1 Challenge the future Flight and Orbital Mechanics Lecture slides
1Challenge the future
Flight and Orbital Mechanics
Lecture slides
Semester 1 - 2012
Challenge the future
DelftUniversity ofTechnology
Flight and Orbital MechanicsLecture hours 3, 4 – Minimum time to climb
Mark Voskuijl
2AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Summary previous lecture
• Performance diagram
• Optimal climb
• Solution low speed aircraft
• Energy height
• Solution high speed aircraft (M>1)
• Example exam question
• Summary
3AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Summary previous lecture
• Performance diagram
• Optimal climb
• Solution low speed aircraft
• Energy height
• Solution high speed aircraft (M>1)
• Example exam question
• Summary
4AE2104 Flight and Orbital Mechanics |
IntroductionQuestion
What is the most efficient way (minimum time) to go
from take-off at sea-level to Mach 1.5 at 15,000 m?
A. Climb at airspeed for max . At 15,000 m accelerate to Mach 2
B. Climb at airspeed for max RC. At 15,000 m accelerate to Mach 2
C. Climb at airspeed for max RC to 11,000 m. Descent and accelerate to Mach 1.2 at 9,000 m, climb and accelerate to Mach 2 at 11,000 m. Decelerated climb to 15,000 m, Mach 1.5
D. Accelerate at sea-level to Mach 1.5, climb to 15,000 m at airspeed for max RC
A. Climb at airspeed for max . At 15,000 m accelerate to Mach 2
B. Climb at airspeed for max RC. At 15,000 m accelerate to Mach 2
C. Climb at airspeed for max RC to 11,000 m. Descent and accelerate to Mach 1.2 at 9,000 m, climb and accelerate to Mach 2 at 11,000 m. Decelerated climb to 15,000 m, Mach 1.5
D. Accelerate at sea-level to Mach 1.5, climb to 15,000 m at airspeed for max RC
5AE2104 Flight and Orbital Mechanics |
15000
11000
9000
0
Altitude [m]
Subsonic
M = 1.2(Supersonic)
M = 2
M = 1.5
IntroductionSolution
6AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Summary previous lecture
• Performance diagram
• Optimal climb
• Solution low speed aircraft
• Energy height
• Solution high speed aircraft (M>1)
• Example exam question
• Summary
7AE2104 Flight and Orbital Mechanics |
Summary previous lecture
• A typical climb (civil subsonic aircraft) is performed at constant
indicated airspeed and at a constant power setting. Therefore, the
true airspeed is actually increasing. Since airspeed is not constant,
it is an unsteady climbing flight
• The climb is almost a straight line. It is therefore a quasi-rectilinear
flight
• Corresponding equations of motion:sin
W dVT D W
g dt
L W
8AE2104 Flight and Orbital Mechanics |
Summary previous lecture
• The rate of climb in an unsteady climb is smaller than in a steady climb
because part of the excess energy is used to accelerate
• You must be able to derive this ratio from the equations of motion
• You must be able to calculate this ratio (see example exam question)
• For more background information: read Ruijgrok – Elements of
airplane performance section 14.2
1; 0
1st
RC dV
V dVRC dh
g dh
9AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Summary previous lecture
• Performance diagram
• Optimal climb
• Energy height
• Solution low speed aircraft
• Solution high speed aircraft (M>1)
• Example exam question
• Summary
10AE2104 Flight and Orbital Mechanics |
Performance diagram
P
V
Pa
a rsteady
P PRC
W
Pr
11AE2104 Flight and Orbital Mechanics |
Aerodynamic forces (year 1) Lift – Drag polar
0
2
LD D
CC C
Ae
12AE2104 Flight and Orbital Mechanics |
Aerodynamic forces (year 1) Lift
0
2
LD D
CC C
AeL W
Drag
So, one part of the drag decreases (!) with airspeed (1/V2) and one part increases with airspeed (V2)
212DD C V S
0
22 21 1
2 2L
D
CD C V S V S
Ae
0
22 21 1
2 22 2 4
4 1 1D
WD C V S V S
S V Ae
0
221
2 212
D
WD C V S
Ae V S
212LC V S W
2
2 1L
WC
S V
13AE2104 Flight and Orbital Mechanics |
Aerodynamic forces (year 1) Drag as a function of airspeed
Aircraft are quite unique in the sense that drag increases when airspeed decreases!
0
0
210 2
2
212
i
D
i
D D D
D C V S
WD
Ae V S
14AE2104 Flight and Orbital Mechanics |
15AE2104 Flight and Orbital Mechanics |
P
V
Pa
Pr
V1 V2
V3
a rsteady
P PRC
W
V3
V1 V2
V4
RCsteady
V
Determine magnitude of difference and divide by aircraft weight (W)
16AE2104 Flight and Orbital Mechanics |
Performance diagramHow does it change with altitude?
Note: this sheet is from the first year lecture
17AE2104 Flight and Orbital Mechanics |
P
V
Pa
Pr
a rsteady
P PRC
W
V3 V4
RCsteady
V
H2
H1
18AE2104 Flight and Orbital Mechanics |
Performance diagramRate of climb as a function of airspeed and altitude
19AE2104 Flight and Orbital Mechanics |
Performance diagram
H
V
RC = 6 m/s
RC = 2 m/s
Rate of climb as a function of airspeed and altitude
20AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Summary previous lecture
• Performance diagram
• Optimal climb
• Energy height
• Solution low speed aircraft
• Solution high speed aircraft (M>1)
• Example exam question
• Summary
21AE2104 Flight and Orbital Mechanics |
Optimal climb
• During climb at constant indicated airspeed, the aircraft is
accelerating. V(H) is fixed
• Easy flight technique for the pilot
• But is this optimal???
• What is optimal?
22AE2104 Flight and Orbital Mechanics |
Optimal climb
• Minimum time to climb (time)
• Minimum fuel consumed during climb (fuel)
• Distance covered during climb (distance)
• Noise during climb (noise)
What is optimal?
23AE2104 Flight and Orbital Mechanics |
Optimal climb
• Minimum time to climb (time)
• Minimum fuel consumed during climb (fuel)
• Distance covered during climb (distance)
• Noise during climb (noise)
We will focus on the first option, but the other three are optimal as
well
What is optimal?
24AE2104 Flight and Orbital Mechanics |
Optimal climb
• Objective: to find the true airspeed as a function of altitude that
yields the minimum time to climb
dH dHRC dt
dt RC
2
1
H
H
dHt dt
RC
0
1
1steady
RC
V dVRC
g dH
2
1
0
1H
steadyH
V dV
g dHt dH
RC
• Time t is not minimal if the integrand is
minimal at every altitude H because the
term dV/dH is in the integrand
• Variational calculus is necessary
25AE2104 Flight and Orbital Mechanics |
Variational calculus?
Choosing maximum RCsteady at sea level seems like a good starting point
2
1
0
1H
steadyH
V dV
g dHt dH
RC
Minimize integrand:• RCsteady maximum• dV/dH <0
RCsteady max
dV/dH<0
• At H = 0, RCst is maximal at V = 100
• The integrand can be minimized more by choosing (dV/dH)1 < 0
• At altitude H > 0 no optimum V can be chosen
• Conclusion: No local optimum but global optimum (consider complete flight path)
26AE2104 Flight and Orbital Mechanics |
Optimal climb
Solution
• Simplify (low speed aircraft)
• Energy method (high subsonic / supersonic aircraft)
2
1
0
1H
steadyH
V dV
g dHt dH
RC
27AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Summary previous lecture
• Performance diagram
• Optimal climb
• Solution low speed aircraft
• Energy height
• Solution high speed aircraft (M>1)
• Example exam question
• Summary
28AE2104 Flight and Orbital Mechanics |
Low speed aircraft
29AE2104 Flight and Orbital Mechanics |
Low speed aircraft
0
0 0
1H H
st
V dV
g dHdHt dH
RC RC
0
H
st
dHt
RC
Assumption: Low speed, low altitude dV/dH is small
Choose maximum steady rate of climb at each altitude
30AE2104 Flight and Orbital Mechanics |
Low speed aircraft
H1H2
V
P RCsteady H
V
sin
a r
a rsteady
W dVT D W
g dt
P P W dVRC V
W g dt
P PRC
W
31AE2104 Flight and Orbital Mechanics |
Low speed aircraft
a r
st
P PRC
W
can be assumed constant as function of airspeed
for propeller aircraft
aP
0,min ,max
3st r L opt DRC P C C Ae
Solution – propeller aircraft
Pa
Pr
V
Pr,min
,
2 1 1opt
L opt
WV
S C
0
Corresponding constant
constant because
e
i i e
V V
V V V
Note: this optimal CL was determined in the first year lectures!
32AE2104 Flight and Orbital Mechanics |
Low speed aircraft
• Conclusion: if the pilot selects the airspeed for the maximum
steady rate of climb at the ground and if he / she keeps the
indicated airspeed constant, then the climb will be optimal
• Optimal in this case means minimum time to climb (assuming
dV/dH is small)
Summary
33AE2104 Flight and Orbital Mechanics |
Low speed aircraftStory – World War I
34AE2104 Flight and Orbital Mechanics |
Low speed aircraft
• Allied airfoils (British / French) versus German airfoils
Story – World War I
• Thin airfoils have low CL,max
• Therefore, German aircraft
were able to fly slower
British / French) versus
German airfoils
35AE2104 Flight and Orbital Mechanics |
Low speed aircraftStory – World War I
Vmin
(thin profiles)Vmin
(thick profiles)
0
,
0 ,
3
0.035
6
0.8
1.26 (large !)
2 1
oL D
D
L opt
E ground
L opt
C C Ae
C
A
e
C
WV V
S C
Pa
Pr
Numerical example
36AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Summary previous lecture
• Performance diagram
• Optimal climb
• Solution low speed aircraft
• Energy height
• Solution high speed aircraft (M>1)
• Example exam question
• Summary
37AE2104 Flight and Orbital Mechanics |
38AE2104 Flight and Orbital Mechanics |
Energy height
• Altitude and flight speed are rapidly interchangeable (exchange
of kinetic energy and potential energy)
• Increasing the total energy is much slower because it depends
on the excess power
2 21 12 2
WEnergy mV mgH V WH
g
V
H
Concept to solve the minimum time to climb problem
Line of constant energy
39AE2104 Flight and Orbital Mechanics |
Energy height
2
2
e
e
EnergyH
W
VH H
g
A
C
D
B
40AE2104 Flight and Orbital Mechanics |
Energy height
Large energy
Small energy
Optimal combination of V and H to increase the energy
V
H
So: “Minimum time to climb” problem approximated by another problem: find Vopt (H) at which minimum time to total energy
41AE2104 Flight and Orbital Mechanics |
Energy height
210 2
E mg H mV
0
sinW dV
T D Wg dt
e a rsteady
dH P PRC
dt W
Equation of MotionEnergy height
212
0
WE WH V
g
2
02e
E VH H
W g
2
0
1
2
edH dH dV
dt dt g dt
0
1sin
dV T D
g dt W
0
sin a rP PV dVV
g dt W
2
0
1
2
a rP PdV dH
g dt dt W
Energy method
42AE2104 Flight and Orbital Mechanics |
Minimum time to climb
es
dHRC
dt
e
st
dHdt
RC
2
1
(time to energy height)
e
e
H
e
stH
dHt
RC
43AE2104 Flight and Orbital Mechanics |
Optimum path subsonic aircraft
(for each constant energy height line, find the maximum steady rate of climb)
44AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Summary previous lecture
• Performance diagram
• Optimal climb
• Solution low speed aircraft
• Energy height
• Solution high speed aircraft (M>1)
• Example exam question
• Summary
45AE2104 Flight and Orbital Mechanics |
Solution high speed aircraft
Altitude h
Exc
ess
pow
er,
ft/s
ec
h1<h2<h3
h1
h3
h2
Mach
46AE2104 Flight and Orbital Mechanics |
Supersonic aircraft
• Dip in excess power is caused
by transonic drag rise:
supersonic aircraft
High subsonic aircraft
0
2
D D LC C M k M C
47AE2104 Flight and Orbital Mechanics |
Solution high speed aircraft
48AE2104 Flight and Orbital Mechanics |
Solution high speed aircraft
• Mig-29: Climb from sea level to 6000 m in less than 1 minute
Example performance
49AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Summary previous lecture
• Performance diagram
• Optimal climb
• Solution low speed aircraft
• Energy height
• Solution high speed aircraft (M>1)
• Example exam question
• Summary
50AE2104 Flight and Orbital Mechanics |
Example exam question
51AE2104 Flight and Orbital Mechanics |
Example exam question
52AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Summary previous lecture
• Performance diagram
• Optimal climb
• Solution low speed aircraft
• Energy height
• Solution high speed aircraft (M>1)
• Example exam question
• Summary
53AE2104 Flight and Orbital Mechanics |
Summary
54AE2104 Flight and Orbital Mechanics |
Questions?