Code_Aster ®
Version
4.0
Titrate:
SDLD29 Transitoire masses spring with 8 ddl
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.01.029-C Page:
1/6
Organization (S): EDF/EP/AMV
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
Document: V2.01.029
SDLD29 - Transitoire masses spring with 8 ddl
and viscous damping nonproportional
Summary:
This problem corresponds to a transitory analysis by modal recombination of a linear discrete system
constituted of 8 degrees of freedom. This system has a non-proportional damping. A force
transient of the crenel type is applied into 1 degree of freedom.
In this problem elements DISCRET with modal masses are tested (M_T_D_N), matrices of rigidity
(K_T_D_L) and matrices of damping (A_T_D_L) in a modeling.
The problem has a reference solution suggested by commission VPCS. Variations with
Code_Aster do not exceed 1,8%.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
Code_Aster ®
Version
4.0
Titrate:
SDLD29 Transitoire masses spring with 8 ddl
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.01.029-C Page:
2/6
1
Problem of reference
1.1 Geometry
U
U
U
2
U
1
3
8
B
With
m
m
m
m
X, U
K
K
K
K
P1
P
P
P
2
3
8
DC
C
C
Cd
Specific masses:
mP = m = m = ...... = m = m
1
P2
P3
P8
Stiffnesses of connection:
kAP1 = kP1P2 = kP2P3 = ...... = kP8B = K
Viscous damping:
cP1P2 = cP2P3 = ...... = CP7P8 = C
cAP1 = DC
CP8B = Cd
1.2
Material properties
Comes out from linear elastic translation
K =
105 NR/m
Specific mass
m =
10 kg
Damping of connection
C =
50 NR/(m/s)
DC =
250 NR/(m/s)
Cd =
25 NR/(m/s)
1.3
Boundary conditions and loadings
Embedded points A and B: U = 0
Loading: Force concentrated not periodical at the P4 point
Fx4
Not P4
Fx4 = F (T) 0 ² T ² 1s
F (T) = 1N = constant
T > 1s
F (T) = 0
1
0
1
T
1.4 Conditions
initial
For T = 0, in any point P
=
I: U = 0,
0.
dt
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
Code_Aster ®
Version
4.0
Titrate:
SDLD29 Transitoire masses spring with 8 ddl
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.01.029-C Page:
3/6
2
Reference solution
2.1
Method of calculation used for the reference solution
Numerical integration (approximate) by the direct method using a diagram of integration
numerical by finished differences, the step of time used must be sufficiently small to obtain one
sufficiently precise solution. With one of the diagrams used (method - Newmark improved), the step
appointed time was of 0.001s.
Method of - Newmark improved (NEWMARK NR. Mr., “A method off computation for structural
dynamics " proceeding ASCE J. Eng. Mech. Div E-3, July 1959, p 67-94) use the diagram of integration
according to:
1 [
M]
1
+
[] 1
C + [K] U
2
(n+2)
T
2t
3
1
=
[(P + + + +
+
-
N2]
[Pn 1] [Pn]) 2 [M] 1 [K] U
M
C
K
U
2
(n+1)
1 [] 1 [] 1 []
2
(N)
3
T
3
+ -
+
-
T
2t
3
Indices N, N + 1, N + 2 respectively indicate the calculations carried out at time tn,
T
= T + T
= T +
[] [] []
N +1
N
and tn +2
N
2t, where T is the increment of appointed time. M, C and K are
respectively the matrices masses, damping and stiffness, U
() is the vector displacement and P
()
the vector forces associated.
Point 4: displacement according to time
2.2
Results of reference
Displacement at the P4 point according to time, cf graph above.
2.3
Uncertainty on the solution
· position of the extremas: T < 0.015
· maximum amplitude: U/U < 0.5%
2.4 References
bibliographical
[1]
Card-index SDLD29/90 of commission VPCS
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A
Code_Aster ®
Version
4.0
Titrate:
SDLD29 Transitoire masses spring with 8 ddl
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.01.029-C Page:
4/6
3 Modeling
With
3.1
Characteristics of modeling
Discrete element of rigidity in translation
y
With
P
P
B
X
1
2
P3
P4
P5
P6
P7
P8
Characteristics of the elements
DISCRET:
with nodal masses
M_T_D_N
and matrices of rigidity
K_T_D_L
and matrices of damping
A_T_D_L
Limiting conditions:
in all the nodes
DDL_IMPO:
(TOUT:“YES” DY: 0. , DZ: 0. )
with the nodes ends
(GROUP_NO: AB DX: 0. )
Names of the nodes:
Not A = N1
P1 = N2
Not B = N10
P2 = N3
P8 = N9
Modal recombination with all the modes (8)
no time used
dt = 1.E3 S
diagram of EULER
3.2
Characteristics of the grid
A number of nodes:
10
A number of meshs and types:
9 SEG2
3.3 Functionalities
tested
Commands
Keys
AFFE_CARA_ELEM
DISCRET
GROUP_MA
“K_T_D_L'
[U4.24.01]
GROUP_NO
“M_T_D_N'
AFFE_CHAR_MECA
DDL_IMPO
TOUT
[U4.25.01]
GROUP_NO
AFFE_MODELE
TOUT
“MECANIQUE”
“DIS_T'
[U4.22.01]
GROUP_NO
“DIS_T'
MODE_ITER_INV
“AJUSTE”
[U4.52.01]
DYNA_TRAN_MODAL
[U4.54.03]
REST_BASE_PHYS
[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:
SDLD29 Transitoire masses spring with 8 ddl
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.01.029-C Page:
5/6
4
Results of modeling A
4.1 Values
tested
Time (S)
Reference
Aster
% Difference
0.09
3.97 E-5
3.95 E-5
0.503
0.18
5.10 E-6
5.03 E-5
1.38
0.27
3.77 E-5
3.77 E-5
0
0.36
7.30 E-6
7.28 E-6
0.293
0.45
3.59 E-5
3.59 E-5
0
0.54
8.81 E-6
8.77 E-6
0.486
0.63
3.47 E-5
3.47 E-5
0.034
0.72
1.01 E-5
1.00 E-5
0.514
0.81
3.36 E-5
3.36 E-5
0
0.91
1.11 E-5
1.14 E-5
2.36
0.99
3.27 E-5
3.26 E-5
0.171
4.2 Remarks
Contents of the file results: displacements.
4.3 Parameters
of execution
Version: NEW3.03
Machine: CRAY C90
System:
UNICOS 8.0
Obstruction memory:
8 megawords
Time CPU To use:
200 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:
SDLD29 Transitoire masses spring with 8 ddl
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.01.029-C Page:
6/6
5
Summary of the results
One obtains a relatively good agreement between the calculated solution and solution VPCS (<0.7%) except with
moment 0.91 (2.4%). The differences are primarily due to the fact that the moments of test are not
given that with 2 significant digits, which does not make it possible to seize the moment sufficiently well of
the extremum.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HI-75/98/040 - Ind A