Note : Les descriptions sont présentées dans la langue officielle dans laquelle elles ont été soumises.
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METHOD FOR DESIGNING A FIN DEPLOYMENT MECHANISM
FIELD OF THE INVENTION
The invention relates to a method for designing a deployment mechanism for
a flight surface on a guided projectile, or a guided airborne body such as a
bomb,
dispenser, munition or missile.
BACKGROUND OF THE INVENTION
Guided airborne bodies such as munitions or missiles are routinely
constructed with flight surface deployment systems to provide control and
extended
range capability. These systems include one or more control mechanisms for
moving a foldable and/or retractable flight surface to move the flight surface
from a
stowed position to an active or deployed position.
Recently, because of volume constraints inside the airborne body, designs of
airborne bodies have required the flight surface and/or control mechanism to
be
stored on, or partially on, the exterior of the airborne body. This has
increased the
complexity of deployment systems, in particular, those in which the flight
surface is
rotated, in sequence, about multiple axes to, for example, position the flight
surface
at a sweep angle or other obliquity relative to the airborne body axis. Such
deployment systems include a complex arrangement of parts such as actuators
and
lock mechanisms to enable achievement of the desired active or deployed
position
of the flight surface.
To reduce the number of rotations and axes about which the surface is
rotated and, consequently, the number of parts of the deployment system,
attempts
have been made to simulate the multiple axes rotations with a single
equivalent axis
and single rotation. Although satisfactory, such attempts have not been
without
difficulty. For example, according to at least one known method, numerous axes
and angles are attempted on a trial and error basis until a suitable single
axis and
single rotation angle are obtained. Such a method is inefficient and tedious.
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Thus, a need exists for an effective method for designing a deployment
mechanism for a flight surface on an airborne body. Such a method preferably
minimizes intrusive volume, the number of parts, and the complexity of the
deployment mechanism. Moreover, such a method preferably simplifies the
manner by which a single axis and single rotation angle may be obtained about
and through which a flight surface is rotatable.
SUMMARY OF THE INVENTION
The present invention concerns a method for designing a deployment
mechanism for a flight surface on an airborne body, comprising the steps of:
determining a stowed position of the flight surface;
determining a deployed position of the flight surface;
identifying a first rotation axis (k) and respective rotation angle (e) and a
second rotation axis (k') and respective rotation angle (e) about and through
which the flight surface is rotatable in sequence to move the flight surface
from the
stowed position to the deployed position, or vice versa; and
using the identified first and second rotation axes (k, k') and rotation
angles
(e, E') to determine a single equivalent rotation axis (kr) and angle (Er),
about and
through which the flight surface can be rotated from the stowed position to
the
deployed position, or vice versa;
wherein the equivalent rotational angle (g) is obtained from the equation:
- ,
s, = 2 cos-t ~cos 2 cos 2- sin 2 sin 2 cos 6l
J
wherein the equivalent rotation axis (kr) is obtained from the equation:
8 S' 8' 8 , 8, g'
sin-cos- sin---cos- sm2 sin 2
kk 2 2+k' 2 2+k'xk
r sin ~ sin 2 sin 2
where 6 is the angle between the rotation axes k and k'.
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The foregoing and other features of the invention are hereinafter more fully
described and particularly pointed out in the claims, the following
description and
the annexed drawings setting forth in detail illustrative embodiments of the
invention, such being indicative, however, of but a few of the various ways in
which the principles of the invention may be employed.
BRIEF DESCRIPTION OF THE DRAWINGS
Fig. I is a flow chart of a method for designing a deployment mechanism
for a flight surface in accordance with the present invention.
Figs. 2-7 schematically illustrate an embodiment of first and second rotation
axes and respective rotation angles about and through which the flight surface
is
rotatable in sequence to move the flight surface from a stowed position to a
deployed position.
Figs. 8 and 9 schematically illustrate the coordinate systems of the first and
second rotation axes shown in Figs. 2-7 transposed relative to one another.
Figs. 10 and 11 schematically illustrate a single equivalent rotation axis and
single equivalent rotation angle based on the first and second rotation axes
and
respective rotation angles of Figs. 2-7, about and through which the flight
surface
may be rotated from the stowed position to the deployed position, or vice
versa, in
accordance with the invention.
Figs. 12-15 schematically illustrate another embodiment of first and second
rotation axes and respective rotation angles about and through which the
flight
surface is rotatable in sequence to move the flight surface from a stowed
position
to a deployed position.
Fig. 16 schematically illustrates the coordinate systems of the first and
second rotation axes shown in Figs. 12-15 transposed relative to one another.
Figs. 17 and 18 schematically illustrate a single equivalent rotation axis and
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single equivalent rotation angle based on the first and second rotation axes
and
respective rotation angles of Figs. 12-15, about and through which the flight
surface
may be rotated from the stowed position to the deployed position, or vice
versa, in
accordance with the invention.
DETAILED DESCRIPTION
Referring now in detail to the drawings, and initially to Fig. 1, there is
shown a
flow chart 100 of a method for designing a deployment mechanism for a flight
surface in accordance with the present invention. Initially, at step 110, a
stowed
position and a deployed position of the flight surface are identified. Figs. 2
and 12,
which are described in greater detail below, show exemplary stowed positions
and
Figs. 7 and 15, which also are described in greater detail below, show
exemplary
deployed positions.
In step 120, a first rotation axis and a respective rotation angle are
identified
about and through which the flight surface may be rotated from the stowed
position
towards the desired deployed position. In step 130, a second, or successive,
rotation axis and a respective rotation angle are identified about and through
which
the flight surface may be rotated to position the flight surface in the
deployed
position. In step 140, a single equivalent rotation axis and angle are
determined
based on the first and second rotation axes and rotation angles, about and
through
which the flight surface may be rotated to achieve the deployed position.
Several advantages are realized by the method according to the present
invention. Because two axes and successive rotations are replaced by a single
equivalent axis and rotation, the amount of parts and, consequently, the
volume
occupied by the deployment mechanism, is minimized. Moreover, by minimizing
the
number of parts, the structure of the deployment mechanism may be simplified.
Also, the method simplifies the manner by which a single axis and single
rotation
angle may be obtained about and through which a flight surface is rotatable.
The following examples demonstrate the method according to the present
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invention and the advantages associated therewith.
Example I
Figs. 2-7 schematically illustrate an embodiment of a flight surface 204
rotated in sequence from a stowed position to a deployed position. To
facilitate
describing the invention, the flight surface 204 is shown transposed in a cube
with
three of its sides marked with, respectively, the letters A, B and C. Fig. 2
shows the
flight surface 204 in a stowed position. As is shown in Fig. 2, an orthogonal
coordinate system (axes i, j and k) is assigned to the flight surface 204, the
coordinate system representing the flight surface 204 in its stowed position.
When
the flight surface 204 is in its stowed position, the flight surface 204 lies
in a plane
parallel to the i and j axes, and a proximal end 206 of the flight surface 204
is
parallel to the j axis and perpendicular to the plane defined by the i and k
axes. The
coordinate system serves as a reference datum from which a first rotation of
the
flight surface 204 is measured.
Referring to Fig. 3, k and e represent the first rotation axis and the
respective
first rotation angle about and through which the flight surface 204 is rotated
from the
stowed position to a position intermediate the stowed position and the desired
deployed position. In the illustrated embodiment, the first rotation angle e
is
measured counterclockwise about the k axis. Once rotated, an orthogonal
coordinate system (axes i1, j1 and k) is assigned to the flight surface 204,
wherein i9
and j9 represent unit vectors obtained by rotating i and j, respectively,
through the
angle e. The i9, j1 and k coordinate system represents the flight surface 204
in the
first, or intermediate, rotated position. The first rotation is one of two
successive
rotations in which the flight surface 204 is moved from its stowed position
(Fig. 2) to
its extended or deployed position (Fig. 7). Fig. 4 shows the flight surface
204
rotated through an exemplary first rotation angle c of 90 degrees
counterclockwise
about the k axis.
Referring now to Fig. 5, a second orthogonal coordinate system (axes i', j'
and k) is assigned to the flight surface 204. The Fig. 5 coordinate system
also
represents the flight surface 204 in its intermediate position but serves as a
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reference datum from which a second rotation of the flight surface 204 is
measured.
As is described in greater detail below, assigning the i', j' and k'
coordinate system
to the intermediate position facilitates analyzing the relationship of the
stowed and
deployed positions relative to the intermediate position of the flight surface
204.
In a manner similar to the identification of the first rotation axis k and
first
rotation angle e, as shown in Fig. 6, a second rotation axis k' and a
respective
second rotation angle E' about and through which the flight surface 204 is
rotated
from the intermediate position to the desired deployed position, are
identified. In the
illustrated embodiment, the second rotation angle e' is measured
counterclockwise
about the k'axis. Once rotated, an orthogonal coordinate system (axes i'1, j'l
and
k) (Figs. 6 and 7) is assigned to the flight surface 204, wherein i'l and j'l
represent
unit vectors obtained by rotating i'and j', respectively, through the angle
e'. The i'1,
j'l and k' coordinate system represents the flight surface 204 in a second
rotated
position (i.e., the deployed position). The second rotation is the second of
the two
successive rotations in which the flight surface 204 is moved from its stowed
position (Fig. 2) to its extended or deployed position (Fig. 7). In Fig. 7,
the flight
surface 204 is shown rotated through an exemplary second rotation angle e'of
105
degrees counterclockwise about the k' axis.
Having identified the first and second rotation axes k and k' and respective
rotation angles e and e' about and through which the flight surface 204 is
successively rotated, a single equivalent rotation axis and rotation angle may
then
determined. One way to determine the single axis and rotation angle is by
application of Euler's Theorem, which provides that a single axis and rotation
angle
may be derived from two axes and respective successive rotations. To
facilitate use
of Euler's Theorem in the present embodiment, the aforementioned i9, j9 and k
intermediate coordinate system (Fig. 4) and the i', j' and k' intermediate
coordinate
system (Fig. 5) are transposed relative to one another according to the manner
shown in Fig. 8. As is shown in Fig. 8, i9 coincides with i', and i9 and i'
are
perpendicular to k and k'. In this way, j1 and j' are in the plane of k and
k'. The
angle 0 represents the angle between k and k'.
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Referring to Fig. 9, assuming that the first and second rotations c and e' are
about mutually perpendicular axes, then the angle 0 between k and k' is 90
degrees
(i.e., axes k and k', are perpendicular). Applying Euler's Theorem, a single
equivalent angle (gr ) may be obtained from the equation:
$r = 2 cos-' ~ cos ~ cos ~- sin 2 sin 2r
cos Bl
l
and a single equivalent rotation axis (kr ) may be obtained from the equation:
s 8' 8' 6 6 s'
sin-cos- sin-cos- sin-sin-
k,.=k 2 2+k' 2 2+k'xk 2 2
sin ~ sin ~ sin ~
where e is the first rotation angle, e' is the second rotation angle, and 6 is
the angle
between the first and second rotation axes k and k'.
Based on the foregoing, for a first rotation angle e of 90 degrees, a second
rotation angle e'of 105 degrees, and an angle 0 of 90 degrees between the
first and
second rotation axes k and k', the single equivalent rotation is:
e r = 129 degrees
and the single equivalent rotation axis is:
k r = 0.477 k + 0.622 k' + 0.622 k' x k
However, from the relationships of the above coordinate systems, the
following equations are applicable:
j1 sin6 = k cos8 - k',
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j'sin8 = k - k' cos6, and
i sinA = k x k'
Thus, transposing the equivalent axis to the coordinate system i, j and k of
the stowed position results in the following single equivalent axis:
k r= 0.622 i- 0.622 j+ 0.477 k
Fig. 10 shows the single equivalent axis k r transposed in the i, j and k
coordinate system with the flight surface 204 in a stowed position. Fig. 11
shows
the flight surface 204 rotated through the single equivalent rotation angle e
, of 129
degrees counterclockwise about the single equivalent axis k, As can be seen by
comparing Figs. 7 and 11, the same result obtains if the flight surface 204 is
successively rotated about and through first and second rotation axes and
respective rotation angles (Figs. 2-7), than if the flight surface 204 is
rotated about
and through a single equivalent rotation axis and rotation angle (Figs. 10 and
11).
Example 2
Figs. 12-15 schematically illustrate another embodiment of a flight surface
304 rotated in sequence from a stowed position to a deployed position. To
facilitate
describing the invention, the flight surface 304 is shown transposed in a cube
with
three of its sides marked with, respectively, the letters A, B and C. Fig. 12
shows
the flight surface 304 in a stowed position. As is shown in Fig. 12, an
orthogonal
coordinate system (axes i, j and k) is assigned to the flight surface 304, the
coordinate system representing the flight surface 304 in its stowed position.
When
the flight surface 304 is in its stowed position, the flight surface 304 lies
in a plane
parallel to the i and j axes, and a proximal end 306 of the flight surface 304
is
parallel to the j axis and perpendicular to the plane defined by the i and k
axes. The
coordinate system serves as a reference datum from which a first rotation of
the
flight surface 304 is measured.
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Referring to Fig. 13, k and E represent the first rotation axis and the
respective first rotation angle about and through which the flight surface 304
is
rotated from the stowed position to a position intermediate the stowed
position and
the desired deployed position. Here, the flight surface 304 is rotated through
an
exemplary first rotation angle e of 75 degrees counterclockwise about the k
axis. An
orthogonal coordinate system (axes i1, j9 and k) is assigned to the flight
surface
304, and represents the flight surface 304 in the first, or intermediate,
rotated
position.
As shown in Fig. 14, a second orthogonal coordinate system (axes i', j' and k)
is assigned to the flight surface 304 which represents the flight surface 304
in its
intermediate position but serves as a reference datum from which a second
rotation
of the flight surface 304 is measured. Referring to Fig. 15, a second rotation
axis k'
and a respective second rotation angle e' about and through which the flight
surface
304 is rotated from the intermediate position to the desired deployed
position, are
identified. Here, the flight surface 304 is shown rotated through an exemplary
second rotation angle e'of 50 degrees counterclockwise about the k' axis. Once
rotated, an orthogonal coordinate system (axes i'l, j'l and k) (Fig. 15) is
assigned to
the flight surface 304, wherein i'l and j'1 represent unit vectors obtained by
rotating
i' and j', respectively, through the angle E. The i'1, j'l and k` coordinate
system
represents the flight surface 304 in a second rotated position (i.e., the
deployed
position).
Having identified the first and second rotation axes k and k' and respective
rotation angles e and e' about and through which the flight surface 304 is
successively rotated, a single equivalent rotation axis and rotation angle may
then
determined. To facilitate use of Euler's Theorem, the i1, j1 and k coordinate
system
(Fig. 13) and the i', j' and k'coordinate system (Fig. 14) are transposed
relative to
one another according to the manner shown in Fig. 16. As is shown in Fig. 16,
i1
coincides with i', and i1 and i' are perpendicular to k and k'. In this way,
j1 and j' are
in the plane of k and k'. The angle 0 represents the angle between k and k'.
Referring to Fig. 16, assuming that the first and second rotations c and E'
are
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about mutually perpendicular axes, then the angle 0 between k and k' is 90
degrees
(i.e., axes k and k', are perpendicular). Applying Euler's Theorem, for a
first rotation
angle c of 75 degrees, a second rotation angle e'of 50 degrees, and an angle 0
of
90 degrees between the first and second rotation axes k and k', the single
equivalent rotation is:
E P = 88 degrees
and the single equivalent rotation axis is:
k r = 0.794 k + 0.482 k' + 0.370 k' x k
From the relationships of the above coordinate systems, transposing the
equivalent axis to the coordinate system i, j and k of the stowed position
results in
the following single equivalent axis:
k r= 0.482 i- 0.370 j+ 0.794 k
Fig. 17 shows the single equivalent axis k r transposed in the i, j and k
coordinate system with the flight surface 304 in a stowed position. Fig. 18
shows
the flight surface 304 rotated through the single equivalent rotation angle e
r of 88
degrees counterclockwise about the single equivalent axis k r. As can be seen
by
comparing Figs. 15 and 18, the same result obtains if the flight surface 304
is
successively rotated about and through first and second rotation axes and
respective rotation angles (Figs. 12-15), than if the flight surface 304 is
rotated
about and through a single equivalent rotation axis and rotation angle (Figs.
17 and
18).
Although the invention has been shown and described with respect to certain
preferred embodiments, equivalent alterations and modifications will occur to
others
skilled in the art upon reading and understanding this specification and the
annexed
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drawings. In particular regard to the various functions performed by the above
described integers (components, assemblies, devices, compositions, etc.), the
terms
(including a reference to a "means") used to describe such integers are
intended to
correspond, unless otherwise indicated, to any integer which performs the
specified
function of the described integer (i.e., that is functionally equivalent),
even though
not structurally equivalent to the disclosed structure which performs the
function in
the herein illustrated exemplary embodiment or embodiments of the invention.
In
addition, while a particular feature of the invention may have been described
above
with respect to only one of several illustrated embodiments, such feature may
be
combined with one or more other features of the other embodiments, as may be
desired and advantageous for any given or particular application.
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