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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
Organization (S): EDF/EP/AMV
Handbook of Référence
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].
Handbook of Référence
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Code_Aster ®
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Titrate:
Modeling of the shocks and friction in transitory analysis
Date:
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Author (S):
G. JACQUART
Key:
R5.06.03-A
Page:
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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 modelled 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
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Titrate:
Modeling of the shocks and friction in transitory analysis
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Page:
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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 model 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
1/2
1/2
1 and 2. D NR is noted
the normal distance enters the structures, FN
force normal reaction of 1 out of 2.
The law of the action and the reaction imposes:
F 2 1
/= - F 1/2
NR
NR
éq 2.1-1
F 2/1
NR
1
D 1/2
NR
2
F 1/2
NR
Appear 2.1-a: Distance normal and normal reaction
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The conditions of unilateral contact, still called conditions of Signorini [bib5], are expressed
following way:
D 1/2 0, F 1/2 0, D 1/2 F 1/2 = 0 and F 2 1/= - F 1/2
NR
NR
NR
NR
NR
NR
éq 2.1-2
F 1/2
NR
D 1/2
NR
Appear 2.1-b: Graphe 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 1/2
N
N
NR
) the normal distance to the support, and Fn reaction of this last (attention!
F = F 2 1/= - F 1/2
N
NR
NR
to see diagram below).
The expression of the conditions of normal contact, expressing the limitation of displacements due to
support is worth:
D 0, F 0, D F
N
N
N
N = 0
Fn
DNN > 0
DNN = 0
(cf feel N
and of Fn)
N
N
Appear 2.1-c: Outdistance normal and normal reaction between a structure and a support
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2.2
Law of friction of Coulomb
The law of Coulomb expresses a tangential limitation of effort F 1/2
T
of tangential reaction of 1 on
1 2
2. That is to say!
/
C the relative speed of 1 compared to 2 in a point of contact and is µ it
coefficient of friction of Coulomb, one has [bib5]:
S = F 1/2 - µ F 1/2, U 1/2
!
= F 1/2
0
, 0, .s
T
NR
T
T
= 0
éq 2.2-1
and the law of the action and the reaction:
F 2 1
/= - F 1/2
T
T
éq 2.2-2
F1/2
T
Ý
U 1/2
T
Appear 2.2-a: Graphe 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 2 1
/= F
T
T is used, the law of friction is expressed in the following way:
S = F - µ F 0, U! = F
, 0, .s
T
N
T
T
= 0
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Modeling of the shocks and friction in transitory analysis
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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 1/2 = F (D 1/2
NR
NR
) 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 between D 1/2
1/2
NR
and FN
:
1
F 1/2 = - D 1/2
if D 1/2 0; F 1/2
NR
NR
NR
NR
= 0
if not
éq 3.1-1
1
If one notes kN = called commonly “stiffness of shock”, one finds the traditional relation,
modelling an elastic shock:
F 1/2 = - K D 1/2
NR
NR
NR
éq 3.1-2
The approximate graph of the law of contact with penalization is as follows:
F 1/2
NR
D 1/2
NR
Appear 3.1-a: Graphe 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 " CN the expression of the normal force of contact is expressed then by:
F 1/2 = - K D 1/2 - C U 1/2
NR
NR
NR
NR
!NR
éq 3.1-3
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where! /
U 1 2
NR
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 1/2
NR
is positive or
+
null. Only the positive part will thus be taken.
expression [éq 3.1-3]:
+
X
= X if X 0
+
X
= 0 if X < 0
The complete relation giving the normal force of contact which is retained for the algorithm is
following:
+
if D 1/2 0 F 1/2 = - K D 1/2 - C U 1/2
!
, F 2 1
/= - F 1/2
NR
NR
NR
NR
NR
NR
NR
NR
if not
F 2 1
/= F 1/2
NR
NR
= 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 1 2
T
=)
0. One thus introduces a univocal relation binding relative tangential displacement
D 1/2
1/2
1/2
T
and the tangential force F
= F (D
T
T
) 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 between D 1/2
1/2
T
and FT:
1
F 1/2 - F 1/2 0 = -
(D 1/2 - D 1/2 0
T
T
T
T
)
éq 3.2-1
1
If one introduces a “tangential stiffness” KT =, one obtains the relation:
F 1 2 = F 1 2 0 - KT (D 1 2 - D 1 2 0
/
/
/
/
T
T
T
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” CT in the expression of the force
tangential. Its final expression is:
F 1/2 = F 1/2 0 - K (D 1/2 - D 1/2 0) - C
T
T U 1/2
!
, F 2 1
/= - F 1/2
T
T
T
T
T
T
T
éq 3.2-3
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It is necessary moreover than this force checks the criterion of Coulomb, that is to say:
U 1/2
!
F 1/2
1/2
µ F
if not one appli
that F 1/2
1/2
T
= - µ F
NR
NR
, F 2 1
/= - F 1/2
T
T
éq 3.2-4
U
T
T
1/2
! T
The approximate graph of the law of friction of Coulomb modelled by penalization is as follows:
F 1/2
T
KT
Ý
U 1/2
T
Appear 3.2-a: Graphe of the law of friction approached by penalization
4
Types of modelled 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]).
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Z
Y
Xloc
X
Be reproduced 4.1.1-a: Structures slim 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.
Zloc
NO1
Y
Play
loc
ORIG_OBST
1
2
Be reproduced 4.1.1-b: Géométrie of the connection node on obstacle plan
Yloc
Are
co-ordinates of
Y, Z
, the origin of this reference mark is it
Z
NOEUD
NO1 in the reference mark (loc
loc)
loc
not ORIG_OBST.
The normal distance D NR in this case, by neglecting rotations of the sections is expressed 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 Zloc between the two
structures.
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Normal vector N in the reference mark (Y, Z
loc
loc) has as components:
sign Y
(loc)
N =
éq 4.1.1-2
0
Other quantities!one, FN!C, FT 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).
Z
X
Y
loc
X
Be reproduced 4.1.2-a: Structures slim with contact node on circular obstacle
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. Are NO1, the node of the connection
considered, geometry of the connection of contact node on circle (called CERCLE in
Code_Aster [bib3]) is described on the figure below.
Zloc
NO1
Yloc
Play
ORIG_OBST
Appear 4.1.2-b: Géométrie of the connection circular node obstacle
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Yloc
Are
co-ordinates of the NOEUD
Y, Z
, of origin
Z
NO1 in the reference mark (loc loc)
ORIG_OBST.
loc
The normal distance D NR, by neglecting rotations of the sections is expressed then by:
2
2
D = - (Y - Y
) + (Z - Z
) + play
NR
loc
ORIGobst
loc
ORIGobst
One poses like normal vector N the vector:
ORIG
- NOEUD1
N
obst
= ORIG - NODE
obst
1
play is a strictly positive distance.
Other quantities!one, FN!C, FT 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).
Zloc
NO1
Yloc
ORIG_OBST
Be reproduced 4.1.3-a: Géométrie of the connection node on discretized concave obstacle
Yloc
Are
co-ordinates of the node
Y, Z
, of origin
Z
NO1 in the reference mark (loc loc)
ORIG_OBST.
loc
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One seeks the facet of contact nearest to node NO1, normal vector N is defined
like the direct orthogonal vector with the facet:
Zloc
N
DNN PNO1
NO1
Yloc
ORIG_OBST
Either PNO1 the projection of node NO1 on the facet, the normal distance D NR in this case is worth:
D
(NO1 PNO
NR =
-
)
1 .n
Other quantities!one, FN!C, FT 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 modelled by beams of section
rectangular on the level of the zones of contact.
Z
Y
Xloc
NOEUD 1
NOEUD2
X
Be reproduced 4.2.1-a: Structures slim with plane contact on plan
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To analyze the conditions of contact, one places oneself in the reference mark perpendicular to the axis Xloc,
direction of neutral fiber of the beams. Are NO1 and NO2, the two nodes of the connection considered,
the geometry of the connection plane contact on plan (called BI_PLAN_Y in Code_Aster [bib3]) is
described on the figure below.
Zloc
NO1
NO2
Y
D1
loc
D2
Be reproduced 4.2.1-b: plane Géométrie of the connection on plan
Yi
loc
Are
co-ordinates of NOEUDi in the reference mark (Y, Z
loc
loc), of origin ORIG_OBST
Zi
loc
(ORIG_OBST can be provided by the user, by defect ORIG_OBST is selected like the medium of
nodes NO1, NO2.
The normal distance D 1/2
NR
in this case, by neglecting rotations of the sections expresses itself then by:
D 1/2 = Y1 - Y2
- D - D
NR
loc
loc
1
2
éq 4.2.1-1
D1 and D2 are strictly positive distances.
The contact in this connection is judicious to take place whatever the shift in Zloc between the two
structures.
The normal vector n1/2 in the reference mark (Y, Z
loc
loc) has as components:
2
1
sign (Y loc - Y loc)
n1/2 =
éq 4.2.1-2
0
Other quantities! /
U 1 2
1/2
1 2
1/2
NR
, FN
! /
C, FT are calculated in a general way [§ 2.4].
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4.2.2 Connections of contact rings on circle
If one considers now two cylinders of circular section, modelled 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:
Xloc
Appear 4.2.2-a: Structures slim with contact rings on circle
One places oneself in the reference mark perpendicular to the axis Xloc 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:
Zloc
R2
NO2
Yloc
NO1
ORIG_OBST
2
R1
1
Appear 4.2.2-b: Géométrie of the connection rings on circle
The normal distance D 1/2
NR
has as an expression:
2
2
D 1/2 = (Y1 - Y2) + (Z1 - Z2) - R - R
NR
loc
loc
loc
loc
1
2
One poses like normal vector of 1 towards 2 the vector:
-
NOEUD2 NOEUD1
n1/2 =
-
NOEUD2 NOEUD1
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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 (T) + F
(X, X! )
T
T
T
ext.
shock
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:
MT
T
T
T
T
! T + C
!T + K
T = ext.
F (T) + CH
F oc (
T,
!T)
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 operator DYNA_TRAN_MODAL [bib1], [U4.54.03]. The type of connection
of shock between the two nodes is specified by a specific command: 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.
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The types of connection with shock with two nodes accepted by command DEFI_OBSTACLE are
described by the key words following:
PLAN_Y, PLAN_Z or CERCLE
BI_PLAN_Y, BI_PLAN_Z or BI_CERCLE (see figure below).
Zloc
Z
Z
loc
loc
Yloc
Yloc
Yloc
PLAN_Y
PLAN_Z
CERCLE
Z
Z
loc
loc
Zloc
Yloc
Yloc
Yloc
BI_PLAN_Y
BI_PLAN_Z
BI_CERCLE
Appear 6.1-a: Géométries of the connections of shock
Prefix BI_ specifies 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 modelled by elements of beam. The contact is treated, as one saw with
[§3.1] and [§3.2] in a plan perpendicular to the Xloc direction of the generator of the cylinders.
To define this change of reference mark completely, one introduces a reference mark local (X, Y, Z
loc
loc
loc).
The Xloc vector is the vector with 3 components provided behind key word NORM_OBST.
Using the first two nautical angles, one passes in a single way of the total reference mark (X, Y, Z) to
a reference mark having Xloc like first basic vector (see [Figure 6.2-a] hereafter). A third
rotation whose angle is provided behind key word ANGL_VRIL gives a single correspondence enters
the principal reference mark and the local reference mark.
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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.
Z=Z1
Z=Z1
X2=
X2=
X
X
loc
loc
Y1=Y2
Y
Y1=Y2
Y
Y1=Y2
Yloc
X1
X1
Z2
Z2
X
Z
Rotation 1 around Z
Rotation 2 around Y1
loc
Rotation 3d' an angle
ANGL_VRIL around Xloc
Appear 6.2-a: Rotations defining the local reference mark
Operand ORIG_OBST makes it possible to define the origin of the reference mark local (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 key words NOEU_1 and NOEU_2, the 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 indicated.
6.4
Definition of dimensions characteristic of the sections
Operand JEU is used for the conditions of contact between a node and an obstacle.
Operands DIST_1 and DIST_2 make it possible 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.
Operand RIGI_NOR is obligatory, it makes it possible to give the value of normal stiffness of shock kN.
The other operands are optional.
Operand AMOR_NOR makes it possible to give the value of normal damping of shock CN.
Operand RIGI_TAN makes it possible to give the value of tangential stiffness KT.
Operand AMOR_TAN makes it possible to give the tangential value of damping of shock CT.
The COULOMB operand makes it possible to give the value of the coefficient of Coulomb.
Handbook of Référence
R5.06 booklet: Dynamics in modal base
HP-61/95/058/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:
18/18
Note:
If a stiffness KT is defined and that key word AMOR_TAN misses, the code calculates one
damping optimized in order to minimize the residual oscillations in adherence [bib7]:
C = 2 (K + K) .m - 2 K .m
T
I
T
I
I
I
I,
where I is the index of the dominating mode in the response of the structure (modal mass more
important).
7 Bibliography
[1]
G. JACQUART - “Méthodes de Ritz in non-linear dynamics - Application with systems
with shock and friction localized " - Rapport EDF DER HP61/91.105
[2]
G. JACQUART - “Command DYNA_TRAN_MODAL” - User's documentation Version 2.6
of Code_Aster - Section [U4.54.03]
[3]
G. JACQUART - “Commande DEFI_OBSTACLE” - Documentation Utilisateur 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 Structures - Séminaire de Formation Code_Aster - Janvier 94 " Note
HP-61/94/189
[5]
Mr. JEAN, J.J. MOREAU “Unilaterality and dry friction in the dynamics off rigid bodies collection”
Proceedings off the Contact Mechanics International Symposium - ED. A. CURNIER - Presses
Polytechnic and Universitaires Romandes - 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 being studied of the vibrations and the wear of the beams of tubes in
transverse flow " - Thèse of doctorate PARIS VI
Handbook of Référence
R5.06 booklet: Dynamics in modal base
HP-61/95/058/A
Outline document