Differential Equations
for Engineers:
the Essentials
Supplement to Class 14B
Homework Assignment 14:
Example Engineering Application of ODEs
Aircraft Flight Performance, Phase 2
Homework Assignment 14
Review:
Refresh your memory by skimming (the guide to the instructor’s or
student’s choice of numerical integration computer programs.)
Work :
We will use the numerical integration computer program to explore
aircraft flight performance. We will add realistic equations for angle
How high can it fly? (ceiling)
How fast can it fly? (max speed)
How long to reach required (minimum time to climb)
altitude and speed?
L
T
D
W
V
Variables for Aircraft Longitudinal Flight
Definition of Terms
=
=
=
V
h
altitude (ft)
speed (ft/sec)
flight path angle (radians)
Definition of Terms (2)
=
=
=
=
W
D
L
T
thrust (lbs)
drag (lbs)
weight (lbs)
lift (lbs)
Definition of Terms (3)
=
=
Temp
Temp
0
Air temperature (degrees R)
Air temperature at sea level (degrees R)
Equations to program
Vh
sin
=
Basic equations of motion:
2
2
2
1
2
1
VSCD
VSCL
Da
La
=
=
Basic lift and drag equations:
gWm /=
Weight and mass:
Mach number:
Equations to program (2)
Constant parameters:
1070
1060
1038.2
2.32
3
0
3
3
0
=
=
=
=
−
xT
xW
x
g
a
slugs/ft^3
ft/sec^2
lbs
lbs
w/o afterburner
Graph 1: Air temperature vs altitude
Coordinates:
(altitude (k ft), TA (deg R))
Equations to program (3)
Equations to program (4)
Graph 2: Air density reference
altitude vs altitude
Coordinates:
(altitude (k ft), (k ft))
)(/
0
hHh
aa e−
=
)(hH
)(hH
Equations to program (5): drag coefficient
M
8.00
1D
C
0
C
D
M
Equations to program (6)
1
Drag coefficient at zero
lift vs Mach number