Code_Aster ®
Version
4.0
Titrate:
SDLD102 transitory Sous structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B Page:
1/12
Organization (S): EDF/EP/AMV
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
Document: V2.01.102
SDLD102 - Transitory Sous structuring:
System 3 masses-4 springs
Summary:
The applicability of this test relates to the dynamics of the structures. It makes it possible to validate the diagram
of integration to step of adaptive time of operator DYNA_TRAN_MODAL [U4.54.03] as well as the calculation of
linear transitory response on a modal basis calculated by under-structuring (for the 4 diagrams
of integration of DYNA_TRAN_MODAL: “EULER”, “DEVOGE”, “NEWMARK” and “ADAPT”). In particular, the case of
the application of a damping reduced to the dynamic modes of the bases of projection of the substructures is
treaty.
It is a question of determining the transitory response of a system made up of 3 masses and 4 springs, embedded with its
ends and subjected to a constant force as from the initial moment. The springs are modelled by
elements of the type “DIS_TR” and masses by elements of the type “DIS_T'.
Three modelings are proposed. In the 2 first, the structure is not deadened. Methods of calculation
transient by under-structuring with interfaces of the type Craig-Bampton (“CRAIGB”) and Mac Neal (“MNEAL”) are
tested. The results of reference which are associated for them result from an analytical calculation. In the third,
one imposes a reduced damping of 1% on the dynamic modes of the bases of projection of
substructures. The transitory equation checked by the complete structure was obtained analytically. Its
resolution, which acts as reference, was carried out by the Maple software.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLD102 transitory Sous structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B Page:
2/12
1
Problem of reference
1.1 Geometry
The studied system is composed of 3 masses (m) and 4 springs (K). The unit is embedded with its
ends.
With
B
x3
x2
x1
1.2
Material properties
Stiffness of the springs: K = 1 NR/Mr.
Specific masses: m = 1 kg.
1.3
Boundary conditions and loadings
F
T
Embedded points A and B.
Application to the point x1 of a constant force F = 1 NR, as from the moment T = 0 S.
1.4 Conditions
initial
Structure initially at rest.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLD102 transitory Sous structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B Page:
3/12
2
Reference solution
2.1
Method of calculation used for the reference solution
2.1.1 Not deadened structure
In this case, the reference solution can be obtained analytically:
1 0 0 X
2 - 1 0 X
1
1
1





m 0 1 0 X + K - 1
2
- 1 X = 0
2
2







0
0
1 X
0 - 1 2 X
0
3
3
The own pulsations of the system mass-arises are worth:

K
K
K
2
2
2
=
-

=

=
+
1
(2 2)
2
2
3
(2 2)
m
m
m
respective modal deformations:
2
1
- 2





=

=

=
1
2
2
0
3
2






2
- 1
- 2
Projected on the basis of clean mode, the transitory equation becomes with like co-ordinates
I
generalized:
8 0 0
4 - 2 2 0
0

+ 2
1
1






m 0 2 0 + 4k
0
1
0
= 2
2
2








0
0
8


0
0
4 + 2 2
- 2
3
3
The system can be solved analytically. One obtains:



2
cos
2 (1 -
T1)
4


1

{(

1
1
T)} =

cos
2 (1 -
T
2)
2m2



2

cos
2 (
T
3
-)
1
4 3


The solution on physical basis is obtained by using the transformation of Ritz:
X


1
2
1
- 2


1
X (T)
= X


2
=
= 2
0
2 2



X

3
2 - 1
2 3
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLD102 transitory Sous structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B Page:
4/12
2.1.2 Structure
deadened
Damping is applied to the clean modes of the bases of projection of the substructures
embedded (reduced damping). In this case, one leads to the transitory equation in co-ordinates
generalized following (bib [1]):
8 0 0
3 - 2 2 0
- 1
1
1




m 0 2 0 + 4 2km
0
1
0

2
2





0
0
8

- 1
0
3 + 2 2

3
3
4 - 2 2 0
0

2
1




+ 4k
0
1
0
= 1
2






0
0
4 + 2 2
- 2
3
This system not being uncoupled, it was solved using the Maple software. One obtained
(= 0.01):
T


-


2

1 -
1

E
cos T

1
4 2


1


T

-

{(
1 1
2

T)}


1 - E
cos T
2m2
2
2


T
-

2 3

E
cos T
3 -

1
4 2


3

with:
3
1
1 =
1
100
1 6
. 5 10 S 2 =
=
and 3 =

4 8
. 5 10 S
2
2
One thus obtains a formulation close to the case not deadened, but in which intervene of
exponential terms which characterize damping.
The solution on physical basis is obtained by using the transformation of Ritz:
X


1
2
1
- 2


1
X (T)
= X


2
=
= 2
0
2 2



X

3
2 - 1
2 3
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLD102 transitory Sous structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B Page:
5/12
2.2
Results of reference
Not deadened structure:
Displacement, speed and acceleration of the node x2 at the moment T = 80 S:
-
X (
)
1
80
= 41700
.
10
m
2
-
-
X (
)
1
80
= ­ 4 3011
.
10
Mr. S 1
2
-
-
X
()
1
80
= 33749
.
10
Mr. S 2
2
Deadened structure:
Displacement of the node x2 at the moment T = 80 S:
-
X
1 m
2 80
() = 4.9867 10
2.3
Uncertainty on the solution
Case not deadened: analytical solution.
Deadened case: semi-analytical solution.
2.4 Reference
bibliographical
[1]
C. VARE - Rapport HP 61/95/025/A - “Mise in work of nonlinear transitory calculation by
under-structuring in Code_Aster ".
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLD102 transitory Sous structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B Page:
6/12
3 Modeling
With
3.1
Characteristics of modeling
The system is divided into 2 substructures:
m
m
m
Substructure 1:
blocking
Substructure 2:
blocking
K
K
NO1
NO3
K
K
NO1
NO3
In situation, the two substructures are connected to the level of the 2nd mass. The dynamic interface
1ère substructure consists of a mass m on the level of node NO3 of the grid and
coincide with the dynamic interface of the 2nd substructure which does not comprise any mass and is
simply blocked on the level of node NO1.
NO1
NO3 NO1
NO3
m
m
Substructure 1
Substructure 2
The clean modes of the complete system are calculated by using the method of calculation modal by
under-structuring with interfaces of the type “Craig-Bampton” (blocked interfaces). Bases of each
substructure are made up of a dynamic mode and a constrained mode.
The transitory response of the system is calculated on the modal basis calculated by under-structuring.
The steps of times used are equal to: 10­2 S in “EULER”, 10­2 S in “NEWMARK”, 10­2 S in
“DEVOGE”, 10-1 S in “ADAPT” (for this last, it acts of the step of initial time of the algorithm and of
no the maximum time of integration).
3.2
Characteristics of the grid of the substructure
A number of nodes: 3
A number of meshs and types: 2 SEG2
3.3 Functionalities
tested
Commands
Keys
NUME_DDL_GENE
BASE
[U4.55.07]
STOCKAGE
“DIAG”
PROJ_MATR_BASE
BASE
[U4.55.01]
NUME_DDL_GENE
MATR_ASSE_GENE
PROJ_VECT_BASE
BASE
[U4.55.02]
NUME_DDL_GENE
VECT_ASSE_GENE
DYNA_TRAN_MODAL
METHODE
“ADAPT”
[U4.54.03]
'EULER
“NEWMARK”
“DEVOGE”
REST_BASE_PHYS
MODE_MECA
[U4.64.01]
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLD102 transitory Sous structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B Page:
7/12
4
Results of modeling A
4.1 Values
tested
Calculation by modal recombination without under-structuring: Method “ADAPT”
Identification
Reference
Aster
% difference
Node x2, displacement (m)
4.1700 10­1
4.1695 10­1
Node x2, speed (Mr. s1)
­ 4.3011 10­1
­ 4.2972 10­1
< 1%
Node x2, acceleration (Mr. s2)
3.3749 10­1
3.3741 10­1
Calculation by under-structuring
Method: “EULER”
Node x2, displacement (m)
4.1700 10­1
4.1480 10­1
Node x2, speed (Mr. s1)
­ 4.3011 10­1
­ 4.2972 10­1
< 1%
Node x2, acceleration (Mr. s2)
3.3749 10­1
3.3823 10­1
Method: “DEVOGE”
Node x2, displacement (m)
4.1700 10­1
4.1700 10­1
Node x2, speed (Mr. s1)
­ 4.3011 10­1
­ 4.3011 10­1
< 1%
Node x2, acceleration (Mr. s2)
3.3749 10­1
4.3749 10­1
Method: “NEWMARK”
Node x2, displacement (m)
4.1700 10­1
4.1711 10­1
Node x2, speed (Mr. s1)
­ 4.3011 10­1
­ 4.3090 10­1
< 1%
Node x2, acceleration (Mr. s2)
3.3749 10­1
3.3763 10­1
Method: “ADAPT”
Node x2, displacement (m)
4.1700 10­1
4.1695 10­1
Node x2, speed (Mr. s1)
­ 4.3011 10­1
­ 4.2972 10­1
< 1%
Node x2, acceleration (Mr. s2)
3.3749 10­1
3.3741 10­1
4.2 Parameters
of execution
Version: 3.4.6
Machine: CRAY C90
Obstruction memory:
8 megawords
Time CPU To use:
39.1 seconds
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLD102 transitory Sous structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B Page:
8/12
5 Modeling
B
5.1
Characteristics of modeling
This modeling is identical to the precedent if they are only the clean modes of the complete system
are calculated by using the method of calculation modal per under-structuring with interfaces of the type
“Mac Neal” (free interfaces). The bases of each substructure are made up of a mode
dynamics and of a mode of fastener.
The transitory response of the system is calculated on the modal basis calculated by under-structuring.
More precisely, the studied substructures have their free interfaces:
Substructure 1:
Blocked NO1
Free NO3
Substructure 2:
Free NO1
Blocked NO3
The steps of times used are worth: 10­2 S in “EULER”, 10­2 S in “NEWMARK”, 10­2 S in “DEVOGE”,
10-2 S in “ADAPT”.
5.2
Characteristics of the grid of the substructure
A number of nodes: 3
A number of meshs and types: 2 SEG2
5.3 Functionalities
tested
Commands
Keys
NUME_DDL_GENE
BASE
[U4.55.07]
STOCKAGE
“DIAG”
PROJ_MATR_BASE
BASE
[U4.55.01]
NUME_DDL_GENE
MATR_ASSE_GENE
PROJ_VECT_BASE
BASE
[U4.55.02]
NUME_DDL_GENE
VECT_ASSE_
REST_BASE_PHYS
MODE_MECA
[U4.64.01]
DYNA_TRAN_MODAL
METHODE
“ADAPT”
[U4.54.03]
“EULER”
“NEWMARK”
“DEVOGE”
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLD102 transitory Sous structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B Page:
9/12
6
Results of modeling B
6.1 Values
tested
Identification
Reference
Aster
% difference
Method: “EULER”
Node x2, displacement (m)
4.1700 10­1
4.1480 10­1
Node x2, speed (Mr. s1)
­ 4.3011 10­1
­ 4.2972 10­1
< 1%
Node x2, acceleration (Mr. s2)
3.3749 10­1
3.3823 10­1
Method: “NEWMARK”
Node x2, displacement (m)
4.1700 10­1
4.1711 10­1
Node x2, speed (Mr. s1)
­ 4.3011 10­1
­ 4.3090 10­1
< 1%
Node x2, acceleration (Mr. s2)
3.3749 10­1
3.3763 10­1
Method: “DEVOGE”
Node x2, displacement (m)
4.1700 10­1
4.1700 10­1
Node x2, speed (Mr. s1)
­ 4.3011 10­1
­ 4.3011 10­1
< 1%
Node x2, acceleration (Mr. s2)
3.3749 10­1
4.3749 10­1
Method: “ADAPT”
Node x2, displacement (m)
4.1700 10­1
4.1695 10­1
Node x2, speed (Mr. s1)
­ 4.3011 10­1
­ 4.2973 10­1
< 1%
Node x2, acceleration (Mr. s2)
3.3749 10­1
3.3742 10­1
6.2 Parameters
of execution
Version: 3.4.6
Machine: CRAY C90
Obstruction memory:
8 megawords
Time CPU To use:
14.8 seconds
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLD102 transitory Sous structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B Page:
10/12
7 Modeling
C
7.1
Characteristics of modeling
The clean modes of the complete system are calculated by using the method of calculation modal by
under-structuring with interfaces of the type “Craig-Bampton” (blocked interfaces). Bases of each
substructure are made up of a dynamic mode and a constrained mode.
With the dynamic mode of each substructure a damping reduced with 1% is associated.
The transitory response of the deadened system is calculated on the modal basis calculated by
under-structuring.
The steps of time taken are equal to: 10­2 S in “ADAPT”, 10­2 S in “EULER”, 10­2 S in “NEWMARK”.
7.2
Characteristics of the grid of the substructure
A number of nodes: 3
A number of meshs and types: 2 SEG2
7.3 Functionalities
tested
Commands
Keys
MACR_ELEM_DYNA
AMOR_REDUIT
[U4.55.05]
NUME_DDL_GENE
BASE
[U4.55.07]
STOCKAGE
“PLEIN”
PROJ_MATR_BASE
BASE
[U4.55.01]
NUME_DDL_GENE
MATR_ASSE_GENE
PROJ_VECT_BASE
BASE
[U4.55.02]
NUME_DDL_GENE
VECT_ASSE_GENE
REST_BASE_PHYS
MODE_MECA
[U4.64.01]
DYNA_TRAN_MODAL
METHODE
“ADAPT”
[U4.54.03]
“EULER”
“NEWMARK”
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLD102 transitory Sous structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B Page:
11/12
8
Results of modeling C
8.1 Values
tested
Identification
Reference
Aster
% difference
Method: “EULER”
Node x2, displacement (m)
4.9867 10­1
4.9637 10­1
< 1%
Method: “NEWMARK”
Node x2, displacement (m)
4.9867 10­1
4.9883 10­1
< 1%
Method: “ADAPT”
Node x2, displacement (m)
4.9867 10­1
4.9863 10­1
< 1%
8.2 Parameters
of execution
Version: 3.4.6
Machine: CRAY C90
Obstruction memory:
8 megawords
Time CPU To use:
11.0 seconds
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLD102 transitory Sous structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B Page:
12/12
9
Summary of the results
The precision on displacement, the speed and the acceleration of the node x2 at the moment T = 80 S is
excellent (relative error < 1%).
This test thus validates the operators of calculation of transitory answer linear on calculated modal basis
by dynamic under-structuring (with and without damping), as well as the diagram of integration with
no the adaptive time of operator DYNA_TRAN_MODAL.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A