Code_Aster
®
Version
6.4
Titrate:
SDND104 - Calculation of the power of wear of a rubbing shoe
Date:
16/05/03
Author (S):
S. LAMARCHE
Key
:
V5.01.104-A
Page:
1/12
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
V5.01.104 document
SDND104 - Calculation of the power of wear of one
mass rubbing under seismic excitation
harmonic
Summary:
One considers a mass in contact rubbing with a rigid plan on which one imposes a vibratory movement of
harmonic type. Friction is modelized by the law of Coulomb. The calculation of the response of the mass is of
nonlinear transitory type. One calculates the power of wear resulting from the phases of slip between
mass and the rigid plan. The calculation of the power of wear being developed in Aster only for calculations
modal, the analysis is carried out on the basis of modal system (commonplace). In order to avoid the numerical problems
resulting from the nullity of the single mode of rigid body of the mass, a spring far from stiff is introduced, flexible
mass at a point interdependent of the vibrating rigid plan.
The reference solution is a quasi analytical calculation of the transitory answer, of which estimates
numerical are programmed with Maple.
Single Aster modeling retained tests the explicit algorithms of integration with constant pitch of Euler
(command 1) and Devogeleare (command 4), as well as the algorithm with variable pitch
ADAPT
(command 2) developed in
order
DYNA_TRAN_MODAL
, for various amplitudes of the harmonic acceleration of excitation
seismic of the rigid plan of support. According to this amplitude, the mode of the response of the mass is of the type
member for any time (stick), successively member and slipping (stick-slip), or always slipping with
inversion of the direction of slip (slipway-slipway).
Account is returned owing to the fact that in the case of a sufficiently low amplitude of excitation (first mode,
permanent adherence), the power of wear is strictly null.
Code_Aster
®
Version
6.4
Titrate:
SDND104 - Calculation of the power of wear of a rubbing shoe
Date:
16/05/03
Author (S):
S. LAMARCHE
Key
:
V5.01.104-A
Page:
2/12
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
1
Problem of reference
1.1 Geometry
The system considered consists of a simple heavy mass posed on a rigid support subjected to
an imposed vibration of type seismic, sinusoidal. The contact, as well as solid friction are
modelized by penalization. The system thus has two degrees of freedom of translation (horizontal and
vertical).
X-ray
RY
RZ
M
F
A very weak spring of stiffness connects the mass to the support in the three directions. This spring is one
artifice of calculation, intended to avoid the nullity of the frequency associated with the rigid mode with translation
horizontal of the mass. Fascinating the Aster results account the presence of this spring are little
different from the results which one would obtain without spring.
1.2
Properties of the model
Stiffness of the spring (according to the three directions):
K = 3.10
- 5
NR/m,
mass:
m = 1 kg,
gravity:
G = 10 m/s2
coefficient of Coulomb:
µ
= 0,1.
1.3
Boundary conditions, conditions initial and loadings
The mass rests on the rigid level with the dimension
Z
=
0
.
The harmonic acceleration imposed on the base has as an equation
has
has
T
=
0
sin ()
. In particular, it is
null at the initial moment. The displacement of the support satisfies the equation
X T
has
T
()
(
/
) sin ()
= -
0
2
, and
thus its movement starts towards the left, with nonnull initial speed
& ()
/
X
has
0
0
= -
.
Initial displacement (with
T
=
0
) of the mass is taken null. The mass is regarded as in state
of adherence at the initial moment. It thus has same nonnull speed as the support with
T
=
0
.
Calculations are carried out for various values of maximum acceleration:
has
0
15
=
m/s
2
,
has
0
1 5
=
,
m/s
2
,
has
0
1 01
=
,
m/s
2
and
has
0
0 99
=
,
m/s
2
and a value of pulsation:
= 2
.
Code_Aster
®
Version
6.4
Titrate:
SDND104 - Calculation of the power of wear of a rubbing shoe
Date:
16/05/03
Author (S):
S. LAMARCHE
Key
:
V5.01.104-A
Page:
3/12
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
2
Reference solution
The reference solution, which is analytical, is calculated in the following way.
That is to say
X T
()
the X-coordinate of the mass in the fixed reference mark and
X T
()
the X-coordinate of the vibrating support in it
even reference mark.
Initially, it is supposed that the mass is adherent on its support. It then remains to it certain
time after the initial moment
T
=
0
. It undergoes of this fact the acceleration imposed by the rigid support, that is to say
&& ()
&& ()
sin
X T
X T
has
T
=
=
0
. The tangential force exerted by the mass on the support is then
F
MX T
my
T
T
= -
= -
&& ()
sin
0
(null at the moment initial, which justifies the starting assumption
that initially, the mass is adherent on its support). The mass remains adherent as long as
F
my
T
F
Mg
T
NR
=
=
0
sin
µ
µ
. If
has
G
0
µ
, the mass thus remains indefinitely adherent on sound
support, and its movement is exactly the same one as the aforementioned. By introducing the coefficient
adimensional
µ
=
G
has
0
, the condition of permanent adherence is written
1
. The curve of acceleration
mass, like support, then takes the following form according to time:
0.2
0.4
0.6
0.8
- 4
- 2
2
4
As for speed, it takes the following form (single primitive of null average):
0.2
0.4
0.6
0.8
- 0.4
- 0.2
0.2
0.4
Code_Aster
®
Version
6.4
Titrate:
SDND104 - Calculation of the power of wear of a rubbing shoe
Date:
16/05/03
Author (S):
S. LAMARCHE
Key
:
V5.01.104-A
Page:
4/12
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
If
G
has
µ
>
0
, there is a smaller time
T
T
=
1
such as
F
my
T
Mg
T
=
=
0
1
sin
µ
. This the smallest
time is necessarily such as
sin
T
0
0
>
, which makes it possible to remove the absolute value in
the preceding expression, and to obtain explicit expression
T
G
has
1
0
1
1
=
=
µ
arcsin
arcsin
. In
private individual,
T
T
1
4
2
4
2
=
=
.
After this moment, the mass slips towards the left compared to the support, therefore it checks the equation
dynamics
&& ()
X T
G
=
µ
, that is to say
& ()
(
)
& ()
X T
G T T
X T
=
-
+
µ
1
1
. Its speed thus increases linearly with
time, while leaving to
T
1
negative value
& ()
cos
X T
has
T
has
1
0
1
0
2
1
= -
= -
-
(indeed,
sin
T
1
=
).
driven G
- driven G
X ''
X ''
x'
X'
T
T1
t2 T3
t4
T
Movement for
>
*
, mode of “stick-slip”, succession of adherence and slip
Necessarily, for a certain value of time
T
2
satisfying
/
/
2
2
2
T
, speed
mass becomes again equal at the speed of the support. At this moment, the movement becomes again adherent if
and only if the acceleration which the mass at the beginning of adherence undergoes is lower in value
absolute with
µ
G
. One examines the translation of this condition in the continuation. One expresses for
to begin the value of
T
2
.
Code_Aster
®
Version
6.4
Titrate:
SDND104 - Calculation of the power of wear of a rubbing shoe
Date:
16/05/03
Author (S):
S. LAMARCHE
Key
:
V5.01.104-A
Page:
5/12
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Time
T
2
satisfied the equation
& ()
& ()
X T
X T
2
2
=
, that is to say
µ
G T
T
has
T
has
T
(
)
cos
cos
2
1
0
1
0
2
-
-
= -
, or
still
(
) cos
cos
T
T
T
T
2
1
1
2
0
-
-
+
=
.
This equation, transcendent, allows the determination of
T
2
according to
T
1
and
, that is to say finally,
taking into account the expression of
T
1
, determination of
T
2
according to the physical parameters of
system
and
. If the acceleration of the support in
T
2
is lower in absolute value than
µ
G
, it
movement remains adherent then up to one moment
T
3
for which the acceleration of the support and
mass reach the value
-
µ
G
, moment which for reasons of clear symmetries on graphs Ci
above, exactly satisfied
T
T
3
1
= +
/
. The mass starts a phase of slip then
up to one moment
T
4
, after which the movement reproduces periodically.
One understands that for sufficiently small values of
, the movement will not be able to become
member as from time
T
2
, because the acceleration of the mass would exceed the threshold
µ
G
. There thus exists
a breaking value
*
such as for
>
*
, the movement of the mass passes without phase
of adherence of a slip to a shift in opposite meaning. A reflection on the continuity of
function response of speed of the mass compared to the parameter
show that for
*
, it
later movement is always slipping (mode of “slipway-slipway”, of alternate directions). For
<
*
, it
movement periodically alternates phases of adherence and slip.
The breaking value
*
admits a simple analytical expression. Indeed, for
=
*
, moments
T
2
and
T
3
are confused. Thus
T
T
T
T
2
1
3
1
- = - =
/
and the equation
(
) cos
cos
T
T
T
T
2
1
1
2
0
-
-
+
=
becomes
*
*
cos
=
=
-
2
2 1
1
2
T
. While passing squared, one
obtains
2
4 4
* 2
* 2
= -
, that is to say
*
,
=
+
2
4
0 537
2
.
For
*
, the movement is only asymptotically periodic. The continuation
()
T
N
moments of
change of direction of slip checks
T
T
N
N
+
-
1
/
when
N
tends towards the infinite one. The figure
below shows the typical pace (broken line) the speed of the mass in the situation of slipway-slipway.
Movement for
*
: mode of “slipway-slipway”, no adherence
Code_Aster
®
Version
6.4
Titrate:
SDND104 - Calculation of the power of wear of a rubbing shoe
Date:
16/05/03
Author (S):
S. LAMARCHE
Key
:
V5.01.104-A
Page:
6/12
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Let us summarize the conclusions:
There is the adimensional coefficient
µ
=
G
has
0
and its breaking value
* such as
*
,
=
+
2
4
0 537
2
.
If
*
< <
1
the established mode is of type “stick-slip”: alternation of phases of adherence and of
slip;
If
<
*
,
the established mode is of type “slipway-slipway”: alternate permanent slip;
If
>
1
,
the established mode is of type “stick”: permanent adherence with the base.
In the results of analytical comparison calculation/Aster which follow, choices of the amplitude
has
0
are such as these three situations are visited. One takes indeed
m
=
1
kg,
G
=
10
m/s
2
,
µ
=
0 1
,
,
has
0
15
=
m/s
2
,
has
0
1 5
=
,
m/s
2
,
has
0
1 01
=
,
m/s
2
and
has
0
0 99
=
,
m/s
2
.
The power of wear is physically null at the time of the phases of adherence.
In Aster, with the operator
DYNA_TRAN_MODAL
used here, adherence is not detected bus
the integration of the movement is made by regularization of the law of friction. The respect of the null result
power of wear during phases of adherence required the introduction of a criterion on
speed of slip, so that in lower part of a certain value, it must regarded as null, and
the adherent movement. One can consult the reference material Opérateur of calculation of
wear/Model of Archard [R7.04.10].
During the phases of slip, the power of wear follows the law
P T
mgV T
U
R
()
()
=
µ
, where
V T
X T
X T
R
()
& ()
& ()
=
-
is the relative speed of slip of the mass on the support. In the situation
mode of stick-slip, for which the movement becomes strictly periodic at the end of one
finished time, the energy of wear during a half-period is exactly
E
mgV T dt
Mg
X T
X T dt
Mg
has
T
G T T
has
T dt
U
R
T
T
T
T
T
T
=
=
-
=
-
-
-
-
()
& () & ()
(
cos
(
(
)
cos
))
1
2
1
2
1
2
0
1
0
1
µ
-
-
-
-
-
=
2
1
2
2
1
1
2
0
)
(
2
))
(sin
1
cos
)
((
T
T
G
T
T
T
T
has
Mg
µ
.
The transcendent formulation of
T
2
apparently does not allow to simplify the expression of this
energy of wear. Power of average wear
P
U
is simply the energy of wear
E
U
above
divided by the half-period of the answer
T/
/
2
=
.
In the case of a movement always slipping (
*
), the interval of integration to be taken is
form
[
]
T T
N
N
,
+
1
with
N
sufficient large, so that
T
T
N
N
+
-
1
that is to say sufficiently near to
limiting value
/
. One can avoid numerical calculation by recursion of this continuation, knowing that
average asymptotic speed is null. Indeed, the continuation
T
N
N
-
/
has a finished limit
.
satisfied properties by
are illustrated on the following figure:
Code_Aster
®
Version
6.4
Titrate:
SDND104 - Calculation of the power of wear of a rubbing shoe
Date:
16/05/03
Author (S):
S. LAMARCHE
Key
:
V5.01.104-A
Page:
7/12
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
The segment of straight line has as an equation
v
G T
W
G T
has
=
-
- =
-
-
µ
µ
(
)
(
)
cos (
)
0
,
and for
T
= +
/
, speed
v
that is to say to take the opposite value
W
has
=
0
cos (
)
, which gives
the equation
µ
G
has
has
/
cos (
)
cos (
)
-
=
0
0
,
that is to say
µ
G
has
=
2
0
cos (
)
;
whose solution is
µ
=
=
1
2
1
2
0
arccos
arccos
G
has
.
Let us note that one finds although for
=
*
, the acceleration of the support calculated at time
T
=
give the limiting value
µ
G
. Indeed
has
has
has
has
has
G
0
0
0
2
0
0
2
1
4
1
1
sin (
)
sin (arccos (
/))
/
(
)
*
* 2
* 2
*
µ
=
=
-
=
- -
=
=
.
In the case of the movement always slipping, the energy of wear during one asymptotic period
is given exactly by the formula
E
mgV T dt
U
R
=
+
()
/
that one can clarify according to preceding calculation, while taking
T
1
=
and
T
2
= +
/
, which gives
E
Mg has T
T
G T
Mg has
G
U
=
-
-
-
=
+
-
-
+
0
2
0
2
2
2
2
1
2
2
2 1 4
2
µ
µ
(cos
sin
)
(
)
(
)
/
,
Code_Aster
®
Version
6.4
Titrate:
SDND104 - Calculation of the power of wear of a rubbing shoe
Date:
16/05/03
Author (S):
S. LAMARCHE
Key
:
V5.01.104-A
Page:
8/12
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
that is to say
E
mga
Mg
has
G
U
=
-
=
-
0
2
2
2
2
0
2
2
2
2
4
4
µ
.
The power of average wear (over one period) asymptotic is then
P
E
mga
mga
U
U
=
=
-
=
-
/
0
2
2
0
2
2
4
4
.
Following the Maple program allows the calculation of the power of exact wear in an interval of
specified time, as well as the layout of the graph showing the convergence of the function speed of
mass towards a periodic function limits, for any value of the physical parameters and of excitation
such that the mode is of slipway-slipway type (
*
), and the exact value of the average power
of wear over one period (the only useful one for what interests us) in the case of the stick-slip.
# This program makes calculation, on the transitory part
# of the beginning of the signal, the power of exact wear,
# until a time specifies at the beginning of program.
Digits:= 20:
pi:= evalf (pi):
T:= 1: # period of the movement of the support
Omega:= 2 * pi/T:
tmin:= 4:
tmax:= 12: # duration of the transient considers
ncycle:= floor (tmax/T) +2: # iteration count of Ti calculation [I] and tf [I]
Nmax:= 100 * ncycle: # to replace the function sin by a line brisee
m:= 1:
G:= 10:
driven:= 0.1:
a0:= 1.5:
eta:= driven * G/a0:
Omega:= 2 * pi/T:
etaetoile:= 2/sqrt (pi^2+4):
Ti [1]:= 1/Omega * arcsin (eta):
dX:= T - > - a0/Omega * cos (Omega * T):
dxmoins [0]:= dX (T):
lignedx:= [Ti [1], dX (Ti [1])] :
Eusure:= 0: # wear is null on the phase of adherence [0, Ti [1]]
#
# Noter that Ti [i+1] is necessarily in the interval [I * T-T/4, I * T+T/2]
# and that tf [I] is necessarily in the interval [I * T-3 * T/4, I * T].
# These two intervals overlap, but there is always tf [I] <ti [i+1].
#
yew eta<etaetoile then # mode of slipway-slipway
for I from 1 to ncycle C
dxplus [I]:= driven * G * (T-Ti [I]) + subs (t=ti [I], dxmoins [i-1]):
tf [I]:= fsolve (dX (T) =dxplus [I], t= (I * T-3 * T/4).(I * T)) :
lignedx:= lignedx, [tf [I], dX (tf [I])] :
tinf:= max (Ti [I], tmin):
tsup:= min (tf [I], tmax):
yew tinf<tsup then
Eusure:= Eusure + int (m * G * (dX (T) - dxplus [I]), t=tinf. .tsup):
fi:
dxmoins [I]:= - driven * G * (t-tf [I]) + subs (t=tf [I], dxplus [I]):
Ti [i+1]:= fsolve (dX (T) =dxmoins [I], t= (I * T-T/4).(T/2+I * T)) :
lignedx:= lignedx, [Ti [i+1], dX (Ti [i+1])] :
tinf:= max (tf [I], tmin):
tsup:= min (Ti [i+1], tmax):
yew tinf<tsup then
Code_Aster
®
Version
6.4
Titrate:
SDND104 - Calculation of the power of wear of a rubbing shoe
Date:
16/05/03
Author (S):
S. LAMARCHE
Key
:
V5.01.104-A
Page:
9/12
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Eusure:= Eusure + int (m * G * (dxmoins [I] - dX (T)), t=tinf. .tsup):
fi:
od:
# courbedX:= stud ([seq ([J * tmax/Nmax, dX (J * tmax/Nmax)], j=0. .Nmax)]):
# courbedx:= stud ([lignedx]):
# with (studs):
# display ([courbedX, courbedx]);
theta:= arccos (pi * eta/2)/Omega:
dxinfini:= T - > driven * G * (T-theta) +dX (theta):
Vginfini:= dxinfini - dX:
Eumoyana:= - int (m * G * Vginfini (T), t=theta.(theta+pi/Omega)) :
Eumoyanaana:= m * G * a0/omega^2 * sqrt (4-eta^2 * pi^2):
Pumoyana:= 2 * Eumoyana/T:
Pumoyanaana:= 2 * Eumoyanaana/T:
Pusure:= Eusure/(tmax-tmin);
elif (eta>etaetoile and eta<1) then # mode of stick-slip
lignedx:= [Ti [1], dX (Ti [1])] :
dxplus [1]:= driven * G * (T-Ti [1]) + subs (t=ti [1], dxmoins [0]):
tf [1]:= fsolve (dX (T) =dxplus [1], t= (T-3 * T/4)..T):
dxplus:= unapply (dxplus [1], T):
Vg:= dxplus - dX:
Have:= - int (m * G * Vg (T), t=ti [1]..tf [1]):
Pusuremoy:= 2 * Have/T;
else # mode of permanent adherence
Have:= 0;
fi:
The Aster solution considered is the calculation of the power of average wear during a phase
going transient from 4 to 11,99 seconds (of 8
/
to 24
/
). The energy of wear for this length of time
transient differs somewhat from the energy of average wear (asymptotic) over this duration (such an amount of in
situation of stick-slip that of slipway-slipway). It is thus appropriate, to precisely compare it with the results
Aster, to make an exact calculation of this energy in the interval of time [4s, 11,99s].
For
has
0
15
=
m/s
2
, the power of average wear asymptotic is 15,1146144886 Watt then
that the power of average wear on the temporal interval [4s, 11,99s] is 15,257521794 Watt.
It is this last value which constitutes the result of reference.
Note:
As a calculation of average power, power of wear calculated on an interval
is not obligatorily increasing with the duration of the interval. If one adds to the interval
one duration over which there is adherence, the power of average wear will be lower.
2.1
Results of reference
Value of max. acceleration a0 (ms
- 2
)
Value of the average power of wear
On the interval [4s, 11,99s], in Watt
15 (slipway-slipway)
15,26709959
1,5 (stick-slip)
0,40906245
1,01 (stick-slip)
2,261641E-4
0,99 (stick)
0
2.2
Uncertainty on the solution
Quasi-analytical solution (presence of transcendent equations solved numerically with one
arbitrary precision).
2.3 References
bibliographical
[1]
B. WESTERMO, F. UDWADIA: Periodic Response off has sliding oscillator system to harmonic
excitation. Earthquake Engineeering and structural dynamics Flight 14.135-146 (1983)
[2] Documentation
Code_Aster [R7.04.10]
Code_Aster
®
Version
6.4
Titrate:
SDND104 - Calculation of the power of wear of a rubbing shoe
Date:
16/05/03
Author (S):
S. LAMARCHE
Key
:
V5.01.104-A
Page:
10/12
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/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.
Calculation is done on modal basis. One locks displacements in Y and Z, the modal base
thus contains that a mode.
One uses the dynamic operator of calculation on modal basis
DYNA_TRAN_MODAL
, with the key word
SHOCK
to modelize nonthe local linearity.
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 OY is
NORM_OBST
: (0., 1., 0.).
The origin of the obstacle is
ORIG_OBST
: (0., 0., 1.), its play which gives the half-spacing between
plans is 0.5.
One places oneself in the relative reference mark (loading mono-support) and one applies a loading in
acceleration with
CALC_CHAR_SEISME
.
One uses a pitch of time of 3.10
5
S for temporal integration to limit the calculating time. It
no time is quite lower than
S
M
K
M
K
NR
4
10
.
7
)
/
/
2
,
/
/
2
min (
-
=
.
The tangential stiffness of friction is taken as large as possible to ensure the stability of
diagram, is
K
T
= 900000 NR/Mr. the value
K
T
= 1000000 NR/m led to a numerical instability.
Normal stiffness K
NR
must be taken equal to 20 NR/m to compensate for the weight exactly of
mass. (the value of the play is of 0,50m). Any other value leads to aberrant results.
3.2
Characteristics of the mesh
A number of nodes: 1
A number of meshs and types: 1 POI1
3.3 Functionalities
tested
Controls
STANDARD DEFI_OBSTACLE
“PLAN_Z”
DYNA_TRAN_MODAL SHOCK
“ADAPT”
“DEVOGE”
“EULER”
POST_DYNA_MODA_T WEAR
4
Results of modeling A
4.1 Values
tested
Identification Reference
Aster
ADAPT
Aster DEVOGE Aster
EULER % difference max
a0 = 15
15,2671
15,2660
15,2665
15,2668
0,007%
a0 = 1,5
0,409062
0,409077
0,409077
0,409077
0,004%
a0 = 1,01
2,26164E-4
2,26164E-4
2,26164E-4 2,26316E-4 0,072%
a0 = 0,99
0
0
0
0
0%
Code_Aster
®
Version
6.4
Titrate:
SDND104 - Calculation of the power of wear of a rubbing shoe
Date:
16/05/03
Author (S):
S. LAMARCHE
Key
:
V5.01.104-A
Page:
11/12
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
5
Summary of the results
The case-test validates the calculation of the power of wear with
POST_DYNA_MODA_T
after a calculation on
DYNA_TRAN_MODAL
, as well on a diagram with variable pitches (
ADAPT
) that on diagrams with pitch
constant (Euler and Devoggeleare). In particular the tangential microcomputer-speeds induced by
model of contact by penalization, at the time of the phases of adherence, are correctly cancelled.
The influence of the added spring remains in on this side precise details obtained.
The tangential stiffness of the contact is the element limiting for a higher precision. Convergence
results towards the reference solution was checked. The tangential stiffness was taken too
large that possible to ensure the stability of the diagram with
dt
=104 S.
The tolerances in the tests-resu are taken just above the found differences.
Code_Aster
®
Version
6.4
Titrate:
SDND104 - Calculation of the power of wear of a rubbing shoe
Date:
16/05/03
Author (S):
S. LAMARCHE
Key
:
V5.01.104-A
Page:
12/12
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Intentionally white left page.