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Organization (S): EDF/EP/AMV
Handbook of Référence
R7.10 booklet: Statistical processing
R7.10.02 document

Postprocessing of modal calculations with shock

Summary:

This document presents the principle of postprocessing of transitory calculations by modal recombination with
non-linearities of shock available in operator POST_DYNA_MODAL.

Two options of postprocessing can be employed, the first usable one for problems of
vibration-wear determines average values and RMS of displacements, forces of shock and power
of wear dissipated on the level of the supports with plays, second is applicable for the fine analysis of the impacts
occurring at the time of transitory stresses, the instantaneous maximum force, duration of the time of shock,
the impulse exchanged, speed before impact are given for each shock.

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Count

matters

1 Introduction ............................................................................................................................................. 3
2 Sizes considered in the vibrations with shocks ........................................................................ 4
2.1 Forces of shock ................................................................................................................................ 4
2.2 Displacements of shock .................................................................................................................... 4
2.3 Secondary sizes ................................................................................................................... 5
2.3.1 Time of shock ....................................................................................................................... 5
2.3.2 Calculated sizes ............................................................................................................... 5
3 modal transitory Postprocessing: option “USURE” .............................................................................. 5
3.1 Statistical processing per blocks ....................................................................................................... 5
3.2 Statistical processing applied to displacements of shock ...................................................... 6
3.3 Statistics for the forces of shock ................................................................................................ 8
3.4 Statistics for times of shock ................................................................................................ 9
3.5 Power of wear .......................................................................................................................... 11
3.6 Structure of data counts POST_DYNA associated with postprocessing option “USURE” .............. 11
3.6.1 Count POST_DYNA_MODAL ................................................................................................... 11
4 modal transitory Postprocessing option “IMPACT” ............................................................................ 12
4.1 Practical current of postprocessing of calculations of core .......................................................... 12
4.2 Calculations for the postprocessing of the impacts .................................................................................. 13
4.3 Structure of data counts POST_IMPACT associated with postprocessing option “IMPACT” ........ 14
4.3.1 Count POST_IMPACT ............................................................................................................ 14
4.3.2 Count IMPACT ....................................................................................................................... 14
4.3.3 Table TOTAL ....................................................................................................................... 14
4.3.4 Count PROBA ......................................................................................................................... 15
5 Conclusion ............................................................................................................................................ 15
6 Bibliography ......................................................................................................................................... 15
Appendix 1 ................................................................................................................................................. 16

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1 Introduction

Numerical developments were carried out in Code_Aster to allow calculation
transient of structures presenting of the vibrations with shock in certain points. In certain cases,
forces of friction can also appear and lead to a phenomenon of localized wear.

That it is about damage by pure impact or impact-friction, the engineer wishes to reach
with the sizes associated with this damage, which requires a specific postprocessing
behind non-linear transitory calculation.

This information of postprocessing is also invaluable when one wishes to validate the module
of non-linear calculation by comparing its results with what can be measured on a test bench
specific. Test routines (MASSIF and MULTICHOC) were implemented to this end
and were the first users of these functionalities of postprocessing.

The objective of this note consists in specifying the sizes to be analyzed in the vibrations with shock and
their specificity. It is then a question of determining the suitable statistical processing to apply to these
signals to release from the instantaneous sizes or the most characteristic averages.

Initially one will in the case of see the processing applied a problem with shock and
friction (option “USURE” of command POST_DYNA_MODAL).

The following chapter will be devoted to the processing applied in the case of a phenomenon of vibration
pure impact, where the sizes of each impact are more finely analyzed (option “IMPACT” of
order POST_DYNA_MODAL).

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2
Sizes considered in the vibrations with shocks

The primary sizes considered in the vibrations with shock are identical that it is about
experimental measurements or of numerical calculation, they relate to the forces of shock and them
displacements on the level of the points of shock. The experimental results present one however
additional difficulty of analysis due to the errors or skews introduced by the systems of measurement.
We will examine the two sizes quoted successively previously.

2.1
Forces of shock

The first concern concerning the structures vibrating with shocks is better to know them
efforts received by the structure at the time of the shocks on its supports with plays or between the structures. These
data are calculated in a temporal way by the algorithm of DYNA_TRAN_MODAL, they are then
filed with a step defined in this same operator. Data of shock having contents
frequential very important one will take care to have a sufficient filing (not to exceed PAS_ARCH: 10).
These forces expressed in a local reference mark with the obstacle (Yloc, Zloc) are traditionally
broken up into a normal part with the obstacle (Fn on the figure below) and a tangential part
(Ft) if friction is taken into account between the structures. The conditions of shock make that the force
normal of shock has a constant sign taken conventionally positive in Code_Aster.

F
Fzloc
Fn
Ft
P
Fyloc
O


2.2
Displacements of shock

Displacements of the structure on the level of its supports with play are another information
important calculated. Its analysis poses however less problems because the spectral contents are
less rich. In the case of circular or described obstacles in a polar way, a polar description of
displacement can be interesting.
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2.3 Sizes
secondaries

2.3.1 Time of shock

The time of contact between the structure and the supports with plays is a characteristic indirect size
movement of vibration with shock. It can be deduced from various ways, on a criterion of
displacement, of force of positive reaction. A concept of time of total shock, broken up into shock
elementary (or rebound) will be introduced in [§3.4].

2.3.2 Sizes
calculated

Other secondary sizes can be important in the analysis of the conditions of shock, it
acts of the impulse at the time of the impact (integral of the exchanged force), the power of wear, the force
maximum at the time of an impact,… Ces sizes are specific to each postprocessing and they will be
specified in the two chapters which follow for postprocessing option USURE and IMPACT.

3
Modal transitory postprocessing: option “USURE”

The characterization of transitional measures is the goal of the processing of the signal. It teaches us that one
signal is entirely determined by the data of all its statistical moments. In practice it
is out of question of calculating every statistical moment, one limits itself in postprocessing to
sizes calculated classically in processing of the signal (average simple, standard deviation and value
RMS). They are characteristic of the signals which one wishes to analyze and compare. Signals
similar must necessarily have these first close statistical moments (the reciprocal one
being false). The statistical sizes selected here are well appropriate to the analysis,
comparison or classification of signals of vibrations under random excitation with not
linearities of shock.

We thus will examine the realized sizes and their calculation, by distinguishing the different ones
sizes quoted in the preceding chapter:

· displacements,
· forces of shock,
· determination of the contact and the time of contact.

Other made up information could be calculated starting from the preceding ones in particular
power of wear.

3.1
Statistical processing per blocks

In order to analyze the stationnarity of the signals and the statistical processing carried out on
signals, one carries out a cutting per blocks of the temporal signals. Thus duration of postprocessing
defined between the initial moment (INST_INIT) and the final moment (INST_FIN) is cut out in a number of
temporal blocks (NB_BLOC) of identical duration. The calculation of the statistics: average,
standard deviation… is carried out for each block, a general value for the signal for the unit
blocks is also calculated.

In the case of a calculation of response of a structure to a random loading, this technique of
calculation per blocks makes it possible to make sure that the transitional stage of calculation is finished and that the value
announced is quite stationary over a time of observation associated with the duration with calculation.
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3.2
Statistical processing applied to displacements of shock

Let us consider the temporal signal Depl_ X (T), which one carries out a filing at a certain frequency
Facquis on NR points. The starting data is thus a vector Depl_ X (I) with NR components.

Average displacement is defined in this case by:

NR
Depl_ X (I)
Depl_ X = 1

NR

This average value characterizes the central value around which the signal of displacement
evolve/move. For displacements, it will thus make it possible to determine if a configuration is observed
centered (displacements with null average), or offset (average nonnull).

The variance of displacement is by definition:

NR
(
2
Depl_ X (I) - Depl_ X)
VAr (Depl_ X) = 1

NR

The standard deviation of displacement is worth then:

(Depl_ X) = Var (Depl_ X)

The standard deviation of a signal characterizes its dispersion around its average value. A weak standard deviation
will concern rather a signal with weak variations of amplitude, a strong standard deviation of the variations more
strong.

For a centered variable i.e. with null average, the standard deviation is equal to average RMS of
signal (Root Mean Square).

For an unspecified variable one defines average RMS of the signal by:

NR
Depl_ X (I) 2
RMS (Depl_ X) =
1

NR

The minimum and maximum absolutes of the signal are also interesting information and very
simple to obtain, who determines the extent of the signal.

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Depl_x
max
Depl_x +
Depl_x
Depl_x -
time
min

Appear 3.2-a: Exemple of signal of displacement and visualization
statistical sizes

A polar representation of the whole of the signals Depl_ X and Depl_ is there also
interesting to analyze an obstacle of circular or close geometry in the case of. Let us be appropriate
to call R radial displacement and angular displacement, equivalents of Depl_ X and
Polar Depl_ there of.

By definition one a:

R (I) = Depl_ X (I) 2 + Depl_ y (I) 2

(
Depl_ there I
I)
()
= Arctg

Depl_ X (I)

This representation makes it possible inter alia things to distinguish:

· orbital movements with permanent contact (average radial displacement about
play and standard deviation of weak radial displacement),
· movements of pure impact (standard deviation of important radial displacement, variation of
weak angular displacement),
· other configurations: orbital movement with impacts…

Note:

In the selected local reference mark for the obstacles of shock, the sizes called here Depl_ X
and Depl_ are there in fact DYloc and DZloc, the axis Xloc having been chosen by convention
perpendicular in the plan of the obstacle.
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In short, the option of postprocessing “USURE” of operator POST_DYNA_MODAL will determine for
local displacements DXloc, DYloc, DZloc like for their polar decomposition R and them
statistical sizes per blocks with the principle stated above:

· average value,
· value RMS,
· standard deviation,
· minimal value,
· maximum value.

3.3
Statistics for the forces of shock

One supposes to lay out as for displacements of a discrete signal on NR points: Fx_
(
shock I).
The signal obtained should be composed of temporal ranges where the force of shock is null (not
contact) and others where the force of shock is significant (effective contact), which is the case at the time of
numerical calculations. In fact, for experimental signals, because of the dynamics of the system of
measure, a level of noise can be observed except period of shock (cf [Figure 3.3-a]). It is thus necessary
only to carry out the statistical processing when the signal leaves the sound level. That requires
the introduction of a threshold of detection (SEUIL_FORCE) which, although superfluous in the field
numerical, was reproduced in the postprocessing of Code_Aster.

Fchoc
Smax
time
Appear 3.3-a: Exemple of signal of force of experimental shock

That is to say the value S max, determining the level max of the noise considered, one then will calculate:

· The number of moments in shock:

Nchoc = card {I/Fx_
(
shock I) > S my}
X

· The average of force of shock over total time:


NR

1
Fx_ shock =
.

Fx_
(
shock I)
NR

I/Fx_choc (I) >S max

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· The average of force of shock brought back to the time of shock is worth:

NR
Fx_ shock = Fx_ shock.

Nchoc

· Average RMS of force of shock over total time is calculated in the following way:

/

1 2
NR

1
RMS (Fx_ shock) =

Fx_
(
shock I) 2


NR


I/Fx_choc (I) >S max


· Average RMS brought back to the time of shock is worth:

NR
RMS (Fx_ shock) = RMS (Fx_ shock).

Nchoc

As for the signals of displacements, one can also be interested to the maximum or
absolute minimum of the signal of force, thus determining its extent. For the normal force, it
minimum is always equal to zero, whereas the tangential force is alternate.

In short, the option of postprocessing “USURE” of operator POST_DYNA_MODAL will determine for
normal and tangential forces of shock statistical sizes per blocks with the principle
statement above:

· average value calculated over the time of shock or total time,
· value RMS calculated over the time of shock or total time,
· maximum value of the signal.

3.4
Statistics for times of shock

The percentage of time of shock is defined by:

% Tchoc = Nchoc/NR

If one looks at information of which one lays out on an experimental system, the signal of force of
shock is adapted the most to determine in a precise way the occurrence of a contact. Like one has it
evoked with the top one tests the need to introduce a maximum level of noise, and to count them
phases of shock when the signal exceeds this threshold (SEUIL_FORCE).
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On the figure below, one can distinguish a concept of given elementary shock like one
successive passage to the top then with the lower part of the threshold, and a more general concept of total shock,
gathering several elementary shocks separated by short moments from return under the threshold.

Fn (T)
DUREE_REPOS
THRESHOLD FORCES
A total shock = 2 shocks
A total shock = 3 shocks

Appear 3.4-a

One thus introduces a time characteristic of rest Tr (DUREE_REPOS); end of a time of shock
total occurring if the signal remains during a time at least higher than Tr at rest. This
concept of time characteristic of Tr rest is well heard rather fuzzy and will have to be given
by the user within sight of the transitory results. It is nevertheless essential because it only allows
to gather a train of constituting very brought closer impact makes only one phase of contact of it.

Concept of elementary time of shock being defined, statistical processing over the time of shock
will consist in determining following information:

· a number of elementary shocks: Nb_ choc_ elem
· a number of total shocks: Nb_ choc_ glob
Nb_ choc_ elem
· a number of elementary shocks per total shock:

Nb_ choc_ glob
· time of average elementary shock:

Nchoc. T

T choc_elem =

Nb_ choc_ elem

· time of average total shock

Nchoc. T

T choc_glob =

Nb_ choc_ glob

· time of maximum total shock the greatest time of total shock noted on the block
analyzed.
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In short, the option of postprocessing “USURE” of operator POST_DYNA_MODAL will determine for
times of shock statistical sizes per blocks with the principle stated above:

· average value of the time of total shock, `
· maximum value of the time of total shock,
· average value of the elementary time of shock,
· the number of total shocks a second,
· the average number of shocks elementary per total shock.

3.5 Power
of wear

The size generally calculated in the vibrations with shock and friction is the power
of wear defined by ARCHARD [bib1], which translates the average power developed by the forces of
friction at the time of the movement. These forces are the engine of wear by friction. Power of wear
in the case of discrete signals is calculated as follows:

NR

(
Fn I) V
. T (I)
P
I/Fn S max
wear =
>

NR

This power can for example be correlated with a wear or removal of matter by
the intermediary of a coefficient of wear KT by a relation of the type: V (T) = K * P
* T
T
wear
where
V (T) is the volume removed for the length of time T.

Other more sophisticated laws of wear can be used in another operator of
postprocessing: POST_USURE describes in [R7.01.10].

3.6 Structure of data counts POST_DYNA associated with postprocessing
option “USURE”

3.6.1 Count
POST_DYNA_MODAL

A structure of the type counts for option USURE of operator POST_DYNA_MODAL gathers them
results previously described.

This table contains the names of the statistical under-tables of results associated different
analyzed sizes: displacements, forces of shock, counting of the shocks and power of wear.

The variables of access of this table are 10:

· for the variables displacement: DEPL_X, DEPL_Y, DEPL_Z, DEPL_RADIAL,
DEPL_ANGULAIRE, which correspond respectively to displacements in X, Y and Z local and
their cylindrical decomposition in the plan of the obstacle.
· for the variables forces of shock: FORCE_NORMALE, FORCE_TANG_1, FORCE_TANG_2,
who correspond respectively to the normal, tangential forces with the obstacle the first
being in the plan of the obstacle, the second orthogonal one in the plan of the obstacle.
· for the variables counting of shock: STAT_CHOC.
· for the variables power of wear: PUIS_USURE.
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Under tables associated with the 10 sizes above, contain a certain number of variables
access for each connection of shock:

· for the variables displacement: MOYEN, ECART_TYPE, RMS, MAXI, MINI, which
correspond respectively to the values average, standard deviations, value RMS or effective,
maximum and minimal value of the corresponding variable displacement.
· for the variables forces of shock: MOYEN_T_TOTAL, MOYEN_T_CHOC, RMS_T_TOTAL,
RMS_T_CHOC, MAXI, which correspond respectively to the values average over time
total, average over the time of shock, value RMS or effective average over total time,
value RMS or effective over the time of shock, maximum value of the variable forces
corresponding.
· for the variables of counting of the shocks: NB_CHOC_S, NB_REBON_CHOC,
T_CHOC_MOYEN, T_CHOC_MAXI, T_CHOC_MINI, T_REBON_MOYEN, % _T_CHOC, which
correspond respectively to the values of the number of shocks a second, of the number of
rebounds by shock, of the time of average shock, time of maximum shock, time of minimal shock,
time of average rebound and percentage of time of shock.
· for the variable power of wear: PUIS_USURE which corresponds to the power of wear
calculated according to ARCHARD.

4
Modal transitory postprocessing option “IMPACT”

4.1
Practical current of postprocessing of calculations of core

The SEPTEN made use, before with the development of postprocessing in Code_Aster,
for its needs for checking of dimensioning, code CLASH [bib2] developed by
BELGONUCLEAIRE. This software calculates the seismic response of a file of assemblies. This code provides
a whole of information detailed for each point of shock and each impact.

Each result consists of a table by point of shock whose example is in appendix 1. It
table comprises following information:

· the moment of the peak of impact,
· the maximum force of impact reached,
· the exchanged impulse, defined as the integral of the force of shock over time,
· total duration of the shock,
· relative speed before impact.

These elements are particularly interesting for the SEPTEN bus in addition to contractual information
very limited, they make it possible to know the number and the composition of the impacts, as well as
essential physical sizes which theirs are associated. Relative speed before impact, the impulse
are for example very invaluable information in the specification of experimental tests of
dynamic buckling of the grids of assemblies.
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4.2
Calculations for the postprocessing of the impacts

One regards as for preceding postprocessing that the conditions of impact are given
as previously by going beyond of a force Smax threshold and one distinguishes in the same way
total shock and elementary shock by the concept of rest period.

Calculation carries out a loop on all non-linearities of shock and an identical processing for
each one.

Then for each identified total shock, one will determine the following sizes:

· Time of beginning of shock: T
such as F
(T) > S
beginning
shock
beginning
max

· Time of end of total shock:

T
such as F
end
shock (Tfin) S
, F
max
shock (T
- T
end
) Smax
and
T
[T, T + T
end
end
rest] Fchoc (T) Smax

where T
is the step of time of integration

· Total duration of the shock: T
= T - T
shock
end
beginning

· Maximum of force at the time of the shock: Fmax = max
()
T [
T
, T
beginning
end] (F
T
shock
)

· The moment of maximum of force of shock,

T
·
end
The impulse exchanged at the time of the shock: I =
F
(T) .dt

shock

t=Tdébut

· Relative normal speed before impact: V
= V (T
- T
shock
beginning
)

· The number of elementary impacts cumulated in the total shock:

NR
= card {T [T, T]/F (T) > S and F
max
(T + T
) S
elementary impacts
beginning
end
shock
shock
max}

In order to synthesize information, one will moreover determine:

· the absolute maximum of force of shock, on a connection of shock given, for the duration
of analysis,

The maximum of force of shock to be more precisely given will not be obtained
like the max in time on the whole of the shocks for each node of shock (to avoid it
skew of the precision of filing) but given in transitory calculation on all the steps
of calculation and filed in the concept tran_gene result. It is this information which will be
used.
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· the average value of the extréma of force of shock like their standard deviation.

· a histogram of the density of probability of the maximum forces of impacts.

This histogram will be relatively summary and will give for NC classes the density of
probability of the maximum force of shock.

The classes will be in the following way defined:


I - 1
I

classify 1.
= F
/
F absolute
max
max
Fmax
F absolute
i= NR
max

C
NR
NR

C
C


4.3 Structure of data counts POST_IMPACT associated with
postprocessing option “IMPACT”

4.3.1 Count
POST_IMPACT

A structure of data of the type counts for option IMPACT of operator POST_DYNA_MODA_T of
Code_Aster is prduite.

The structure of result will be a subscripted table by the names of connections of shock, of type
POST_IMPACT, container of the names of tables which it contains.

The contents of each cell of this table are a name of table stored in CHARACTER * 24. Three type
tables are contained a table known as IMPACT, a table known as GLOBAL and a table known as PROBA.

It thus has 3 parameters: IMPACT, GLOBAL and PROBA. The variable of access corresponds to the name
connection of shock considered.

4.3.2 Count
IMPACT

Table “IMPACT” is of type TABL_IMPACT and has 6 parameters of access: INST, F_MAX,
T_CHOC, IMPULS, V_IMPACT, NB_IMPACT.
The contents of each cell of this table are a REAL * 8.

4.3.3 Count
GLOBAL

The table `GLOBAL `is of type TABL_FMAX and has 3 parameters of access:

· F_MAX_ABS, which gives access the absolute maximum of force of shock on all the shocks
noted,
· F_MAX_MOY, which gives access the average value of maximum of force of shock
noted,
· F_MAX_ETYP, which gives access the standard deviation extrema of forces of shock.

The contents of each cell of this table are a REAL * 8.
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4.3.4 Count
PROBA

The table `PROBA `is of type TABL_HISTO and has 3 parameters of access:

· DEBUT, which gives access the value of minimal force of class I,
· FIN, which gives access the value of maximum force of class I,
· PROBA, which gives access the density of probability of the variable forces maximum for
classify I.

The contents of each cell of this table are a REAL * 8.

5 Conclusion

One presented in this document the methods of postprocessing applicable to the transients with
shock calculated by modal synthesis on structures with play. According to concerns', one can
to carry out a postprocessing directed towards a diagnosis of the wear undergone by the components at the time of
shocks, a whole of important statistical sizes are then given. If the concern
rather relate to the impacts and their level, another option allows a detailed analysis of each
impact.

These two functionalities make it possible to synthesize the transitory results obtained by integration
temporal, to classify by level of severity of the digital simulations different or from
to compare ends of validation of the calculated and measured sizes.

6 Bibliography

[1]
ARCHARD J.F and HIRST W.: The wear off metals under unlubrificated conditions - Proc. Roy.
Plowshare (1956).
[2]
J.P. FABRY, A. DECAUWERS: Code CLASH - Etude Sismique of a line of assemblies
REP.
Handbook of Référence
R7.10 booklet: Statistical processing
HP-51/96/078/A

Code_Aster ®
Version
3.0
Titrate:
Postprocessing of modal calculations with shocks


Date:
08/01/01
Author (S):
G. JACQUART Key
:
R7.10.02-A Page
: 16/16

Appendix 1
--------------------------------------------------------------------------------
ASTER 3.05.29 CONCEPT TT CALCULATES THE 20/12/95 A 17:32:16 OF TYPE TABL_POST_DYNA
TOTAL IMPACT
NO1 TT _I_NO1 TT _G_NO1
PROBA
NO1 TT _P_NO1

----->
IMPRESSION OF THE TABLE: TT _I_NO1
MOMENT F_MAX IMPULSE T_CHOC V_IMPACT
CHOC 1 1.55000E-02 9.95269E+03 1.98051E+02 3.10000E-02 - 1.00000E+00
CHOC 2 3.61000E-01 9.95478E+03 1.98093E+02 3.15000E-02 - 1.00031E+00
NB_IMPACT
CHOC 1 1.00000E+00
CHOC 2 1.00000E+00

----->
IMPRESSION OF THE TABLE: TT _G_NO1
F_MAX_ABS F_MAX_MOY F_MAX_ETYPE
NO1 9.95478E+03 9.95373E+03 1.04759E+00

----->
IMPRESSION OF THE TABLE: TT _P_NO1
BEGINNING FINE PROBA
CLAS 1 9.95269E+03 9.95295E+03 5.00000E-01
CLAS 2 9.95295E+03 9.95321E+03 0.00000E+00
CLAS 3 9.95321E+03 9.95347E+03 0.00000E+00
CLAS 4 9.95347E+03 9.95373E+03 0.00000E+00
CLAS 5 9.95373E+03 9.95400E+03 0.00000E+00
CLAS 6 9.95400E+03 9.95426E+03 0.00000E+00
CLAS 7 9.95426E+03 9.95452E+03 0.00000E+00
CLAS 8 9.95452E+03 9.95478E+03 5.00000E-01

--------------------------------------------------------------------------------

Handbook of Référence
R7.10 booklet: Statistical processing
HP-51/96/078/A

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