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EDF/IMA/MN
Manual of Reference
R5.03 booklet: Nonlinear mechanics
Document: R5.03.40
Static and dynamic modeling of the beams
in great rotations
Summary:
This note gives a mechanical formulation of the beams in great displacements and great rotations but
with an elastic behavior. The essential difficulty of the analysis of rotations holds so that they are not
not commutable and a vector space, but a variety does not constitute.
At any moment, the configuration of a cross-section of beam is defined by the vector-displacement of sound
center of gravity and the vector-rotation of the system of the main axes of inertia compared to a position of
reference. As in classical theory of the beams, the interior efforts are reduced to their resultant and them
moment on the locus of centers of the sections. The associated deformations are defined.
The linearization of the interior efforts compared to displacements leads to the matrix of usual rigidity, which
is symmetrical, and, because of great displacements and rotations, with the geometrical matrix of rigidity, which is
unspecified.
The linearization of the inertias carries out, for the translatory movement, with the matrix of usual mass
who is symmetrical and, for the rotational movement, with a matrix much more intricate and without any
symmetry.
The diagram of temporal integration is that of Newmark.
This modeling was tested on five problems of reference: three of statics and two of dynamics.
This work was undertaken within the framework of the PPRD MEKELEC (M7-90-01) whose objective was to develop
tools of modeling for the components of the lines and the posts. The goal of the modeling presented in
this note is the dynamic study of the conductors fitted with spacers (for the lines) or with descents on
equipment (for the posts) and subjected to the forces of Laplace resulting from currents of short-circuit.
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Contents
1 Notations ................................................................................................................................................ 4
2 Introduction ............................................................................................................................................ 6
3 Kinematics of a beam in finished rotations ......................................................................................... 6
4 Vector and operator of rotation ............................................................................................................ 7
4.1 Vector-rotation ............................................................................................................................... 7
4.2 Operator of rotation ...................................................................................................................... 8
5 Passage of the local axes to the general axes ..................................................................................... 10
6 interior Efforts, deformations and law of behavior ...................................................................... 11
6.1 Interior efforts ............................................................................................................................ 11
6.2 Variation of curvature in a point of the locus of centers ............................................................ 11
6.3 Virtual work in the beam and vector of the deformations ............................................................ 12
6.4 Law of behavior ..................................................................................................................... 14
7 elementary Inertias ............................................................................................................... 15
8 Equation of the movement and course of a calculation ........................................................................... 16
8.1 Equation of the movement not deadened ............................................................................................. 16
8.2 Course of a calculation ................................................................................................................ 17
9 Linearization of the equations of the movement ........................................................................................ 17
9.1 Matrices of rigidity ........................................................................................................................ 18
9.2 Matrices of inertia ........................................................................................................................... 20
9.2.1 Differentiation of the inertia of translation
With!!X
O
................................................................ 20
9.2.2 Differentiation of the inertia of rotation
I
I
!
+
.................................................. 20
9.2.2.1 Terms coming from the differentiation from
I
........................................................ 21
9.2.2.2 Terms coming from the differentiation from
and
!
................................................ 21
10 Implementation by finite elements ...................................................................................................... 22
10.1 Stamp interior deformation and efforts ................................................................................. 22
10.2 Matrices of rigidity ...................................................................................................................... 23
10.3 Inertias ............................................................................................................................ 24
10.4 Stamp inertia ........................................................................................................................... 24
10.5 External forces data ........................................................................................................ 25
10.6 Linear system of iteration .......................................................................................................... 25
10.7 Update of displacement, speed and acceleration ................................................... 26
10.7.1 Translatory movement .................................................................................................. 26
10.7.2 Rotational movement ....................................................................................................... 26
10.8 Update of the vector variation of curvature .............................................................................. 27
10.9 Initialization before the iterations ................................................................................................... 28
10.9.1 Translatory movement .................................................................................................. 28
10.9.2 Rotational movement ....................................................................................................... 29
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11 diagrammatic Organization of a calculation ................................................................................................ 30
11.1 Static calculation .............................................................................................................................. 30
11.2 Dynamic calculation ......................................................................................................................... 31
12 Use by Code_Aster .............................................................................................................. 32
13 Digital simulations ..................................................................................................................... 32
13.1 Embedded right beam subjected to one moment concentrated in end (case-test SSNL103) ..... 32
13.2 Arc embed-ball joint charged at the top ........................................................................................ 34
13.3 Arc circular of 45° embedded and subjected in end to a force perpendicular to its plan..35
13.4 Movement of an bracket ........................................................................................................... 36
13.5 Setting in rotation of an arm of robot .............................................................................................. 38
14 Bibliography ....................................................................................................................................... 40
Appendix 1 Some definitions and results concerning the antisymmetric matrices of command 3 ............. 41
Appendix 2 Processing of the forces of damping .................................................................................. 42
Appendix 3 Algorithm of Newmark in great rotations ........................................................................ 44
Appendix 4 Calculation of the differentials of Fréchet ....................................................................................... 46
Appendix 5 Complements on the calculation of the matrices of rigidity ................................................................ 47
Appendix 6 Principle of the iterative calculation of rotations .................................................................................... 48
Appendix 7 Need for transport in a space of reference for the relative vectorial operations
with the rotational movement ............................................................................................................... 51
Appendix 8 Use of the quaternions in modeling of great rotations [bib14] [bib15] ................. 52
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1 Notations
symbol of the vector product.
operator of multiplication by the vector
on the left.
Q
derived from
Q
compared to the curvilinear X-coordinate.
!Q
derived from
Q
compared to time.
S
curvilinear X-coordinate on the locus of centers of the sections.
“U
antisymmetric matrix of command 3 associated of axial vector U.
1
stamp unit of command 3.
Df
X
.
directional derivative of F in the direction
X
.
D
DS1
stamp diagonal
DIAG D
ds
D
ds
D
ds
,
,
.
AD INTERIM
,
,
1 2
3
or
surface and moments of inertia compared to the main axes
1 2
3
,
or of the section
straight line.
B
stamp deformation.
C
stamp behavior.
E G
,
Young modulus and rigidity to shearing, density.
E
I I
=
1 3
,
general axes of co-ordinates.
()
E
I
I
S
=
1 3
,
main axes of inertia of the section of X-coordinate
S
in position of reference.
()
F
S T
,
linear external force exerted on the beam.
()
F
S T
,
force in the beam with the X-coordinate
S
and at the moment
T
.
()
F
S T
,
() ()
R
T
S T
S T
,
,
F
.
F
ext.
forces external data with the nodes.
F
F
iner
int
,
inertias and efforts interior to the nodes.
I
tensor of inertia a length unit of beam in deformed position, expressed
in the general axes.
J
tensor of inertia a length unit of beam in position of reference, expressed
in the general axes.
()
m
S T
,
linear external moment exerted on the beam.
()
m
S T
,
moment in the beam with the X-coordinate
S
and at the moment
T
.
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()
M
S T
,
() ()
R
T
S T
S T
,
,
m
.
NR
I
function of form relating to the node
I
.
()
R S T
,
operator or matrix, in general axes, of rotation of the cross-section
of X-coordinate
S
, of the configuration of reference to that at the moment
T
.
()
R
O
S
rotation making pass from the general axes to the main axes of inertia of
section of X-coordinate
S
in configuration of reference.
()
R
early
S, T
()
()
R
R
S T
S
O
,
.
()
SO3
group operators of rotation in space with 3 dimensions.
()
T
I
I
S T
,
,
=
1 3
main axes of inertia of the section of X-coordinate
S
at the moment
T
.
()
X
O
,
S T
position, at the moment
T
, of the center of the cross-section of X-coordinate
S
.
X
T
O
'
-
1
.
E
R
T
.
R
0
0
R
early
early
.
()
S T
,
()
()
X
O
S T
S T
S
T
,
,
O
: position of the section of X-coordinate at the moment, defined by
vector position
center and the vector rotation.
X
()
S
()
()
X
O
S
S
: virtual displacement with the X-coordinate
S
.
()
S
()
()
X
O
S
S
: correction of displacement to the X-coordinate
S
.
()
S T
,
defining vector, with the X-coordinate
S
and at the moment
T
, variation of curvature by
report/ratio with the configuration of reference.
X
R
T
.
()
S T
,
vector rotation, at the moment
T
, of the section of X-coordinate
S
compared to its
position of reference.
I
I
N
-
1,
vector rotation enters the moment
I
-
1
and the iteration
N
moment
I
.
I
N
angular velocity of a section of beam calculated with the iteration
N
moment
I
.
Q Q
,
-
1
operator of passage of a vector rotation to the associated quaternion and its reverse.
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2 Introduction
The essential difficulty of the mechanics of the beams in great displacements lies in
formulation of rotations. The rotation of a section compared to a configuration of reference is
defined by vector-rotation ([bib3], [bib4] and [bib5]). The quaternions are used for the update
this vector.
In [bib4] and [bib5], the increment of rotation is expressed in the configuration of reference (diagram
Lagrangian total). The calculation of the matrices of mass is complicated and cannot besides be completely
concluded its. But finally, all the matrices used are symmetrical.
In [bib1] with [bib3], the increment of rotation is expressed in the last calculated configuration
(updated Lagrangian diagram). It is this diagram which we chose. The calculation of the matrices
be completed without excessive difficulty but they are not symmetrical.
With the difference of [bib3], we expressed speeds and the angular accelerations in the axes
Generals and not in local axes. The matrices are thus more intricate, but one avoids
the ambiguity which appears with the connection of two beams not colinéaires.
3
Kinematics of a beam in finished rotations
E
3
S
(A)
P
E
2
E
1
E
2
E
1
E
3
(b)
P'
T
3
T
2
T
1
X
O
S
Position of reference
Position at the moment
T
Appear 3-a: Evolution of a section of beam
Let us follow the evolution of a section of beam of its initial position - or of reference - [3-a] (A) with its
position deformed at the moment
T
[3-a] (b).
The cross-section of the center
P
beam in position of reference is identified by the X-coordinate
curvilinear
S
P
of
on the locus of centers (or neutral fiber). One attaches to this section the trihedron
orthonormé
E E E
E
1
2
3
1
:
is the unit tangent of the locus of centers in
P; E
E
2
3
and
are
directed along the main axes of inertia of the section.
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As in [bib1] with [bib5], one makes the assumption that during the movement the sections initially
straight lines remain plane and do not change a form.
Moment 0 at the moment
T
:
·
P
comes in
P
and the position of
P
is defined by the vector
()
X
O
S T
,
;
·
the orthonormé trihedron
E E E
1
2
3
becomes the orthonormé trihedron
T T T T
T
1 2 3
2
3
.
and
are
always directed along the main axes of inertia of the section and
T
1
is always normal
unit with this section. But
T
1
is not inevitably tangent with the locus of centers in
P
:
in other words, there can be, in deformed position, a slip due to shearing, like
in the model of Timoshenko.
The state of the section at the moment
T
is thus defined by:
·
vector
()
X
O
S T
,
, which gives the position of the center of gravity;
·
the vector-rotation which makes pass from the trihedron
E E E
1
2
3
with the trihedron
T T T
1 2
3
, and which is defined in
[§4.1].
The whole of these two vectors constitutes the vector
()
S T
,
.
4
Vector and operator of rotation
Appendix 1 gives preliminary results concerning the antisymmetric matrices of command 3.
4.1 Vector-rotation
Let us suppose that, in the system of general axes
P E E E
1 2
3
[4.1-a], the point
M
results from
M
by the rotation of angle
around the axis passing by
P
and of unit vector
U
. Let us pose:
=
U,
vector-rotation is called making pass from
M
M
with
.
According to the formula of Euler-Rodrigues [bib6] p. 186 and [bib7]:
(
)
PM
PM
PM
PM
=
+
+ -
sin
cos
U
U
U
1
.
M
Me
U
E
3
E
1
E
2
P
Appear 4.1-a: Representation of a finished rotation
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In general, the vector-rotation of the product of two rotations is not the geometrical sum of
vector-rotation components. The case 2D is a particularly simple exception:
vector-rotation, perpendicular to the plan, are added algebraically.
4.2
Operator of rotation
Taking into account [éq An1-3], the preceding equation is written:
(
)
[
]
PM
PM
=
+
+ -
1
U
U
sin
“
cos
“
.
1
2
The expression between hooks defines the operator of rotation
R
making pass from
PM
PM
with
:
(
)
R
1
U
U
=
+
+ -
sin
“
cos
“.
1
2
éq 4.2-1
One calls “parameters of Euler” of rotation the four following numbers:
E
E
U
E
U
E
U
0
1
1
2
2
3
3
2
2
2
2
=
=
=
=
cos
sin
sin
sin
éq 4.2-2
One has obviously:
E
E
E
E
02
12
22
32
1
+
+
+
=
.
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Let us pose:
E
U
=
=
E
E
E
1
2
3
2
sin
.
By using the relation [éq A1-4], one easily puts the expression [éq 4.2-1] of
R
in the form:
(
) (
)
R
1
=
-
+
+
2
1
2
02
0
E
E
T
E E
E ".
éq 4.2-3
In addition, substitute
sin
cos
and
, with the second member of [éq 4.2-1], by their developments
in whole series, it comes:
() ()
() ()
R
1
U
U
=
+
-
+
+
+ -
-
+
+
-
+
+
+ -
-
-
-
3
5
1
2
1
2
4
6
1
2
3
5
1
2
1
2
4
6
1
2
!
!
!
“
!
!
!
! “
#
#
#
p
p
p
p
p
p
2
maybe, while using [éq A1-5] and [éq A1-6], the exponential shape of the operator of rotation:
()
()
()
()
R
1
U
U
U
U
=
+
+
+
+
+
=
=
“
“
!
“
!
exp
“
exp
“.
2
2
#
#
p
p
éq 4.2-4
It appears on [éq 4.2-4] that when
0
,
R
+
1
.
éq 4.2-5
()
R
U
=
exp
~
is obviously not calculated by the development [éq 4.2-4], but by
the expression [éq 4.2-1].
Since
“
“
U
U
T
= -
, the transposition of all the terms of the second member of [éq 4.2-4] gives:
()
[
]
()
exp
“
exp
“
T
=
-
, that is to say:
éq 4.2-6
R
R
T
=
-
1
and:
R R
1
T
=
.
éq 4.2-7
The operators of rotation, orthogonal according to the equation [éq 4.2-7], form a group compared to
the operation of multiplication - noncommutative in 3D - called group of Dregs and indicated by
()
SO3
(Special Orthogonal group).
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5
Passage of the local axes to the general axes
The components of the vectors are expressed in the general axes
E E E
1 2
3
[3-a].
matrices of the operators who connect them are thus valid only in these axes. But mechanics
beams is formulated much more simply in the main axes of inertia buildings
T T T
1 2
3
in current configuration. One is thus brought to make the change of axes of the general trihedron
E E E
1 2 3
with the local trihedron
T T T
1 2 3
by the product
R
early
of two rotations:
·
rotation
R
O
,
invariable, which brings the general axes
(
)
E E E
1 2
3
on the local axes in
position of reference
(
)
E E E
1
2
3
;
·
rotation
R
, depend on the time, which brings the trihedron
(
)
E E E
1
2
3
on the local trihedron in
current deformed position
(
)
T T T
1 2 3
, that is to say:
R
R R
early
O
=
.
éq 5-1
Being given a vector
v
, of components known in the general trihedron, its components in
local trihedron are the components in the general trihedron of the vector:
V
R
=
early
T
v.
éq 5-2
One can thus replace calculations relating to vectors expressed in local axes in
current configuration, by same calculations relating to the same vectors turned of
R
early
T
and
expressed in general axes. In other words, this rotation
R
early
T
allows to replace calculations in
local axes of the current configuration, by same calculations in general axes.
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6
Interior efforts, deformations and law of behavior
One places oneself within the framework of the theory of the beams. The efforts and the deformations are defined by
their end cells on the line of the centers of gravity of the sections. Thus, virtual work in
the beam is calculated by a simple curvilinear integral along this line.
6.1 Efforts
interiors
One calls interior efforts on the section of center
P
and of normal
T
1
the efforts which exerts on
this section the part of the beam located in the direction of
T
1
[6.1-a]. These efforts form one
torque of which end cells in
P
are: the resultant
F
, resulting moment
m
.
F
m
P "
P'
T
1
Appear 6.1-a: Section of beam reduced to the locus of centers
If
F
m
and
are respectively the force and the moment outsides given per unit of length not
deformation -
F
m
and
are supposed to be independent of the configuration, i.e. “conservative” or
“not-follower” -, the static balance of the section of beam
P P
of length
ds
is written:
F
F
m
F
m
S
S
S
+
=
+
+
=
0
X
0
O
.
éq 6.1-1
6.2
Variation of curvature in a point of the locus of centers
Local axes of the section of X-coordinate
S
result from the general axes by the relation [éq 5-1]:
()
()
()
T
R
R
E
I
S
S
S
I
=
.
.
O
éq 6.2-1
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While deriving compared to
S
:
(
)
T
R R
R R E
I
I
'
'
'
,
=
+
O
O
maybe, by reversing the relation [éq 6.2-1]:
(
)
T
R R
R R R R
T
I
T
T
I
'
'
'
=
+
T
O
O
.
Let us pose:
R R
'
“
T
=
éq 6.2-2
The matrix
“
because derivation is antisymmetric compared to
S
of [éq 4.2-7] gives:
“
“
+
=
T
0
.
It is checked that the matrix
“
O
defined by:
“
'
O
O
O
= R R R R
T
T
is also antisymmetric.
Therefore, while utilizing axial vectors
and
O
:
(
)
T
T
I
I
'
O
=
+
.
·
ds
O
is the vector rotation which makes pass from
()
(
)
T
I
I
S
S ds
with
T
+
when the beam undergoes one
uniform rotation
(
)
R'
=
0
of its position of reference to its current position.
()
O
S
characterize the curvature of the configuration of reference to the X-coordinate
S
.
·
ds
is the increase in rotation of
()
(
)
T
I
I
S
S ds
with
T
+
had with the variation of
R
along
the beam. It is this vector
who characterizes the variation of curvature between the configuration
of reference and current deformed configuration.
6.3
Virtual work in the beam and vector of the deformations
This paragraph is the extension to the three-dimensional case of a step made in [bib8] on the beams
plane.
Configuration of the beam at the moment
T
is defined [§3] by the field
()
S T
,
:
()
()
()
S T
S T
S T
,
,
,
O
=
X
.
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Let us calculate work
W
linear external forces
F
m
and
in following virtual displacement:
()
()
()
S
S
S
=
X
O
.
One has obviously:
(
)
W
ds
S
S
=
+
F
m
.
.
X
O
1
2
.
The equilibrium equations [éq 6.1-1] make it possible to replace the external forces
F
m
and
by
interior efforts
F
m
and
:
W
S
S
S
ds
S
S
=
-
-
-
F
m
F
.
.
.
X
X
O
O
1
2
,
maybe, by integration by parts of the first two terms:
[
]
(
)
[
]
W
ds
S
S
= -
+
+
+
F
m
F
m
F
.
.
.
.
.
.
X
X
X
O
O
'
O
'
'-
1
2
1
2
If one indicates by
W
ext.
the total work of the external forces on the beam - length and in ends -
the preceding equation is written:
(
)
[
]
(
)
[
]
W
ds
ds
S
S
S
S
ext.
O
O
O
O
=
+
+
+
=
-
+
F
m
F
m
F
m
.
.
.
.
.
.
'
.
'
'
X
X
X
X
1
2
1
2
1
2
éq 6.3-1
According to the theorem of virtual work for the continuous mediums, the second member is work
virtual of the interior efforts, which one notes
W
int
. According to the idea of [bib8], let us seek to put them
coefficients of
F
m
and of
in the form of the virtual increase in two vectors.
Let us make the assumption that the vector
X
O
'
tangent with the locus of centers but not inevitably length
unit, does not differ from the unit vector
T
1
, normal with the section, that by infinitely small of the 1
Er
command (of
the command of
). That implies that the beam lengthens little and undergoes a weak slip. Then:
(
)
[
]
W
ds
S
S
int
.
.
=
-
+
F
m
X
T
O
'
1
1
2
.
Indeed, in virtual displacement:
·
X
O
'
increases
X
O
'
;
·
the unit vector
T
1
turn of
;
·
vector
, defining the variation of curvature [§6.2], increases
'
.
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One thus has:
[
]
W
ds
S
S
int
.
.
,
=
+
F
m
1
2
éq 6.3-2
with:
=
-
X
T
O
'
1
.
éq 6.3-3
In the equation [éq 6.3-2],
F
m
and
thus seem respectively a torque
efforts or generalized stresses and the torque of the associated deformations.
defines lengthening and the slip;
the variation of curvature [§6.2] defines.
It is observed that, if
must be small, on the other hand it does not have there a limitation for
. Majority of the beams
enter within this framework.
The relation [éq 6.3.1] is written:
(
)
W
ds
S
S
int
;
.
=
B
F
m
1
2
éq 6.3-4
with:
B
1
1
=
D
ds
D
ds
X
0
O
'
éq 6.3-5
B
matrix of deformation is called.
6.4
Law of behavior
According to [§5], the components in local axes of the generalized stress and deformation are
components in general axes of the vectors:
F
M
F
M
=
=
and
such as:
E
X
E
X
T
T
F
m
éq 6.4-1
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One supposes that the law of behavior is elastic and that, in local axes, it with same
expression that for a beam of Timoshenko:
[
]
F
M
=
DIAG
E
X
EA GA GA GI I.E.(INTERNAL EXCITATION)
I.E.(internal excitation)
,
,
,
,
,
2
3
1
2
3
éq 6.4-2
With
With
2
3
and
being two surfaces depending on the size and the form of the section.
One poses:
[
]
C
=
DIAG EA GA GA GI I.E.(INTERNAL EXCITATION) I.E.(INTERNAL EXCITATION)
,
,
,
,
,
.
2
3
1
2
3
C
matrix of behavior is called.
7
Elementary inertias
Inertias applied to an element
ds
form a torque which admits, in the center of gravity:
·
a general resultant,
()
-
With ds
S T
!! ;
X
O
·
one equal moment resulting contrary to the absolute velocity of the elementary kinetic moment
H
.
To express the angular velocity, let us proceed as to [§6.2] and derive the relation:
T
R
I
I
=
E,
compared to
T
, by holding account that
E
I
does not depend on time. One obtains:
!
!
.
T
R R T
I
T
I
=
éq 7-1
Let us pose:
!
“.
R R
T
=
éq 7-2
By deriving the relation [éq 4.2-7] compared to
T
, it is seen that the matrix
“
is antisymmetric. If one
indicate by
the axial vector of this matrix, the relation [éq 7.1] is written:
()
()
! ,
.
T
T
I
I
S T
S T
=
is thus the angular Flight Path Vector of the section of beam of X-coordinate
S
at the moment
T
.
The elementary kinetic moment has as an expression:
H
=
=
ds
ds
T
I
R J R
éq 7-3
where
J
is the tensor of inertia in the configuration of reference:
[
]
J
R
R
=
O
O
DIAG
I
I
I
1
2
3
,
,
.
T
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While deriving compared to time:
(
)
!
!
!
!
H
=
+
+
ds
ds
T
T
I
R J R
R J R
.
But:
!
!
“
R J R
R R R J R
I
I
T
T
T
,
,
,
,
=
=
=
and:
R J R
R J R R R
!
!
T
T
T
=
=
0
because, according to the equation [éq A1-2]:
R R
!
“
T
= -
=
0
.
From where moment of the inertias of the element:
-
= -
-
!
!
H
ds
ds
I
I
.
The virtual work of the inertias thus has as an expression:
W
With
ds
S
S
iner
O
O
= -
+
!!
!
.
.
X
I
I
X
1
2
éq 7-4
8
Equation of the movement and course of a calculation
8.1
Equation of the movement not deadened
If one adds the inertias to the external forces, the weak form of the equations of the movement,
in other words the virtual work of the forces not balanced in the beam is written:
(
)
(
)
(
)
(
)
W
W
W
W
,
!!! ;
;
,
! !! ;
;
.
int
=
-
-
iner
ext.
éq 8.1-1
With balance
W
is null, for all
.
(
)
W
int
;
is given by the equation [éq 6.3-4] where the torque of efforts
F
m
, has as an expression,
according to [éq 6.4-2]:
F
m
=
C
T
éq 8.1-2
The torque of deformations generalized of the second member of [éq 8.1-2] results from the position
current:
is given by [éq 6.3-3] and
rise from [éq 6.2-2].
(
)
W
iner
,
! !!
;
is given by the equation [éq 7-4].
(
)
W
ds
S
S
ext.
O
;
.
=
+
F
m
1
2
X
work of the concentrated forces
éq 8.1-3
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If the external forces given are conservative, i.e. independent of the configuration,
W
ext.
does not depend on
.
8.2
Course of a calculation
·
In the dynamic case, one seeks the fields of displacements, speeds and accelerations
in a discrete succession of moments:
T T
T
T
T
I
I
N
1 2
1
,
,
.
#
#
-
·
In the static case, one splits the total load in increments of load which one adds
successively from zero to reconstitute the truck load. With each stage of
loading, named by “urgent” abuse, one calculates the field of displacements.
Knowing the state of the structure at the moment
T
I
-
1
, one deduces his state at the moment from it
T
I
by prediction
correction:
·
In the static case (
STAT_NON_LINE
), the prediction consists in calculating the answer of
structure with
I
- ème increment of load, by preserving its behavior at the moment
T
I
-
1
.
·
In the dynamic case, one must initially initialize the fields at the moment
T
I
, by formulas
rising from the algorithm of temporal integration used: in the operator
DYNA_NON_LINE
,
it is the algorithm of Newmark [§A3]. Then one applies the increase in load enters
T
T
I
I
-
1
and
with the behavior in initialized situation.
In the predicted state, the equilibrium equation is generally not satisfied and one must correct them
displacements by iterations resting on linearized equations.
9
Linearization of the equations of the movement
Let us suppose calculated the state of the structure to the iteration
N
moment
I N
.
=
1
corresponds to the phase of
prediction. The weak form of the equilibrium equations is, with this iteration [éq 8.1-1]:
(
)
(
)
(
)
(
)
W
W
W
W
in
in
in
in
in
in
in
in
,
! !! ;
;
,
! !! ;
;
int
=
-
-
iner
ext.
.
éq 9-1
·
If this quantity is rather small, within the meaning of the criterion of stop [bib10], one considers that this
N
-
ième iteration gives the state of the structure to the moment
I
.
·
If not, corrections of displacement are calculated
in
+
1
such as:
(
) (
) (
)
(
)
(
)
L W
W
DW
in
in
in
in
in
in
in
in
in
in
in
in
in
+
+
+
=
+
=
+
+
+
+
1
1
1
1
0
,
,
;
,
! !! ;
,
! !! ;
.
.
.
.
éq 9-2
(
)
DW
in
in
in
in
,
! !! ;
.
+
1
is the differential of Fréchet of
(
)
W
in
in
in
,
! !! ;
in the direction
in
+
1
[An4].
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9.1
Matrices of rigidity
They result from the differentiation of Fréchet from
(
)
W
int
;
in the direction
. According to
equations [éq 6.3-4] and [éq 6.4-1]:
(
)
{}
W
ds
S
S
int
;
.
.
=
B
F
M
1
2
That is to say:
{}
[
]
{} (
)
{}
D W
D
ds
D
ds
D
ds
S
S
S
S
S
S
int
.
.
.
.
.
.
.
.
=
+
+
B
B
B
1
2
1
2
1
2
F
M
F
M
F
M
éq 9.1-1
However, according to the equation [éq 6.3-5]:
{}
B
X
X
=
+
O
'
O
'
'
.
Thus [éq A4-2]:
{}
D B
X
.
.
=
O
'
0
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In addition [éq A4-5]:
D
.
.
^
^
=
R
R
early
early
0
0
Lastly, according to [§6.4]:
D
D
T
T
T
F
M
=
=
C
C
B
C
.
,
^
^
-
0
0
because [éq 6.3-2] and [éq 6.3-5]:
D
=
.
,
B
and [éq A4.6]:
D
early
T
early
T
R
R
.
.
^
= -
It is shown [A5] that:
·
the sum of the first two integrals of the second member of [éq 9.1-1] can be put
in the form:
T
T
S
S
ds
E
1
2
;
·
the third integral can be written:
T
T
T
S
S
T
T
S
S
ds
ds
B
C
B
B
+
1
2
1
2
Z
,
with:
=
D
ds
D
ds
1
1
1
0
0
0
;
E
0
0
0
0
0
=
-
-
F
m
F
F
““
;
'
X
O
éq 9.1-2
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(
)
(
)
Z
0
0
=
-
-
R C R
R C
R
R C R
R C
R
early
early
T
early
early
T
early
early
T
early
early
T
1
1 1
2
2 1
F
m
v
v
éq 9.1-3
C
C
1
2
and
being two submatrices of
C
:
[
]
[
]
C
C
1
2
3
2
1
2
3
=
=
DIAG
DIAG
EA GA GA
GI I.E.(INTERNAL EXCITATION) I.E.(INTERNAL EXCITATION)
,
,
,
,
,
.
T
E
is appeléematrice of rigiditegeometric
;
éq 9.1-4
B
C
B
T
T
matrix of material rigidity is called
.
éq 9.1-5
Lastly, during the differentiation a matrix appears which does not appear in [bib2]:
B
T
Z
that we call complementary matrix.
éq 9.1-6
9.2 Matrices
of inertia
They result from the differentiation of Fréchet from
W
iner
[éq 7-4] in the direction
. More
precisely, one places oneself in the configuration of
N
- ième iteration of the moment
I
and one differentiates
in the direction
I
N
+
1
.
9.2.1 Differentiation of the inertia of translation
With!!X
O
One has immediately, according to the equations [éq An4-4] on the one hand and [éq An3.1-4] on the other hand:
(
)
D
With
With
With
I
N
I
N
I
N
I
N
!!
.
!!
,
,
,
,
X
X
T
X
O
O
O
O
+
+
+
=
=
1
1
2
1
X
éq 9.2.1-1
9.2.2 Differentiation of the inertia of rotation
I
I
!
+
According to the equation [éq 7-3]:
I
R J R
=
T
,
J
being a constant tensor.
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9.2.2.1 Terms coming from the differentiation from
I
According to the equations [éq A4-3] and [éq A4-4], these terms are:
()
()
-
+
-
+
+
I
I
I
I
!
!
“
“
“
“
I
N
I
N
1
,
éq 9.2.2.1-1
()
()
-
+
-
+
+
I
I
I
I
!
!
“
“
“
“
v
v
I
N
in
1
all sizes appearing in the hook
being taken with the iteration
N
moment
I
.
9.2.2.2 Terms coming from the differentiation from
and
!
According to the expressions [éq A3.2-3] and [éq A3.2-4], speed and angular acceleration with the iteration
N
moment
I
are:
(
)
(
)
I
N
I
N
in
T
in
I
N
iT
I
I
N
I
I
N
I
N
I
N
in
T
in
in
I
T
I
I
N
I
I
N
T
T
=
+
-
=
+
-
-
-
-
-
--
-
-
-
-
--
R R
R R
R R
R R
1
1
1
1
11
1
1
2
1
1
11
,
,
,
,
,
,
!
!
.
1
A variation
vector-rotation can affect only the sizes relating to this iteration
N
moment
I
, since, in the two preceding relations, the other sizes are fixed.
In other words, only are to be differentiated
R
in
I
I
N
and
-
1,
, increment of vector-rotation of the moment
I
-
1
with the iteration
N
moment
I
.
However [éq A4-5]:
D
in
I
N
I
N
I
N
R
R
.
.
^
+
+
=
1
1
And, according to [bib3]:
(
)
D
I
I
N
I
N
I
I
N
I
N
-
+
-
+
=
1
1
1
1
,
,
.
,
T
with:
()
T
2
=
+
-
-
1
2
2
1
2
2
T
T
tan
“.
1
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Terms coming from differentiation from
and
!
are thus:
(
)
(
)
(
)
(
)
(
)
[
]
-
-
-
+
+
-
-
-
+
-
-
-
-
-
-
--
-
-
I
R R
I
I
R R
I
I
I
R R
R R
T
,
,
,
,
,
,
,
,
,
,
,
!
“
“
I
N
I
N
in
T in
I
N
I
N
I
N
I
N
I
N
I
N
T in
I
N
I
N
I
N
I
N
I
N
I
N
I
T
I
I
N
I
I
N
in
iT
I
T
T
1
1
1
1
2
1
1
11
1
1
1
()
I
N
,
éq 9.2.2.2-1
combination of the three preceding matrices being to multiply by
I
N
+
1
.
10 Implementation by finite elements
One gives below the representation in finite elements of the matrices of [§9]. These matrices appear
in expressions to be integrated along the beam. One thus calculates them at the points of Gauss.
NR NR
I
J
,
… are the values taken, at the point of Gauss considered, by the relative functions of form
with the nodes
I J
,
….
The matrices of rigidity connect the increase in displacement of the node
J
with the increase in force
intern with the node
I
for the element
E
.
10.1 Stamp interior deformation and efforts
The matrix of deformation has as a continuous expression [éq 6.3-5]:
B
1 X
1
=
D
ds
D
ds
O
'
0
.
In finite elements, the contribution of the displacement of the node
I
with the deformation at the point of Gauss
considered is obtained by multiplying the 6 components of this displacement by the matrix:
()
B
1
X
1
I
H
I
I
I
NR
NR
G
NR
=
'
'
“
O
'
0
.
éq 10.1-1
The superscript
H
indicate that it is about the discretized shape of the matrix
B
.
According to [éq 6.3-4]:
F
B
E
int I
E
I
HT
ds
=
F
m
éq 10.1-2
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is the interior effort applied to the node
I
element
E
and due to the stress field generalized
F
m
in the element. This stress is calculated according to the equation [éq 8.1-2]. But it is necessary to notice
that:
·
on the one hand [éq 6.3-3]:
R
R
X
E
early
early
O
T
T
=
-
!
;
1
·
in addition, the vector
is updated at each iteration, as it is indicated to [§10.8].
10.2 Matrices of rigidity
The expression continues matrix of material rigidity [éq 9.1-5] is:
B
C
B
T
T
.
One deduces some, for the finite element
E
, the matrix connecting the increase in displacement of the node
J
with
the increase in force internal to the node
I
:
S
B
C
B
chechmate I J
E
I
HT
T
jh
E
ds
=
.
éq 10.2-1
One calculates numerically
C
T
at the point of current Gauss.
The expression continues geometrical matrix of rigidity [éq 9.1-4] is:
T
E,
where:
=
D
ds
D
ds
1
1
1
0
0
0
,
and
E
is given by the equation [éq 9.1-2].
One deduces some, as for the matrix of material rigidity:
S
géom I J
E
ihT
jh
E
ds
=
E
,
éq 10.2-2
where:
I
H
I
I
I
NR
NR
NR
=
'
'
'
,
1
1
1
0
0
0
and where
E
is calculated numerically at the point of current Gauss.
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The expression continues complementary matrix of rigidity [éq 9.1-5] is:
B
T
Z,
where
Z
is given by the equation [éq 9.1-3].
One deduces some:
S
B
compl I J
E
I
HT
J
E
NR
ds
=
Z
,
where
Z
is calculated numerically at the point of current Gauss.
10.3 Forces
of inertia
According to [éq 7.3-1]:
(
)
F
X
I
I
iner
O
I
E
I
I
E
NR
With
NR
ds
= -
+
!!
!
éq 10.3-1
is the inertia of the element
E
with the node
I
.
10.4 Stamp
of inertia
Let us pose:
I
0
0
M
I
O iner
=
=
With
NR NR
ds
I J
E
I
J
E
1
J
and
O iner
O iner
.
It is seen, according to [éq 7-4] and [éq 9.2.1-1], that
1
2
T
E
M
O iner
is well the matrix of inertia of the element
E
for the translatory movement (diagonal submatrix
To 1
I
of
O iner
). But according to
[éq 9.2.2.1-1] and [éq 9.2.2.2-1], it is not the matrix of inertia of rotation.
Nevertheless, the matrix
M
O iner
, assembly of
M
O iner
E
, is used to calculate the initial acceleration of
beam when it leaves its position of reference with a null initial speed
(
)
O
=
0
. Indeed,
one has then, according to [éq 7.3-1]:
()
()
F
F
M
X
ext.
iner
O iner
O
O
T
T
=
= -
=
=
0
0
!!
!
,
since, in position of reference:
I
J
=
.
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Let us call
J
the sum of the two matrices [éq 9.2.2.1-1] and [éq 9.2.2.2-1] and let us pose:
I
J
M
I
belch
belch
belch
=
=
0 0
0
'
and
I J
E
I
J
E
NR NR
dS
Also let us pose:
I
J
M
I
O belch
O belch
O belch
=
=
0
0
0
and
I J
E
I
J
E
NR NR
ds
The matrix of complete inertia of an element of beam is obviously:
M
M
M
M
iner
O iner
O belch
belch
E
E
E
E
T
T
=
-
+
1
1
2
2
.
10.5 External forces data
According to [éq 8.1-3]:
F
ext. I
E
I
I
E
NR
NR
ds
=
F
m
éq 10.5-1
is the force applied to the node
I
element
E
who is equivalent to the external forces distributed.
10.6 Linear system of iteration
By discretization in finite elements, the equation [éq 9-1] gives:
(
)
W
H
=
-
-
F
F
F
int
.
iner
ext.
.
In addition, by supposing that the external forces are conservative, one has, according to [§10.2] and
[§10.4]:
D W
T
T
H
.
.
=
+
+
+
-
+
S
S
S
M
M
M
chechmate
geom
compl
belch
O belch
O iner
1
1
2
2
.
The relation [éq 9-2], having to be checked by all
, thus leads, with the iteration
N
moment
I
, with
linear system following in
in
+
1
:
[
S
S
S
M
M
M
X
F
F
F
chechmate,
geom,
compl,
belch
O
O iner
O + 1
+ 1
ext.,
iner,
I
N
I
N
I
N
I
N
belch
I
N
I
N
I
I
N
I
N
T
T
+
+
+
-
+
=
-
+
,
,
int,
.
1
1
2
2
éq 10.6-1
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In Code_Aster, the matrices elementary
M
O iner
E
,
who are independent of displacement, are
assemblies only once to constitute the total matrix
M
O iner
,
who is in particular used to calculate
initial acceleration [§10.4].
Three matrices of elementary rigidity
S
S
S
chechmate
geom
compl
E
E
E
,
and
, the matrix of inertia of rotation
M
belch
E
,
who depend all the four on displacement, and inertia of corrective rotation stamps it
-
1
2
T
E
M
O belch
,
who is invariable, with each iteration, combined then are assembled for
to constitute a pseudo matrix of total rigidity
~.
K
in
The linear system [éq 10.6-1] thus becomes:
~
.
,
,
int,
,
K
M
X
F
F
F
in
I
N
N
I
I
N
I
N
T
+
=
-
+
+
1
2
1
O iner
O
+ 1
ext.
iner
I
éq 10.6-2
In the case of a static problem, the preceding system is simplified in:
[]
K
X
F
F
in
I
N
N
I
I
N
O
I
ext.
,
,
int,
+
+
=
-
1
1
,
éq 10.6-3
where
K
I
N
is the assembly of the only matrices of rigidity to the iteration
N
“moment”
I
[§8.2]:
S
S
S
chechmate
geom
compl
,
,
,
.
I
N
I
N
I
N
+
+
10.7 Update of displacement, speed and acceleration
The processing of the translatory movement is conventional; that of the rotational movement is done with
the aid of the quaternions [A8].
10.7.1 Translatory movement
One applies the formulas [éq A3.1-3] and [éq A3.1-4].
10.7.2 Rotational movement
The sizes to be updated are:
·
on the one hand, vector-rotation, speed and angular acceleration;
·
in addition, for later calculations, the matrix of rotation and the increment of vector-rotation
moment
I
-
1
with the current iteration of the moment
I
[§A6].
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The update of vector-rotations rests on the following property of the quaternions [§A8.4]: “it
quaternion of the product of two rotations is equal to the product of the quaternions of rotations
components ".
Thus while posing [§An8.6]:
(
)
(
)
()
()
X
X
I
N
I
N
O
O
,
,
,
X
X
=
=
+
Q
Q
1
it comes:
(
) ()
[
]
in
X
X
+
-
=
1
1
Q
O
O
,
,
.
X
X
$
éq 10.7.2-1
In addition, according to the equation [éq An6-2], if:
()
(
)
y
I
I
N
O
,
,
,
y
Q
=
-
1
then:
(
) ()
[
]
I
I
N
X
y
- +
-
=
11
1
,
,
,
.
Q
y
O
O
O
X
éq 10.7.2-2
The update of the matrix of rotation is immediate [§4.2]:
()
R
in
in
+
+
=
1
1
exp
“
,
who is calculated according to [éq 4.2-1].
Finally speed and the angular acceleration are updated by the relations [éq A3.2-3] and [éq A3.2-4].
10.8 Update of the vector variation of curvature
The vector
,
who defines the deformation of rotation [§6.2], should be calculated only at the points of Gauss.
In Code_Aster, it is treated by means of computer like a “internal variable”.
According to [éq 6.2-2]:
()
“
.
.
I
N
I
N
in T
D
ds
=
R
R
And, according to [éq 4.2-6]:
“
exp
exp
.
^
^
I
N
I
N
I
N
in T
I
N
D
ds
+
+
+
=
-
1
1
1
R
R
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That is to say:
“
exp
exp
exp
“exp
.
^
^
^
^
I
N
I
N
I
N
I
N
I
N
I
N
D
ds
+
+
+
+
+
=
-
+
-
1
1
1
1
1
In the preceding equation, the matrix of the first member is antisymmetric by construction;
second matrix of the second member is obviously antisymmetric; thus the first matrix of
second member is too.
One shows in [bib2] Appendix B, that the axial vector of this last matrix is:
(
)
(
)
(
)
=
+ -
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
sin
sin
.
sin
.
'
'
'
I
N
I
N
I
N
I
N
I
N
I
N
I
N
I
N
I
N
I
N
I
N
in
I
N
in
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
1
2
1
2
1
2
Thus:
I
N
I
N
I
N
I
N
+
+
+
=
+
-
1
1
1
AXIAL exp
“
exp
.
^
^
10.9 Initialization before the iterations
In the dynamic case, if the loading is constant in time, the iterations cannot
to start at the moment
I
who if one initializes some of the fields of displacement, speed and acceleration with
values different from those of the moment
I
-
1
. These initializations are done as follows.
10.9.1 Translatory movement
X
X
O
O
O
,
,
.
I
I
=
-
1
Then, according to the equation [éq An3.1-1]:
!!
!
!!
.
,
,
,
X
X
X
O
O
O
O
I
I
I
T
= -
+
-
-
-
1
2
1
2
1
1
According to the equation [éq An3.1-2]:
()
[
]
!
!
!!
!!
.
,
,
,
,
X
X
X
X
O
O
O
O
O
O
I
I
I
I
T
=
+
-
+
-
-
1
1
1
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10.9.2 Rotational movement
One takes expressions similar to the preceding ones:
I
I
O
=
-
1
,
éq 10.9.2-1
!
!
I
I
I
T
O
= -
+
-
-
-
1
2
1
2
1
1
,
éq 10.9.2-2
()
[
]
I
I
I
I
T
O
O
=
+
-
+
-
-
1
1
1
!
! .
éq 10.9.2-3
The second members of [éq 10.9.2-2] and [éq 10.9.2-3] have a direction because all the vectors which appear in it
are in the tangent vector space with
()
SO
I
3
1
in
R
-
.
Like consequences of the equation [éq 10.9.2-1]:
R
R
I
I
O
=
-
1
and:
I
I
-
=
1,
.
O
0
One sees that at the first moment
()
I
=
1
, initializations require the knowledge of acceleration
initial
!!
!
,
X
O
O
whose calculation is indicated to [§10.4].
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11 diagrammatic Organization of a calculation
This paragraph shows how the concepts presented to [§10] in unfolding are articulated of one
calculation.
11.1 Calculation
statics
ITER = 0
ITER = ITER + 1
A max. number of iterations
reached
?
yes
calculation is stopped
not
Calculation of the interior forces [§10.1]
Test of stop: the forces are they
balanced, except for a tolerance?
yes
end
not
Calculation of the matrices of rigidity [§10.2]
Resolution of the linear system with
corrections of displacements [éq 10.6-3]
Update of displacements [§10.7]
Calculation of the external forces [§10.5]
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11.2 Calculation
dynamics
ITER = 0
ITER = ITER + 1
A max. number of iterations
reached
?
yes
calculation is stopped
not
Calculation of the interior forces [§10.1]
Calculation of the inertias [§10.3]
Test of stop: the forces are they
balanced, except for a tolerance?
yes
no next time
not
Calculation of the pseudo matrix of rigidity
Combination of and of
Resolution of the linear system with
corrections of displacements
Update of displacements, speeds
and accelerations
Calculation of the matrix of inertia
M
Calculation of initial acceleration
Loop on the pitches of time
Calculation of the forces external at the moment running [§10.5]
Prediction of displacements, speeds and accelerations [§10.9]
[§10.4]
O iner
[§10.6]
~
K
I
N
~
K
I
N
1
T
2
M
O iner
[éq 10.6-2]
[§10.7]
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12 Use by Code_Aster
This paragraph indicates how intervene the beams in great displacements in
controls of Code_Aster.
Order
Key word factor
Key word
Argument
AFFE_MODELE
AFFE
PHENOMENON
“MECHANICAL”
MODELING
“POU_D_T_GD”
AFFE_CARA_ELEM
BEAM
SECTION
“GENERAL”
'RIGHT-ANGLED
“CIRCLE”
STAT_NON_LINE
and
DYNA_NON_LINE
COMP_ELAS
RELATION
DEFORMATION
“ELAS_POUTRE_GD”
“GREEN”
13 Simulations
numerical
One gives five digital simulations below resting on the formulation presented in this
note. The three first relate to static problems, the two last on problems
dynamic.
13.1 Embedded right beam subjected to one moment concentrated in
end (case-test SSNL103)
That is to say
M
this moment. The beam is the seat only one constant moment and the equation [éq 6.4-2] shows
that variation of curvature
X
is also constant. The beam thus becomes deformed in circle of
radius:
R
E I
=
3
M
,
in a plan perpendicular to the vector moment.
The figure [13.1-a] shows the deformations of a beam length unit, of which
E I
3
2
=
and subjected
at the moments
,
.
2
4
and
The beam is cut out in 10 finite elements of the 1
Er
command. One applies from the start the final moment and
convergence is reached in 3 iterations.
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M
=
M
=
2
M
=
4
Appear 13.1-a: Beam subjected to one moment in end
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13.2 Arc embed-ball joint charged at the top
The figure [13.2-a] shows the deformations of an arc of 215° of opening, embedded on the right, ball joint with
left and subjected to an increasing force concentrated at the top. The solution of this problem is
data in [bib11] for an initial radius of 100 and following characteristics of beam:
EA
GA
I.E.(internal excitation)
=
=
×
=
5 10
10
7
6
and
.
The arc is modelized by 40 elements of the 1
Er
command. One made grow the force up to 890 per eight
increments of 100, an increment of 50 and four increments of 10. Beyond buckling appears,
i.e. displacement continues to grow under a force which decrease brutally. The algorithm
described here does not allow to take into account such a phenomenon, and diverges for a force from 900. Da
Deppo, in [bib11], locates the force criticizes to 897.
There is the table of comparative results according to:
Force
Vertical displacement
point
of application
Displacement
horizontal of the point
of application
Our calculations
890
110.5
60.2
Da Deppo
897
113.7
61.2
F = 400
F = 700
F = 890
Appear 13.2-a: Arc embed-ball joint charged in the node (Da Deppo)
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13.3 Arc circular of 45° embedded and subjected in end to a force
perpendicular in its plan
The problem is three-dimensional. It was proposed in [bib12]. The figure [13.3-a] shows three
successive configurations of the beam of initial radius 100, square section and characteristics:
EA
GA
GA
GI
I.E.(internal excitation)
I.E.(internal excitation)
=
=
= ×
=
=
=
×
10
5 10
8.333 10
7
1
2
6
1
2
3
5
,
,
.
.
50
40
30
20
10
0
10
20
30
Z
X
Y
50
40
30
20
10
60
70
P = 600 lb
P = 300 lb
To (29.3 70.7 0
CONFIGURATION
CONFIGURATION
To (22.5 59.2 39.5
With (15.9 47.2 53.4)
INITIAL
FINALE
Appear 13.3-a: Three configurations of the beam (extracted from [bib12])
We modelized the beam by 8 elements of the 1
Er
command. The force grows by increments of 20. One has
following comparative results, for the co-ordinates of the point of application of the force, in
configurations of the figure [13.3-a].
FORCE
X
Y
Z
300
Our calculations
22.3
58.9
40.1
BATHE-BOLOURCHI
22.5
59.2
39.5
600
Our calculations
15.7
47.3
53.4
BATHE-BOLOURCHI
15.9
47.2
53.4
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13.4 Movement of an bracket
It is about a three-dimensional dynamic problem dealt with in [bib3].
An bracket consists of a post and a cross-piece length 10 [13.4-a] (A). The foot of
post is embedded and one applies to the connection a not-following force, perpendicular to the plan of
bracket at rest [13.4-a] (b).
10
F
Z
y
X
10
(A)
F
50
0
1
2
T
(b)
Appear 13.4-a: Bracket subjected to a dynamic force
perpendicular in its plan
The characteristics of the elements provided by [bib3] are as follows:
EA
GA
GA
GI
I.E.(internal excitation)
I.E.(internal excitation)
With
I
I
I
=
=
=
=
=
=
=
=
=
=
1
2
6
1
2
3
3
2
3
1
10
10
1
10
20
,
,
,
;
.
It is noticed that these data do not make it possible to identify a material and a section of beam, because
one has at the same time:
I.E.(internal excitation)
I
E
EA
With
E
2
2
2
6
10
10
=
=
=
=
and
.
One can thus deal with this problem only by imposing, by program, a characteristic: one chose
to impose the product
EA
.
The post and the cross-piece are modelized each one by 4 elements of the 1
Er
command and duration of the analysis
comprise 120 equal pitches of 0.25.
The figures [13.4-b] and [13.4-c] give the evolution of the component according to
X
displacement
respectively for the connection and the end of the cross-piece. In cartridge, one reproduced them
corresponding curves data in [bib3] for two modelings.
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24.0 26.0 28.0 30.0 32.0 times
2.0
4.0.6.0.8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0
4.0
3.0
2.0
1.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
displacement
100
Displacement
80
60
40
20
00
20
40
60
80
100
0.0 30 60 90.120.150 180.210.240 270 300
Time
AB
C
D
Appear 13.4-b: Bracket of SIMO - Displacement of the connection
perpendicular to the initial plan
10.00
8.75
7.50
6.25
5.00
3.75
2.50
1.25
0.00
- 1.25
- 2.50
- 3.75
- 5.00
- 6.25
- 7.50
- 8.75
- 10.00
2.0.4.0.6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0
displacement
Time
time
100
Displacement
80
60
40
20
00
20
40
60
80
100
0.0 30 60 90.120.150 180.210.240 270
AB
C
D
300
Appear 13.4-c: Bracket of SIMO - Displacement of the end
perpendicular to the initial plan
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13.5 Setting in rotation of an arm of robot
An arm of robot
OA
is put moving in the plan
E E
1 2
by a rotation
()
T
imposed on sound
center 0 [13.5-a]. One wants to calculate the displacement of the end
With
in a system of axes
E E
1 2
'
'
involved in rotation
()
T
.
10
E
2
E
1
With
O
(T)
10 elements of beam of the 1
Er
command
(A)
Time
0.0
0.0
27.0
54.0
81.0
108.0
135.0
6.0 12.0 18.0 24.0 30.0
angle
(
T
)
(radians)
(b)
Appear 13.5-a: Arm of robot subjected to an imposed rotation
Characteristics of material:
EA
GA
I.E.(internal excitation)
With
I
=
×
=
×
=
×
=
=
×
-
2 8 10
1 10
1 4 10
1 2
6 10
7
2
7
4
4
,
,
,
,
,
,
The pitch of time evolves/moves of 0,05 at the beginning of the analysis with 0,001 with the end.
The figures [13.5-b] and [13.5-c] give the evolution of following displacement
E
1 '
and according to
perpendicular direction. One reproduced, in cartridge, the corresponding curve of [bib3]. When
rotational speed becomes constant, the arm undergoes a permanent lengthening due to the centrifugal force
and it is subjected to an oscillation of bending of low amplitude.
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0.00125
0.00000
0.00125
0.00250
0.00375
0.00500
0.00625
0.00750
0.00875
0.01000
0.01125
0.01250
0.01375
0.01500
0.01625
0.01750
0.01875
1.0.2.0.3.0 4.0.5.0.6.0 7.0.8.0.9.0 10.011.012.013.014.015.016.017.018.0
0.0040
0.0000
- 0.0008
- 0.0056
- 0.0104
- 0.0152
- 0.0200
0.0
6.0
12.0
18.0
24.0
30.0
Time
Tip Displacement U
2
(L, I)
0.02000
5.14 X 10
- 4
~
Appear 13.5-b: following Displacement
E
1 '
0.04
0.00
- 0.04
- 0.08
- 0.12
- 0.16
- 0.20
- 0.24
- 0.28
- 0.32
- 0.36
- 0.40
- 0.44
- 0.48
- 0.52
- 0.56
- 0.60
1.0.2.0.3.0 4.0.5.0.6.0 7.0.8.0.9.0 10.011.012.013.014.015.016.017.018.0
0.10
0.00
0.06
- 0.22
- 0.38
- 0.54
- 0.70
0.0
6.0
12.0
18.0
24.0
30.0
Time
Tip Displacement U
2 (L, I)
~
Appear 13.5-c: Following displacement
E
2
'
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14 Bibliography
[1]
J.C. SIMO: With finite strain beam formulation. The three-dimensional dynamic problem. I leaves.
Comput. Meth. appl. Mech. Engng. 49, 55-70 (1985).
[2]
J.C. SIMO and L. VU-QUOC
: With three-dimensional finite-strain rod model.
Leaves II:computational aspects. Comput. Meth. appl. Mech. Engng. 58, 79-116 (1986).
[3]
J.C. SIMO and L. VU-QUOC: One the dynamics in space off rods undergoing broad motions. With
geometrically exact approach. Comput. Meth. appl. Mech. Engng. 66, 125-161 (1988).
[4]
A. CARDONA and Mr. GERADIN: With beam finite element nonlinear theory with finite rotations.
Int. J. Numer. Meth. Engng. 26, 2403-2438 (1988).
[5]
A. CARDONA: Year integrated approach to mechanism analysis. Thesis, University of Liege
(1989).
[6]
J. DUKE and D. BELLET: Mechanics of the real solids. Elasticity. Cepadues-editions (2
E
edition,
1984).
[7]
H. CHENG and K.C. GUPTA: Year historical note one finite rotations. Newspaper off Applied
Mechanics 56, 139-145 (1989).
[8]
E. REISSNER: One one-dimensional finite-strain beam theory: the planes problem. Newspaper off
Applied Mathematics and Physics 23, 795-804 (1972).
[9]
H. HUTS: Run of general Mechanics. Dunod (1961).
[10]
Mr. AUFAURE: Dynamic nonlinear algorithm of Code_Aster. Note HI-75/95/044/A.
[11]
D.A. DADEPPO and R. SCHMIDT: Instability off clamped-hinged circular arches subjected to
load does not have. Trans. ASME 97 (3) (1975).
[12]
K.J. BATHE and S. BOLOURCHI: Broad displacement analysis off three-dimensional beam
structures. Int. J. Numer. Meth. Engng. 14, 961-986 (1979).
[13]
K.J. BATHE: Finite element procedures in engineering analysis. Prentice-hall (1982).
[14]
D. HENROTIN: Report/ratio of training course at the Institute Montefiore (University of Liege). Communication
deprived.
[15]
P. of CASTELJAU: Quaternions. Hermès (1987).
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Appendix
1 Some definitions and results concerning
antisymmetric matrices of command 3
Any vector
U
of command 3 and components
U, U, U
X
y
Z
one can associate the antisymmetric matrix
“U
of command
3 following:
“U
=
0
- U
U
U
0
- U
- U
U
0
Z
y
Z
X
y
X
.
éq A1-1
Conversely, any antisymmetric matrix of command 3 can be written in the form [A1-1] and him can thus
to associate a vector
U
. This vector is called the axial vector of the matrix.
One sees without difficulty that:
“U U
=
0
,
éq A1-2
in other words the axial vector is clean vector of antisymmetric associated for the eigenvalue 0.
·
Whatever the vector
v:
“U v
U v
=
;
éq A1-3
(
)
““
U v
v U
U. v 1
=
-
T
.
éq A1-4
·
If
U
is unit:
(
)
(
)
“
“
“
;
.
U
U U
1
U
U
2
3
=
-
= -
T
symmetrical matrix
antisymmetric matrix
From where:
()
(
)
“
“
;
U
U
2
1
2
1
p
p
=
-
-
symmetrical matrix
éq A1-5
()
(
)
“
“
.
U
U
2
1
1
1
p
p
-
-
= -
antisymmetric matrix
éq A1-6
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Appendix 2 Processing of the forces of damping
These forces are not taken into account currently in Code_Aster.
A2.1 Assumptions and end cells in the center of the section
One makes the assumption that the element of volume
FD
bordering a point
M
interior with the beam is subjected to one
force damping being composed of two parts:
·
for the translatory movement of the section of X-coordinate
S
which belongs
M
:
()
D
S T FD
F
1
1
= -
!
,
;
X
O
·
for the rotational movement velocity angular
around the center
P
section:
()
D
S T
P M FD
F
2
2
= -
,
.
Integrated on the volume of beam length
ds
, these forces admit for end cells in
P
:
·
the force:
()
D
With
S T ds
F
= -
1
!
,
;
X
O
·
moment:
()
()
D
ds
P M
S T
P M D
S T ds
m
= -
= -
2
2
section
,
,
.
I
A2.2 elementary Forces of damping
The virtual work of the forces of damping is:
W
With
ds
S
S
amor
O
O
= -
1
1
2
!
.
.
X
I
X
2
If damping in account is taken,
W
amor
must be cut off with the second member from the equation [éq 9-1].
It results from this, as with [§10.3] for the inertias, that the contribution of the element
E
with the force
of damping to the node
I
is:
F
X
I
amor
O
I
E
I
I
E
NR
With
NR
ds
= -
1
2
!
.
If damping in account is taken, these forces must be added to the second member of the equation
[éq 10.6-1].
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A2.3 Matrices of damping
According to the relations [éq A4-3] and [éq A4-7] giving the differentials of Fréchet of
!X
O
and of
in the direction
, and given that ([éq A4-5] and [éq A4-6]):
(
)
()
()
D
D W
With
ds
ds
S
S
S
S
I
I
I
0
0
I
0
0
0
I
.
“
,
.
.
!
.
.
=
-
+
= -
-
-
amor
1
2
1
2
1
2
2
1
One deduces the matrices from them from damping of the element
E
:
·
except for the sign, relation between the increase speed to the node
J
and the increase in force
of damping to the node
I
:
C
1
I
amor I J
E
I
J
E
NR NR
With
ds
=
1
2
0
0
;
·
except for the sign, relation between the increase in displacement to the node
J
and the increase in
force damping with the node
I
:
()
S
I
amor I J
E
I
J
E
NR NR
ds
=
-
0
0
0
2
.
In the hook of the first member of the equation [éq 10.6.1], it is necessary to add:
·
stamp
T C
amor
;
·
stamp
S
amor
.
The matrix
C
amor
E
is symmetrical, but the matrix
S
amor
E
is antisymmetric.
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Appendix 3 Algorithm of Newmark in great rotations
Conventional A3.1 Diagram of Newmark in translation
The state of the locus of centers (displacement, speed and acceleration) is supposed to be known at the moment
I
-
1.
The algorithm of Newmark [bib10] and [bib13], rests on the following developments of the displacement and of
speed, with the iteration
N
+
1
moment
I
:
(
)
[
]
X
X
X
X
X
O
O
O
O
O
,
,
,
,
,
!
!!
!!
I
N
I
I
I
I
N
T
T
+
-
-
-
+
=
+
+
-
+
1
1
1
2
1
1
2
1 2
2
éq A3.1-1
()
[
]
!
!
!!
!!
,
,
,
,
X
X
X
X
O
O
O
O
I
N
I
I
I
N
T
+
-
-
+
=
+
-
+
1
1
1
1
1
éq A3.1-2
and
are the parameters of Newmark which, in the case of the “rule of the trapezoid”, are worth
1
4
1
2
and
: hooks
in the equations [éq A3.1-1] and [éq A3.1-2] are then the arithmetic mean of
!!
!!
.
,
,
X
X
O
O
I
I
N
-
+
1
1
and of
Récrivant each one of these relations to the iteration
N
moment
I
and cutting off member with member from the relations
the preceding ones, it comes:
(
)
(
)
!!
!!
!
!
.
,
,
,
,
,
,
,
,
X
X
X
X
X
X
X
X
O
O
O
O
O
O
O
O
I
N
I
N
I
N
I
N
I
N
I
N
I
N
I
N
T
T
+
+
+
+
=
+
-
=
+
-
1
2
1
1
1
1
and
If one poses:
X
X
X
O
O
O
,
,
,
,
I
N
I
N
I
N
+
+
=
+
1
1
éq A3.1-3
one thus has:
!!
!!
!
!
,
,
,
,
,
,
X
X
X
X
X
X
O
O
O
O
O
O
I
N
I
N
I
N
I
N
I
N
I
N
T
T
+
+
+
+
=
+
=
+
1
2
1
1
1
1
and
éq A3.1-4
A3.2 Diagram of Newmark adapted to great rotations
To unify calculations, one would like to be able to write, in rotation, the following similar relations:
(
)
(
)
in
in
I
I
N
I
I
N
I
N
I
N
I
I
N
I
I
N
T
T
+
- +
-
+
- +
-
=
+
-
=
+
-
1
11
1
1
2
11
1
1
,
,
,
,
!
!
,
and
where
I
I
N
I
I
N
-
- +
1
11
,
,
and
are respectively the vectors increment of rotation of the section considered, enters
the moment
I
-
1
and iterations
N
N
and
+
1
moment
I
[A6].
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But these relations are not exact because one cannot combine a speed (or an acceleration)
angular with the iteration (
N
+
1
) of the moment
I
with the homologous size with the iteration
N
and of vector-rotation
counted starting from the configuration at the moment
I
-
1
[A7]. One can combine only after having brought by
rotation (one says “transported”) all sizes in the same configuration. One chooses to simplify
configuration of reference.
Thus let us pose:
in
in
T
in
in
in T
I
N
+
+
+
=
=
1
1
1
R
R
,
,
éq A3.2-1
With
With
in
I
N
T
I
N
I
N
I
N T
I
N
+
+
+
=
=
1
1
1
R
R
,
,
!
!
éq A3.2-2
I
I
N
iT
I
I
N
I
I
N
I
T
I
I
N
- +
-
- +
-
-
-
=
=
11
1
11
1
1
1
,
,
,
,
R
R
The algorithm of Newmark in rotation results in the following relations:
·
for the operator of rotation, [éq A6-1]:
R
R
in
I
N
in
+
+
=
1
1
exp
;
^
·
for the increment of vector-rotation since the moment
I
-
1
, [éq A6-2]:
(
)
(
)
exp
“
exp
exp
“
;
,
^
,
I
I
N
I
N
I
I
N
- +
+
-
=
11
1
1
·
for the angular velocity:
(
)
in
in
I
I
N
I
I
N
T
+
- +
-
=
+
-
1
11
1
,
,
;
·
for the angular acceleration:
(
)
With
With
in
I
N
I
I
N
I
I
N
T
+
- +
-
=
+
-
1
2
11
1
1
,
,
.
The two preceding relations give, by opposite transport on the configuration
N
I
+
1
and taking into account
relations [éq A3.2-1] and [éq A3.2-2]:
(
)
in
I
N
in T
I
N
in
I
T
I
I
N
I
I
N
T
+
+
+
-
- +
-
=
+
-
1
1
1
1
11
1
R
R
R
R
,
,
,
éq A3.2-3
(
)
!
!
.
,
,
,
in
in
I
N T
I
N
I
N
iT
I
I
N
I
I
N
T
+
+
+
-
- +
-
=
+
-
1
1
2
1
1
11
1
1
R
R
R
R
éq A3.2-4
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Appendix 4 Calculation of the differentials of Fréchet
One will have to use thereafter the identity following, known as hook of Dregs, whose demonstration is immediate. If
“
“
With
B
and
are two antisymmetric matrices of the 3
E
command, of axial vectors
With
B
and
, then one has, for all
vector
v
:
(
)
(
)
““
““
WITH B B A v
WITH B
v
-
=
.
éq A4-1
In other words: is the axial vector of the antisymmetric matrix
““
““
WITH B B A
-
.
Differential of Fréchet of the function is called
F
in the direction
X
, quantity:
()
(
)
D
D
D
F
F
F
X
X
X
X
X
.
.
lim
.
=
=
+
grad
0
It is the main part of the increase in
F
corresponding to the increase
X
variable
X
.
Here the catalog of the differentials of Fréchet intervening in this note.
(
)
()
D
D
D
X
X
X
X
X
X
O
'
O
O
O
O
O
'
.
lim
'
'
=
+
=
=
0
.
éq A4-2
D! .
!
X
X
X
O
O
O
=
.
éq A4-3
D!!.
!!
X
X
X
O
O
O
=
.
éq A4-4
D
D
D
R
R
R
.
lim
exp
^
^
=
=
0
éq A4-5
D
D
D
R
R
R
T
T
T
.
lim
exp
^
^
=
-
= -
0
éq A4-6
because, according to the equation [éq 4.2-6]:
exp
exp
.
^
^
=
-
T
“
!
.
=
R R
T
However:
D
D
D
! .
lim
exp
!
!
^
^
R
R
R
R
=
=
+
·
0
.
Therefore, according to [éq A4-6]:
(
)
(
)
D
D
D
T
T
“.
! .
!
.
!
“
“
.
^
^
^
-
=
+
=
+
R
R
R
R
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Maybe, by using the hook of Dregs [éq A4-1]:
D
.
!
=
-
.
éq A4-7
Appendix 5 Complements on the calculation of the matrices of rigidity
This appendix develops calculations of [§9.1].
A5.1 Stamps geometrical rigidity
{}
[
]
{}
[
]
(
)
(
)
D
D
B
B
X
X
X
.
.
.
.
.
.
'.
^
^
F
M
F
M
F
F
m
+
=
+
+
+
O
'
O
'
O
'
.
By rearranging the terms and by using the vectorial identity:
()
(
)
a. B C
turnover B
=
,
the second preceding member is written:
()
-
-
+
X
X
X
O
'
O
'
.
'.
.
. ““
'
F
m
F
F
.
+
=
.
O
E
A5.2 Matrices of material and complementary rigidity
As it is shown with [§9.1]:
D
T
T
F
M
=
-
.
.
^
^
C
B
C
0
0
However:
=
-
C
1
T
F
m
.
Consequently:
{}
{}
B
B
C
B
.
.
.
D
T
F
M
=
éq A5.2-1
{}
-
B
C
C
.
.
^
^
T
T
0
0
1
F
m
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The term:
{}
B
C
B
.
T
conduit immediately with the matrix of material rigidity.
In addition:
(
)
(
)
^
^
.
0
0
= -
-
-
-
C
R
C
R
R
C
R
1
1 1
2 1
T
T
T
F
m
F
m
early
early
early
early
The second term of the equation [éq A5.2-1] is thus written:
(
)
(
)
B
0
0
.
R
C R
R
C
R
R
C R
R
C
R
early
early
early
1 1
early
early
early
early
2
1
early
1
2
T
T
T
T
F
m
.
While indicating by
Z
the matrix between hooks of the expression which precedes,
B Z
T
is the matrix of rigidity
complementary.
Appendix 6 Principle of the iterative calculation of rotations
This appendix visualizes the operation of exponentiation, defines the vectors increment of rotation which intervene
in appendix 3 and the relation between them gives.
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R
I
-
1
SO3
()
R
I
N
R
I
N
+
1
1
+
^
I
N
+
1
(
)
R
I
N
1
+
^
I
-
1, I
N
(
)
R
I
-
1
1
+
^
I
-
1, I
N
+
1
(
)
R
I
-
1
Appear intervening A6-a: Representation of the operators in
the rotation of the sections.
: symbol of exponential projection
The curved surface of the figure [. A6-a] represents the unit
()
SO3
operators of rotation
R
. One has
appeared tangent spaces in
()
SO
I
3
1
in
R
-
, rotation calculated at the moment
I
-
1
, and in
R
in
,
N
- ième iteration
in the calculation of rotation at the moment
I
.
That is to say
I
-
1
the vector rotation corresponding to
R
I
-
1
. There is a vector increment of rotation
I
I
N
-
1,
such
that:
(
)
R
R
in
I
I
N
I
=
-
-
exp
“
.
,
1
1
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If the equilibrium equation of the beam is not satisfied, at the moment
I
, by
R
in
, a correction is sought
I
N
I
I
N
in
+
-
+
1
1
1
of
,
.
is obtained by linearization, while replacing in the equilibrium equation, according to
equivalence [éq 4.2-5]:
exp
.
^
^
I
N
in
I
N
I
N
+
+
+
1
1
R
R
by
1
The second of the preceding expressions is not an operator of rotation, but is in space
tangent with
()
SO
in
3
in
R
[A6-a]. Having calculated
I
N
I
N
+
R
1
1
,
+
results some while projecting by exp on
()
SO3
:
R
R
I
N
I
N
I
N
+
+
=
1
1
exp
.
^
éq A6-1
The increment of swing angle
I
I
N
- +
11,
making pass from
R
R
I
-
1
with
I
N + 1
is such as:
(
)
R
R
I
N
I
I
N
I
+
- +
-
=
1
11
1
exp
“
.
,
Vector-rotation not being additive, one does not have:
=
I
I
N
I
N
I
I
N
,
- +
+
-
+
11
1
1
,
,
but one a:
(
)
(
)
exp
exp
“
exp
“
,
^
,
=
I
I
N
I
N
I
I
N
- +
+
-
11
1
1
.
éq A6-2
With [§10.7.2], one solves the preceding equation compared to
I
I
N
- +
11,
by using the properties of the quaternions.
Increments of swing angle
I
I
N
I
I
N
-
- +
1
11
,
,
and
are used to calculate the corrections speed and
of acceleration of
N
I
N
I
+
with
1
.
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Appendix 7 Need for transport in a space of reference
for the vectorial operations relative to
rotational movement
The sizes kinematics, speeds and accelerations, in a given configuration are in
tangent vector space with
()
SO3
at the point defined by the rotation of the configuration compared to the position of
reference. One can combine the sizes relating to two distinct configurations only after having them
transported in the same vector space taken for reference. The example which follows will make include/understand this
need.
Let us examine the angular velocity of a gyroscope at three moments
T T
T
1 2
3
,
and
[A7-a]. Let us suppose that one passes from
configuration 1 with configuration 2 by the rotation of angle -
around
E
1
, and of configuration 1 with
configuration 3 by the rotation of angle
-
/2
around the same axis. In 1, the angular velocity is carried by
the axis of the gyro and has as components (0, 0,
) in the general axes
E E E
1 2
3
. In 2, the angular velocity is
also carried by the axis and has as general components (0, 0, 3
). One wants to determine the components
general speed in 3, knowing that this speed is the average speeds into 1 and 2.
That the angular velocity in 3 is the average angular velocities into 1 and 2 does not mean that it is
average in the general axes, but in the axes related to the gyro. In the example, since the gyro turns
around its axis, in 1 with speed
, in 2 with speed 3
, then in 3 it turns around this axis with
speed 2
. Therefore, in 3, the general components angular velocity are (0, 2
, 0).
Taking into account [§5], one obtains the preceding result in “transporting” the angular Flight Path Vector of
configuration 2 on configuration 1, taken for reference, i.e. while making turn this vector of the angle
around
E
1
. Its general components are then (0, 0, 3
). One makes the average of this vector and the vector
angular velocity in 1, to obtain the vector of components (0, 0, 2
). One “transports” finally this last
vector on configuration 3 by making it turn of
-
/2
around
E
1
, and one leads well to the vector
angular velocity of general components (0, 2
, 0).
1
2
3
O
E
3
E
2
E
1
1
3
0
0
2
0
0
3
0
2
0
Appear A7-a: Evolution angular velocity of a gyroscope
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Appendix
8 Use of the quaternions in modeling of
great rotations [bib14] [bib15]
A8.1 Definitions
A quaternion, noted
()
()
Q
or
X
O
, X
, is defined by the whole of a scalar
X
O
and of a vector with three
dimensions
(
)
X X X X
1
2
3
:
()
X,
X
X
X
X
O
O
X
=
+
+
+
1
2
3
,
,
,
and
being three numbers satisfying the relations:
2
2
2
1
=
=
= -
= -
=
= -
=
= -
=
;
;
;
.
Combined quaternion
Combined, noted *, from a quaternion is obtained by changing the sign of the vectorial part:
()
(
)
X
X
O
O
,
*
,
.
X
X
=
-
Purely vectorial quaternion
A quaternion is known as purely vectorial when its scalar part is null and that it is thus of the form:
()
0, X
.
A quaternion
()
v
is purely vectorial if and only if:
() ()
()
v
v
0
+
=
*
.
A8.2 Elements of algebra of the quaternions
Multiplication
By applying the definition, one shows immediately that the multiplication, noted
$
, of two quaternions is:
() ()
(
)
X
y
X
X
y
O
O
O
O
O
O
,
,
y
,
X
y
X. y
y
X X y
$
=
-
+
+
.
It is checked that this multiplication is:
·
associative
:
() ()
[
]
()
() () () ()
X
y
Z
X
Z
y
Z
O
O
O
O
O
O
O
,
,
,
,
,
,
,
X
y
Z
X
Z
y
Z
+
=
+
$
$
$
;
·
noncommutative bus:
() () () ()
(
)
X
y
y
X
O
O
O
O
,
,
,
,
X
y
y
X
X y
$
$
-
=
0 2
,
.
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Normalizes of a quaternion, noted
.
:
()
() ()
X
X
X
X
X
X
X
O
O
O
02
12
22
32
,
,
,
X
X
X
2
=
=
+
+
+
$
*
.
A quaternion is unit if its standard is equal to the unit. It is checked that:
() ()
[
]
() ()
Q
Q
Q
Q
1
2
2
1
$
$
*
*
*.
=
A8.3 Representation of a rotation by a quaternion
If
()
v
1
is a purely vectorial quaternion and
()
U
a unit quaternion, then the quaternion
()
v
2
such as:
()
()
()
()
v
U
v
U
2
1
=
$
$
*
éq A8.3-1
is purely vectorial and has even standard that
()
v
1
. Indeed:
() ()
()
()
()
()
()
()
()
() ()
[
]
()
()
() ()
()
()
() ()
()
()
()
v
v
U
v
U
U
v
U
U
v
v
U
0
v
v
U
v
U
U
v
U
v
2
2
1
1
1
1
2
2
1
1
1
2
+
=
+
=
+
=
=
=
*
*
*
*
*
;
*
*
*
*
.
*
$
$
$
$
$
$
$
$
$
$
$
$
If one poses:
()
() ()
()
v
v
1
2
=
=
0,
0,
X
y
;
,
it is seen that, by the relation [éq A8.3-1], the unit quaternion
()
U
defines an orthogonal transformation of
vector
X
on the vector
y
. This transformation is the rotation whose matrix is defined by [éq A8.5-1].
Opposite rotation is defined by the quaternion
()
U *
, because:
()
()
()
() ()
()
() ()
()
U
v
U
U
U
v
U
U
v
*
*
*
.
$
$
$
$
$
$
2
1
1
=
=
A8.4 successive Rotations
Let us subject the purely vectorial quaternion
()
v
1
two successive rotations defined by the quaternions
unit
() ()
U
U
1
2
and
:
()
() () ()
v
U
v
U
2
1
1
1
=
$
$
*
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and:
()
() () ()
() () () () ()
[
]
v
U
v
U
U
U
v
U
U
3
2
2
2
2
1
1
2
1
=
=
$
$
$
$
$
$
*
*
éq A8.4-1
One sees on the last member of [éq A8.4-1] that the unit quaternion defining the product of two
rotations is equal to the product of the quaternions of these rotations.
A8.5 matric Expressions
That is to say
()
Q
a quaternion:
()
()
Q
=
X
O
, X
.
Let us pose:
()
Q
=
X
O
X,
()
Q
is the formed vector of the four components of the quaternion.
Let us define two matrices built on
()
Q
:
()
()
With
X
X
1 X
B
X
X
1 X
Q
Q
=
-
+
=
-
-
X
X
X
X
T
T
O
O
O
O
“
“.
and
One checks without sorrow, according to the rule of multiplication, that:
() ()
()
()
()
()
Q
Q
Q
Q
Q
Q
1
2
2
1
1
2
$
=
=
With
B
.
Now let us take the rotation defined by the unit quaternion
()
()
U
=
E
O
, E
.
If:
()
()
()
()
v
U v
U
2
1
=
*
.
Then:
()
() ()
()
v
v
U
U
2
1
=
With
B
T
.
In addition:
() ()
With
B
R
U
U
T
T
=
1 0
0
,
with:
R
E E
1
E E
=
+
+
+
T
E
E
O
O
2
2
2
““
,
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maybe, by using the relation [éq A1-4]:
(
) (
)
R
1
E E
E
=
-
+
+
2
1
2
2
E
E
T
O
O
“
.
éq A8.5-1
By bringing the preceding relation closer to the equation [éq 4.2-3], one sees that the components of the quaternion
unit defining a rotation than the parameters of Euler of this rotation are not other.
A8.6 Passage of a vector-rotation to the associated quaternion and vice versa
The operator
Q
who makes pass from a vector rotation
in the associated quaternion is defined by the relations
[éq 4.2-2]:
E
O
=
cos 12
éq A8.6-1
E
=
sin 12
éq A8.6-2
The operator
Q
-
1
the reverse of a trigonomitric function is less simple bus does not have a single determination.
But one notices on [éq 10.7.2-1] and [éq 10.7.2-2] that
Q
-
1
is used to calculate vector-rotation
+
+
+
+
N 1
deduced from
vector-rotation
N
by the correction
+
+
+
+
N
1
. In general:
+
+
+
+
N
N
1
<<
.
We thus adopted the following strategy: among all the determinations of the vector
+
+
+
+
N 1
that is taken
to which the module is closest to:
+
+
+
+
N
N
1
+
.
In the plane case, this strategy is rigorous [§ 4.1], but in the three-dimensional case it is not to it parce-
that:
+
+
+
+
+
+
+
+
N
N
N
1
1
+
.
From [éq A8.6-1], one fires:
()
1
2
2
1
1
+
+
+
+
N
O
E
K
= ±
+
-
cos
.
The adopted strategy leads to only one determination of
1
2
1
+
+
+
+
N
.
[éq A8.6-2] gives then:
+
+
+
+
+
+
+
+
+
+
+
+
N
N
N
1
1
1
2
1
2 1
2
=
sin
.
E
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