Code_Aster ®
Version
5.0
Titrate:
SDND100 Lâcher 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

Organization (S): EDF/RNE/AMV

Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
V5.01.100 document

SDND100 - Lâcher 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.

Handbook of Validation
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Code_Aster ®
Version
5.0
Titrate:
SDND100 Lâcher 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

1
Problem of reference

1.1 Geometry

K
m
Uo
Z
Y
G
m
N
K
=45°
X


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/s2
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: r0 = 0,85 mm according to the direction.

Null initial speed.

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Code_Aster ®
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Titrate:
SDND100 Lâcher 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

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 = µ F with F = - mgsign (R
N
N
&)
R
(T =)
0 = R 0
0

R

& (T =)
0 = 0

It is shown [bib1] that the solution of the differential equation is written:

µ F
µ F
R T
N
=
+ R
N
()
(-
) cos T
K
0
K
0

N
The amplitude of the extrema, which all come the tn+ =
1
, obeys the law of following recurrence:
0



µ F
R
(T
N
) = (-)
1
1 R
N
-
0
cos T
n+

K
0







R (T
)
µ F
with N =,
1 2,…, NR
n+1
N
such as
<

R
K R

0
0

R (T
)
µ F
+
The movement stops when
N 1
N
<
with the position R (T
)
R
K R
n+1.
0
0

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) established above).

2.3
Uncertainty on the solution

Analytical solution.

2.4 References
bibliographical

[1]
F. AXISA - Méthodes of analysis in nonlinear dynamics of the structures: non-linearities of
contact - Cours IPSI from the 28 to May 30, 1991
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HT-62/01/012/A

Code_Aster ®
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Titrate:
SDND100 Lâcher 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

3 Modeling
With

3.1
Characteristics of modeling

An element of the type DIS_T on a mesh POI1 is used to model the system.
Conditions of relations between degrees of freedom are employed to force the movement to be
one-way in the direction:

LIAISON_DDL: (NOEUD: 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/kN.
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. 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
JEU: 0.5

Tangential stiffness of shock: KT = 400.000 NR/m: it is large in front of the stiffness of the oscillator
so that the phase of stop is modelled correctly.

No the time used for temporal integration: 5.10­4s.

3.2
Characteristics of the grid

A number of nodes: 1
A number of meshs and types: 1 POI1

3.3 Functionalities
tested

Commands
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

METHODE
“EULER”
REST_BASE_PHYS SHOCK

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Code_Aster ®
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Titrate:
SDND100 Lâcher 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

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) Référence Déplacement difference %
Aster
DY = r2 cos45
X 10­2 ­ 4.596E4 ­ 4.595E4 ­ 0.02
DY = r3 cos45
2 X 10­2 3.182E4 3.181E4 ­ 0.045
DY = r4 cos45
3 X 10­2 ­ 1.768E4 ­ 1.767E4 ­ 0.07
DY = r5 cos45
4 X 10­2 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



Handbook of Validation
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HT-62/01/012/A

Code_Aster ®
Version
5.0
Titrate:
SDND100 Lâcher 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

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 modelled by a node and an element of the type POI1. Node NO2 is supposed
blocked, it materializes the plan of friction. One imposes conditions of relations between degrees of
freedom with the node NO1 (which models the shoe) so that the movement is one-way in
direction.

LIAISON_DDL: (NOEUD: 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 parameters DIST_1 and DIST_2 which 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/kN.
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. Sachant that both
nodes NO1 and NO2 are geometrically confused, one chooses for example DIST_1 = DIST_2 =
N/2.

Tangential stiffness of shock: KT = 400.000 NR/m: it is large in front of the stiffness of the oscillator
so that the phase of stop is modelled correctly.

No the time used for temporal integration: 5.10­4 S.

5.2
Characteristics of the grid

A number of nodes: 2
A number of meshs and types: 2 POI1

5.3 Functionalities
tested

Commands
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
METHODE
“EULER”
CHOC
NOEUD_1
NOEUD_2
REST_BASE_PHYS

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Code_Aster ®
Version
5.0
Titrate:
SDND100 Lâcher 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

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) Référence Déplacement
difference %
Aster
DY = r2 cos45
X 10­2 ­ 4.596E4 ­ 4.595E4 ­ 0.02
DY = r3 cos45
2 X 10­2 3.182E4 3.181E4
­ 0.029
DY = r4 cos45
3 X 10­2 ­ 1.768E4 ­ 1.767E4 ­ 0.018
DY = r5 cos45
4 X 10­2 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



Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A

Code_Aster ®
Version
5.0
Titrate:
SDND100 Lâcher 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

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 key word ORIENTATION of operator AFFE_CARA_ELEM of
following way:

ORIENTATION:(MAILLE: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/kN.
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 KT = 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 step of time of 5.10­4 S for temporal integration.

7.2
Characteristics of the grid

A number of nodes: 1
A number of meshs and types: 1 POI1

7.3 Functionalities
tested

Commands


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


Handbook of Validation
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HT-62/01/012/A

Code_Aster ®
Version
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Titrate:
SDND100 Lâcher 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

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) Référence
displacement
difference %
Aster
DY = r2 cos45
X 10­2 ­ 4,585E04
­ 4,58552E04 0,011
DY = r3 cos45
2 X 10­2 3,173E04
3,17331E04 0,01
DY = r4 cos45
3 X 10­2 ­ 1,754E04
­ 1,75481E04 0,046
DY = r5 cos45
4 X 10­2 3,550E05
3,54945E05 ­ 0,016

8.2 Parameters
of execution

Version:
STA 5.02


Machine:
SGI ORIGIN2000

Time CPU To use:
94 seconds



Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A

Code_Aster ®
Version
5.0
Titrate:
SDND100 Lâcher 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

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 step 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.

Handbook of Validation
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