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Code_Aster
®
Version
4.0
Titrate:
Examination of the random answers
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R7.10.01-B
P
age:
1/20
Manual of Reference
R7.10 booklet: Statistical processing
HP-52/96/008/B
Organization (S):
EDF/EP/AMV
Manual of Reference
R7.10 booklet: Statistical processing
Document: R7.10.01
Examination of the random answers
Summary:
The introduction of a “stochastic approach of seismic calculation” to solve a problem of mechanics
vibratory under random excitation requires a particular postprocessing.
The control
POST_DYNA_ALEA
[U4.76.02] allows, starting from the spectral concentration of power of one
interspectre-answer, to evaluate its standard deviation, its apparent frequency, the distribution of its peaks. It allows
also, in a first approach, to calculate the useful function of Vanmarcke in the case of an analysis
seismic.
NB:
This control also makes it possible to carry out the statistical estimates for any type of interspectre
of response to a random excitation not necessarily seismic (for example: effect of the swell or
of a turbulent flow).
background image
Code_Aster
®
Version
4.0
Titrate:
Examination of the random answers
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R7.10.01-B
P
age:
2/20
Manual of Reference
R7.10 booklet: Statistical processing
HP-52/96/008/B
Contents
1 Introduction ............................................................................................................................................ 3
2 Spectrum - Interspectre - Interspectrale Matrix ....................................................................................... 3
2.1 Processing of the signal - Conventions selected .................................................................................. 3
2.1.1 Introduction ............................................................................................................................. 3
2.1.2 Transformation of Fourier ...................................................................................................... 4
2.2 Concept of Power - Spectral concentration of Power ................................................................. 4
2.2.1 Power of a signal - Spectrum of Power of a signal .................................................... 4
2.2.2 Power of interaction - Spectral concentration of interaction of two signals - Interspectre ..... 5
2.2.3 Stamp interspectrale ............................................................................................................. 5
2.3 Establishment in Aster ................................................................................................................... 5
3 Recalls on the statistical laws [bib4] ................................................................................................... 6
3.1 ........................................................................................................................................ Definitions 6
3.2 Assumptions in random dynamics ............................................................................................... 6
3.2.1 Stationary processes with null average - variance ............................................................. 7
3.2.2 Ergodicity ............................................................................................................................... 7
3.3 Spectral concentration of power ...................................................................................................... 7
3.4 Spectral moments ......................................................................................................................... 8
4 measurements of going beyond of threshold and reliability ............................................................................. 8
4.1 Spectral moments and characteristic parameters ....................................................................... 8
4.1.1 Formulas of Rice ................................................................................................................... 9
4.2 Distributions of the positive peaks ........................................................................................................ 10
4.2.1 Signal with broad band: law of Gauss or normal law .............................................................. 11
4.2.2 Signal with narrow tape: law of Rayleigh ................................................................................ 11
4.2.3 Calculation of the values in Code_Aster ............................................................................... 12
4.3 Seismic answer: law of Vanmarcke [bib8] ................................................................................ 12
4.3.1 Assumption of independent crossings ................................................................... 12
4.3.2 Law of Vanmarcke ................................................................................................................ 13
4.3.3 Establishment in Code_Aster ......................................................................................... 14
5 Remarks .......................................................................................................................................... 15
6 Bibliography ........................................................................................................................................ 15
Appendix 1 Conventions for the Spectral concentrations of Power ...................................................... 16
Appendix 2 Transformation of Hilbert ....................................................................................................... 20
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Code_Aster
®
Version
4.0
Titrate:
Examination of the random answers
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R7.10.01-B
P
age:
3/20
Manual of Reference
R7.10 booklet: Statistical processing
HP-52/96/008/B
1 Introduction
For a structure subjected to a random excitation of type swells, turbulent flow, or
seism… the loading is not known in a deterministic way, but is generally described by
probabilistic or spectral information like the spectral concentration of power. For
linear structures it is possible to use a method of calculation stochastic which allows
to determine the spectral concentrations of power of response to these random excitations.
The operator
POST_DYNA_ALEA
has as a function to carry out the statistical analyzes of the density
spectral of power of answer. It thus provides probabilistic information of the answer of
structure. Statistical calculations of parameters are carried out on the basis of calculation of the moments
spectral of the spectral concentration of power considered.
These statistical parameters are: the standard deviation, the apparent frequency, distribution of the peaks. It is
also possible in the operator to calculate, a first approach, the function of
Useful VANMARCKE in the case of a seismic analysis
.
Note:
The operator
POST_DYNA_ALEA
, conceived initially for the seismic approach, after a calculation with
the operator
DYNA_ALEA_MODAL
[U4.56.06] ([bib1], [bib2]), can also carry out postprocessings
of the operator
DYNA_SPEC_MODAL
developed by department TTA within the framework of
resorption of FLUSTRU. This operator carries out the calculation of the response of a structure of the type
uniformly tube Steam Generator excited by a transverse flow.
2
Spectrum - Interspectre - Interspectrale Matrix
2.1
Processing of the signal - Conventions selected
2.1.1 Introduction
A signal can have two representations: a temporal representation of the form
()
X
T
=
F
or one
frequential representation of the form
()
X
F
=
F
. These two representations are connected between them
by the Transformation of Fourier.
There exists in the numerical field and the experimental field various manners of calculating
spectral sizes relating to a temporal signal
()
X T
(dimensional representation or not,
factor 1/2
or not for the Transformation of FOURIER).
However, if various definitions of the DSP (cf [§2.2.2] and [Annexe1]) starting from the Transformation of
Fourier of the signal do not change anything with the calculation carried out by
CALC_INTE_SPEC
[U4.56.03], it is important in
revenge, in the calculations carried out by the operator of postprocessing
POST_DYNA_ALEA
, that them
data are coherent so that the results produced by this operator are with dimension
physique of the starting signal.
It is also necessary to know, for a quantitative comparison between calculation and experiment,
which are the conventions adopted for the calculation of the spectral quantities. The whole of these
conventions is recalled in [Annexe1] for each type of signals. We give again only them here
general formulas.
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Code_Aster
®
Version
4.0
Titrate:
Examination of the random answers
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R7.10.01-B
P
age:
4/20
Manual of Reference
R7.10 booklet: Statistical processing
HP-52/96/008/B
2.1.2 Transformation of Fourier
For the Transformation of FOURIER into frequency
()
F
of a signal (of unit U), expressed out of U/Hz
we adopt the following definition:
()
()
X
X
E
F
T
dt
I ft
=
-
-
+
2
The reverse transformation is expressed then by:
()
()
X
X
E
T
F
df
I ft
=
+
-
+
2
One can also express the Transformation of Fourier into pulsation
(
)
=
2
F
, by the definition
following:
()
()
X
X
E
p
I T
T
dt
=
-
-
+
1
2
The reverse transformation is expressed by:
()
()
X
X
E
T
D
p
I T
=
+
-
+
What leads to equivalence:
()
(
)
()
X
X
X
p
p
F
F
=
=
2
1
2
2.2
Concept of Power - Spectral concentration of Power
2.2.1 Power of a signal - Spectrum of Power of a signal
Just like the signal itself, the power of the signal can be expressed according to the time or of
the frequency:
·
the instantaneous temporal power is simply called power:
()
()
()
p
X. X *
T
T
T
=
where
()
X * T
is the complex quantity combined of
()
X T
.
·
the frequential power is commonly called spectral concentration of power or
spectrum:
() () () ()
S
X
.X *
X
xx
F
F
F
F
=
=
2
This definition is not possible that when the transform of Fourier of the signal exists.
One can then express the total energy of the signal by
()
()
E
F df
F
df
xx
=
=
-
+
-
+
S
X
2
The expression of this DSP for the various types of signals is given in [Annexe1]. One will see
later on [§3.3] another definition - equivalent according to the theorem of Wiener-Kinchine - but
more general, of the spectral concentration of power based on the statistical approach.
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Code_Aster
®
Version
4.0
Titrate:
Examination of the random answers
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R7.10.01-B
P
age:
5/20
Manual of Reference
R7.10 booklet: Statistical processing
HP-52/96/008/B
2.2.2 Power of interaction - Spectral concentration of interaction of two signals -
Interspectre
·
One defines also the instantaneous power of interaction of two signals
()
()
X T
y T
and
:
() () ()
()
() ()
()
()
p
X .y *
p
X *
.y
p
p *
xy
yx
xy
yx
T
T
T
T
T
T
T
T
=
=
=
and
connected by
·
If the two signals admit a transform of Fourier
()
()
X
Y
F
F
and
, one can express
frequential power of interaction or interspectre by
() () ()
S
X
.Y *
XY
F
F
F
=
·
If
two signals are real then the power of interaction
()
()
() ()
p
p
X.
xy
yx
T
T
T y T
=
=
is real. But there is no reason so that
()
S
XY
F
that is to say also real; on the other hand
()
S
XY
F
is complex with square symmetry, namely:
even real part and odd imaginary part or even module and odd phase
·
If
() ()
X
Y
F
F
=
, one speaks then about autospectre.
2.2.3 Stamp
interspectrale
A matrix interspectrale of command
NR
is a matrix
NR, NR
complex, whose each term depends on
the frequency in the form of a function of
F
. The diagonal terms are the autospectres, them
extra-diagonal terms are the interspectres between the points considered (each line or column
representing a point in physical mesh or a mode in modal calculation). Handled interspectres
in practice being square, only them
(
)
NR NR 1
2
+
terms of triangular higher (or lower)
are sufficient to define the matrix interspectrale completely.
2.3 Establishment
in
Aster
The matrices interspectrales handled by the operator
POST_DYNA_ALEA
consist of
complex functions of the frequency:
()
S
XY
F
.
These matrices are stored in tables of concept
tabl_intsp
.
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Code_Aster
®
Version
4.0
Titrate:
Examination of the random answers
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R7.10.01-B
P
age:
6/20
Manual of Reference
R7.10 booklet: Statistical processing
HP-52/96/008/B
3
Recalls on the statistical laws [bib4]
3.1 Definitions
T
discrete parameter
()
T
N N 1, NR
=
or continuous (time or a variable of space).
()
X T
random process.
At every moment
T
N
a random variable is associated
X
N
, random variable of realization
X
N
.
Then
()
()
(
)
X
1
T
X
X T
N
N
N, NR
=
=
=
is a realization of the process
()
X T
, process made up of
NR
random variables a priori independent.
Each variable
X
N
is characterized by its function of distribution
()
(
)
F
Prob
N
N
N
X, T
X
X
=
or
by its density of probability
()
()
p
F
X
X, T
X, T
N
N
N
=
.
The random process is also characterized by its functions moments, the first two moments
have a particular importance. It is about the expectation or average
()
µ
T
noted
too
()
[]
E
T
X
and for any couple
()
T T
1 2
,
function of autocorrelation
()
R,
T T
1 2
or
()
R
T T
XX 1 2
,
noted too
() ()
[
]
E
T
T
X
X
1
2
.
()
()
[]
()
()
() ()
[
]
(
)
µ
T
T
X X T dx
T T
T
T
X X
X T
X T
dx dx
XX
=
=
=
=
E X
p,
R
,
E X
X
p
,
;
,
1
2
1 2
1 2
1
1
2
2
1
2
One defines also a function of intercorrelation for two processes
()
()
X T
Y T
and
.
()
() ()
[
]
(
)
R
,
E X
Y
p
,
;
,
1
2
XY
T T
T
T
X y
X T
y T
dx Dy
1 2
1 2
1
1
2
2
1
2
=
=
The “spreading out” of the process is characterized by the variance:
()
()
()
(
)
[
]
µ
2
2
T
T
T
=
-
E X
For a process with null average (
µ
= 0.), the variance which characterizes the “intensity then of
phenomenon " (square of the standard deviation or average quadratic value) is equal to the function
of autocorrelation at time
T T
T
= =
1
2
:
()
() ()
[
]
()
()
2
2
T
T
T
T, T
X
X T dx
XX
=
=
=
E X
X
R
p,
3.2
Assumptions in random dynamics
Very classically several assumptions are posed within the framework of random dynamics. One
admits as well as the studied processes are stationary, with average null and ergodic.
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Code_Aster
®
Version
4.0
Titrate:
Examination of the random answers
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R7.10.01-B
P
age:
7/20
Manual of Reference
R7.10 booklet: Statistical processing
HP-52/96/008/B
3.2.1 Stationary processes with null average - variance
A process is known as stationary if the whole of its “probabilistic characteristics” is invariant
during a translation T
0
parameter T. What implies:
()
()
(
)
()
()
µ
T
Cte
T T
T
T
XX
XX
XX
XX
=
=
-
=
=
-
R
,
R
R
R
1 2
2
1
For a process with null average
()
2
0
=
R
XX
.
3.2.2 Ergodicity
This concept comes from a reasoning of Gibbs (1839-1903) for whom time from observation from one
physical phenomenon can be regarded as infinite in front of the scale of time on the level
molecular. The system passes then by all the possible states while remaining longest possible,
or while generally passing, in the states which are most probable, so that the average
temporal becomes equal to the statistical average on the states, i.e. the hope
mathematics. This is prolonged for the functions of correlation and intercorrelation.
()
()
(
) ()
µ
=
=
-
+
-
+
+
-
+
lim
X
R
lim
X
X
/
/
XX
/
/
T
T
T
T
T
T
T
T dt
T
T
T dt
1
1
2
2
2
2
Note:
For the continuation of the document one will suppose that the random process is stationary with average
null and ergodic. The whole of the developments carried out in Code_Aster checks these
assumptions.
3.3
Spectral concentration of power
Within the framework of this statistical approach, one can give a very general definition of the density
spectral of power or DSP. One will retain for Code_Aster
expressed following definitions
in frequency or pulsation:
()
()
()
()
()
()
()
()
S
G
S
G
XX
XX
p XX
p XX
F
D
F
D
D
D
XX
I F
XX
I F
XX
I
XX
I
=
=
=
=
-
+
-
+
-
-
+
-
+
-
R
E
R
E
R
E
R
E
.
.
2
0
2
0
1
2
1
2
;
;
who lead to the following relations:
()
()
()
()
()
()
G
G
S
G
S
G
p XX
XX
XX
XX
XX
p
p XX
=
=
=
1
2
2
2
F
F
F
One can show that
()
G
XX
F
, which is equal to the Transformation of Fourier of
()
R
XX
T
, is real,
positive. One will refer to [Annexe1] who contains all conventions adopted to ensure
coherence of the results.
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Code_Aster
®
Version
4.0
Titrate:
Examination of the random answers
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R7.10.01-B
P
age:
8/20
Manual of Reference
R7.10 booklet: Statistical processing
HP-52/96/008/B
3.4 Moments
spectral
One calls spectral moments the following quantities (which one defined in pulsation):
()
()
I
I
p XX
I
XX
D
F df
=
=
-
+
-
+
S
S
One has in particular:
0
=
=
=
2
2
2
4
2
! !
!!!!
XX
XX
who are the standard deviations of
X
and of its
first derived.
These moments are systematically calculated until command 4; using the key word
MOMENT
it is
possible to ask the calculation of the higher modes. In Code_Aster, calculation is carried out for
a DSP expressed according to the frequency
F
.
Code_Aster calculates the spectral moments on the basis of field of definition of the functions
such as they are provided to him.
4
Measurements of going beyond of threshold and reliability
The conventional methods give access only the maximum of displacement (or of
acceleration) by summation “adapted” of the maxima on each mode. Essential interest of
the stochastic approach of random vibratory calculation lies in the statistical knowledge of
response of the structure which can thus be converted into statistical data of reliability. It
titrate, two modes of ruin can be taken into account:
·
ruin by going beyond of threshold: this type of ruin occurs when the response of the system
exceed a limiting value. That amounts seeking the probability that the values of the process
remain in lower part of an extreme value (peak Factor or factor of peaks) during the duration
of observation T.
·
ruin by fatigue or accumulation of damage.
This second approach could also be treated starting from the first calculated statistical elements
in
POST_DYNA_ALEA
. It is carried out in the control
POST_FATI_ALEA
[U4.67.05]
[R7.04.02].
Within the framework of the studies under seismic excitations, we are interested primarily in
problem of going beyond of threshold. From where initially the calculation of a certain number of
statistical parameters which make it possible to characterize the signal to study (spectral moments and
formulas of Rice [§4.1]), fitted with these characteristics we will be able to then estimate the probabilities of
going beyond of threshold using conventional models of probability [§4.2], as well as a criterion of
reliability (law of Vanmarcke [§4.3]).
4.1
Spectral moments and characteristic parameters
The spectral moments are defined by:
()
I
I
XX
F df
=
-
+
S
The infinite whole of these spectral moments characterize the interspectre perfectly and allow
thus to determine a certain number of numerical results. In the particular case of an oscillator with 1
ddl or of a signal with only one peak, the first three spectral moments are enough to find
the autospectre
S
XX
. It is the case which we retain in Code_Aster since it is supposed that them
values are distributed according to a law of GAUSS.
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Code_Aster
®
Version
4.0
Titrate:
Examination of the random answers
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R7.10.01-B
P
age:
9/20
Manual of Reference
R7.10 booklet: Statistical processing
HP-52/96/008/B
4.1.1 Formulas of Rice
For a random signal such as definite previously: stationary with null average (centered) and
ergodic, one supposes moreover than the measured values are distributed according to a normal law profile
of type Gauss (cf [§4.2.1]).
The analysis of a stationary Gaussian random loading has the advantage of leading to
simple analytical expressions [bib8] - known under the name of formulas of Rice - and to represent
many real phenomena.
The following statistical parameters are obtained as from the various spectral moments connected to
different derivative from
X
(cf [§3.4]):
·
Standard deviation
:
X
=
0
Note:
If only the positive part of the spectrum is provided, Code_Aster multiplies by 2 the 1
Er
moment
spectral
X
=
2
0
.
A extremum (maximum or minimum) of amplitude
X
is defined by the probability of having a derivative
null
!X
=
0
associated a derivative second
!!
X
unspecified.
·
Average extrema numbers a second:
NR
E
X
X
=
=
1
1
4
2


!!
!
The going beyond of a level
X
0
is defined by the probability of having
X
X
=
0
with a slope
!X
unspecified: one thus counts the passages of this level with the positive and negative slopes.
Taking into account the assumptions of Gaussian laws, the number of passage by
X
0
and a second
express yourself by:
NR
E
X
X
X
X
0
0
2
2
1
2
=
-

!
X
What leads to the following expressions:
·
A number of goings beyond of level with positive slope a second:
NR
NR
X
X
0
0
1
2
+
=
·
A number of passages by zero (
X
0
= 0) a second:
NR
X
X
0
2
0
1
1
=
=


!
·
A number of passages by zero with positive slope a second:
NR
NR
0
0
2
0
1
2
1
2
+
=
=
NR
0
+
represent an average statistical frequency of passage by zero with positive slope.
In the case of a “simple” signal, i.e. with only one peak,
NR
0
+
, a number of passages by zero
can be compared to a frequency connect also noted
F
E
. In the case much more general
of an unspecified signal, the physical interpretation of the value
NR
0
+
is more of doubtful validity!
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Code_Aster
®
Version
4.0
Titrate:
Examination of the random answers
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R7.10.01-B
P
age:
10/20
Manual of Reference
R7.10 booklet: Statistical processing
HP-52/96/008/B
The factor of irregularity translates the frequential pace of the signal. Ranging between 0 and 1, it tends towards 1
when the process is with narrow tape, on the other hand it tightens towards 0 for a broad band process.
Its expression is:
I
NR
NR
E
X
X X
=
=
=
0
2
2
2
0
4
!
!!
Three parameters -
NR
NR I
E
0
,
,
- characterize the signal entirely. One can, in particular, estimate
the average number of positive peaks a second:
(
)
NR
I NR
peak
E
+
=
+
1 4 1
/
.
The whole of these parameters is calculated and stored in a “printable” table on the file
RESULT
using the control
IMPR_TABLE
.
4.2
Distributions of the positive peaks
One of main knowledge interesting the designers of structures starting from his answer
estimated at a random excitation is the determination of the goings beyond of threshold and in particular them
probabilities of goings beyond of certain critical points.
The formulas of Rice (preceding paragraph) make it possible to know the average rate of
crossings of certain levels. The following approach makes it possible to give a law of probability of
presence of such or such peak. One is thus interested in maximum positive of the answer.
A maximum occurs when
()
()
!
!!
X T
X T
=
<
0
0
with
. One is thus interested in the density of probability
joint
(
)
() () ()
p X X
X T
X T X T X T
!
!!,
,
! !!
=
0
of
. (It is necessary thus that the process is twice derivable,
what is acquired when one admits a Gaussian distribution of the signal.)
This density of probability of the positive peaks makes it possible for example to calculate the proportion of peaks
understood between A and B (or probability that the next peak lies between A and b) which is worth:
(
)
p X
X T dx
has
B
!!,
0
The stationary Gaussian signal, being centered compared to its average value (null in analysis
seismic), the distribution of the peaks is symmetrical compared to this average. One is thus interested in
distribution of the positive peaks. In the general case, the distribution of the peaks of amplitude
X
positive
is written in the form [bib5]:
()
(
)
()
()
p
I
I E
I E
E
dt
X
p
X
I
X
I
I
peak
I
T
peak
X
X
X
X
+
-
-
-
-
-
+
=
+
-
+




<
=
=
=
-


X
X
X
X
X
X
X
X
2
2
1
1
0
0
1
2
2
1
2
2
2
2
2
2
2
2
2
2
If
then
with
!
!!
It is about the so known formula under the name of LONGUET-HIGGINS [bib7], whose step is
also clarified in [bib11]. We present, Ci after, the chart of this formula
for 4 values of
I
.
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0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
0,00
0,40
0,80
1,20
1,60
2,00
2,40
2,80
3,20
3,60
4,00
I = 0 law of Gauss
I = 0.4
I = 0.7
I = 1 law of Rayleigh
Amplitude/Standard deviation
Probability
Appear 4.2-a: Distribution of peaks of standardized positive amplitude
compared to the standard deviation of the signal
This distribution of the positive peaks is in the case of simplified the signals for which the factor
of irregularity is worth
I
I
=
=
0
1
or
.
4.2.1
I 0
=
Signal with broad band: law of Gauss or normal law
In the case of a broad band signal, the positive peaks are distributed according to a law of GAUSS:
()
p
X
E
peak
X
X
X
+
-
=
2
2
2
2
2
2
4.2.2
I 1
=
Signal with narrow tape: law of Rayleigh
In the case of a signal with narrow tape, the positive peaks are distributed according to a law of RAYLEIGH:
()
p
X
X E
peak
X
X
X
+
-
=
2
2
2
2
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4.2.3 Calculation of the values in Code_Aster
The values of these two laws are calculated in Code_Aster under the key words factor
RAYLEIGH
or
GAUSS
.
From the standard deviation
X
=
0
calculated previously one calculates the values of probability of the peaks
()
[
]
p
X
X
peak
X
+
for
0 6
,
with a pitch by defect of
6
200
X
.
If the user wishes to refine his analysis, it can provide the values
VALMIN
and
VALMAX
field of
variation of
X
. It can also provide the step value of calculation, if not the aforementioned will be taken to the 200
ème
bandage reserve.
The figure [Figure 4.2-a] shows that the field chosen by defect, until
6
X
, totality covers well
values of
X
with nonnull probabilities.
4.3
Seismic answer: law of Vanmarcke [bib8]
In the case of the response to a seism of a primary structure (i.e. excited at its base by
ground) having a dominating mode, IE which answers (taking into account the exiting frequencies) on one
only mode, one uses the law of reliability of VANMARCKE [bib8] which makes it possible to estimate, over one duration of
operation T probability that the process exceeds the threshold of ruin.
The concept of dominating mode is very important here, if the structure answers on several modes
formulate in its current expression is not appropriate more.
That is to say
()
X T
the response to a Gaussian white Gaussian noise, of a slightly deadened linear oscillator. One defines
probability
()
W T
that the process remains in the field of security.
()
W T
represent the fraction
of sample which did not cross the threshold of ruin after one duration
T
; it is a measurement of reliability.
It can be written in the form
()
()
{
}
()
()
W
Prob
T
;
:
dW
D
T
X
X
T
T
p T
T
T
=
<
<
= -
0
1
0
is
density of probability of crossing of the threshold.
For the high values of
T
one will take:
()
p T
With
With
T
1
=
-
E
where
depends on the initial conditions
and
is the limiting rate of decrease.
4.3.1 Assumption of independent crossings
With the assumption that the goings beyond of threshold with a positive slope are events
independent, the number of crossings on
[[
0, T
constitute a process of Poisson of rate
of arrival
NR
NR
X
X
0
0
2
=
+
(a number of going beyond of
X
0
defined in [§4.1.1]). Probability that
N
passages occur over the duration
T
is written by application of the law of Poisson (see [§4] [bib8]):
[[
{
}
(
)
P
, T
!
N
E
NR
T NR
T
N
X
X
N
passages on
0
0
0
=
-
The structure is “reliable” if the threshold is not exceeded during the duration
T
. Reliability
()
W T
corresponds
thus with
N
=
0
passage from where
()
W T
E
NR
T
X
=
-
0
.
The limiting rate of decrease is thus worth here
=
=
+
NR
NR
X
X
0
0
2
.
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4.3.2 Law of Vanmarcke
For a Gaussian stationary process, probability of exceeding the value
X
is worth [§4.1.1]:
NR
NR E
X
X
X
=
-
0
2
2
2
; one deduces from it that the probability that the initial value of the envelope is lower
with the threshold
X
is:
1
1
0
2
2
2
-
= -
-
NR
NR
E
X
X
.
One then combines this expression with the limiting law of decrease obtained with the assumption of
independent crossings, which leads to the expression of reliability:
()
(
)
(
)
W T
Ae
E
T
S
NR T
E
E
hs
S
=
= -
-
-
-
-
-
-
1
2
0
2
2
2
1
1
E
/
with
NR
0
2
0
1
=
rate of passage by 0 and
T
duration of observation
where
S
X
H
=
=
=
-
0
1 2
12
0
2
2
1
.
is an estimator of bandwidth of the DSP of
X
.
This relation has the immense advantage of providing an explicit estimator of reliability according to
reduced value of the threshold S, the number of equivalent semi-cycles
NR
0
, and of the parameter of width of
bandage
.
NB:
“The agreement between the estimator and simulations can be improved if one replaces
by
1 2
.
[bib8] “correction” introduced into the formula written above compared to the expression of the rate
of decrease limits given in the preceding paragraph.
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The following chart is carried out in the case of a process of frequency
connect 15 Hz, that is to say
NR
0
= 30 Hz, for one duration of observation of
T
= 1s. the estimator of width
of tape
is taken equal to 0.30. As in the preceding illustrations, the amplitude is standardized
compared to the standard deviation.
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
Vanmarcke
Amplitude/Standard deviation
Reliability
Appear 4.3.2-a: Evolution of the reliability according to the law of Vanmarcke
according to the amplitude of the standardized process
compared to the standard deviation of the signal
Recall:
This statistical analysis is carried out starting from relatively restrictive assumptions, namely
that the process must be with “narrow tape”; it will thus have to be checked that the factor of irregularity
I
is not too different from 1 and that the signal comprises only one main peak.
4.3.3 Establishment in Code_Aster
It is exactly the expression of
()
W T
who is established in the operator
POST_DYNA_ALEA
, under
key word factor VANMARCKE. The field of definition of the calculation of the function of reliability is here too
by defect
[
]
0 6
,
X
with a pitch of
6
200
X
. As for the laws of GAUSS and RAYLEIGH
[§ 4.2.3], it can be restricted by the user.
The calculation of reliability is made for one duration
T
(in S) of operation: it is taken by defect with
T
= 10 S what is appropriate well for a seismic calculation.
The figure [Figure 4.3.2-a] shows well that for
()
6
X
W T
,
tends towards asymptote 1, the process has
surely exceeded the threshold of ruin.
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5 Remarks
This postprocessing is carried out on interspectres stored in
tabl_intsp
. It provides
statistical elements of the response of the structure which can thus be converted into information
statistics of reliability, or to be useful then for calculations of damage by fatigue
(
POST_FATI_ALEA
).
6 Bibliography
[1]
C. DUVAL: Schedule of conditions of a module of calculation of dynamic response under
random excitations for Code_Aster - Report/ratio EDF HP61/91.177
[2]
C. DUVAL: Dynamic response under random excitations in Code_Aster: Principles
theoretical and examples of use - Report/ratio EDF HP61/92.148
[3]
J. MAX and coll: Methods and techniques of processing of the signal and applications to the measurements
physics
[4]
Francoise BROUAYE: “The modeling of uncertainties” Chapter 5 - Collection of the Studies
and Search of EDF ESE n°8 - Eyrolles
[5]
P. MORILHAT: “Thermal Faience manufacturing. Statistical estimate of the mechanical damage” - EDF
July 90 Report/ratio HP 14/90.07
[6]
P. LABBE and H. NOAH: “Stochastic approach for the seismic design off nuclear power seedling
equipment ". Nuclear Engineering and Design 129 (1991) 367-379
[7]
D.E. CARTWRIGHT & Mr. S. LONGUET-HIGGINS: The statistical distribution off maximum the
off random function has - Proceedings off the Royal Society off London - Series A Flight 237 (1956)
[8]
A. PREUMONT “random Vibrations and analyzes spectral” - Press polytechnic and
French academics Edition 1990 - In particular Chapter 10
[9]
R.J. GIBERT: “Vibrations of the structures. Interactions with the fluids. Sources of excitation
random " - CH 17: general concepts on the random processes and the answer of
linear systems - CH 20: Seismic excitations of the structures. Collection of the Studies and
Search of EDF ESE n°69 - Eyrolles 1988
[10]
R.W. CLOUGH and J. PENZIEN in “Dynamics off structures” 4th part: Vibrations
random, 5th part: Analyze response of structures to the seisms - Mc Graw Hill
1975
[11]
Y.K. FLAX & G.Q CAI: Structural Probabilistic dynamics (advanced theory and applications)
Mc Graw Hill
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Appendix
1
Conventions for the Spectral concentrations of
Power
A1.1 Introduction
In order to preserve coherence necessary for the whole of calculations and the comparisons with
the experiment (cf [§2.3] and [§3.3]), we develop hereafter the two assemblies of definitions
coherent with calculations of random answer and postprocessing such as they were retained for
Aster:
·
the first starting from spectral data expressed according to the frequency. It is this
together which is coherent with the calculation carried out in the operator
CALC_INTE_SPEC
[U4.56.03].
·
the second starting from spectral data expressed according to the pulsation.
These two assemblies return validates postprocessing such as it is expressed in
POST_DYNA_ALEA
.
We will each time specify the unit in which the various quantities are expressed
handled according to the unit U of the signal of reference. The explanations given are brief.
One will be able for more details to refer to the reference [bib10].
A1.2 Types of signals and definition of the power
We consider four types of signals:
·
signals of finished energy,
·
signals
periodicals,
·
signals of finished power and signals deterministic,
·
random satisfying the assumption of ergodicity and stationary signals.
In random dynamic calculation the signals are random. For the interpretation of results
experimental, the signals are either periodic, or of finished power (deterministic).
We define for each type of signal an energy quantity which is either an energy, or
a power and which we will indicate in the following paragraphs under the single term of
power:
·
signals of finished energy are defined by their energy
E
expressed out of U
2
S:
()
E
X T dt
=
< +
-
+
2
éq An1.2-1
·
periodic signals are defined by the power
P
signal expressed out of U
2
:
()
[]
P
T
X T dt
T
=
1
2
éq An1.2-2
T
indicate the period of the signal.
[]
T
is an interval length
T
.
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·
signals of finished power are defined by the average power
P
signal
expressed out of U
2
:
()
P
T
X T dt
T
T
T
=




< +
+
-
+
lim
/
/
1
2
2
2
éq An1.2-3
·
random signals are defined by the average power
P
signal expressed out of U
2
:
()
[]
()
P E
T
T
X T dt
T
T
T
=
=




< +
+
-
+
X
lim
/
/
2
2
2
2
1
éq An1.2-4
One is useful oneself here of the assumption of ergodicity which subtends that the average statistics and temporal
carried out on a realization of a process are identical.
A1.3 Autocorrelations
Taking into account the statistical recalls carried out in the body text one has for each type of
signals previously definite:
·
Autocorrelation
signals of finished energy, expressed out of U
2
/Hz:
()
() (
)
R
X T X T
dt
XX
=
+
éq An1.3-1
·
Autocorrelation
periodic signals, expressed out of U
2
:
()
() (
)
[]
R
T
X T X T
dt
XX
T
=
+
1
éq An1.3-2
·
Autocorrelation
signals of finished power, expressed out of U
2
:
()
() (
)
R
T
X T X T
dt
XX
T
T
T
=
+
+
-
+
lim
/
/
1
2
2
éq An1.3-3
·
Autocorrelation
random signals, expressed out of U
2
:
()
() (
)
[
]
() (
)
R
E
T
T
T
X T X T
dt
XX
T
T
T
=
+
=
+
+
-
+
X
X
lim
/
/
1
2
2
éq An1.3-4
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A1.4 Definition of the spectral concentration of power.
A1.4.1 Expression in frequency
One defines the spectral concentration of power by:
()
()
()
()
S
F
R
D
G
F
R
D
XX
XX
I F
XX
XX
I F
=
=
-
-
+
-
+
E
E
2
2
0
or
éq An1.4.1-1
The mechanic being interested only in the positive values of the frequency and time, the function
G
XX
is more often used.
One can show, if the Transformations of Fourier of the signals exist, that this
definition is equivalent (theorem of Wiener-Kinchine) to the definitions of the spectral concentration of
power following.
·
For the signals of finished energy:
()
()
G
F
X F
XX
=
2
/
expressed out of U
Hz
2
2
éq An1.4.1-2
·
For the periodic signals:
()
(
)
()
(
)
If
then
X F
C
F
N F
G
F
C
F
N F
N
N
N
XX
N
N
N
=
-
=
-
=
=+
=
=+
0
2
0
éq An1.4.1-3
()
G
F
XX
express yourself out of U
2
/Hz.
F
0
is the reverse of the period of the signal.
C
N
coefficient of the Dirac function functions.
·
For the signals of finished power:
()
[]
()
G
F
T X
F
XX
T
T
=




+
lim 1
2
out of U
Hz
2
/
éq An1.4.1-4
where
[]
X
T
indicate the restriction of
()
[
]
X T
T/; T/
with
-
2
2
.
·
For the random signals:
()
[]
()
G
F
E T X F
XX
T
T
=




+
lim
1
2
out of U
Hz
2
/
éq An1.4.1-5
where
[]
X
T
indicate the restriction of
()
[
]
X T
T
T
with
-
/; /
2
2
.
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Link between the DSP and the power.
With the definitions given above for the spectral concentrations of power, one has for
all signals, the relation:
()
P
G
F df
XX
=
-
+
éq An1.4.1-6
This relation is established by using the theorem of PARSEVAL.
A1.4.2 Expression in pulsation
In pulsation, one defines the spectral concentration of power by:
()
()
G
R
D
XX
XX
I
'
E
=
-
-
+
1
2
éq An1.4.2-1
Just as for the expression in frequency, one can show, if the Transformations
of Fourier of the signals exist, that this definition is equivalent (theorem of Wiener-Kinchine)
with the definitions of the spectral concentration of power following
·
For the signals of finished energy:
()
()
G
X
XX
'
'
=
2
2
/
expressed out of U
Hz
2
2
éq An1.4.2-2
·
For the periodic signals:
()
(
)
()
(
)
If
then
X
C
N
G
C
N
N
N
N
XX
N
N
N
'
'
=
-
=
-
=
=+
=
=+
0
2
0
éq An1.4.2-3
()
G
XX
'
express yourself out of U
2
/Hz, and
0
=
2
T
where
T
is the period of the signal.
C
N
coefficient of the Dirac function functions.
·
For the signals of finished power:
()
[]
()
G
T X
XX
T
T
'
'
lim
=




+
2
2
out of U
Hz
2
/
éq An1.4.2-4
[]
X
T
'
indicate the restriction of
()
[
]
X T
T/; T/
with
-
2
2
.
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·
For the random signals:
()
[]
()
G
E T X
XX
T
T
'
'
lim
=




+
2
2
out of U
Hz
2
/
éq An1.4.2-5
[]
X
T
'
indicate the restriction of
()
[
]
X T
with
-
T/; T/
2
2
.
Link between the DSP and the power.
In the same way, there is for all the signals the relation - which rises from the theorem of PARSEVAL -:
()
P
G
D
XX
=
-
+
'
éq An1.4.2-6
A1.4.1 Relation between DSP in frequency and DSP in pulsation
For the four types of signals:
()
()
G
G
F
XX
XX
'
=
1
2
éq An1.4.3-1
Appendix 2 Transformation of Hilbert
That is to say
()
X T
a real signal of transform of Fourier
()
X
.
That is to say
()
H
the transfer function:
()
()
H
sign


=
= -



J
J
J
>
<
=
0
0
0
0
()
H
transform
()
X T
in its transform of Hilbert noted
()
X T
. The system of transfer function
()
H
product a dephasing of +90° for the positive frequencies and ­ 90° for the frequencies
negative. It follows theorem of convolution that
()
X T
can also be defined like the convolution of
()
X T
by the corresponding impulse response, that is to say
()
H T
T
=
1/
.
()
X T
is also real, one second application of the transform of Hilbert restores the initial signal,
changed of sign and cut down by its possible continuous component.
Example:
()
()
X T
With
T
X T
With
T
=
= -
cos
sin
This property is at the base of the use of the transform of Hilbert to define the envelope of one
process in narrow tape.