SDNS01 - Nonparametric probabilistic model… parametric dune flexbeam with nonlocalized linearities of shock

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Titrate:

SDNS01 - Nonparametric probabilistic model

Date:

01/07/03

Author (S):

S. CAMBIER, C.DESCELIERS

Key

V5.06.001-A

Page:

1/8

Manual of validation

V5.06 booklet: Nonlinear dynamics of the hulls and plates

HT-66/03/008/A

Organization (S):

EDF-R & D/AMA

Manual of Validation
V5.06 booklet: Nonlinear dynamics of the hulls and plates
Document: V5.06.001

SDNS01 - Nonparametric probabilistic model -
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 stop 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 the key word

CALC_AMOR_GENE

COMB_MATR_ASSE

thus testing this

functionality.

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Titrate:

SDNS01 - Nonparametric probabilistic model

Date:

01/07/03

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S. CAMBIER, C.DESCELIERS

Key

V5.06.001-A

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HT-66/03/008/A

0.50 m

0.23

0.15

0.06

0.21

0.31

0.40

0.15

Stop

Problem of reference

1.1 Geometry

Thickness of the plate: E = 0.0004 Mr.

Play enters the stops: play = ±0.002 Mr.

1.2

Material properties

Plate:

Poisson's ratio: 0.3
Young modulus: 2.1 10

NR/m

Density: 7800 kg/m

Concentrated stiffness: 2.388 10

NR/m

Mass concentrated: 4 kg
Stiffness of shock: 25000 NR/m

Concentrated stiffness

Mass
concentrated

Impulse load

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Titrate:

SDNS01 - Nonparametric probabilistic model

Date:

01/07/03

Author (S):

S. CAMBIER, C.DESCELIERS

Key

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1.3

Boundary conditions and loadings

The flexbeam is in simple on 3 edges and free support on its 4

edge GOLD. Degrees of

freedoms locked are thus:

on COp and QR, displacements according to X

, X

and rotations according to X

, X

on PQ, displacements according to X

, X

and rotations according to X

, X

on GOLD, displacements according to X

, X

The plate is subjected to a vertical impulse load E (T) on 9 nodes of the plate according to
direction X

. The loading E (T) is such as, for t<0 and t>2t

, E (T) =0 and for 0

E (T) = (

(t-t

))

- 1

{sin {(

/2) (t-t

)}- sin {(

/2) (t-t

)}}.

with t1= 2

40 rad/S,

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.

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 mesh 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 modelized 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

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Titrate:

SDNS01 - Nonparametric probabilistic model

Date:

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S. CAMBIER, C.DESCELIERS

Key

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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 mesh
whose pitch is constant and is worth 0.01m in directions X

and X

. 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 X

and X

Eigen frequencies of the dynamic system linearized (the plate without the stops of shock but
with the concentrated masses and stiffnesses) are F

=1.94, F

=10.28, F

=15.47,…, F

=53.5, F

=66,1,

=68.9,…, F

=198.3, F

=206.0, F

=208.9,…, F

=330.9, F

=336.3,…, F

100

=670.8, F

120

=817.6Hz.

Modeling:

DIS_T

The concentrated masses and the concentrated stiffness are modelized 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

min

max

min

max

min

max

has

and

where

=0.04,

min

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 the Results of modelings).

The degree of freedom observed is D.D.L J

stop

corresponding to displacement in translation according to Z of

node or are the elastic stops. 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, the key word

INIT

must take the value

“YES”

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.

Code_Aster

Version

6.4

Titrate:

SDNS01 - Nonparametric probabilistic model

Date:

01/07/03

Author (S):

S. CAMBIER, C.DESCELIERS

Key

V5.06.001-A

Page:

5/8

Manual of validation

V5.06 booklet: Nonlinear dynamics of the hulls and plates

HT-66/03/008/A

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). The 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 the random generator

GENE_VARI_ALEA

with the 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). The key word

DELTA

thus takes value 0.01.

Resolution of the probabilistic nonlinear dynamic system.

The 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 N

achievements of each random variable (stiffness of shock + matrices) what corresponds to 5 iteration of
method of digital simulation of Monte Carlo (cf paragraph Comments of the Results of
modelings).

The temporal interval of the study is T= [0,4] S, with a pitch of 5. 10

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.J

stop

by the operator

RECU_FONCTION

. The spectrum of answer standardized is

then built by the 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 N

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 aid only of the operator

CALC_FONCTION

and of the key words

WRAP

POWER

and

COMB

. At the end of N

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 the key word

NORMALIZES

of the 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 Goes up
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 mesh

A number of degrees of freedom: 6009
A number of finite elements: 2000 QUA4 and 2 DIS_T

3.3 Functionalities

tested

Controls

Key word

factor

GENE_MATR_ALEA

GENE_VARI_ALEA

CALC_FONCTION POWER

CALC_FONCTION NORMALIZES

Code_Aster

Version

6.4

Titrate:

SDNS01 - Nonparametric probabilistic model

Date:

01/07/03

Author (S):

S. CAMBIER, C.DESCELIERS

Key

V5.06.001-A

Page:

6/8

Manual of validation

V5.06 booklet: Nonlinear dynamics of the hulls and plates

HT-66/03/008/A

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
max

4.4433958494950E+02 4.4433958494950E+02

Estimate of the envelope
min

1.2534278720661E+02 1.2534278720661E+02

Estimate of the average

3.2330416710925E+02

Estimate of the moment
of command 2

1.2260792008492E+02 1.2260792008492E+02

Estimate of the L2 Standard
average

2.0657959602609E+03 2.0657959602609E+03

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.

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Version

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Titrate:

SDNS01 - Nonparametric probabilistic model

Date:

01/07/03

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S. CAMBIER, C.DESCELIERS

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V5.06 booklet: Nonlinear dynamics of the hulls and plates

HT-66/03/008/A

5 Modeling

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 agent with one
straight line modal reduced depreciation of 4%.

5.2

Characteristics of the mesh

A number of degrees of freedom: 6009
A number of finite elements: 2000 QUA4 and 2 DIS_T

5.3 Functionalities

tested

Controls

Key word

factor

GENE_MATR_ALEA

GENE_VARI_ALEA

CALC_FONCTION POWER

CALC_FONCTION NORMALIZES

COMB_MATR_ASSE CALC_AMOR_GENE

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
max

2.7570639015302E+02 2.7570639015302E+02

Estimate of the envelope
min

9.3629561795277E+01 9.3629561795277E+01

Estimate of the average

1.9641139264813E+02

Estimate of the moment
of command 2

4.3869049613023E+04 4.3869049613023E+04

Estimate of the L2 Standard
average

1.2384277999571E+03

6.2 Comments

Same comments as for modeling A.