Numerical Continuation Analysis of a Dual-sidestay
Main Landing Gear Mechanism
James A. C. Knowles
∗
Oxford Brookes University, Oxford, OX33 1HX, UK
Bernd Krauskopf
†
The University of Auckland, Auckland 1142, NZ
Mark H. Lowenberg
‡
and Simon A. Neild
‡
University of Bristol, Bristol, BS8 1TR, UK
P. Thota
§
Airbus in the UK, Bristol, BS99 7AR, UK
A model of a three-dimensional dual-sidestay landing gear mechanism is presented and
employed in an investigation of the sensitivity of the downlocking mechanism to attachment
point deflections. A motivation for this study is the desire to understand the underlying
nonlinear behaviour, which may prevent a dual-sidestay landing gear from downlocking
under certain conditions. The model formulates the mechanism as a set of steady-state
constraint equations. Solutions to these equations are then continued numerically in state
and parameter space, providing all state parameter dependencies within the model from
a single computation. The capability of this analysis approach is demonstrated with an
investigation into the effects of the aft sidestay angle on retraction actuator loads. It was
found that the retraction loads are not significantly affected by the sidestay plane angle,
but the landing gear’s ability to be retracted fully is impeded at certain sidestay plane
angles. This result is attributed to the landing gear’s geometry, as the locklinks are placed
under tension and cause the mechanism to lock. Sidestay flexibilities and attachment
point deflections are then introduced to enable the downlock loads to be investigated. The
investigation into the dual sidestay’s downlock sensitivity to attachment point deflections
yields an underlying double hysteresis loop, which is highly sensitive to these deflections.
Attachment point deflections of a few millimetres were found to prevent the locklinks from
automatically downlocking under their own weight, hence requiring some external force
to downlock the landing gear. Sidestay stiffness was also found to influence the downlock
loads, although not to the extent of attachment point deflection.
∗
Lecturer in Mechanical Engineering, Department of Mechanical Engineering and Mathematical Sciences, Wheatley Campus,
Wheatley, Oxford, OX33 1HX, UK
†
Professor of Applied Mathematics, Department of Mathematics, Private bag 92019, Auckland 1142, NZ
‡
Readers in Dynamics, Faculty of Engineering, Queen’s Building, University Walk, Bristol, BS8 1TR, UK
§
Model Developer – Physical Systems, Airbus in the UK, New Filton House, Filton, Bristol, BS99 7AR, UK
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I. Introduction
Conventional Main Landing Gears (MLGs) have a single-sidestay to support the shock strut when the
gear experiences side loads (e.g. under high-speed cornering on the ground). Some MLGs may have a second
structural drag stay to support the shock strut against aerodynamic drag loading. Others feature an angled
sidestay to absorb both types of loads (lateral ground and aerodynamic drag loads). With the increasing use
of new materials (such as carbon fibre composites) in new aircraft primary structural elements, landing gear
designs are having to evolve to meet new design constraints. Whilst composite materials offer large potential
weight savings due to their high strength, they are not as good as metals at absorbing point loads. This
provides a challenge when integrating the landing gear into a carbon fibre wing-box section, because the
attachment points (where the landing gear meets the wing-box) transfer very large loads into the airframe.
In order to be able to integrate the landing gear into a carbon fibre wing, the loads at the attachment
points must be reduced. One solution could be to increase the number of landing gears on the aircraft, thus
reducing the load on each gear when the aircraft is manoeuvring on the ground. An alternative solution,
which has been adopted by both Boeing
1
and Airbus
2
for the main landing gears on both of their latest
aircraft, is to add a second sidestay into the mechanism; this is referred to as a dual-sidestay main landing
gear (DSS MLG). The presence of two sidestays spreads the loads transfered from the gear to the wingbox,
allowing the DSS MLG to be integrated into a carbon fibre wing.
Whilst DSS MLGs provide a solution to integrating a landing gear into a composite wing structure, the
nature of the DSS mechanism presents challenges in itself due to its sensitivity to changes in MLG parameters,
such as attachment point positions and aerodynamic drag. The mechanism is particularly sensitive to these
parameters around the downlock point, which is the state of the landing gear defined as separating the
‘unlocked’ and ‘downlocked’ states. In this state the two locklink links align with one another, and at the
same time the upper and lower sidestay links are also very close to aligning. The reasons for the sensitivity
of DSS MLGs near the downlock point are not fully understood.
The literature on landing gear mechanism analysis is limited and relatively old,
3, 4
and it focuses on
the kinematic aspects of the landing gear mechanism from a preliminary design perspective. There are
currently no examples in the public domain of DSS MLG mechanism modelling. The vast majority of
previous work into landing gear modelling has tended to focus on capturing the landing gear properties
under ground loading
5–8
by building relatively complex dynamic models using dynamic simulation software
packages (such as Dymola or ADAMS). These models are very good at capturing many different aspects of
the physical system, and they can provide quantitatively accurate results for a specific system of interest.
On the other hand, complex dynamic models are less suitable to developing an understanding of general
underlying nonlinear behaviour. This is because the model parameters that can be simulated continuously
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within the model are often limited to externally applied forces. If, for example, the MLG geometry was to
be investigated, the model would need to be adjusted and multiple time histories conducted for different
(discrete) geometries. Not only is this a time-consuming process, but areas of highly nonlinear behaviour
may be missed if a relevant geometry is not simulated.
The approach presented here expresses the mechanism as a set of steady-state constraint equations, which
are solved simultaneously with the method of numerical continuation. Tools from Bifurcation Theory,
9–11
including numerical continuation, have been used to help understand nonlinear problems in Aerospace ap-
plications before.
12–14
For all of these applications, numerical continuation was shown to provide significant
advantages over alternative analysis methods. This paper outlines a mechanism modelling approach that
enables the use of numerical continuation methods to analyse DSS MLG mechanisms. The following section
briefly describes the model; it is self-contained and builds on our previous work modelling single-stay NLG
and MLG mechanisms.
12, 15
A formulation validation is then presented by starting from the case of a single-
sidestay MLG and then ‘rotating out’ an extra sidestay. Subsequently, continuation results for a DSS MLG
are presented with an emphasis on downlock sensitivity to sidestay flexibility. The final section presents
some concluding remarks and offers an outlook on future model advancements that could be introduced to
increase the applicability of the results.
II. Model details
The model used in this work was derived using newtonian mechanical principles. Because
of this, the equations within this section are presented in their entirety.
.
.
A
O
B
C
D
E
G
H
I
J
K
L
2
L
3
L
1
L
4
L
5
L
6
L
7
L
8
L
9
X
Z
A, G
O
B,H
C, I
D, J
E,K
L
2
, L
6
L
3
, L
7
L
1
L
4
, L
8
L
5
, L
9
Y
Z
A
G
O
B,H
ˆn
a
ˆn
f
Y
X
α
f
α
a
Figure 1: Three-view of a symmetrical DSS MLG arrangement, with joints, locklinks and sidestay plane
normal vectors shown.
Figure 1 shows the DSS MLG geometry considered, with a main vertical shock strut supported by two
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folding sidestays. For the landing gear considered in this work, the rake angle is taken to be zero. The
sidestays are attached to the shock strut at points slightly offset from the shock strut centreline. The other
end of each sidestay is attached to the airframe at points A and G. The locklinks, attached between the
sidestay joints and the shock strut, lock the gear in position when deployed. Locklink configurations differ
from landing gear to landing gear; one of the locklinks on the Boeing 787 DSS MLG, for example, is attached
to the sidestays and the airframe (rather than the sidestays and the shock strut as considered here). The
model of the DSS MLG considered here is a development of the single sidestay MLG model formulation
presented previously;
15
as such, the notation follows a similar convention. As Figure 1 shows, the DSS MLG
mechanism consists of nine links, which are initially assumed to be rigid bodies with uniformly distributed
mass along their lengths. Each link, L
i
, is connected to another link or the aircraft structure via rotational
joints; the majority of the joints labelled in Figure 1 are planar joints, with the exception of joints A, B,
G and H which are spherical joints that allow connected bodies to rotate about the joint freely in three-
dimensions. The X-axis is defined as the shock strut rotation axis, with the shock strut rotation joint at the
global co-ordinate origin point O. The gear is defined to retract in the positive (Y ,Z)-plane and the Z-axis
is aligned with the global gravity vector, positive down.
Due to the presence of two sidestays, two transformation matrices are required to define the two sidestay
rotation planes — a fore and aft plane. These two transformation matrices are defined in terms of two
normal vectors ˆn
f
and ˆn
a
for the fore and aft sidestay planes, respectively, as given by:
ˆn
f
= OA × OB (1a)
ˆn
a
= OG × OH (1b)
The two sidestay local co-ordinate systems can now be defined with two rotation matrices. The fore
rotation matrix T
f
describes rotations about the global origin point O, which aligns the local fore x-axis
(x
f
) with ˆn
f
by a rotation over α
f
about the global Y -axis, followed by a rotation through β
f
about the
intermediate z-axis:
T
f
=
cos β
f
cos α
f
−sin β
f
cos β
f
sin α
f
sin β
f
cos α
f
cos β
f
sin β
f
sin α
f
−sin α
f
0 cos α
f
. (2)
The aft rotation matrix T
a
is also a transformation about the global origin point O, but one that aligns
the local aft x-axis (x
a
) with ˆn
a
by a rotation over α
a
about the global Y -axis followed by a rotation through
β
a
about the intermediate z-axis:
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T
a
=
cos β
a
cos α
a
−sin β
a
cos β
a
sin α
a
sin β
a
cos α
a
cos β
a
sin β
a
sin α
a
−sin α
a
0 cos α
a
. (3)
The two local co-ordinate systems are therefore related to the global (X, Y, Z) co-ordinates as follows:
x
f
y
f
z
f
= T
f
X
Y
Z
, (4a)
x
a
y
a
z
a
= T
a
X
Y
Z
. (4b)
The equations are formulated by considering each link L
i
within the mechanism as an individual rigid
body in static equilibrium. This method has been previously introduced,
12, 15
and is now extended to the
case of a DSS MLG.
A. Link description and co-ordinate systems
Figure 2 depicts the general naming convention used for each link within the landing gear mechanism in local
fore (a) and aft (b) co-ordinates. Each link is described in terms of seven elements, L
i
= {X
i
, Y
i
, Z
i
, ˆn, θ
i
, L
i
, m
i
},
where:
• L
i
is the i
th
link;
• X
i
, Y
i
, Z
i
are the global Cartesian co-ordinates which describe the position of L
i
’s centre of gravity
(cg);
• ˆn is the normal vector to L
i
’s plane of rotation, i.e. perpendicular to the page in Figure 2;
• θ
i
is the local rotation of L
i
relative to the local y-axis
a
;
• L
i
is the length of L
i
;
• m
i
is the mass of L
i
, assumed to be evenly distributed along L
i
.
a
For the main strut L
1
a global rotation Θ
1
is used to define the link: see Figure 3 for graphical representation. The
corresponding local rotations θ
f
1
and θ
a
1
(shown in Figure 2) are functions of Θ
1
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