Code_Aster
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Version
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Titrate:
SDND100 To release of a shoe rubbing with friction of the Coulomb type
Date:
14/09/01
Author (S):
Fe WAECKEL, G. DEVESA
Key
:
V5.01.100-C
Page:
1/10
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
Organization (S):
EDF/RNE/AMV
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
V5.01.100 document
SDND100 - To release of a rubbing shoe
with friction of the Coulomb type
Summary
One considers the one-way system with a degree of freedom made up of a mass in rubbing contact of type
Coulomb on a rigid level, and of a spring attaching it to a fixed point. The mass is released in a position
initial except balance. It oscillates until the complete stop at the end of a finished time.
The first two modelings correspond to the transitory response by modal recombination of the shoe
rubbing, the third corresponds to its direct transitory answer. Three calculations are compared with the solution
analytical.
Code_Aster
®
Version
5.0
Titrate:
SDND100 To release of a shoe rubbing with friction of the Coulomb type
Date:
14/09/01
Author (S):
Fe WAECKEL, G. DEVESA
Key
:
V5.01.100-C
Page:
2/10
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
1
Problem of reference
1.1 Geometry
K
m
Uo
Y
X
Z
G
K
m
=45°
N
Direction of displacement:
= 45° in plan XY
1.2
Material properties
Stiffness of the spring:
K = 10.000 NR/m
Specific mass:
m = 1 kg
Gravity:
G = 10 m/s
2
Coefficient of Coulomb:
µ
= 0,1
1.3
Boundary conditions and loadings
The system rests on the plan Z = 0 on which it can slip with a coefficient of friction of
Coulomb of
µ
= 0 1
,
.
1.4 Conditions
initial
Initial displacement of the mass:
R
0
= 0,85 mm according to the direction
.
Null initial speed.
Code_Aster
®
Version
5.0
Titrate:
SDND100 To release of a shoe rubbing with friction of the Coulomb type
Date:
14/09/01
Author (S):
Fe WAECKEL, G. DEVESA
Key
:
V5.01.100-C
Page:
3/10
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
2
Reference solution
2.1
Method of calculation used for the reference solution
For a system without damping, the differential equation to solve is written:
m R K R
F
F
Mg
R
R T
R
R T
N
N
&&
(&)
(
)
& (
)
+
=
= -
=
=
=
=
µ
with
sign
0
0
0
0
0
It is shown [bib1] that the solution of the differential equation is written:
R T
F
K
R
F
K
T
N
N
()
(
) cos
=
+
-
µ
µ
0
0
The amplitude of the extrema, which all come them
T
N
N
+
=
1
0
, obeys the law of following recursion:
R T
R
F
K
T
N
NR
R T
R
F
K R
N
N
N
N
N
(
) ()
cos
,…,
(
)
+
+
= -
-
=
<
1
0
0
1
0
0
1
1 2
µ
µ
with
such as
The movement stops when
R T
R
F
K R
N
N
(
)
+
<
1
0
0
µ
with the position
R T
N
(
)
+1
.
2.2
Results of reference
Values of displacements in the direction
for the moments of change of sign speed
(
() ()
()
R T
R T
R T
1
2
5
,
,…,
benches above).
2.3
Uncertainty on the solution
Analytical solution.
2.4 References
bibliographical
[1]
F. AXISA - Methods of analysis in nonlinear dynamics of the structures: non-linearities of
contact - Course IPSI from the 28 to May 30, 1991
Code_Aster
®
Version
5.0
Titrate:
SDND100 To release of a shoe rubbing with friction of the Coulomb type
Date:
14/09/01
Author (S):
Fe WAECKEL, G. DEVESA
Key
:
V5.01.100-C
Page:
4/10
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
3 Modeling
With
3.1
Characteristics of modeling
An element of the type
DIS_T
on a mesh
POI1
is used to modelize the system.
Conditions of relations between degrees of freedom are employed to force the movement to be
one-way in the direction
:
LIAISON_DDL: (NODE: NO1
DDL
:
(“DX”
“DY”)
COEF_MULT
:
(0.707
- 0.707)
COEF_IMPO
:
0.)
An obstacle of the type
PLAN_Z
(two parallel plans separated by a play) is used to simulate the plan
of slip. One chooses to take for generator of this plan axis OY, that is to say
NORM_OBST: (0.,
1., 0.)
. The origin of the obstacle is
ORIG_OBST: (0., 0., 1.)
. It remains to define its play which gives it
half-spacing enters the plans.
So that there is a force of reaction of the plan on the system it is necessary that this last is slightly
inserted in the plane obstacle of a distance
N
such as:
F
K
N
N
N
=
.
Like
F
Mg
N
=
, one has then
N Mg K
N
=
/
.
One considered a normal stiffness of shock of 20 NR/m (fictitious stiffness which has direction only for
to generate a force of reaction of the plan on the system), one thus has
N
= 0,5. The obstacle
PLAN_Z
having
for origin Z = 1 and the solid being in Z = 0; a play of 0,5m will create a depression
N
= 0,5 m from where
PLAY: 0.5
Tangential stiffness of shock:
K
T
= 400.000 NR/m: it is large in front of the stiffness of the oscillator
so that the phase of stop is modelized correctly.
No the time used for temporal integration: 5.10
4
S.
3.2
Characteristics of the mesh
A number of nodes: 1
A number of meshs and types: 1
POI1
3.3 Functionalities
tested
Controls
AFFE_CHAM_NO SIZE
“DEPL_R”
PROJ_VECT_BASE VECT_ASSE
PROJ_MATR_BASE MATR_ASSE
STANDARD DEFI_OBSTACLE
“PLAN_Y”
“PLAN_Z”
DYNA_TRAN_MODAL DEPL_INIT_GENE
METHOD
“EULER”
REST_BASE_PHYS SHOCK
Code_Aster
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Version
5.0
Titrate:
SDND100 To release of a shoe rubbing with friction of the Coulomb type
Date:
14/09/01
Author (S):
Fe WAECKEL, G. DEVESA
Key
:
V5.01.100-C
Page:
5/10
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
4
Results of modeling A
4.1 Values
tested
Values of displacements (in meters) in the direction
for the moments of change of sign
speed over the period of time (0; 0.3 S).
Identification moment
(S) Reference Displacement
Aster
difference %
DY = r2 cos45
X 10
2
4.596E4 4.595E4 0.02
DY = r3 cos45
2
X 10
2
3.182E4 3.181E4 0.045
DY = r4 cos45
3
X 10
2
1.768E4 1.767E4 0.07
DY = r5 cos45
4
X 10
2
3.536E5 3.550E5 0.41
One presents Ci below the evolution of displacement and speed at point NO1
Displacement of point NO1
Speed of point NO1
4.2 Parameters
of execution
Version:
STA 5.02
Machine:
SGI ORIGIN2000
Time CPU To use:
2.21 seconds
Code_Aster
®
Version
5.0
Titrate:
SDND100 To release of a shoe rubbing with friction of the Coulomb type
Date:
14/09/01
Author (S):
Fe WAECKEL, G. DEVESA
Key
:
V5.01.100-C
Page:
6/10
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
5 Modeling
B
5.1
Characteristics of modeling
In modeling B, one regards the shoe and the plan as two mobile structures. Each
structure is then modelized by a node and an element of the type
POI
1
. Node NO2 is supposed
locked, it materializes the plan of friction. One imposes conditions of relations between degrees of
freedom with the node NO1 (which modelizes the shoe) so that the movement is one-way in
direction
.
LIAISON_DDL: (NODE: NO1
DDL
:
(“DX”
“DY”)
COEF_MULT
:
(0.707
- 0.707)
COEF_IMPO
:
0.)
An obstacle of the type
BI_PLAN_Z
(two mobile parallel plans separated by a play) is used for
to simulate the slip surface. One chooses to take for generator of this plan axis OY, that is to say
NORM_OBST: (0., 1., 0.).
By defect, the origin of the obstacle is located at semi distance from the nodes
NO1 and NO2. It remains to define the parameters
DIST_1
and
DIST_2
who represent the thickness of
matter around the nodes of shock.
So that there is a force of reaction of the plan on the system it is necessary that this last is slightly
inserted in the plane obstacle of a distance
N
such as:
F
K
N
N
N
=
.
Like
F
Mg
N
=
, one has then
N Mg K
N
=
/
.
One considered a normal stiffness of shock of 20 NR/m (fictitious stiffness which has direction only for
to generate a force of reaction of the plan on the system), one thus has
N
= 0,5 Mr. Knowing that both
nodes NO1 and NO2 are geometrically confused, one chooses for example
DIST_1
=
DIST_2
=
N
/2.
Tangential stiffness of shock:
K
T
= 400.000 NR/m: it is large in front of the stiffness of the oscillator
so that the phase of stop is modelized correctly.
No the time used for temporal integration: 5.10
4
S.
5.2
Characteristics of the mesh
A number of nodes: 2
A number of meshs and types: 2 POI1
5.3 Functionalities
tested
Controls
AFFE_CHAM_NO SIZE
“DEPL_R”
PROJ_VECT_BASE VECT_ASSE
PROJ_MATR_BASE MATR_ASSE
STANDARD DEFI_OBSTACLE
“BI_PLAN_Z”
DYNA_TRAN_MODAL DEPL_INIT_GENE
METHOD
“EULER”
SHOCK
NOEUD_1
NOEUD_2
REST_BASE_PHYS
Code_Aster
®
Version
5.0
Titrate:
SDND100 To release of a shoe rubbing with friction of the Coulomb type
Date:
14/09/01
Author (S):
Fe WAECKEL, G. DEVESA
Key
:
V5.01.100-C
Page:
7/10
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
6
Results of modeling B
6.1 Values
tested
Values of displacements (in meters) in the direction of the oscillator for the moments of
change of sign speed over the period of time (0; 0.3 S).
Identification moment
(S) Reference Displacement
Aster
difference %
DY = r2 cos45
X 10
2
4.596E4 4.595E4
0.02
DY = r3 cos45
2
X 10
2
3.182E4 3.181E4
0.029
DY = r4 cos45
3
X 10
2
1.768E4 1.767E4 0.018
DY = r5 cos45
4
X 10
2
3.536E5 3.543E5 0.205
6.2 Parameters
of execution
Version:
STA 5.02
Machine:
SGI ORIGIN2000
Time CPU To use:
2.3 seconds
Code_Aster
®
Version
5.0
Titrate:
SDND100 To release of a shoe rubbing with friction of the Coulomb type
Date:
14/09/01
Author (S):
Fe WAECKEL, G. DEVESA
Key
:
V5.01.100-C
Page:
8/10
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
7 Modeling
C
7.1
Characteristics of modeling
This modeling corresponds to the direct transitory response of the rubbing shoe.
The normal direction of contact is the local axis X which corresponds in the case test to total axis Z. It
slip surface is the local plan (Y, Z) that is to say the plan (X, Y) in the total reference mark. One thus directs
the element of shock to a node, with the key word ORIENTATION of operator AFFE_CARA_ELEM of
following way:
ORIENTATION:(MESH:EL1 CARA: “VECT_X_Y”
VALE: (0. 0. - 1. 0. 1. 0. ))
To be able to obtain a force of reaction of the plan on the system it is necessary that this last is slightly
inserted in the plane obstacle of a distance
N
such as:
F
K
N
N
N
=
.
The reaction balances the weight of the shoe, one thus has:
F
Mg
N
=
i.e.
N Mg K
N
=
/
.
One considered a normal stiffness of shock of 20 NR/m (fictitious stiffness which has direction only for
to generate a force of reaction of the plan on the system), one thus has
N
= 0,5 from where DIST_1 = 0.5.
The tangential stiffness of shock considered is
K
T
= 400.000 NR/m, the coefficient of Coulomb is worth 0,1.
The law of behavior of shock is thus in the following way defined in DEFI_MATERIAU:
DIS_CONTACT: (RIGI_NOR: 20.
DIST_1: 0.5
RIGI_TAN: 400000.
COULOMB: 0.1)
One uses a pitch of time of 5.10
4
S for temporal integration.
7.2
Characteristics of the mesh
A number of nodes: 1
A number of meshs and types: 1 POI1
7.3 Functionalities
tested
Controls
DEFI_MATERIAU DIS_CONTACT
AFFE_CARA_ELEM ORIENTATION VECT_X_Y
AFFE_CHAR_MECA LIAISON_DDL
AFFE_CHAM_NO
DYNA_NON_LINE ETAT_INIT DEPL_INIT
COMP_INCR
RELATION
DIS_CHOC
HHT
RECU_FONCTION
Code_Aster
®
Version
5.0
Titrate:
SDND100 To release of a shoe rubbing with friction of the Coulomb type
Date:
14/09/01
Author (S):
Fe WAECKEL, G. DEVESA
Key
:
V5.01.100-C
Page:
9/10
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
8
Results of modeling C
8.1 Values
tested
Values of displacements in the direction of the oscillator for the approximate moments of
change of sign speed over the period of time (0; 0.2 S).
Identification moments
(S) Reference
displacement
Aster
difference %
DY = r2 cos45
X 10
2
4,585E04
4,58552E04 0,011
DY = r3 cos45
2
X 10
2
3,173E04
3,17331E04 0,01
DY = r4 cos45
3
X 10
2
1,754E04
1,75481E04 0,046
DY = r5 cos45
4
X 10
2
3,550E05
3,54945E05 0,016
8.2 Parameters
of execution
Version:
STA 5.02
Machine:
SGI ORIGIN2000
Time CPU To use:
94 seconds
Code_Aster
®
Version
5.0
Titrate:
SDND100 To release of a shoe rubbing with friction of the Coulomb type
Date:
14/09/01
Author (S):
Fe WAECKEL, G. DEVESA
Key
:
V5.01.100-C
Page:
10/10
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
9
Summary of the results
The analytical solution of the problem with friction is reproduced with a very good precision
(<0.5%). That asks for nevertheless the use of a parameter of tangent stiffness raised enough by
report/ratio with the rigidity of the system as well as a pitch of relatively reduced time of integration.
On this example, direct nonlinear calculation is much more expensive in calculating times, factor
20, that that on modal basis.