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Organization (S): EDF-R & D/AMA

Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
V5.01.104 document

SDND104 - Calcul 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 modelled 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 step of Euler
(command 1) and Devogeleare (command 4), as well as the algorithm with variable step 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.
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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
modelled by penalization. The system thus has two degrees of freedom of translation (horizontal and
vertical).

RZ
RY
M
X-ray
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 sin (T
). In particular, it is
0
null at the initial moment. The displacement of the support satisfies equation X (T) = - (A/2
) sin (T
), and
0
thus its movement starts towards the left, with nonnull initial speed &
X ()
0 = - has/.
0

The 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:

= 15 m/s 2 has, has = 15
, m/s 2, has = 1 0
, 1 m/s 2 and has = 0 9
, 9 m/s 2
0
0
0
0

and a value of pulsation: = 2.

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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
&
X (T) = &X (T) = has sin T
. The tangential force exerted by the mass on the support is then
0
F = - mx& (T) = - my sin T
(null at the moment initial, which justifies the starting assumption
T
0
that initially, the mass is adherent on its support). The mass remains adherent as long as
F = my sin T

µ
µ
. If G has
µ, the mass thus remains indefinitely adherent on sound
0
F = Mg
T
NR
0
support, and its movement is exactly the same one as this one. By introducing the coefficient
µg
adimensional =
, the condition of permanent adherence is written 1. The curve of acceleration
a0
mass, like support, then takes the following form according to time:

4
2
0.2
0.4
0.6
0.8
- 2
- 4


As for speed, it takes the following form (single primitive of null average):

0.4
0.2
0.2
0.4
0.6
0.8
- 0.2
- 0.4

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If a0 > G
µ, there exists a smaller time T = T such as F = my sin T

µ
. This the smallest
0
1 =
Mg
1
T
time is necessarily such as sint > 0, which makes it possible to remove the absolute value in
0
1
G
µ
1
the preceding expression, and to obtain expression clarifies T =
arcsin
= arcsin. In
1

has


0
T
2

private individual, T
=
=
1
4
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, is &x (T) = G
µ (T - T)
. Its speed thus increases linearly with
1
+ &x (T) 1
has
has
time, while leaving to T the negative value X T
0
= -
T
0
2
= -
1 - (indeed,
1
& ()
cos
1
1


sint =).
1


X ''
driven G
X ''
T
- driven G
X'
x'
T1
t2 T3
t4
T


Movement for > *, mode of “stick-slip”, succession of adherence and slip

Necessarily, for a certain value of time T satisfactory/
2 T
, speed
2

2/
2
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 µg. One examines the translation of this condition in the continuation. One expresses for
to begin the value of T.
2
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has
has
Time T satisfies equation X T
= X T, is µg (T - T)
0
-
cos T
0
= -
cos T, or
2
& ()
2
& ()
2
2
1

1

2
still (T - T) - cost + cost = 0.
2
1
1
2

This equation, transcendent, allows the determination of T according to T and, that is to say finally,
2
1
taking into account the expression of T, the determination of T according to the physical parameters of
1
2
system and. If the acceleration of the support in T is lower in absolute value than µg, it
2
movement remains adherent then up to one moment T for which the acceleration of the support and of
3
mass reach the value - µg, moment which for reasons of clear symmetries on graphs Ci
above, exactly satisfied T = T +/. The mass starts a phase of slip then
3
1
up to one moment T, after which the movement reproduces periodically.
4

It is understood that for sufficiently small values of, the movement will not be able to become
member as from time T, because the acceleration of the mass would exceed the threshold µg. There thus exists
2
a value criticizes * 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 shows 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 value criticizes * admits a simple analytical expression. Indeed, for = *, the moments T
2
and
T are confused. Thus T - T = T - T =/and the equation
3
2
1
3
1
(T - T) - cost + cost = 0
2
becomes * = 2
T = 2 1
. While passing squared, one
1
- *
cos
2
1
1
2
2
obtains 2 * 2
4

4 * 2
= -
, that is to say * =
,
0.537.
2 + 4

For *, the movement is only asymptotically periodic. The continuation (T) of the moments of
N
change of direction of slip checks T
/when N tends towards the infinite one. The figure
1 - T

n+
N
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
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Let us summarize the conclusions:
µg
2
There is the adimensional coefficient =
and its value criticizes * such as * =
,
0.537.
has
2
0
+ 4


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, the choices of the amplitude have
0
are such as these three situations are visited. One takes m indeed = 1 kg, G = 10 m/s 2, µ = 0 1
,

= 15 m/s 2 has, has = 15
, m/s 2, has = 1 0
, 1 m/s 2 and has = 0 9
, 9 m/s 2.
0
0
0
0

The power of wear is physically null at the time of the phases of adherence.
In Aster, with 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/Modčle d' Archard [R7.04.10].

During the phases of slip, the power of wear follows the law P (T) = m
µ Steam Generator (T), where
U
R
V (T) = x& (T) - X & (T) is the relative speed of slip of the mass on the support. In the situation
R
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

T
T
T
2
2
2
has
has
E =
mgV (T) dt

= Mg
X & (T

) - x& (T) dt = Mg (- 0 cos T
- (G
µ (T - T)
0


1
-
cos T))dt
U
R
T
T
T
1
1
1
1



a0
1
G
µ

=
2
Mg ((t2 - 1t) cos 1t - (sin T
2 -)) -
(t2 - 1t).


2


The transcendent formulation of T apparently does not make it possible to simplify the expression of this
2
energy of wear. The power of average wear P is simply the energy of wear E above
U
U
divided by the half-period from the answer T/2 =/.

In the case of a movement always slipping (*), the interval of integration to be taken is
form [T, T
with N sufficiently large, so that T
that is to say sufficiently near to
1 - T
N
n+1]
n+
N
value limits/. One can avoid numerical calculation by recurrence of this continuation, knowing that
average asymptotic speed is null. Indeed, the continuation T - N/has a finished limit.
N
satisfied properties by are illustrated on the following figure:
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The segment of straight line has as an equation
has
v = G
µ (T -) - W = G
µ (T -) - 0
cos
(
),

has
and for T = +/, speed v is to take the opposite value W = 0


cos (
), which gives
the equation
has
has
µ
G/- 0
0
cos
(
) = cos
(
),
that is to say
µ
G = 2a cos (
);
0

whose solution is
1
µg
1



=

arccos


= arccos
.
2a0

2
Let us note that one finds although for = *, the acceleration of the support calculated at time T =
give the value limits µg. Indeed

sin has (
) = sin has (arccos (*/2)) = has
* 2
2
1 -
/4 = has 1 - 1
(
* 2
-
) = has *




= G
µ.
0
0
0
0
0

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, by taking T = and T = +/, which gives
1
2

has
1
G
/
2
2
2
µ
2

µ
0
2 +
has
G

E = Mg
(T cos -
sin T
) -
(T -)
Mg 0 (
1
)
,
U


2


=
+
-
-
2


2

4
2
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that is to say

mga
Mg
E =
0
2
2
2
2
2
2
4
4
µ
.
2
-
=
has
2
0 -
G
U



The power of average wear (over one period) asymptotic is then

E
mga
mga
4
P
U
0
2
2
0
2
U =
=
4 - =
.
2 -
/




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
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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 A = 15 m/s 2, the power of average wear asymptotic is 15,1146144886 Watt then
0
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 Vol 14.135-146 (1983)
[2] Documentation

Code_Aster [R7.04.10]
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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.
Calculation is done on modal basis. One blocks 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
CHOC to model 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 step of time of 3.10­5 S for temporal integration to limit the calculating time. It
no time is quite lower than min (2/K/M, 2/K/M
4
)
10
.
7
-
=
S
NR
.
The tangential stiffness of friction is taken as large as possible to ensure the stability of
diagram, is KT = 900000 NR/Mr. the value KT = 1000000 NR/m led to a numerical instability.
Normal stiffness kN 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 grid

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

3.3 Functionalities
tested

Commands


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
Aster DEVOGE Aster
EULER % difference max
ADAPT
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%
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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 steps (ADAPT) as on diagrams with step
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.
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Key S. LAMARCHE
:
V5.01.104-A Page:
12/12

Intentionally white left page.
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A

Outline document