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Lateral-Directional Aircraft Dynamics
Under Static Moment Nonlinearity
Tiauw Hiong Go
*
Nanyang Technological University,
Singapore 639798, Republic of Singapore
DOI: 10.2514/1.39025
I. Introduction
T
HIS Note specifically focuses on the aircraft lateral-
directional dynamics in the presence of nonlinearity in lateral
moments with respect to sideslip (static lateral moments). Such
nonlinearity is often encountered during flight in the regimes in
which the aerodynamic properties are nonlinear: for example,
during high-angle-of-attack flight. In the previous works by the
author [1–3], which focus on the multiple-degree-of-freedom
wing-rock dynamics at high angles of attack, the results imply that
by itself, the nonlinearity in the static lateral moments [such as that
considered here (cubic polynomial with respect to sideslip)] does
not lead to wing rock. Those analyses, however, consider only the
situation in which the nonlinearity is relatively weak. In [4], cubic
nonlinearity of static lateral moments with respect to sideslip is
shown to cause wing rock in a multimode system in certain
conditions. Because of these seemingly discrepant findings, the
topic is revisited here.
This Note presents detailed analysis of the lateral-directional
motion when the cubic type of static lateral moment nonlinearity is
present in the system. The equations of motion as derived in [3] are
used as the basis for the analysis. To gain physical insight into the
problem, an analytical approach using the multiple-time-scales
(MTS) method is used (see, for example, [5,6]). Both weak and
strong nonlinearity cases are considered.
II. Equations of Motion
Figure 1 shows an example of nonlinear variations in rolling and
yawing moment coefficients C
l
and C
n
with respect to the sideslip
angle . The variations shown in the figure are typical for fighter
aircraft at high angles of attack, although the strength of the
variations may vary with configurations and flight conditions. Such
nonlinearity can be represented by adding cubic terms in the
expression for the lateral moments. All other nonlinearities are
neglected in the current analysis, and thus the lateral moments can be
expressed as follows:
L=I
xx
L
L
1
3
L
p
p L
r
r L
_
_
N=I
zz
N
N
1
3
N
p
p N
r
r N
_
_
(1)
where p is roll rate; r is yaw rate; is sideslip angle;
_
is sideslip rate;
I
xx
and I
zz
are the moments of inertia of the aircraft about its body x
and z axes, respectively; L and N are the total rolling and yawing
moments, respectively; L
i
and N
i
, where i , p, r, and
_
are the
usual linear stability derivatives; and L
1
and N
1
are the coefficients
of the cubic sideslip terms, which represent the nonlinearity. In this
formulation, the coupling between the longitudinal and the lateral-
directional modes is neglected and the lateral-directional modes are
analyzed separately.
In [3], the nonlinear equations describing the attitude dynamics of
a rigid aircraft with a conventional configuration with three rotational
degrees of freedom have been derived. In the derivation, the
trajectory of the center of mass of the aircraft is assumed to be straight
and horizontal and is not affected by its attitude motion. The lateral-
directional part of the equations are used as the basis of the analysis
here and will not be rederived. With the preceding simplifying
assumption on the nonlinearity, the equations of motion can be
written as follows:
!
2
1
~
1
_
~
2
~
2
_
~
3
~
1
~
3
~
1
_
~
2
_
~
3
(2)
where is the roll angle and
!
2
1
1=1 n
1
n
3
L
n
1
N
s N
n
3
L
c
~
1
1=1 n
1
n
3
n
3
L
r
N
r
L
r
n
1
N
r
t
L
_
n
1
N
_
s n
3
L
_
N
_
c
~
2
g=Vc=1 n
1
n
3
N
r
n
3
L
r
L
r
n
1
N
r
t
~
2
g=Vc 1=1 n
1
n
3
L
p
L
r
t n
1
N
p
N
r
ts
N
p
N
r
t n
3
L
p
L
r
tc
~
1=1 n
1
n
3
L
1
n
1
N
1
s n
3
L
1
N
1
c
~
1
1=1 n
1
n
3
L
n
1
N
~
3
g=VL
r
n
1
N
r
=1 n
1
n
3
~
1
1=1 n
1
n
3
L
p
n
1
N
p
L
r
n
1
N
r
t
~
2
1=1 n
1
n
3
L
_
n
1
N
_
L
r
n
1
N
r
=c
~ 1=1 n
1
n
3
L
1
n
1
N
1
(3)
where g is the gravitational acceleration; V is the nominal airspeed;
n
1
and n
3
are defined as I
xz
=I
xx
and I
xz
=I
zz
, respectively; and c, s,
and t denote cos
0
, sin
0
, and tan
0
, respectively, where
0
is the
nominal angle of attack of the flight. Note that the coefficients of the
nonlinear lateral moment terms L
1
and N
1
appear only in
~
and ~ .
III. Weak Nonlinearity Case
As in [3], Eq. (2) is parameterized by imposing several
assumptions. First, only the case in which the damping terms are
Received 10 June 2008; revision received 20 August 2008; accepted for
publication 25 August 2008. Copyright © 2008 by Tiauw Hiong Go.
Published by the American Institute of Aeronautics and Astronautics, Inc.,
with permission. Copies of this paper may be made for personal or internal
use, on condition that the copier pay the $10.00 per-copy fee to the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include
the code 0731-5090/09 $10.00 in correspondence with the CCC.
*
Assistant Professor, School of Mechanical and Aerospace Engineering,
50 Nanyang Avenue. Senior Member AIAA.
JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS
Vol. 32, No. 1, January–February 2009
305