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coordinate system-based origin permits constraints established in the kinematic reference
frame, e.g., assembly clearance envelope, retraction path, and swept volume, be translated
into the aircraft reference frame and checked for interference with surrounding structures.
6.3.1. The Pivot Axis and Its Direction Cosines
In the determination of the alignment of the landing gear pivot axis, it is assumed that
the axle/piston centerline intersection is brought from its deployed position to a given
location within the stowage volume. For wing-mounted assemblies, the retracted position
of axle/piston centerline intersection is assumed to coincide with the center of the stowage
volume. In the case of fuselage-mounted assemblies with a forward-retracting scheme, the
retracted position is assumed to be at the center of the cross-sectional plane located at the
forward third of the stowage length.*
* Note: to reduce structural cut-away, many forward retracting gears have shrink mechanisms. In
particular, it appears that the Airbus A 330 and A340 aircraft may have shrink struts on the main gear.
This consideration is neglected in the current analysis, but probably should be considered.
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y
z
x
Aircraft reference frame
Kinematic reference frame
y
z
x
Tires in the deployed position
xgear
ygear
zgear
Figure 6.1 Relationships between the aircraft and kinematic reference frames
6.3.1.1. The Fuselage-mounted Assembly
For fuselage-mounted assemblies with a forward retracting-scheme, the pivot axis is
defined by the cross product of the space vectors corresponding to the deployed and
retracted position of a point location on the truck assembly. As shown in Fig. 6.2, the cross
product of two vectors (V1 and V2) representing the deployed and retracted positions of a
given point location, here taken as the axle/piston centerline intersection, is orthogonal to
both vectors, i.e., in the direction of the pivot axis. Thus,
V = V1 ´ V2 (6.1)
From standard vector operation, the direction cosines of the fuselage-mounted assembly is
given as
l
X
X Y Z
m
Y
X Y Z
n
Z
X Y Z
=
+ +
=
+ +
=
2 2 2 2 2 2 2 + 2 + 2
(6.2)
and the angle between the two vectors, i.e., the angle of retraction (f full) in this case, is
calculated using the expression
49
cosf full = l1l2 + m1m2 + n1n2 (6.3)
where li, mi, and ni are the respective direction cosines of the deployed and retracted space
vectors.
y
V, pivot axis
V1
V2
z
x
ffull
Figure 6.2 Fuselage-mounted assembly pivot axis alignment
6.3.1.2. The Wing-mounted Assembly
The determination of the wing-mounted assembly pivot axis involves the deployed and
retracted positions of two points on the assembly. Essentially, the problem consists of
bringing the line segment between the two points from its deployed position to its retracted
position [31]. For ease of visualization, a twin-wheel configuration is used here to illustrate
the procedure involved in determining the alignment of the desired pivot axis. Identical
procedure is used for other configurations as well.
As shown in Fig. 6.3, the axle/piston centerline intersection is selected as the first point
(point A), while the second point (point B) is conveniently located at a unit distance along
the axle, inboard from the first point location. retracted positions of the first and second
points are given as point A’ and B’, respectively.
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V1
V2
V3
V4
Ur
Point A
Point B
Point A’
Point B’
y
z
x
Figure 6.3 Vector representation of the wing-mounted landing gear
Of the four point positions required in the analysis, the positions of point A and A’ are
readily determined from the geometry of the landing gear and the stowage volume,
respectively. From simple vector algebra
V2 = V1 + $j (6.4)
where subscripts 1 and 2 denote the space vector corresponding to the deployed positions
of points A and B, respectively. Similarly,
V4 = V3 + Ur (6.5)
where subscript 3 and 4 denote the retracted positions of point A and B, respectively, and
Ur defines the orientation of the unit vector in its retracted position and is unknown.
To solve for Ur, it is assumed that no devices are used to shorten the length of the strut
during the retraction process, i.e., that the magnitudes of V2 and V4 remain constant,
X1 (Y ) Z (X XU) (Y YU) (Z ZU)
2
1
2
1
2
3
2
3
2
3
2 + +1 + = + + + + + (6.6)
　
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