Code_Aster
®
Version
5.0
Titrate:
Seismic SDLD103 Response of a system 3 masses and 4 springs
Date:
30/08/01
Author (S):
Fe WAECKEL
Key
:
V2.01.103-B
Page:
1/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
Organization (S):
EDF/RNE/AMV
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
V2.01.103 document
SDLD103 - Seismic response of a system
3 masses and 4 springs multimedia
Summary
The problem consists in analyzing the response of a mechanical structure of embed-embedded beam type and
not deadened, modelized by a system 3 masses and 4 springs and subjected to a seismic loading
unspecified.
One tests the discrete element in traction and rotation, the calculation of the clean modes and the static modes and calculation
transitory response by modal superposition of a structure subjected to a accélérogramme of translation
(modeling A) or of rotation (modeling B).
The results obtained are in very good agreement with the results of reference (analytical results).
Code_Aster
®
Version
5.0
Titrate:
Seismic SDLD103 Response of a system 3 masses and 4 springs
Date:
30/08/01
Author (S):
Fe WAECKEL
Key
:
V2.01.103-B
Page:
2/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
1
Problem of reference
1.1 Geometry
The beam is modelized by a whole of 4 springs and 3 specific masses.
NO2
3
m
2
m
m 1
1
m
2
Z
1
X
K 1
2
3
4
K
K
K
NO3
NO4
NO5
NO1
1.2
Material properties
Stiffness of connection: K = K
1
= K
2
= K
3
= K
4
= 104 NR/m;
specific mass: m = m
1
= m
2
= m
3
= 10 kg.
1.3
Boundary conditions and loadings
Boundary conditions:
Only authorized displacements are the translations according to axis X.
Points NO1 and NO5 are embedded: dx = Dy = dz = drx = dry = drz = 0.
The other points are free in translation according to direction X: Dy = dz = drx = dry = drz = 0.
Loading:
The points of anchoring NO1 and NO5 each one are subjected to a transverse acceleration
()
1
2
T
At
=
with
has
= 2. 10
5
m/s
4
in NO1 and
()
2
T
= 0 m/s
2
out of NO5.
1.4 Conditions
initial
The system is at rest: with
()
()
T
dx
dx dt
=
=
=
0
0
0
0
0
,
,
/
in any point.
Code_Aster
®
Version
5.0
Titrate:
Seismic SDLD103 Response of a system 3 masses and 4 springs
Date:
30/08/01
Author (S):
Fe WAECKEL
Key
:
V2.01.103-B
Page:
3/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
2
Reference solution
2.1
Method of calculation used for the reference solution
The problem consists in calculating the response of a system to five degrees of freedom subjected to two
accelerations
()
1 T
and
()
2 T
distinct of an unspecified form. It is explained in detail in the reference
[bib2].
One calculates the Eigen frequencies initially F
I
, standardized associated clean vectors
compared to the modal mass
Ni
and static modes
system (analytical values). One
calculate then the generalized response of the system multimedia while solving analytically
the integral of Duhamel [bib1]. Lastly, one restores on the physical basis the vector of displacements
relative (on the active degrees of freedom)
X
R
, which allows us, after having calculated the vector of
displacements of drive
X
E
, to calculate the vector of absolute displacements
X
X
X
has
R
E
=
+
.
2.2
Results of reference
·
Calculation of the three Eigen frequencies F
I
, of the associated clean vectors standardized compared to
modal mass
Ni
and of the static modes
system
(
)
(
)
F
m/K
,
F
m/K
,
F
m/K
,
1
2
3
1
2
2
2
2
3 85
1
2
2
7 12
1
2
2
2
2
9 3
=
+
=
=
=
=
-
=
Hz
Hz
Hz
,
NR
m
=
-
-
-
1
2
2
1
2
2
1
2
1
0
1
and
=
1
4
3 1
2 2
1 3
.
·
Calculation of the generalized response of the multimedia system
The fundamental equation of dynamics, in the relative reference mark on the active degrees of freedom is written:
(
)
MR. X
K X
MR. M
X
&&
&&
R
R
S
+
=
+
xs
with
&&X
S
At
=
2
0
, the vector of the accelerations imposed on
level of the various points of anchoring.
The equation of the movement projected on the basis of dynamic mode standardized compared to
modal mass
NR
is written, by considering only the active degrees of freedom:
()
()
&&
&&
Q
K Q
M
4
G
T
S
T
T
m T has
NR
+
= -
= -
+
-
X
2
2
2
2
2
2
.
Code_Aster
®
Version
5.0
Titrate:
Seismic SDLD103 Response of a system 3 masses and 4 springs
Date:
30/08/01
Author (S):
Fe WAECKEL
Key
:
V2.01.103-B
Page:
4/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
The response of this linear system, at one moment T, then consists in calculating the integral of Duhamel:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
Q
cos
cos
cos
()
T
m has
K
T
T
m K
T
T
m K
T
T
m K
= -
+
+
+
-
+
-
-
+
-
-
3
2
1
2
2
2
3
4
3 2 2
2
2
1
1
2
3 2 2
2
2
1
.
·
Calculation of displacement relating to the active degrees of freedom:
X
R
Ni
I
Q
=
I
, that is to say:
(
)
(
) (
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
X
R
m has
K
T
m
K
T
T
T
m K
T
m
K
T
T
m K
T
m
K
T
T
T
m K
= -
+
+
+
+
-
+
+
+ -
+
+
+
-
+
-
8
7
10 7 2
1
1
10 7 2
1
8
10 2 14
1
10 2 14
1
5
10 7 2
1
1
10 7 2
1
2
1
2
3
2
1
3
2
1
2
3
cos
cos
cos
cos
cos
cos
cos
cos
.
·
Calculation of displacements of drive to the active degrees of freedom:
X
X
E
S
T has
=
=
4
48
3
2
1
.
·
Calculation of absolute displacements to the active degrees of freedom:
X
X
X
has
R
E
=
+
.
2.3
Uncertainty on the solution
No if one calculates the integral of Duhamel analytically [bib1].
2.4 References
bibliographical
[1]
J.S. PRZEMIENIECKI: Theory off matrix structural analysis. New York, Mac Graw-Hill, 1968,
pages 351-357.
[2]
Fe WAECKEL: Documentations use and validation of the developments carried out for
to calculate the seismic response of multimedia structures. HP-52/96/002
Code_Aster
®
Version
5.0
Titrate:
Seismic SDLD103 Response of a system 3 masses and 4 springs
Date:
30/08/01
Author (S):
Fe WAECKEL
Key
:
V2.01.103-B
Page:
5/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
3 Modeling
With
3.1
Characteristics of modeling
The elements are modelized by discrete elements with 3 degrees of freedom
DIS_T
.
2
Z
1
X
K 1
2
3
4
K
K
K
NO1
NO2
NO3
NO4
NO5
Node NO1 is subjected to an imposed acceleration
()
1 T
, node NO5 with
()
2 T
. It is calculated
relative displacement of nodes NO2, NO3 and NO4 compared to their static deformation, them
displacement of drive and their absolute displacement.
Temporal integration is carried out with the algorithms of Euler (not of time: 10
- 3
second), of
Devogelaere (not of time: 10
- 3
second) and with an algorithm with pitch of adaptive time.
3.2
Characteristics of the mesh
The mesh consists of 5 nodes and 4 discrete elements (
DIST_T
).
3.3 Functionalities
tested
Controls
Keys Doc. V5
AFFE_MODELE GROUP_MA
“MECHANICAL”
“DIS_T'
[U4.41.01]
DISCRETE AFFE_CARA_ELEM
NODE
M_T_D_N
[U4.42.01]
NET
K_T_D_L
AFFE_CHAR_MECA DDL_IMPO
[U4.44.01]
MACRO_MATR_ASSE
[U4.61.21]
MODE_ITER_INV CALC_FREQ
ADJUST
[U4.52.04]
CALC_FONC_INTERP
[U4.32.01]
MODE_STATIQUE DDL_IMPO
[U4.52.14]
CALC_CHAR_SEISME NODE
[U4.63.01]
MACRO_PROJ_BASE
[U4.63.11]
DYNA_TRAN_MODAL EXCIT
MULT_APPUI
“YES”
[U4.53.21]
METHOD
EULER
DEVOGE
ADAPT
REST_BASE_PHYS MULT_APPUI
“YES”
[U4.63.21]
MULT_APPUI
“NOT”
RECU_FONCTION RESU_GENE
[U4.32.03]
Code_Aster
®
Version
5.0
Titrate:
Seismic SDLD103 Response of a system 3 masses and 4 springs
Date:
30/08/01
Author (S):
Fe WAECKEL
Key
:
V2.01.103-B
Page:
6/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
4
Results of modeling A
4.1
Values tested of modeling A
4.1.1 Displacements
relative
nodes NO2, NO3 and NO4
·
Relative displacements of node NO2 with the numerical algorithm of integration of Euler:
Time (S)
Reference
Code_Aster Error
(%)
0,1 8,47734E01
8,47725E01
0,001
0,3 1,55202E+01
1,55201E+01
0
0,5 4,36449E+01
4,36450E+01
0
0,7 8,50830E+01
8,50832E+01
0
1,0 1,74790E+02
1,74790E+02
0
·
Relative displacements of node NO2 with the numerical algorithm of integration of Devogelaere:
Time (S)
Reference
Code_Aster Error
(%)
0,1 8,47734E01
8,47734E01
0
0,3 1,55202E+01
1,55202E+01
0
0,5 4,36449E+01
4,36449E+01
0
0,7 8,50830E+01
8,50830E+01
0
1,0 1,74790E+02
1,74790E+02
0
·
Displacements relating of node NO2 with the numerical algorithm of integration to pitch of time
adaptive:
Time (S)
Reference
Code_Aster Error
(%)
0,1 8,47734E01
8,47761E01
0,003
0,3 1,55202E+01
1,55201E+01
0
0,5 4,36449E+01
4,36450E+01
0
0,7 8,50830E+01
8,50832E+01
0
1,0 1,74790E+02
1,74790E+02
0
·
Relative displacements of node NO3 with the numerical algorithm of integration of Euler:
Time (S)
Reference
Code_Aster Error
(%)
0,01 9,87666E10
7,32629E05
0
*
0,02 2,49501E07
1,46526E04
0
*
0,03 6,25468E06
2,19789E04
0
*
0,04 6,05829E05
2,93052E04
0
*
0,05 3,47191E04
3,66314E04
0
*
0,06 1,42349E03
1,32757E02
0,012
*
0,07 4,62144E03
2,61852E02
0,022
*
0,08 1,26245E02
3,90946E02
0,026
*
0,09 3,01825E02
5,20040E02
0,022
*
0,1 7,68449E01
7,68420E01
0,004
0,3 1,76923E+01
1,76922E+01
0
0,5 4,99310E+01
4,99311E+01
0
0,7 9,70711E+01
9,70714E+01
0
1,0 1,99722E+02
1,99722E+02
0
* absolute error
Code_Aster
®
Version
5.0
Titrate:
Seismic SDLD103 Response of a system 3 masses and 4 springs
Date:
30/08/01
Author (S):
Fe WAECKEL
Key
:
V2.01.103-B
Page:
7/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
·
Relative displacements of node NO3 with the numerical algorithm of integration of Devogelaere:
Time (S)
Reference
Code_Aster Error
(%)
0,1 7,68449E01
7,68449E01
0
0,3 1,76923E+01
1,76923E+01
0
0,5 4,99310E+01
4,99310E+01
0
0,7 9,70711E+01
9,70711E+01
0
1,0 1,99722E+02
1,99722E+02
0
·
Displacements relating of node NO3 with the numerical algorithm of integration to pitch of time
adaptive:
Time (S)
Reference
Code_Aster Error
(%)
0,1 7,68449E01
7,68462E01
0,002
0,3 1,76923E+01
1,76922E+01
0
0,5 4,99310E+01
4,99311E+01
0
0,7 9,70711E+01
9,70715E+01
0
1,0 1,99722E+02
1,99722E+02
0
·
Relative displacements of node NO4 with the numerical algorithm of integration of Euler:
Time (S)
Reference
Code_Aster Error
(%)
0,1 4,09632E01
4,09604E01
0,007
0,3 1,10372E+01
1,10371E+01
0
0,5 3,12415E+01
3,12416E+01
0
0,7 6,05833E+01
6,05835E+01
0
1,0 1,24803E+02
1,24804E+02
0
·
Relative displacements of node NO4 with the numerical algorithm of integration of Devogelaere:
Time (S)
Reference
Code_Aster Error
(%)
0,1 4,09632E01
4,09632E01
0
0,3 1,10372E+01
1,10372E+01
0
0,5 3,12415E+01
3,12415E+01
0
0,7 6,05833E+01
6,05833E+01
0
1,0 1,24803E+02
1,24803E+02
0
·
Displacements relating of node NO4 with the numerical algorithm of integration to pitch of time
adaptive:
Time (S)
Reference
Code_Aster Error
(%)
0,1 4,09632E01
4,09630E01
0
0,3 1,10372E+01
1,10371E+01
0
0,5 3,12415E+01
3,12416E+01
0
0,7 6,05833E+01
6,05835E+01
0
1,0 1,24803E+02
1,24804E+02
0
4.1.2 Absolute displacements of nodes NO2, NO3 and NO4
·
Absolute displacements of node NO2 with the numerical algorithm of integration of Euler:
Time (S)
Reference
Code_Aster Error
(%)
0,1 4,02266E01
4,02275E01
0,002
0,3 8,57298E+01
8,57299E+01
0
0,5 7,37605E+02
7,37605E+02
0
0,7 2,91617E+03
2,91617E+03
0
1,0 1,23252E+04
1,23252E+04
0
Code_Aster
®
Version
5.0
Titrate:
Seismic SDLD103 Response of a system 3 masses and 4 springs
Date:
30/08/01
Author (S):
Fe WAECKEL
Key
:
V2.01.103-B
Page:
8/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
·
Absolute displacements of node NO2 with the numerical algorithm of integration of Devogelaere:
Time (S)
Reference
Code_Aster Error
(%)
0,1 4,02266E01
4,02266E01
0
0,3 8,57298E+01
8,57298E+01
0
0,5 7,37605E+02
7,37605E+02
0
0,7 2,91617E+03
2,91617E+03
0
1,0 1,23252E+04
1,23252E+04
0
·
Absolute displacements of node NO2 with the numerical algorithm of integration to pitch of time
adaptive:
Time (S)
Reference
Code_Aster Error
(%)
0,1 4,02266E01
4,02239E01
0,007
0,3 8,57298E+01
8,57299E+01
0
0,5 7,37605E+02
7,37605E+02
0
0,7 2,91617E+03
2,91617E+03
0
1,0 1,23252E+04
1,23252E+04
0
·
Absolute displacements of node NO3 with the numerical algorithm of integration of Euler:
Time (S)
Reference
Code_Aster Error
(%)
0,1 6,48847E02
6,49134E02
0,044
0,3 4,98077E+01
4,98078E+01
0
0,5 4,70902E+02
4,70902E+02
0
0,7 1,90376E+03
1,90376E+03
0
1,0 8,13361E+03
8,13361E+03
0
·
Absolute displacements of node NO3 with the numerical algorithm of integration of Devogelaere:
Time (S)
Reference
Code_Aster Error
(%)
0,1 6,48847E02
6,48847E02
0
0,3 4,98077E+01
4,98077E+01
0
0,5 4,70902E+02
4,70902E+02
0
0,7 1,90376E+03
1,90376E+03
0
1,0 8,13361E+03
8,13361E+03
0
·
Absolute displacements of node NO3 with the numerical algorithm of integration to pitch of time
adaptive:
Time (S)
Reference
Code_Aster Error
(%)
0,1 6,48847E02
6,48714E02
0,021
0,3 4,98077E+01
4,98078E+01
0
0,5 4,70902E+02
4,70902E+02
0
0,7 1,90376E+03
1,90376E+03
0
1,0 8,13361E+03
8,13361E+03
0
·
Absolute displacements of node NO4 with the numerical algorithm of integration of Euler:
Time (S)
Reference
Code_Aster Error
(%)
0,1 7,03506E-03
7,06261E-03
0
0,3 2,27128E+01
2,27129E+01
0
0,5 2,29175E+02
2,29175E+02
0
0,7 9,39833E+02
9,39833E+02
0
1,0 4,04186E+03
4,04186E+03
0
Code_Aster
®
Version
5.0
Titrate:
Seismic SDLD103 Response of a system 3 masses and 4 springs
Date:
30/08/01
Author (S):
Fe WAECKEL
Key
:
V2.01.103-B
Page:
9/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
·
Absolute displacements of node NO4 with the numerical algorithm of integration of Devogelaere:
Time (S)
Reference
Code_Aster Error
(%)
0,1 7,03506E03
7,03504E03
0
0,3 2,27128E+01
2,27128E+01
0
0,5 2,29175E+02
2,29175E+02
0
0,7 9,39833E+02
9,39833E+02
0
1,0 4,04186E+03
4,04186E+03
0
·
Absolute displacements of node NO4 with the numerical algorithm of integration to pitch of time
adaptive:
Time (S)
Reference
Code_Aster Error
(%)
0,1 7,03506E03
7,03655E03
0
0,3 2,27128E+01
2,27129E+01
0
0,5 2,29175E+02
2,29175E+02
0
0,7 9,39833E+02
9,39833E+02
0
1,0 4,04186E+03
4,04186E+03
0
4.2 Parameters
of execution
Version: STA 5.02
Machine: SGI Origin 2000
Time CPU to use: 16,4 seconds
Code_Aster
®
Version
5.0
Titrate:
Seismic SDLD103 Response of a system 3 masses and 4 springs
Date:
30/08/01
Author (S):
Fe WAECKEL
Key
:
V2.01.103-B
Page:
10/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
5 Modeling
B
It is same modeling as the preceding one except for loading which is one
accélérogramme of rotation.
5.1
Characteristics of modeling
The elements are modelized by discrete elements with 3 degrees of freedom
DIS_T
.
2
Z
1
X
K 1
2
3
4
K
K
K
NO1
NO2
NO3
NO4
NO5
Node NO1 is subjected to an imposed acceleration
1
(T), node NO5 with
2
(T). It is calculated
relative displacement of nodes NO2, NO3 and NO4 compared to their static deformation, them
displacement of drive and their absolute displacement.
Temporal integration is carried out with the algorithm of Euler (not of time: 10
3
second).
5.2
Characteristics of the mesh
The mesh consists of 5 nodes and 4 discrete elements (
DIST_TR
).
5.3 Functionalities
tested
Controls
Keys Doc. V5
AFFE_MODELE GROUP_MA
“MECHANICAL”
“DIS_T'
[U4.41.01]
DISCRETE AFFE_CARA_ELEM NODE M_TR_D_N
[U4.42.01]
NET
K_TR_D_L
AFFE_CHAR_MECA DDL_IMPO
[U4.44.01]
MACRO_MATR_ASSE
[U4.61.21]
MODE_ITER_INV CALC_FREQ
ADJUST
[U4.52.04]
CALC_FONC_INTERP
[U4.32.01]
MODE_STATIQUE DDL_IMPO
[U4.52.14]
CALC_CHAR_SEISME NODE
[U4.63.01]
MACRO_PROJ_BASE
[U4.63.11]
DYNA_TRAN_MODAL EXCIT
MULT_APPUI
“YES”
[U4.53.21]
METHOD
EULER
REST_BASE_PHYS MULT_APPUI
“YES”
[U4.63.21]
MULT_APPUI
“NOT”
RECU_FONCTION RESU_GENE
[U4.32.03]
Code_Aster
®
Version
5.0
Titrate:
Seismic SDLD103 Response of a system 3 masses and 4 springs
Date:
30/08/01
Author (S):
Fe WAECKEL
Key
:
V2.01.103-B
Page:
11/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
6
Results of modeling B
6.1
Values tested of modeling B
6.1.1 Displacements
relative
nodes NO2, NO3 and NO4
·
Relative displacements of node NO2:
Time (S)
Reference
Code_Aster Error
(%)
0,1 8,47734E01
8,47725E01
0,001
0,3 1,55202E+01
1,55201E+01
0
0,5 4,36449E+01
4,36450E+01
0
0,7 8,50830E+01
8,50832E+01
0
1,0 1,74790E+02
1,74790E+02
0
·
Relative displacements of node NO3:
Time (S)
Reference
Code_Aster Error
(%)
0,1 7,68449E01
7,68420E01
0,004
0,3 1,76923E+01
1,76922E+01
0
0,5 4,99310E+01
4,99311E+01
0
0,7 9,70711E+01
9,70714E+01
0
1,0 1,99722E+02
1,99722E+02
0
·
Relative displacements of node NO4:
Time (S)
Reference
Code_Aster Error
(%)
0,1 4,09632E01
4,09604E01
0,007
0,3 1,10372E+01
1,10371E+01
0
0,5 3,12415E+01
3,12416E+01
0
0,7 6,05833E+01
6,05835E+01
0
1,0 1,24803E+02
1,24804E+02
0
6.1.2 Absolute displacements of nodes NO2, NO3 and NO4
·
Absolute displacements of node NO2:
Time (S)
Reference
Code_Aster Error
(%)
0,1 4,02266E01
4,02275E01
0,002
0,3 8,57298E+01
8,57299E+01
0
0,5 7,37605E+02
7,37605E+02
0
0,7 2,91617E+03
2,91617E+03
0
1,0 1,23252E+04
1,23252E+04
0
Code_Aster
®
Version
5.0
Titrate:
Seismic SDLD103 Response of a system 3 masses and 4 springs
Date:
30/08/01
Author (S):
Fe WAECKEL
Key
:
V2.01.103-B
Page:
12/12
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-62/01/012/A
·
Absolute displacements of node NO3:
Time (S)
Reference
Code_Aster Error
(%)
0,01 9,87666E10
7,32627E05
0
*
0,02 2,49501E07
1,46525E04
0
*
0,03 6,25468E06
2,19788E04
0
*
0,04 6,05829E05
2,93051E04
0
*
0,05 3,47191E04
3,66313E04
0
*
0,06 1,42349E03
1,32757E02 0,012
*
0,07 4,62144E03
2,61852E02 0,022
*
0,08 1,26245E02
3,90946E02 0,026
*
0,09 3,01825E02
5,20040E02 0,022
*
0,10 6,48847E02
6,49134E02 0,044
0,30 4,98077E+01
4,98078E+01 0
0,50 4,70902E+02
4,70902E+02 0
0,70 1,90376E+03
1,90376E+03 0
1,0 8,13361E+03
8,13361E+03
0
* absolute error
·
Absolute displacements of node NO4:
Time (S)
Reference
Code_Aster Error
(%)
0,1 7,03506E-03
7,06264E-03 0
0,3 2,27128E+01
2,27129E+01 0
0,5 2,29175E+02
2,29175E+02 0
0,7 9,39833E+02
9,39833E+02 0
1,0 4,04186E+03
4,04186E+03 0
6.2 Parameters
of execution
Version: STA 5.02
Machine: Sgi Origin 2000
Time CPU to use: 14,3 seconds
7
Summary of the results
The results obtained with Code_Aster are in conformity with the results of reference (the error is in
general lower than 0,03%).