Lecture #AC–3
Aircraft Lateral Dynamics
Spiral, Roll, and Dutch Roll Modes
Copy right 2003 by Jon at h an H ow
1
Spring 2003 16.61 AC 3–2
Aircraft Lateral Dynamics
? Using a procedure similar to the longitudinal case, we can develop the equa-
tions of motion for the lateral dynamics
˙x = Ax + Bu , x =
?
?
?
?
?
?
?
?
v
p
r
φ
?
?
?
?
?
?
?
?
,u=
?
?
δ
a
δ
r
?
?
and
˙
ψ = r sec θ
0
A =
?
?
?
?
?
?
?
?
?
?
?
?
?
Y
v
m
Y
p
m
Y
r
m
? U
0
g cosθ
0
(
L
v
I
prime
xx
+ I
prime
zx
N
v
)(
L
p
I
prime
xx
+ I
prime
zx
N
p
)(
L
r
I
prime
xx
+ I
prime
zx
N
r
)0
(I
prime
zx
L
v
+
N
v
I
prime
zz
)(I
prime
zx
L
p
+
N
p
I
prime
zz
)(I
prime
zx
L
r
+
N
r
I
prime
zz
)0
01tanθ
0
0
?
?
?
?
?
?
?
?
?
?
?
?
?
where
I
prime
xx
=(I
xx
I
zz
? I
2
zx
)/I
zz
I
prime
zz
=(I
xx
I
zz
? I
2
zx
)/I
xx
I
prime
zx
= I
zx
/(I
xx
I
zz
? I
2
zx
)
and
B =
?
?
?
?
?
?
?
?
(m)
?1
00
0(I
prime
xx
)
?1
I
prime
zx
0 I
prime
zx
(I
prime
zz
)
?1
000
?
?
?
?
?
?
?
?
·
?
?
?
?
?
Y
δ
a
Y
δ
r
L
δ
a
L
δ
r
N
δ
a
N
δ
r
?
?
?
?
?
2
Spring 2003 16.61 AC 3–3
? The code gives the numerical values for all of the stability derivatives. Can
solve for the eigenvalues of the matrix A to find the modes of the system.
?0.0331± 0.9470i
?0.5633
?0.0073
– Stable, but there is one very slow pole.
? There are 3 modes, but they are a lot more complicated than the longi-
tudinal case.
Slow mode -0.0073 ? Spiral Mode
Fast real -0.5633 ? Roll Damping
Oscillatory ?0.0331± 0.9470i ? Dutch Roll
Can look at normalized eigenvectors:
Spiral Roll Dutch Roll
β 0.0067 -0.0197 0.3269 -28
?
?p -0.0009 -0.0712 0.1198 92
?
?r 0.0052 0.0040 0.0368 -112
?
φ 1.0000 1.0000 1.0000 0
?
Not as enlightening as the longitudinal case.
3
Spring 2003 16.61 AC 3–4
Lateral Modes
Roll Damping -welldamped.
– As the plane rolls, the wing going down has an increased α
(wind is e?ectively “coming up” more at the wing)
– Opposite e?ect for other wing.
– There is a di?erence in the lift generated by both wings
→ more on side going down
– The di?erential lift creates a moment that tends to restore the equi-
librium
– After a disturbance, the roll rate builds up exponentially until the restor-
ing moment balances the disturbing moment, and a steady roll is estab-
lished.
Spring 2003 16.61 AC 3–5
Spiral Mode - slow, often unstable.
– From level flight, consider a disturbance that creates a small roll angle
φ>0
– This results in a small side-slip v (vehicle slides downhill)
– Now the tail fin hits on the oncoming air at an incidence angle β
→ extra tail lift → yawing moment
– The positive yawing moment tends to increase the side-slip
→ makes things worse.
– If unstable and left unchecked, the aircraft would fly a slowly diverging
path in roll, yaw, and altitude ? it would tend to spiral into the ground!!
? Can get a restoring torque from the wing dihedral
? Want a small tail to reduce the impact of the spiral mode.
5
Spring 2003 16.61 AC 3–6
Dutch Roll - damped oscillation in yaw, that couples into roll.
? Frequency similar to longitudinal short period mode, not as well damped
(fin less e?ect than the horizontal tail).
? Do you know the origins on the name of the mode?
? Consider a disturbance from straight-level flight
→ Oscillation in yaw ψ (fin provides the aerodynamic sti?ness)
→ Wings moving back and forth due to yaw motion result in oscillatory
di?erential Lift/Drag (wing moving forward generates more lift)
→ Oscillation in roll φ that lags ψ by approximately 90
?
? Forward going wing is low
Oscillating roll → sideslip in direction of low wing.
? Damp the Dutch roll mode with a large tail fin.
6
Spring 2003 16.61 AC 3–7
Aircraft Actuator Influence
? Transfer functions dominated by lightly damped Dutch-roll mode.
– Note the rudder is physically quite high, so it also influences the A/C
roll.
– Ailerons influence the Yaw because of the di?erential drag
? Impulse response for the two inputs:
– Rudder input
a51 β shows a very lightly damped decay.
a51 p, r clearly excited as well.
a51 φ oscillates around 2.5
?
? Dutch-roll oscillations are clear.
? Spiral mode ultimately dominates φ → 0 after 250 sec.
– Aileron input
a51 Large impact on p
a51 Causes large change to φ
a51 Very small change to remaining variables.
a51 Influence smaller than Rudder.
? Lateral approximate models are much harder to make (see discussion in
Etkin and Reid). Not worth discussing at length.
7
Spring 2003 16.61 AC 3–8
0 2 4 6 8 10 12 14 16 18 20
?4
?3
?2
?1
0
1
2
time sec
Rudder Impulse
Beta
P
R
Phi
Figure 1: Rudder impulse to flight variables. The rudder excites all modes. Dutch
roll oscillations dominate initially. The spiral mode dominates longer term.
8
Spring 2003 16.61 AC 3–9
0
2
4
6
8
10
12
14
16
18
20
?0.25
?0.2
?0.15
?0.1
?0.05
0
0.05
0.1
time sec
Aileron Impulse
BetaP R Phi
Figure 2: Aileron impulse to flight variables. Response primarily in φ.
9