Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
1/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
Organization (S):
EDF/EP/AMV
Manual of Reference
R5.06 booklet: Dynamics in modal base
Document: R5.06.03
Modeling of the shocks and friction in
analyze transitory by modal recombination
Summary:
This document describes the physical laws of contact with friction between structures and the modeling which in is
made in the transitory algorithm of analysis by modal recombination of Code_Aster
DYNA_TRAN_MODAL
[U4.54.03]. For the various non-linear connections of contact usable, one details the calculation of the sizes
defining the conditions of contact.
The diagrams of use used are described in [R5.06.04].
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
2/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
Contents
1 Introduction ............................................................................................................................................ 3
2 Relations of contact between two structures ............................................................................................ 3
2.1 Relation of unilateral contact .......................................................................................................... 3
2.2 Law of friction of Coulomb ......................................................................................................... 5
3 approximate Modeling of the relations of contact between 2 structures by penalization .......................... 6
3.1 Model of normal force of contact ............................................................................................... 6
3.2 Model of tangential force of contact .......................................................................................... 7
4 Types of modelized connections of contact .............................................................................................. 8
4.1 Connections between a node and an indeformable obstacle ..................................................................... 8
4.1.1 Connections of contact node on plane obstacle .......................................................................... 8
4.1.2 Connections of contact node on concave circular obstacle .................................................. 10
4.1.3 Connections of contact node on concave obstacle discretized by segments .......................... 11
4.2 Connections between two nodes of two deformable structures ....................................................... 12
4.2.1 Connections of plane contact on plan ......................................................................................... 12
4.2.2 Connections of contact rings on circle ................................................................................... 14
5 Use of the localized non-linear forces of shock and friction in modal recombination ........ 15
6 Precision on the use of non-linearities of shock with friction .................................................. 15
6.1 Definition of the type of connection of shock ............................................................................................. 15
6.2 Definition of the local reference mark for the conditions of contact .............................................................. 16
6.3 Definition of the nodes of the connections ................................................................................................. 17
6.4 Definition of dimensions characteristic of the sections .............................................................. 17
6.5 Definition of the parameters of contact ............................................................................................ 17
7 Bibliography ........................................................................................................................................ 18
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
3/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
1 Introduction
The problems of shock with friction which interest EDF relate to for example modeling
tubular vibrations of structures maintained by supports with plays, or separated by plays
weak and thus being able to come into contact. Tubes of the steam generators, pencils of
control rods, the assemblies of fuel are examples of structures of which one
wish to modelize the vibrations.
The major consequence of the vibrations in the presence of play is to cause shocks as well as
friction enters the structure and its supports or between the structures from where risks of wear. It
document describes the type of non-linearities introduced by the presence of these plays, as well as
modeling used to take them into account in the algorithm of modal recombination.
2
Relations of contact between two structures
Two relations govern the contact between two structures:
·
the relation of unilateral contact which expresses the non-interpenetrability between the solid bodies,
·
the relation of friction which governs the variation of the tangential stresses in the contact. One
will retain for these developments a simple relation: the law of friction of
Coulomb.
2.1
Relation of unilateral contact
Are two structures
1
and
2
. One notes
D
NR 1 2
/
the normal distance enters the structures,
F
N1 2
/
force normal reaction of
1 on
2.
The law of the action and the reaction imposes:
F
F
NR
NR
2 1
1 2
/
/
= -
éq 2.1-1
1
2
D
NR
1/2
F
NR
1/2
F
NR
2/1
Appear 2.1-a: Outdistances normal and normal reaction
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
4/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
The conditions of unilateral contact, still called conditions of Signorini [bib5], are expressed
following way:
D
F
D
F
and F
F
NR
NR
NR
NR
NR
NR
1 2
1 2
1 2
1 2
2 1
1 2
0
0
0
/
/
/
/
/
/
,
,
=
= -
éq 2.1-2
D
NR
1/2
F
NR
1/2
Appear 2.1-b: Graph of the relation of unilateral contact
This graph translates a relation force-displacement which is not differentiable. It is thus not usable
in a simple way in a dynamic calculation algorithm.
If one restricts the study with the case of a tubular structure in the presence of an indeformable support, one notes
(
)
D D
D
N
N
NR
=
1 2
/
the normal distance to the support, and
F
N
reaction of this last (attention!
F
F
F
N
NR
NR
=
= -
2 1
1 2
/
/
to see diagram below).
The expression of the conditions of normal contact, expressing the limitation of displacements due to
support is worth:
D
F
D
F
N
N
N
N
=
0
0
0
,
,
D
N
> 0
D
N
= 0
N
N
F
N
(cf feel N
and of
F
N
)
Appear 2.1-c: Outdistance normal and normal reaction between a structure and a support
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
5/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
2.2
Law of friction of Coulomb
The law of Coulomb expresses a tangential mechanical force limiting
F
T1 2
/
of tangential reaction of
1
on
2
. That is to say
!
/
U
T1 2
the relative speed of
1
compared to
2
in a point of contact and is
µ
coefficient of friction of Coulomb, one has [bib5]:
S
F
S
NR
= F
U
F
T
T
T
1 2
1 2
1 2
1 2
0
0
0
/
/
/
/
,
!
,
,
.
-
=
=
µ
éq 2.2-1
and the law of the action and the reaction:
F
F
T
T
2 1
1 2
/
/
= -
éq 2.2-2
Ý
U
T
1/2
F
T
1/2
Appear 2.2-a: Graph of the law of friction of Coulomb
The graph of the law of Coulomb is him also nondifferentiable and is thus not simple to use in
a dynamic algorithm.
If one restricts the study with the case of a tubular structure in the presence of an indeformable support, only
tangential stress
F
F
T
T
2 1
/
=
is used, the law of friction expresses itself in the following way:
S
F
S
N
= F
U
F
T
T
T
-
=
=
µ
0
0
0
,
!
,
,
.
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
6/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
3
Approximate modeling of the relations of contact between 2
structures by penalization
3.1
Model of normal force of contact
The principle of the penalization applied to the graph of the figure [Figure 2.1-b] consists in introducing one
univocal relation
(
)
F
D
NR
NR
1 2
1 2
/
/
F
=
by means of a parameter
. The graph of
F
must tend towards
the graph of Signorini when
tends towards zero [bib6].
One of the possibilities consists in proposing a linear relation enters
D
NR 1 2
/
and
F
N1 2
/
:
F
D
if D
F
NR
NR
NR
NR
1 2
1 2
1 2
1 2
1
0
0
/
/
/
/
;
= -
=
if not
éq 3.1-1
If one notes
K
NR
=
1
called commonly “stiffness of shock”, one finds the conventional relation,
modelizing an elastic shock:
F
K
D
NR
NR
NR
1 2
1 2
/
/
= -
éq 3.1-2
The approximate graph of the law of contact with penalization is as follows:
D
NR
1/2
F
NR
1/2
Appear 3.1-a: Graph of the relation of unilateral contact approached by penalization
To take account of a possible loss of energy in the shock, one introduces a “damping of
shock "
C
NR
The expression of the normal force of contact is expressed then by:
F
K
D
C
U
NR
NR
NR
NR
NR
1 2
1 2
1 2
/
/
/
!
= -
-
éq 3.1-3
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
7/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
where
!
/
U
NR 1 2
is the relative normal speed of
1
compared to
2
. To respect the relation of
Signorini (not of blocking), one must on the other hand check a posteriori that
F
N1 2
/
is positive or
null. Only the positive part will thus be taken
.
+
expression [éq 3.1-3]:
X
X
X
X
X
+
+
=
=
<
if
if
0
0
0
The complete relation giving the normal force of contact which is retained for the algorithm is
following:
if
D
F
K
D
C
U
F
F
NR
NR
NR
NR
NR
NR
NR
NR
1 2
1 2
1 2
1 2
2 1
1 2
0
/
/
/
/
/
/
!
,
= -
-
= -
+
if not
F
F
NR
NR
2 1
1 2
0
/
/
.
=
=
éq 3.1-4
3.2
Model of tangential force of contact
The graph describing the tangential force with law of Coulomb is not-differentiable for the phase
of adherence
(
)
!
/
U
T1 2
0
=
. One thus introduces a univocal relation binding relative tangential displacement
D
T1 2
/
and the tangential force
(
)
F
D
T
T
1 2
1 2
/
/
F
=
by means of a parameter
. The graph of
F
must
to tend towards the graph of Coulomb when
tends towards zero [bib6].
One of the possibilities consists in writing a linear relation enters
D
T1 2
/
and
F
T1 2
/
:
(
)
F
F
D
D
T
T
T
T
1 2
1 2 0
1 2
1 2 0
1
/
/
/
/
-
= -
-
éq 3.2-1
If one introduces a “tangential stiffness”
K
T
=
1
, the relation is obtained:
(
)
F
F
D
D
T
T
T
T
1 2
1 2 0
1 2
1 2 0
/
/
/
/
=
-
-
K
T
éq 3.2-2
For numerical reasons, related to the dissipation of parasitic vibrations [bib7] in phase
of adherence, one is brought to add a “tangential damping”
C
T
in the expression of the force
tangential. Its final expression is:
(
)
F
F
D
D
U
F
F
T
T
T
T
T
T
T
1 2
1 2 0
1 2
1 2 0
1 2
2 1
1 2
/
/
/
/
/
/
/
!
,
=
-
-
-
= -
K
C
T
T
éq 3.2-3
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
8/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
It is necessary moreover than this force checks the criterion of Coulomb, that is to say:
F
F
U
U
F
F
T
T
T
T
T
T
1 2
1 2
1 2
1 2
1 2
1 2
2 1
1 2
/
/
/
/
/
/
/
/
!
!
,
= -
= -
µ
µ
F
F
NR
NR
if not one applies
éq 3.2-4
The approximate graph of the law of friction of Coulomb modelized by penalization is as follows:
Ý
U
T1/2
F
T1/2
K
T
Appear 3.2-a: Graph of the law of friction approached by penalization
4
Types of modelized connections of contact
Like it was specified in the paragraph [§2.2], the developments presented here relate to the setting in
work of non-linear connections with unilateral contact and friction between 1 node and an obstacle or
between 2 nodes given.
The nodes in contact are supposed to belong to two slim structures of beam type or to one
beam and an indeformable obstacle. The nodes on which will carry the condition of contact are
presumedly carried by the average line of the beams.
4.1
Connections between a node and an indeformable obstacle
4.1.1 Connections of contact node on plane obstacle
One considers a slim structure represented by elements of the beam type. Its displacement
is limited in a point by the presence of an obstacle made up of two infinite half-planes in
direction Y (see [Figure 4.1.1-a]).
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
9/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
X
loc
X
Y
Z
Be reproduced 4.1.1-a: slim Structures with contact node on plan
To analyze the conditions of contact, one places oneself in the reference mark perpendicular to the axis Xloc,
direction of neutral fiber or a generator of the beam. That is to say NO1, the node of the connection
considered on the beam, the geometry of the connection contact node on plan (called
PLAN_Y
in
Code_Aster [bib3]) is described on the figure below.
NO1
Y
loc
ORIG_OBST
Play
Z
loc
2
1
Be reproduced 4.1.1-b: Geometry of the connection node on obstacle plan
Are
Y
Z
loc
loc
co-ordinates of
NODE
NO1
in the reference mark
(
)
Y
Z
loc
loc
,
, the origin of this reference mark is it
not
ORIG_OBST
.
The normal distance
D
NR
in this case, by neglecting rotations of the sections expresses itself then by:
D
Y
play
NR
loc
= -
+
éq 4.1.1-1
The contact in this connection is judicious to take place whatever the shift in
Z
loc
between the two
structures.
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
10/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
The normal vector
N
in the reference mark
(
)
Y
Z
loc
loc
,
has as components:
N
=
sign Y
loc
(
)
0
éq 4.1.1-2
Other quantities
!U
NR
,
F
NR
,
!U
T
,
F
T
are calculated in a general way as specified with [§3].
4.1.2 Connections of contact node on concave circular obstacle
One considers a hurled structure, represented by elements of the beam type. Its displacement
is limited in a point by the presence of an obstacle made up of a bored infinite plan of a circular hole
(see figure below).
X
Y
Z
X
loc
Be reproduced 4.1.2-a: slim Structures with contact node on circular obstacle
To analyze the conditions of contact, one places oneself in the reference mark perpendicular to the axis
X
loc
,
direction of neutral fiber or a generator of the beam. Are NO1, the node of the connection
considered, geometry of the connection of contact node on circle (called
RING
in
Code_Aster [bib3]) is described on the figure below.
NO1
Y
loc
ORIG_OBST
Play
Z
loc
Appear 4.1.2-b: Geometry of the connection circular node obstacle
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
11/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
Are
Y
Z
loc
loc
co-ordinates of NODE NO1 in the reference mark
(
)
Y
Z
loc
loc
,
, of origin
ORIG_OBST
.
The normal distance
D
NR
, by neglecting rotations of the sections expresses itself then by:
(
) (
)
D
Y
Y
Z
Z
play
NR
loc
ORIGobst
loc
ORIGobst
= -
-
+
-
+
2
2
One poses like normal vector
N
the vector:
N
ORIG
NODE
ORIG
NODE
obst
obst
=
-
-
1
1
play
is a strictly positive distance.
Other quantities
!U
NR
,
F
NR
,
!U
T
,
F
T
are calculated in a general way as specified with [§3].
4.1.3 Connections of contact node on concave obstacle discretized by segments
One considers a hurled structure, represented by elements of the beam type. Its displacement
is limited in a point by the presence of an obstacle made up of a bored infinite plan of a hole of form
concave unspecified being able to be discretized in polar co-ordinates by segments (see figure
below).
NO1
Y
loc
ORIG_OBST
Z
loc
Be reproduced 4.1.3-a: Geometry of the connection node on discretized concave obstacle
Are
Y
Z
loc
loc
co-ordinates of the node
NO1
in the reference mark
(
)
Y
Z
loc
loc
,
, of origin
ORIG_OBST
.
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
12/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
The breakage of contact nearest to the node is sought
NO1
,
the normal vector
N
is defined
like the direct orthogonal vector with the breakage:
NO1
Y
loc
ORIG_OBST
N
DNN
PNO1
Z
loc
That is to say PNO1 the projection of node NO1 at the breakage, the normal distance
D
NR
in this case is worth:
(
)
D
NO
PNO
NR
=
-
1
1 .n
Other quantities
!U
NR
,
F
NR
,
!U
T
,
F
T
are calculated in a general way as specified with [§3].
4.2
Connections between two nodes of two deformable structures
4.2.1 Connections of plane contact on plan
The contacts between assemblies fuel, on the level of the grids of mixture, constitute one
example of plane contact on plan (see [Figure 4.2.1-a]).
One thus considers two hurled structures, being able to be modelized by beams of section
rectangular on the level of the areas of contact.
X
Y
Z
X
loc
NODE 1
NOEUD2
Be reproduced 4.2.1-a: slim Structures with plane contact on plan
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
13/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
To analyze the conditions of contact, one places oneself in the reference mark perpendicular to the axis
X
loc
,
direction of neutral fiber of the beams. Are NO1 and NO2, the two nodes of the connection considered,
geometry of the connection plane contact on plan (called
BI_PLAN_Y
in Code_Aster [bib3]) is
described on the figure below.
NO1
NO2
D1
D2
Y
loc
Z
loc
Be reproduced 4.2.1-b: Geometry of the connection plan on plan
Are
Y
Z
iloc
iloc
co-ordinates of NOEUDi in the reference mark
(
)
Y
Z
loc
loc
,
, of origin
ORIG_OBST
(
ORIG_OBST
can be provided by the user, defect
ORIG_OBST
is selected like the medium of
nodes NO1, NO2.
The normal distance
D
NR 1 2
/
in this case, by neglecting rotations of the sections expresses itself then by:
D
Y
Y
D
D
NR
loc
loc
1 2
1
2
1
2
/
=
-
-
-
éq 4.2.1-1
D
1
and
D
2
are strictly positive distances.
The contact in this connection is judicious to take place whatever the shift in
Z
loc
between the two
structures.
The normal vector
N
1 2
/
in the reference mark
(
)
Y
Z
loc
loc
,
has as components:
N
1 2
2
1
0
/
(
)
=
-
sign Y
Y
loc
loc
éq 4.2.1-2
Other quantities
!
/
U
NR 1 2
,
F
N1 2
/
,
!
/
U
T1 2
,
F
T1 2
/
are calculated in a general way [§ 2.4].
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
14/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
4.2.2 Connections of contact rings on circle
If one considers now two cylinders of circular section, modelized by elements of
beam. The connection of contact between two nodes of the average lines is supposed to take place enters
two circles as shown in the figure following:
X
loc
Appear 4.2.2-a: slim Structures with contact rings on circle
One places oneself in the reference mark perpendicular to the axis
X
loc
parallel with a generator of the cylinders.
Are NOEUD1 and NOEUD2, the two nodes of the connection considered, the geometry of the connection contact
ring on circle (called
BI_CERCLE
in Code_Aster [bib3]) is described on the geometry
below:
NO1
NO2
R1
R2
1
2
ORIG_OBST
Y
loc
Z
loc
Appear 4.2.2-b: Geometry of the connection rings on circle
The normal distance
D
NR 1 2
/
has as an expression:
(
) (
)
D
Y
Y
Z
Z
R
R
NR
loc
loc
loc
loc
1 2
1
2
2
1
2
2
1
2
/
=
-
+
-
-
-
One poses like normal vector of
1
towards
2
the vector:
N
1 2
/
=
-
-
NOEUD2 NOEUD1
NOEUD2 NOEUD1
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
15/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
5
Use of the localized non-linear forces of shock and
friction in modal recombination
The non-linear forces expressed above are explicit functions of the position and
speed of the nodes to which the conditions of contact relate.
One chooses to use the technique of pseudo-forces to solve the projected dynamic problem. If it
direct dynamic system is written:
MX
CX
KX
F
F
X X
!!
!
()
(
,
! )
T
T
T
ext.
shock
T
T
T
+
+
=
+
The technique of pseudo-forces consists in projecting on the basis of linear system and maintaining them
non-linear forces with the second member.
The projected dynamic system takes the form:
T
T
T
T
T
T
T ext.
T shock
T
T
T
M
C
K
F
F
!!
!
()
(
,
! )
+
+
=
+
The projected problem is integrated numerically by an explicit diagram.
6
Precision on the use of non-linearities of shock with
friction
Non-linearities of shock between a structure and an obstacle or two structures were
introduced into the algorithms of modal recombination of Code_Aster: an algorithm of Euler
of command 1 and Devogelaere of command 4 [bib4] [R5.06.04].
These algorithms are used by the operator
DYNA_TRAN_MODAL
[bib1], [U4.54.03]. The type of connection
of shock between the two nodes is specified by a specific control:
DEFI_OBSTACLE
[U4.21.07].
6.1
Definition of the type of connection of shock
The type of connection of shock is a generic concept, which does not comprise any physical information
like a distance or unspecified dimension. The type of connection specifies simply the form
geometrical of the connection considered.
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
16/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
Types of connection with shock with two nodes accepted by the control
DEFI_OBSTACLE
are
described by the key words following:
PLAN_Y
,
PLAN_Z
or
RING
BI_PLAN_Y
,
BI_PLAN_Z
or
BI_CERCLE
(see figure below).
PLAN_Y
PLAN_Z
Z
loc
RING
Y
loc
Y
loc
Z
loc
Y
loc
Z
loc
Y
loc
BI_PLAN_Y
BI_PLAN_Z
BI_CERCLE
Z
loc
Y
loc
Z
loc
Y
loc
Z
loc
Appear 6.1-a: Geometries of the connections of shock
The prefix
BI_
specify that it is about a connection with two nodes.
6.2
Definition of the local reference mark for the conditions of contact
Treated structures, being regarded as cylindrical slim (circular section or
rectangular), are modelized by elements of beam. The contact is treated, as one saw with
[§3.1] and [§3.2] in a plan perpendicular to the direction
X
loc
generator of the cylinders.
To define this change of reference mark completely, a local reference mark is introduced
(
)
X
Y
Z
loc
loc
loc
,
,
.
The vector
X
loc
is the vector with 3 components provided behind the key word
NORM_OBST
.
Using the first two nautical angles, one passes in a single way of the total reference mark
(
)
X Y Z
,
with
a reference mark having
X
loc
like first basic vector (see [Figure 6.2-a] hereafter). A third
rotation whose angle is provided behind the key word
ANGL_VRIL
give a single correspondence enters
the main reference mark and the local reference mark.
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
17/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
Note:
the orientation of this local reference mark is important because it is in this reference mark that they are analyzed
conditions of contact, and are provided the local positions of the nodes of shock.
X
Y
Z=Z1
X1
Y1=Y2
Y
Z=Z1
X1
Y1=Y2
Z2
Y1=Y2
X2=X
loc
Z2
Z
loc
Rotation 1 around Z
Rotation 2 around Y1
Rotation 3d' an angle
ANGL_VRIL around
X
loc
Y
loc
X2=X
loc
Appear 6.2-a: Rotations defining the local reference mark
The operand
ORIG_OBST
allows to define the origin of the local reference mark
(
)
Orig X
Y
Z
loc
loc
loc
,
,
,
. This
operand is optional and in theory will not be used in the case of the shocks between two nodes.
code considers whereas the origin is located in the middle of the segment connecting the two nodes.
6.3
Definition of the nodes of the connections
One specifies, behind the key words
NOEU_1
and
NOEU_2
, names of the two nodes of the structures on
which will carry the conditions of shock. If it is about a connection between a node and an obstacle, only
NOEU_1
is well informed.
6.4
Definition of dimensions characteristic of the sections
The operand
PLAY
is used for the conditions of contact between a node and an obstacle.
Operands
DIST_1
and
DIST_2
allow to specify dimensions characteristic of
sections of the structures surrounding the nodes of shock. In the case of the connections plan on plan, it are
thicknesses of matter surrounding the node of shock in the direction considered.
In the case of connections rings on circle, it acts of the radii of the sections surrounding the nodes of
shock.
6.5
Definition of the parameters of contact
The parameters stiffnesses and damping of shock were introduced with the §3.1 and §3.2, one specifies them here
key words allowing to define them for a given connection.
The operand
RIGI_NOR
is obligatory, it allows to give the value of normal stiffness of shock
K
NR
.
The other operands are optional.
The operand
AMOR_NOR
allows to give the value of normal damping of shock
C
NR
.
The operand
RIGI_TAN
allows to give the value of tangential stiffness
K
T
.
The operand
AMOR_TAN
allows to give the tangential value of damping of shock
C
T
.
The operand
COULOMB
allows to give the value of the coefficient of Coulomb.
Code_Aster
®
Version
3.0
Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
12/07/95
Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
18/18
Manual of Reference
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
Note:
If a stiffness
K
T
is defined and that the key word
AMOR_TAN
misses, the code calculates one
damping optimized in order to minimize the residual oscillations in adherence [bib7]:
(
)
C
K
K
m
K m
T
I
T
I
I
I
I
=
+
-
2
2
.
.
,
where I is the index of the dominating mode in the response of the structure (modal mass more
important).
7 Bibliography
[1]
G. JACQUART - “Methods of Ritz in non-linear dynamics - Application to systems
with shock and friction localized " - Report/ratio EDF DER HP61/91.105
[2]
G. JACQUART - “Control
DYNA_TRAN_MODAL
“- User's documentation Version 2.6
of Code_Aster - Section [U4.54.03]
[3]
G. JACQUART - “Control
DEFI_OBSTACLE
“- User's documentation Version 3.0 of
Code_Aster - Section [U4.21.07]
[4]
P. ORSERO, J.R. LEVESQUE, C. VARE, G. JACQUART, Mr. AUFAURE “Support of the course
Dynamics of the Structures - Seminar of Code_Aster Formation - January 94 " Notes
HP-61/94/189
[5]
Mr. JEAN, J.J. MOREAU “Unilaterality and dry friction in the dynamics off rigid bodies collection”
Proceedings off the International Mechanics Contact Symposium - ED. A. CURNIER - Presses
Polytechnic and University French - Lausanne, 1992, p 31-48
[6]
J.T. ODEN, J.A.C. MARTINS “Models and computational methods for dynamic friction
phenomena " - Computational Methods Appl. Mech. Engng. 52, 1992, p 527-634
[7]
B. BEAUFILS “Contribution to the study of the vibrations and the wear of the beams of tubes in
cross-flow " - Thesis of doctorate PARIS VI