Code_Aster ®
Version
5.0
Titrate:
SDLD104 - Extrapolation of local measurements on a complete model
Date:
04/03/02
Author (S):
S. AUDEBERT, P. Key HERMANN
:
V2.01.104-A Page:
1/12
Organization (S): EDF/RNE/AMV, CS IF
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
V2.01.104 document
SDLD104 - Extrapolation of local measurements
on a complete model (discrete)
Summary:
It is about a test of linear dynamics discrete.
The goal is to test command PROJ_MESU_MODAL in the case of a discrete system. This command
allows to project experimental dynamic transitory answers in a certain number of points on
a modal base of a numerical modeling.
This test contains 2 modelings:
·
projection is done on a basic concept modal of type [mode_meca],
·
projection is done on a basic concept modal of type [base_modale].
For 2 modelings, provided experimental measurements are identical and make it possible to test
seek nodes in opposite, the taking into account of a local orientation and the processing of one
sampling in constant time or not, for measurements in displacement.
In both cases, the reference solution is analytically given (by Maple); projection is
realized in the favorable configuration where the number of modes is equal to the number of measurements.
The answers in displacement obtained after projection are identical to displacements of reference
provided in data.
The values speeds and the accelerations deduced from the displacements obtained after projection are
close relations of those obtained analytically. The weak noted variations are due to the errors of approximation
generated by the determination via a linear diagram in time of speeds and accelerations.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Code_Aster ®
Version
5.0
Titrate:
SDLD104 - Extrapolation of local measurements on a complete model
Date:
04/03/02
Author (S):
S. AUDEBERT, P. Key HERMANN
:
V2.01.104-A Page:
2/12
1
Problem of reference
1.1
Description of the system
We consider the system represented by the diagram below:
K
K
K
m
m
X1
X2
1.2
Masses and rigidity
The three springs are of identical rigidity: K = 1000 NR/Mr.
The two masses are equal to m = 10 kg.
1.3
Boundary conditions and loading
The two ends are embedded.
The loading is a thrust load in traction applied to the mass m1, sinusoidal according to
time, of pulsation.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Code_Aster ®
Version
5.0
Titrate:
SDLD104 - Extrapolation of local measurements on a complete model
Date:
04/03/02
Author (S):
S. AUDEBERT, P. Key HERMANN
:
V2.01.104-A Page:
3/12
2
Reference solutions
2.1
Method of calculation used for the reference solution
The analytical solution of this problem is presented below.
·
Modes and frequencies of vibration:
The following system characterizes the dynamics of the masses:
MX & + 2kx - kx
1
1
2 = 0
éq 2.1-1
MX
& + 2kx - kx
2
2
1 = 0
What is equivalent to the following system:
(
m X & + X
1
&2) + K (X + X
1
2) = 0
éq 2.1-2
(
m X & - X
1
&2) + K
3 (X - X
1
2) = 0
The 2 Eigen frequencies of the system are thus given by:
K
K
=
and
3
1
2 =
éq
2.1-3
m
m
and the associated modal deformations are:
1
1
= and
1
2 =
éq
2.1-4
1
- 1
The generalized matrices are:
1
1m 01
1 2m
0
M = TM =
=
1 -
1 0
m 1 -
1
0
2
m
éq
2.1-5
1
1 2k - k1
1 2k
0
K = TK =
=
1 -
1 - K
2k 1 -
1
0 6k
·
Transitory answer:
1
The sinusoidal effort is applied to the first mass: F =
(
sin T)
0
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Code_Aster ®
Version
5.0
Titrate:
SDLD104 - Extrapolation of local measurements on a complete model
Date:
04/03/02
Author (S):
S. AUDEBERT, P. Key HERMANN
:
V2.01.104-A Page:
4/12
The checked dynamic system is as follows:
MX
& + KX = F
éq 2.1-6
While projecting on the basis of clean mode, we obtain:
T & + T = T
M
K
F
éq
2.1-7
That is to say:
2m
0 & 2k
0 1 1 1
1
1
+
=
(
sin T
)
éq
2.1-8
0
2m & 0 6k 1 - 10
2
2
We thus end to the following uncoupled system:
1
m
& + K = if (
N T
)
1
1
2
éq
2.1-9
1
m
& + K
3 =
(
sin T
)
2
2
2
The solution of this system is given by:
sin T
(T)
1
= 1
With
(
cos 1t) + 1
B if (
N 1t)
()
+ 2
(m2 - 2
1
)
éq
2.1-10
sin
(
T
T)
2
= 2
With
(
cos 2t) + 2
B S (
in 2t)
()
+
2
(m2 - 2
2
)
Displacements in physical space are obtained by the formula of Ritz:
x1
1
1 1
+
1
2
X = = =
=
éq
2.1-11
x2
1 -
1 2
-
1
2
One deduces the expressions from them from:
X (T) and X (T)
1
2
éq
2.1-12
sin T
1
1
X (T) = A cos
sin
cos
sin
1
1
(T1) + B1 (T1) + A2 (t2) + B2 (t2)
()
+
+
2m2
2
2
2
1
-
2 -
sin T
1
1
X (T) = A cos
sin
cos
sin
2
1
(T1) + B1 (T1) - A2 (t2) + B2 (t2)
()
+
-
2m2
2
2
2
1
-
2 -
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Code_Aster ®
Version
5.0
Titrate:
SDLD104 - Extrapolation of local measurements on a complete model
Date:
04/03/02
Author (S):
S. AUDEBERT, P. Key HERMANN
:
V2.01.104-A Page:
5/12
At the initial moment, the system is at rest, from where final expressions of X (T)
1
and X (T)
2
:
(
sin T
) -
S (
in T
1)
(
sin T
) -
S (
in T
2)
1
X (T)
1
2
1
=
+
2m
2
2
2
2
1 -
2 -
éq
2.1-13
(
sin T
) -
S (
in T
1)
(
sin T
) -
S (
in T
2)
1
X (T)
1
2
2
=
-
2m
2
2
2
2
1 -
2 -
Speeds of the two masses are calculated by deriving displacements compared to time:
(
cos T
) - C (
bone T
1)
(
cos T) - C (
bone
T
2)
X (T)
&
1
=
+
2m
2
2
2
2
1
-
2 -
éq
2.1-14
(
cos T
) - C (
bone T
1)
Co (
S T) - C (
bone
T
2)
X (T)
&
2
=
-
2m
2
2
2
2
1
-
2 -
Accelerations of the two masses are calculated by deriving speeds compared to time:
(
sin T
) - S
1
(
in
T
1)
if (
N T) -
S
2
(
in
T
2)
&x (T)
1
=
+
2m
2
2
2
2
1
-
2 -
éq 2.1-15
(
sin T
) - S
1
(
in
T
1)
(
sin T) -
S
2
(
in
T
2)
&x (T)
2
=
-
2m
2
2
2
2
1
-
2 -
2.2
Results of reference
The comparison of the results relates to displacements, speeds and accelerations along the axis of
two masses, at five different moments.
2.3
Uncertainty on the solution
The reference solution is exact.
The discrete model represents perfectly the problem arising (the modal base is complete; there is not
thus not of approximation related to a possible modal truncation). The number of modes of the base
of modal projection is equal to the number of measurements, therefore the solution of the inversion is exact (by
opposition to an approximate solution of a generalized opposite problem). If the search of the nodes in
opposite is good, the displacements obtained after projection must be in perfect adequacy
with the experimental values. Speeds and accelerations are determined by derivation of
modal contributions identified via a diagram of linear approximation in time, thus being able
to generate some errors.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Code_Aster ®
Version
5.0
Titrate:
SDLD104 - Extrapolation of local measurements on a complete model
Date:
04/03/02
Author (S):
S. AUDEBERT, P. Key HERMANN
:
V2.01.104-A Page:
6/12
3 Modeling
With
3.1
Characteristics of modeling and the grids
·
Numerical grid:
The numerical grid is carried out directly with format ASTER. It comprises 4 nodes and 3
discrete meshs.
K
m K
m
K
X
N1
N2
N3
N4
(x=0.) (x=0.1) (x=0.2) (x=0.3)
·
Experimental grid:
The grid of measurement includes/understands only 2 specific elements and 2 nodes:
X
N2 N3
(x=0.12) (x=0.18)
3.2
Characteristics of measurements
Provided experimental measurements are:
·
With the N3 node:
The data are the displacements axial, multiplied by ( 2/2), and applied in the direction
X. the local orientation indicated in the command file is (45. 0. 0.)
The sampling of time is constant: initial time is 0 S, the step of time is 103 S and it
a many moments are 1001 (i.e until a final time of 1 S).
·
With the node N2:
The data are the axial displacements, applied in direction X.
The sampling of time is variable: every moment is indicated of 0 S to 1 S, by step of
103 S (1001 moments on the whole).
The values result from the analytical calculation carried out with Maple.
3.3
Characteristics of the modal base
The two only modes are stored in a concept of the type [mode_meca] created by the command
MODE_ITER_SIMULT. Their Eigen frequencies are identical to the analytical Eigen frequencies.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Code_Aster ®
Version
5.0
Titrate:
SDLD104 - Extrapolation of local measurements on a complete model
Date:
04/03/02
Author (S):
S. AUDEBERT, P. Key HERMANN
:
V2.01.104-A Page:
7/12
3.4 Functionalities
tested
Commands
AFFE_CARA_ELEM
DISCRET
CARA
'M_T_D_N
“K_T_D_L'
ORIENTATION
“ANGL_NAUT”
REPERE
“LOCAL”
AFFE_CHAR_MECA
DDL_IMPO
TOUT
NOEUD
AFFE_MODELE
TOUT
“MECHANICAL” “DIST_T'
ASSE_MATRICE
CALC_MATR_ELEM OPTION
“MASS_MECA”
“RIGI_MECA”
MODE_ITER_SIMULT CALC_FREQ
OPTION
“BANDE”
NUME_DDL
NUME_DDL_GENE
PROJ_MATR_BASE
PROJ_MESU_MODAL
MESURE
REGULARIZATION
REST_BASE_PHYS TOUT_CHAM
“OUI”
TEST_RESU NOM_CHAM
“DEPL”
CRITERE
“VITE”
“ACCE”
“RELATIF”
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Code_Aster ®
Version
5.0
Titrate:
SDLD104 - Extrapolation of local measurements on a complete model
Date:
04/03/02
Author (S):
S. AUDEBERT, P. Key HERMANN
:
V2.01.104-A Page:
8/12
4
Results of modeling A
4.1 Values
tested
Identification Reference
Code_Aster difference
with T = 0.1 S
1.745 104 1.745
104 0.01
%
with T = 0.3 S
6.797 104 6.797
104 0.01
%
DEPL_X
with the node N2
with T = 0.5 S
1.217 103 1.217
103 0.01
%
(m) (mass
1)
with T = 0.7 S
5.214 104 5.214
104 0.01
%
with T = 0.9 S
9.031 104 9.031
104 0.00
%
with T = 0.1 S
9.154 106 9.154
106 0.00
%
with T = 0.3 S
6.414 104 6.414
104 0.00
%
DEPL_X
with the N3 node
with T = 0.5 S
8.636 104 8.636
104 0.00
%
(m) (mass
2)
with T = 0.7 S
1.107 104 1.107
104 0.03
%
with T = 0.9 S
1.633 103 1.633
103 0.02
%
with T = 0.1 S
4.586 103 4.616
103 0.65
%
with T = 0.3 S
7.598 103 7.663
103 0.85
%
VITE_X
with the node N2
with T = 0.5 S
1.581 104 8.000
105 7.81
105 m/s
(m/s) (mass
1)
with T = 0.7 S
9.382 103 9.354
103 0.30
%
with T = 0.9 S
7.481 103 7.537
103 0.75
%
with T = 0.1 S
4.328 104 4.405
104 1.79
%
with T = 0.3 S
3.671 103 3.640
103 0.84
%
VITE_X
with the N3 node
with T = 0.5 S
1.539 102 1.536
102 0.20
%
(m/s) (mass
2)
with T = 0.7 S
2.453 102 2.457
102 0.15
%
with T = 0.9 S
1.899 102 1.912
102 0.68
%
with T = 0.1 S
6.112 102 6.100
102 0.20
%
with T = 0.3 S
1.306 101 1.300
101 0.46
%
ACCE_X
with the node N2
with T = 0.5 S
1.571 101 1.600
101 1.85
%
(m/s2) (mass
1) with T = 0.7 S
5.657 102 5.800
102 2.53
%
with T = 0.9 S
1.124 101 1.130
101 0.53
%
with T = 0.1 S
1.562 102 1.618
102 3.58
%
with T = 0.3 S
6.031 102 6.223
102 3.18
%
ACCE_X
with the N3 node
with T = 0.5 S
5.102 102 5.374
102 5.33
%
(m/s2) (mass
2) with T = 0.7 S
7.428 102 7.043
102 5.19
%
with T = 0.9 S
2.364 101 2.263
101 4.28
%
Note:
Speed with the node N2 at the moment T = 0.5 S being relatively close to zero, the comparison
is realized for this case in absolute value.
4.2 Parameters
of execution
Version: STA5 (5.05)
Machine: CLASTER
Obstruction memory: 100 Mo
Time CPU To use: 9.05 seconds
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Code_Aster ®
Version
5.0
Titrate:
SDLD104 - Extrapolation of local measurements on a complete model
Date:
04/03/02
Author (S):
S. AUDEBERT, P. Key HERMANN
:
V2.01.104-A Page:
9/12
5 Modeling
B
5.1
Characteristics of modeling and the grids
·
Numerical grid:
The numerical grid is carried out directly with format ASTER. It comprises 4 nodes and 3
discrete meshs.
K
m K
m
K
X
N1
N2
N3
N4
(x=0.) (x=0.1) (x=0.2) (x=0.3)
·
Experimental grid:
The grid of measurement includes/understands only 2 specific elements and 2 nodes:
X
N2
N3
(x=0.12) (x=0.18)
5.2
Characteristic of measurements
Provided experimental measurements are:
·
With the N3 node:
The data are the displacements axial, multiplied by ( 2/2), and applied in the direction
X. The local orientation indicated in the command file is (45. 0. 0.)
The sampling of time is constant: initial time is 0 S, the step of time is 103 S and it
a many moments are 1001 (i.e until a final time of 1 S).
·
With the node N2:
The data are the axial displacements, applied in direction X.
The sampling of time is variable: every moment is indicated of 0 S to 1 S, by step of
103 S (1001 moments on the whole).
The values result from the analytical calculation carried out with Maple.
5.3
Characteristics of the modal base
The two only modes are stored in a concept of the type [base_modale], created by the command
DEFI_BASE_MODALE. The interface, of Craig-Bampton type, is placed on the degree of freedom in
displacement following X of the node N2 (corresponding to the mass m 1). The modal base thus contains
a dynamic mode (with blocked N2) and a static mode.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Code_Aster ®
Version
5.0
Titrate:
SDLD104 - Extrapolation of local measurements on a complete model
Date:
04/03/02
Author (S):
S. AUDEBERT, P. Key HERMANN
:
V2.01.104-A Page:
10/12
5.4 Functionalities
tested
Commands
AFFE_CARA_ELEM
DISCRET
CARA
'M_T_D_N
“K_T_D_L'
ORIENTATION
“ANGL_NAUT”
REPERE
“LOCAL”
AFFE_CHAR_MECA
DDL_IMPO
TOUT
NOEUD
AFFE_MODELE
TOUT
“OUI”
PHENOMENE
“MECANIQUE”
MODELISATION
“DIST_T'
ASSE_MATRICE
CALC_MATR_ELEM OPTION
“MASS_MECA”
“RIGI_MECA”
TRADITIONAL DEFI_BASE_MODALE
DEFI_INTERF_DYNA INTERFACES
MODE_ITER_SIMULT CALC_FREQ
OPTION
“BANDE”
NUME_DDL
NUME_DDL_GENE
PROJ_MATR_BASE
PROJ_MESU_MODAL
MESURE
REGULARIZATION
REST_BASE_PHYS TOUT_CHAM
“OUI”
TEST_RESU NOM_CHAM
“DEPL”
CRITERE
“VITE”
“ACCE”
“RELATIF”
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Code_Aster ®
Version
5.0
Titrate:
SDLD104 - Extrapolation of local measurements on a complete model
Date:
04/03/02
Author (S):
S. AUDEBERT, P. Key HERMANN
:
V2.01.104-A Page:
11/12
6
Results of modeling B
6.1 Values
tested
Identification Reference
Code_Aster difference
with T = 0.1 S
1.745 104 1.745
104 0.01
%
with T = 0.3 S
6.797 104 6.797
104 0.01
%
DEPL_X
with the node N2
with T = 0.5 S
1.217 103 1.217
103 0.01
%
(m) (mass
1)
with T = 0.7 S
5.214 104 5.214
104 0.01
%
with T = 0.9 S
9.031 104 9.031
104 0.00
%
with T = 0.1 S
9.154 106 9.154
106 0.00
%
with T = 0.3 S
6.414 104 6.414
104 0.00
%
DEPL_X
with the N3 node
with T = 0.5 S
8.636 104 8.636
104 0.00
%
(m) (mass
2)
with T = 0.7 S
1.107 104 1.107
104 0.03
%
with T = 0.9 S
1.633 103 1.633
103 0.02
%
with T = 0.1 S
4.586 103 4.616
103 0.65
%
with T = 0.3 S
7.598 103 7.663
103 0.85
%
VITE_X
with the node N2
with T = 0.5 S
1.581 104 8.000
105 7.81
105 m/s
(m/s) (mass
1)
with T = 0.7 S
9.382 103 9.354
103 0.30
%
with T = 0.9 S
7.481 103 7.537
103 0.75
%
with T = 0.1 S
4.328 104 4.405
104 1.79
%
with T = 0.3 S
3.671 103 3.640
103 0.84
%
VITE_X
with the N3 node
with T = 0.5 S
1.539 102 1.536
102 0.20
%
(m/s) (mass
2)
with T = 0.7 S
2.453 102 2.457
102 0.15
%
with T = 0.9 S
1.899 102 1.912
102 0.68
%
with T = 0.1 S
6.112 102 6.100
102 0.20
%
with T = 0.3 S
1.306 101 1.300
101 0.46
%
ACCE_X
with the node N2
with T = 0.5 S
1.571 101 1.600
101 1.85
%
(m/s2) (mass
1) with T = 0.7 S
5.657 102 5.800
102 2.53
%
with T = 0.9 S
1.124 101 1.130
101 0.53
%
with T = 0.1 S
1.562 102 1.618
102 3.58
%
with T = 0.3 S
6.031 102 6.223
102 3.18
%
ACCE_X
with the N3 node
with T = 0.5 S
5.102 102 5.374
102 5.33
%
(m/s2) (mass
2) with T = 0.7 S
7.428 102 7.043
102 5.19
%
with T = 0.9 S
2.364 101 2.263
101 4.28
%
Note:
Speed with the node N2 at the moment T = 0.5 S being relatively close to zero, the comparison
is realized for this case in absolute value.
6.2 Parameters
of execution
Version: STA5 (5.05)
Machine: CLASTER
Obstruction memory: 100 Mo
Time CPU To use: 9.24 seconds
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Code_Aster ®
Version
5.0
Titrate:
SDLD104 - Extrapolation of local measurements on a complete model
Date:
04/03/02
Author (S):
S. AUDEBERT, P. Key HERMANN
:
V2.01.104-A Page:
12/12
7
Summary of the results
For two modelings, the answers in displacement obtained after projection are identical
with the displacements of reference calculated analytically with Maple and provided in data.
Values speeds and the accelerations deduced from the displacements obtained after projection
are close to those obtained analytically. The weak noted variations are due to the errors
of approximation generated by the determination by a linear diagram in time speeds and
accelerations.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Outline document