Differential Equations
for Engineers: Math 241
Supplement to Class 18
Systems of First Order ODEs
Definition of Terms and
Derivation of Equations for
Aircraft Longitudinal Dynamics Example
Aircraft Dynamics: Equilibrium
Straight and level flight equilibrium:
Lift
Drag
Lift = Weight
Thrust = Drag
0
L
W
T
0
D
Aircraft Dynamics: Perturbations
Longitudinal perturbations from straight and level flight:
u
V
Aircraft
centerline
Appendix: Definition of Terms
=
=
=
W
m
g
Acceleration of gravity
Aircraft mass
Aircraft weight
Appendix: Definition of Terms (2)
=
=
=
=
T
L
V
V
S
0
Plan area of aircraft wings
Air density
Aircraft speed
Aircraft speed at equilibrium
Appendix: Definition of Terms (3)
=
=
u
Fractional increase in speed
Increase in angle of attack
Appendix: Definition of Terms (4)
=
=
=
l
h
I
p
Moment of inertia, pitch axis
Distance between center of pressure and center of gravity
Distance between center of tails and center of gravity
Appendix: Definition of Terms (5)
L
m
CSV
L
2
0
=
Appendix: Basic Equations
sincos
WDT
dt
dV
m
−−=
Equations of motion:
Aerodynamic forces:
Appendix: Deriving Linearized
Equations of Motion for Perturbed Flight
We start with the basic equations on the previous chart and assume
equilibrium straight and level flight. Assuming the equilibrium
angle of attack is small,
Appendix: Deriving Linearized
Equations of Motion for Perturbed Flight (2)
Define
Then
Now
dt
du
V
dt
dV
uVVVV
0
00 )1(
=
+=+=
Appendix: Deriving Linearized
Equations of Motion for Perturbed Flight (3)
Hence the linearized equation of motion along the aircraft flight path is
Dmg
dt
dV
m
−−=
In words, the aircraft decelerates when (a) the component of gravity is
larger, or (b) drag increases due to an increase in airspeed or angle of
attack
Appendix: Deriving Linearized
Equations of Motion for Perturbed Flight (4)
The equation of motion for perturbations in the axis perpendicular to the
flight path is:
Now
We will compute that
VSCL
L
2/
2
=
L
T
Appendix: Deriving Linearized
Equations of Motion for Perturbed Flight (5)
Hence the linearized equation of motion in the vertical direction is
In words, the aircraft accelerates upward when the lift force increases
due to an increase in angle of attack or airspeed.