
W
L W
sin
T
D
Maximum excess thrust
42 Flight Mechanics
Solutions Steepest climb for jet aircraft
max at Dmin This corresponds to the following condition:
L
D
L D
C
C
C C Ae
0
max
Assumption: T is independent of airspeed
T
D
Maximum excess thrust
43 Flight Mechanics
Contents
1. Summary horizontal flight
2. Introduction climbing and descending flight
3. Equations of motion straight and symmetric flight
4. Gliding flight
5. Climbing flight
6. Example calculations
44 Flight Mechanics
Example 1 Climbing performance of the Beach King Air
Two engine propeller aircraft CD = CD0 + kCL
2 CD0 = 0.02 k = 0.04 W = 60 [kN] S = 28.2 [m2]
Maximum power available (741 kW) can be assumed independent of airspeed. The aircraft is performing a steady symmetrical climb Question: What is the maximum rate of climb of this aircraft at sea-level ( = 1.225 [kg/m3] and what is the corresponding airspeed
45 Flight Mechanics
Example 1 Solution
2
Power equation:
1sin
2a rP P dV
VW g dt
741000 2060008.9 [m/s]
60000a rP P
RCW
steady flight
sina rP PV RC
W
max r,min
2
3
Maximum power available is independent of airspeed (741 kW)
RC at P
2r
DrD
LL
P DVCW
P WCD W S C
C
0
3
,min 2
max
0.023 3 1.22
0.04
DLr L
D
CCP C
kC
2 1 60000 2 153 [m/s] (=190 km/h)
28.2 1.225 1.22L
WL W V
S C
0
2 20.02 0.04 0.41 0.08D D LC C kC
3 31 12 2
0.08 1.225 53 28.2 206 [kW]r DP DV C V S