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Version
6.4
Titrate:
SDNS01 - Nonparametric probabilistic Modčle
Date:
01/07/03
Author (S):
S. CAMBIER, C.DESCELIERS Key
:
V5.06.001-A Page:
1/8
Organization (S): EDF-R & D/AMA
Handbook of Validation
V5.06 booklet: Nonlinear dynamics of the hulls and plates
Document: V5.06.001
SDNS01 - Nonparametric probabilistic Modčle -
parametric of a flexbeam with not
localized linearities of shock
Summary:
This case-test relates to the nonparametric and parametric probabilistic models of uncertainties in
linear dynamics with possibly of nonthe localized linearities. The mechanical model used is one
rectangular plate with an elastic thrust of shock. Random generators of matrices and variables
random (operators GENE_MATR_ALEA and GENE_VARI_ALEA) are tested and validated in this case test.
statistical postprocessings (CALC_FONCTION) are also tested.
This case-test has two modelings. The first uses a damping proportional. The second
use a definite reduced damping by key word CALC_AMOR_GENE of COMB_MATR_ASSE testing thus this
functionality.
Handbook of validation
V5.06 booklet: Nonlinear dynamics of the hulls and plates
HT-66/03/008/A
Code_Aster ®
Version
6.4
Titrate:
SDNS01 - Nonparametric probabilistic Modčle
Date:
01/07/03
Author (S):
S. CAMBIER, C.DESCELIERS Key
:
V5.06.001-A Page:
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1
Problem of reference
1.1 Geometry
X
3
X2
Q
P
Concentrated stiffness
Impulse load
0.23
Mass
0.15
concentrated
Stop
O
R
X1
0.06
0.15
0.40
0.21
0.31
0.50 m
Thickness of the plate: E = 0.0004 Mr.
Play enters the thrusts: play = ±0.002 Mr.
1.2
Material properties
Plate:
Poisson's ratio: 0.3
11
Young modulus: 2.1 10 NR/m2
Density: 7800 kg/m2
Concentrated stiffness: 2.388 107 NR/m
Mass concentrated: 4 kg
Stiffness of shock: 25000 NR/m
Handbook of validation
V5.06 booklet: Nonlinear dynamics of the hulls and plates
HT-66/03/008/A
Code_Aster ®
Version
6.4
Titrate:
SDNS01 - Nonparametric probabilistic Modčle
Date:
01/07/03
Author (S):
S. CAMBIER, C.DESCELIERS Key
:
V5.06.001-A Page:
3/8
1.3
Boundary conditions and loadings
The flexbeam is in simple on 3 edges and free support on its 4th edge GOLD. Degrees of
freedoms blocked are thus:
· on COp and QR, displacements according to X1, X2, X3 and rotations' according to X1, X3.
· on PQ, displacements according to X1, X2, X3 and rotations' according to X2, X3.
· on GOLD, displacements according to X1, X2.
The plate is subjected to a vertical impulse load E (T) on 9 nodes of the plate according to
X3 direction. The loading E (T) is such as, for t<0 and t>2t1, E (T) =0 and for 0t2t1:
E (T) = ((T-T1))- 1 {sin {(c+/2) (T-T1)}- sin {(C/2) (T-T1)}}.
with t1= 2/, =2×40 rad/S, C =2×20 rad/S.
The energy of the function E (T) is mainly distributed in the frequential tape [0,60] Hz, which
contains 8 elastic modes of the linearized dynamic system.
1.4 Conditions
initial
The dynamic system is initially at rest.
2
Reference solution
2.1
Method of calculation used for the reference solution
We study the transitory response of a nonlinear dynamic system subjected to a load
impulse determinist due to a shock on the structure. Nonthe linearity of the system is due to one
butted elastic of high rigidity comprising a certain play. Spectra of frequency response
standardized are used in order to study the transitory response of this system. Equations of
dynamics are discretized by the method with the finite elements. The grid of the structure is supposed
sufficient fine to collect all the dynamic phenomena of this mechanical system in term of
field of displacement for the impulse loading considered. Random uncertainties of
dynamic system are modelled by using the nonparametric probabilistic model
uncertainties. Consequently, the transitory answer is a nonstationary stochastic process
whose statistical estimates are evaluated.
The results of reference are given in the form of graphs in the article referred below,
consultable on http://www.resonance-pub.com.
2.2 Reference
bibliographical
[1]
C. SOIZE: Not linear dynamical Systems with Nonparametric Model off Random
Uncertainties ", Uncertainties in Engineering Mechanics (2001) 1 (1), 1-38,
http://www.resonance-pub.com
Handbook of validation
V5.06 booklet: Nonlinear dynamics of the hulls and plates
HT-66/03/008/A
Code_Aster ®
Version
6.4
Titrate:
SDNS01 - Nonparametric probabilistic Modčle
Date:
01/07/03
Author (S):
S. CAMBIER, C.DESCELIERS Key
:
V5.06.001-A Page:
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3 Modeling
With
3.1
Characteristics of modeling
Modeling: DKT
The average model with the finite elements of the plate consists of a regular rectangular grid
whose step is constant and is worth 0.01m in the directions X1 and X2. There are thus 41 nodes in
width and 51 nodes in the length. Consequently, all the finite elements are identical and
each one is an element plates with 4 nodes. This average model finite elements comprises 2000 elements
stop and m=6009 degrees of freedom, by counting only the translations in Z and rotations according to X1
and X2).
Eigen frequencies of the dynamic system linearized (the plate without the thrusts of shock but
with the concentrated masses and stiffnesses) are f1=1.94, f2=10.28, f3=15.47,…, f8=53.5, f9=66,1,
f10=68.9,…, f30=198.3, f31=206.0, f32=208.9,…, f50=330.9, f51=336.3,…, f100=670.8, f120=817.6Hz.
Modeling: DIS_T
The concentrated masses and the concentrated stiffness are modelled by elements DIST_T.
Damping
The matrix of damping [D] of the model average finite element is defined as being one
linear combination of the average matrices finite elements of mass [M] and stiffness [K]. One thus has
[D] =a [M] + B [K] with
2maxmin
2
has =
and
B =
,
max + min
max + min
where =0.04, min=4 rad/S and min=200 rad/S.
Small-scale model and ddl observed
For this case test, the model finite elements is projected on the first 5 elastic modes of
structure linearized, which constitutes the data of the model reduces average. It should be noted that 5 first
modes are not enough to obtain convergence compared to the number of modes (cf paragraph
Comments of Résultats of modelings).
The degree of freedom observed is D.D.L jstop corresponding to displacement in translation according to Z of
node or are the elastic thrusts. It is the node of co-ordinates (0.31, 0, 0). The spectrum of
response standardized for this D.D.L is built for a tape of frequential analysis J=2 [1, 100]
rad/S and of which the frequential resolution is of 0.5Hz.
Achievements of the random matrices of the nonparametric probabilistic model parametric
The matrices of masses, stiffnesses and dissipation of the model reduces average are replaced by
achievements of the random matrices of mass, stiffnesses and of dissipation according to the model
nonparametric probabilist. For that, we use the generator of random matrices
GENE_MATR_ALEA. At the time of the first call to this generator, key word INIT must take the value
“OUI” in order to initialize the generator of random variable of uniform law of Python. Thereafter, INIT
will be able to take its default value (INIT= `NON'). The initialization of the generator of random variable
of Python is to be done only one and only once by study, in theory, except if the user wishes
explicitly to re-use the same random pseudo sequence.
Handbook of validation
V5.06 booklet: Nonlinear dynamics of the hulls and plates
HT-66/03/008/A
Code_Aster ®
Version
6.4
Titrate:
SDNS01 - Nonparametric probabilistic Modčle
Date:
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Author (S):
S. CAMBIER, C.DESCELIERS Key
:
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The level of dispersion of the random matrices of the nonparametric probabilistic model is controlled
by a parameter of dispersion fixed at 20% (=0.2). Key word DELTA thus takes value 0.2.
Lastly, for each hard copy of the random matrices of mass, stiffness and dissipation, it is necessary
to inform the corresponding average matrix of the model reduces average via the key word
MATR_MOYEN.
The stiffness of shock is it also made random due to uncertainties. To build a realization
stiffness of shock following a law gamma, we use random generator GENE_VARI_ALEA
with key word TYPE= `GAMMA'. We suppose that the possible whole of the values for
achievements of the random stiffness of shock is the interval [0, + [, that the average value of the unit
achievements of the stiffnesses of shock corresponds to the stiffness of shock of the average model finite elements.
The level of dispersion of the achievements of the stiffness of shock is controlled by a parameter 'fixed at
1% (=0.01). Key word DELTA thus takes value 0.01.
Resolution of the probabilistic nonlinear dynamic system.
Operator DYNA_TRAN_MODAL is used to build the transitory response of the dynamic system
nonlinear for each realization of the random stiffness of random shock and the matrices of mass,
of stiffness and dissipation. It should be noted that we carry out for this case test only ns=5
achievements of each random variable (stiffness of shock + matrices) what corresponds to 5 iteration of
method of digital simulation of Monte Carlo (cf Commentaires paragraph of Résultats of
modelings).
The temporal interval of the study is T= [0,4] S, with a step of 5. 105 S. the diagram of integration
temporal selected is EULER.
Construction of the statistical estimates.
After each call to DYNA_TRAN_MODAL, we have a realization of the process
stochastic of generalized displacements. It is thus possible to build acceleration with the node
of shock following D.D.L. jstop by operator RECU_FONCTION. The spectrum of answer standardized is
then built by operator CALC_FONCTION.
These two operations are classically carried out at the time of deterministic studies and give us here
a realization of the stochastic process of the spectrum of answer standardized. It is then about
to build statistical estimates of the NS achievements of this last. Estimates considered
in this case test are the envelopes min and max as well as the average and the moment of command two of
standardized spectra of answer. With each hard copy (iteration of Monte Carlo) we build these
estimates with the assistance only of operator CALC_FONCTION and key words ENVELOPPE,
PUISSANCE and COMB. At the end of the NS iterations of Monte Carlo, we have the estimates
required statistics relating to the stochastic process of the spectrum of answers standardized. Finally
L2 of the average normalizes is calculated by key word NORME of operator CALC_FONCTION. Interest
to evaluate such a standard is to allow studies of convergence according to the number of modes of
model reduces average random and according to the iteration count of the numerical method of Monte
Carlo. This standard is calculated here to check that the functionality goes, but convergence is not
not reached to save time CPU.
3.2
Characteristics of the grid
A number of degrees of freedom: 6009
A number of finite elements: 2000 QUA4 and 2 DIS_T
3.3 Functionalities
tested
Commands
Key word
factor
GENE_MATR_ALEA
GENE_VARI_ALEA
CALC_FONCTION POWER
CALC_FONCTION NORMALIZES
Handbook of validation
V5.06 booklet: Nonlinear dynamics of the hulls and plates
HT-66/03/008/A
Code_Aster ®
Version
6.4
Titrate:
SDNS01 - Nonparametric probabilistic Modčle
Date:
01/07/03
Author (S):
S. CAMBIER, C.DESCELIERS Key
:
V5.06.001-A Page:
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4
Results of modeling A
4.1
Values of reference Aster
The initial validity of the case test was established by comparison with the bibliographical reference given in
[§3.2].
One tests the following values in nonregression (cf comments):
Statistics on the values of the spectrum of response to 50Hz with the ddl observed (cf modeling)
Identification References
Aster %
Difference
Estimate of the envelope
4.4433958494950E+02 4.4433958494950E+02
0
max
Estimate of the envelope
1.2534278720661E+02 1.2534278720661E+02
0
min
Estimate of the average
3.2330416710925E+02
3.2330416710925E+02
0
Estimate of the moment
1.2260792008492E+02 1.2260792008492E+02
0
of command 2
Estimate of Norme L2
2.0657959602609E+03 2.0657959602609E+03
0
average
4.2 Comments
The various statistical estimates are not converged here. Only 5 simulations of
Monte Carlo were made. One would have needed 700 at least of them. Moreover, it is necessary to increase the number of
modes with 50 to obtain the convergence of the model projected in the frequency band considered.
Calculations being then too long for a case test, we preferred to voluntarily degenerate them
two convergences after validation of those on a complete study. After convergence, them
statistical estimates calculated starting from Aster correspond very exactly to the results given
by the standard commodity.
Handbook of validation
V5.06 booklet: Nonlinear dynamics of the hulls and plates
HT-66/03/008/A
Code_Aster ®
Version
6.4
Titrate:
SDNS01 - Nonparametric probabilistic Modčle
Date:
01/07/03
Author (S):
S. CAMBIER, C.DESCELIERS Key
:
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5 Modeling
B
5.1
Characteristics of modeling
Only the modeling of damping changes compared to modeling A.
Damping
The matrix of damping [D] of the model average finite element is defined as correspondent with one
straight line modal reduced depreciation of 4%.
5.2
Characteristics of the grid
A number of degrees of freedom: 6009
A number of finite elements: 2000 QUA4 and 2 DIS_T
5.3 Functionalities
tested
Commands
Key word
factor
GENE_MATR_ALEA
GENE_VARI_ALEA
CALC_FONCTION POWER
CALC_FONCTION NORMALIZES
COMB_MATR_ASSE CALC_AMOR_GENE
6
Results of modeling B
6.1
Values of reference Aster
One tests the following values in nonregression, with 50Hz:
Statistics on the values of the spectrum of response to 50Hz with the ddl observed (cf modeling)
Identification References Aster %
Difference
Estimate of the envelope
2.7570639015302E+02 2.7570639015302E+02
0
max
Estimate of the envelope
9.3629561795277E+01 9.3629561795277E+01
0
min
Estimate of the average
1.9641139264813E+02
1.9641139264813E+02
0
Estimate of the moment
4.3869049613023E+04 4.3869049613023E+04
0
of command 2
Estimate of Norme L2
1.2384277999571E+03
1.2384277999571E+03
0
average
6.2 Comments
Same comments as for modeling A.
Handbook of validation
V5.06 booklet: Nonlinear dynamics of the hulls and plates
HT-66/03/008/A
Code_Aster ®
Version
6.4
Titrate:
SDNS01 - Nonparametric probabilistic Modčle
Date:
01/07/03
Author (S):
S. CAMBIER, C.DESCELIERS Key
:
V5.06.001-A Page:
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7
Summary of the results
The results obtained are completely in conformity with those of the bibliographical reference [§2.2] obtained
entirely in Matlab.
Handbook of validation
V5.06 booklet: Nonlinear dynamics of the hulls and plates
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