Code_Aster
®
Version
4.0
Titrate:
SDLD102 Under transitory structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B
Page:
1/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
Organization (S):
EDF/EP/AMV
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
Document: V2.01.102
SDLD102 - Under transitory 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 pitch of adaptive time of the 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 modelized 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.
Code_Aster
®
Version
4.0
Titrate:
SDLD102 Under transitory structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B
Page:
2/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
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.
Code_Aster
®
Version
4.0
Titrate:
SDLD102 Under transitory structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B
Page:
3/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
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:
m
X
X
X
K
X
X
X
1
0
0
0
1
0
0
0
1
2
1
0
1
2
1
0
1
2
1
0
0
1
2
3
1
2
3
+
-
-
-
-
=
The own pulsations of the system mass-arises are worth:
(
)
(
)
1
2
2
2
3
2
2
2
2
2
2
=
-
=
=
+
K
m
K
m
K
m
respective modal deformations:
1
2
3
2
2
2
1
0
1
2
2
2
=
=
-
=
-
-
Projected on the basis of clean mode, the transitory equation becomes
I
with like co-ordinates
generalized:
m
K
8
0
0
0
2
0
0
0
8
4
4
2 2
0
0
0
1
0
0
0
4
2 2
2
2
2
1
2
3
1
2
3
+
-
+
=
+
-
The system can be solved analytically. One obtains:
()
{}
(
)
(
)
(
)
T
m
T
T
T
=
-
-
-
1
2
2
4
1
1 1
2
4
1
1
2
1
2
2
2
3
2
3
cos
cos
cos
The solution on physical basis is obtained by using the transformation of Ritz:
()
X T
X
X
X
=
=
=
-
-
1
2
3
1
2
3
2
1
2
2
0
2
2
1
2
Code_Aster
®
Version
4.0
Titrate:
SDLD102 Under transitory structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B
Page:
4/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
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]):
m
km
K
8
0
0
0
2
0
0
0
8
4
2
3
2 2
0
1
0
1
0
1
0
3
2 2
4
4
2 2
0
0
0
1
0
0
0
4
2 2
2
1
2
1
2
3
1
2
3
1
2
3
+
-
-
-
+
+
-
+
=
-
This system not being uncoupled, it was solved using the Maple software. One obtained
=
0.01
(
)
:
()
{}
T
m
E
T
E
T
E
T
S
S
T
T
T
-
-
-
=
=
=
=
-
-
-
1
2
2
4
1
1
1
2
4
1
1 65 10
1
100
2
4 85 10
1
2
1
2
2
2
3
2
3
1
3
2
2
3
1
1
2
3
cos
cos
cos
.
.
with:
and
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 T
X
X
X
=
=
=
-
-
1
2
3
1
2
3
2
1
2
2
0
2
2
1
2
Code_Aster
®
Version
4.0
Titrate:
SDLD102 Under transitory structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B
Page:
5/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
2.2
Results of reference
Not deadened structure:
Displacement, speed and acceleration of the node
X
2
at the moment T = 80 S:
()
()
()
X
m
X
m S
X
m S
2
1
2
1
1
2
1
2
80
4 1700 10
80
4 3011 10
80
3 3749 10
=
=
=
-
-
-
-
-
.
.
.
.
.
Deadened structure:
Displacement of the node
X
2
at the moment T = 80 S:
X
2
80
()
=
4.9867 10
-
1
m
2.3
Uncertainty on the solution
Case not deadened: analytical solution.
Deadened case: semi-analytical solution.
2.4 Reference
bibliographical
[1]
C. VARE - Report/ratio HP 61/95/025/A - “Implementation of nonlinear transitory calculation by
under-structuring in
Code_Aster
“.
Code_Aster
®
Version
4.0
Titrate:
SDLD102 Under transitory structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B
Page:
6/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
3 Modeling
With
3.1
Characteristics of modeling
The system is divided into 2 substructures:
Substructure 1:
m
K
m
K
NO1
NO3
blocking
m
K
K
Substructure 2:
NO1
NO3
blocking
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 mesh and
coincide with the dynamic interface of the 2
ème
substructure which does not comprise any mass and is
simply locked on the level of node NO1.
Substructure 1
m
m
NO1
NO3 NO1
NO3
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” (locked 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 pitches of times used are equal to: 10
2
S in
“EULER”
, 10
2
S in
“NEWMARK”
, 10
2
S in
“DEVOGE”
, 10
- 1
S in
“ADAPT”
(for this last, it is about the pitch of initial time of the algorithm and of
no the maximum time of integration).
3.2
Characteristics of the mesh of the substructure
A number of nodes: 3
A number of meshs and types: 2 SEG2
3.3 Functionalities
tested
Controls
Keys
NUME_DDL_GENE
BASE
[U4.55.07]
STORAGE
“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
METHOD
“ADAPT”
[U4.54.03]
'EULER
“NEWMARK”
“DEVOGE”
REST_BASE_PHYS
MODE_MECA
[U4.64.01]
Code_Aster
®
Version
4.0
Titrate:
SDLD102 Under transitory structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B
Page:
7/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
4
Results of modeling A
4.1 Values
tested
Calculation by modal recombination without under-structuring: Method “ADAPT”
Identification
Reference
Aster
% difference
Node
X
2
, displacement (m)
4.1700 10
1
4.1695 10
1
Node
X
2
, speed (Mr. S
1
)
4.3011 10
1
4.2972 10
1
< 1%
Node
X
2
, acceleration (Mr. S
2
)
3.3749 10
1
3.3741 10
1
Calculation by under-structuring
Method: “EULER”
Node
X
2
, displacement (m)
4.1700 10
1
4.1480 10
1
Node
X
2
, speed (Mr. S
1
)
4.3011 10
1
4.2972 10
1
< 1%
Node
X
2
, acceleration (Mr. S
2
)
3.3749 10
1
3.3823 10
1
Method: “DEVOGE”
Node
X
2
, displacement (m)
4.1700 10
1
4.1700 10
1
Node
X
2
, speed (Mr. S
1
)
4.3011 10
1
4.3011 10
1
< 1%
Node
X
2
, acceleration (Mr. S
2
)
3.3749 10
1
4.3749 10
1
Method: “NEWMARK”
Node
X
2
, displacement (m)
4.1700 10
1
4.1711 10
1
Node
X
2
, speed (Mr. S
1
)
4.3011 10
1
4.3090 10
1
< 1%
Node
X
2
, acceleration (Mr. S
2
)
3.3749 10
1
3.3763 10
1
Method: “ADAPT”
Node
X
2
, displacement (m)
4.1700 10
1
4.1695 10
1
Node
X
2
, speed (Mr. S
1
)
4.3011 10
1
4.2972 10
1
< 1%
Node
X
2
, acceleration (Mr. S
2
)
3.3749 10
1
3.3741 10
1
4.2 Parameters
of execution
Version: 3.4.6
Machine: CRAY C90
Overall dimension memory:
8 megawords
Time CPU To use:
39.1 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLD102 Under transitory structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B
Page:
8/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
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:
Locked NO1
Free NO3
Substructure 2:
Free NO1
Locked NO3
Substructure 1:
The pitches of times used are worth: 10
2
S in
“EULER”
, 10
2
S in
“NEWMARK”
, 10
2
S in
“DEVOGE”
,
10
- 2
S in
“ADAPT”
.
5.2
Characteristics of the mesh of the substructure
A number of nodes: 3
A number of meshs and types: 2 SEG2
5.3 Functionalities
tested
Controls
Keys
NUME_DDL_GENE
BASE
[U4.55.07]
STORAGE
“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
METHOD
“ADAPT”
[U4.54.03]
“EULER”
“NEWMARK”
“DEVOGE”
Code_Aster
®
Version
4.0
Titrate:
SDLD102 Under transitory structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B
Page:
9/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
6
Results of modeling B
6.1 Values
tested
Identification
Reference
Aster
% difference
Method: “EULER”
Node
X
2
, displacement (m)
4.1700 10
1
4.1480 10
1
Node
X
2
, speed (Mr. S
1
)
4.3011 10
1
4.2972 10
1
< 1%
Node
X
2
, acceleration (Mr. S
2
)
3.3749 10
1
3.3823 10
1
Method: “NEWMARK”
Node
X
2
, displacement (m)
4.1700 10
1
4.1711 10
1
Node
X
2
, speed (Mr. S
1
)
4.3011 10
1
4.3090 10
1
< 1%
Node
X
2
, acceleration (Mr. S
2
)
3.3749 10
1
3.3763 10
1
Method: “DEVOGE”
Node
X
2
, displacement (m)
4.1700 10
1
4.1700 10
1
Node
X
2
, speed (Mr. S
1
)
4.3011 10
1
4.3011 10
1
< 1%
Node
X
2
, acceleration (Mr. S
2
)
3.3749 10
1
4.3749 10
1
Method: “ADAPT”
Node
X
2
, displacement (m)
4.1700 10
1
4.1695 10
1
Node
X
2
, speed (Mr. S
1
)
4.3011 10
1
4.2973 10
1
< 1%
Node
X
2
, acceleration (Mr. S
2
)
3.3749 10
1
3.3742 10
1
6.2 Parameters
of execution
Version: 3.4.6
Machine: CRAY C90
Overall dimension memory:
8 megawords
Time CPU To use:
14.8 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLD102 Under transitory structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B
Page:
10/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
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” (locked 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 pitches of time taken are equal to: 10
2
S in
“ADAPT”
, 10
2
S in
“EULER”
, 10
2
S in
“NEWMARK”
.
7.2
Characteristics of the mesh of the substructure
A number of nodes: 3
A number of meshs and types: 2 SEG2
7.3 Functionalities
tested
Controls
Keys
MACR_ELEM_DYNA
AMOR_REDUIT
[U4.55.05]
NUME_DDL_GENE
BASE
[U4.55.07]
STORAGE
“FULL”
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
METHOD
“ADAPT”
[U4.54.03]
“EULER”
“NEWMARK”
Code_Aster
®
Version
4.0
Titrate:
SDLD102 Under transitory structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B
Page:
11/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
8
Results of modeling C
8.1 Values
tested
Identification
Reference
Aster
% difference
Method: “EULER”
Node
X
2
, displacement (m)
4.9867 10
1
4.9637 10
1
< 1%
Method: “NEWMARK”
Node
X
2
, displacement (m)
4.9867 10
1
4.9883 10
1
< 1%
Method: “ADAPT”
Node
X
2
, displacement (m)
4.9867 10
1
4.9863 10
1
< 1%
8.2 Parameters
of execution
Version: 3.4.6
Machine: CRAY C90
Overall dimension memory:
8 megawords
Time CPU To use:
11.0 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLD102 Under transitory structuring
Date:
01/09/99
Author (S):
G. ROUSSEAU, C. VARE
Key:
V2.01.102-B
Page:
12/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
9
Summary of the results
Precision on displacement, the speed and the acceleration of the node
X
2
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 the operator
DYNA_TRAN_MODAL
.