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Titrate:
SDLD21 - Système mass-arises to 8 ddl with viscous damping device
Date:
23/06/03
Author (S):
O. NICOLAS
Key: V2.01.021-C Page:
1/10
Organization (S): EDF-R & D/AMA
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
V2.01.021 document
SDLD21 - Système mass-arises to 8 ddl
with viscous damping device
Summary:
This one-way problem consists in carrying out a harmonic analysis of a mechanical structure
composed of a whole of mass-springs with damping devices viscous and subjected to an excitation
sinusoidal. This test of mechanics of the structures corresponds to a dynamic analysis of a discrete model
having a linear behavior. It includes/understands three modelings.
Via this problem, one tests the discrete elements in translation (mass, arises, damping device),
the definition of a force of specific excitation harmonic, the operator of calculation modal (MODE_ITER_SIMULT
[U4.52.03]) into quadratic and the operator of harmonic calculation of answer (DYNA_LINE_HARM [U4.54.02]).
In addition, several operators of postprocessing are tested: RECU_FONCTION [U4.62.03], TEST_FONCTION
[U4.72.02], RECU_CHAMP [U4.62.01].
Results obtained (field of displacement, speed and acceleration for various frequencies of excitation)
are in concord with the results of guide VPCS.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.0
Titrate:
SDLD21 - Système mass-arises to 8 ddl with viscous damping device
Date:
23/06/03
Author (S):
O. NICOLAS
Key: V2.01.021-C Page:
2/10
1
Problem of reference
1.1 Geometry
U1
U2
U3
U8
K
K
K
K
With
m
m
m
m
B
X, U
P1
P2
P3
P8
C
C
C
C
Specific masses:
mP = m = m = ...... = m = m
1
P2
P3
P8
Stiffnesses of connection:
kAP1 = kP1P2 = kP2P3 = ...... = kP8B = K
Viscous damping:
cAP1 = cP1P2 = cP2P3 = ...... = cP8B = C
1.2
Material properties
Comes out from linear elastic translation
K =
105 NR/m
Specific mass
m =
10 kg
One-way viscous damping
C =
50 NR/(m/s)
1.3
Boundary conditions and loadings
Boundary conditions:
Embedded points A and b: (U = 0).
Loading: Force concentrated sinusoidal of variable frequency at the P4 point
Not P
=
4
Fx
F
4
0 sin T
= 2 F 5 Hz F 40 Hz
F0 = constant = 1 NR
Other points P
F = 0
I
xi
1.4 Conditions
initial
Without object for the study of the permanent harmonic mode.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.0
Titrate:
SDLD21 - Système mass-arises to 8 ddl with viscous damping device
Date:
23/06/03
Author (S):
O. NICOLAS
Key: V2.01.021-C Page:
3/10
2
Reference solution
2.1
Method of calculation used for the reference solution
The system of differential equations of the second command coupled is form:
M u&+ C U & + K U = F
2 - 1
10
- 1 2
- 1
10
-
with M
1
2
.
=
.
C = 50
.
.
.
10
10
.
.
- 1
- 1 2
2 - 1
- 1 2
- 1
5
-
+
1
2
.
K = 10
.
.
.
.
.
- 1
- 1 2
The solution with a harmonic excitation F = F0 E J T j2 = - 1
(
) is form U =u0 ejt, it
who leads to: K - M2 + J C
(
) u0 =F0
This system can be solved for all, either directly, or by using the modal transformation with
to leave the real clean modes obtained by the conservative system associated K - m2
(
) =0.
It admits N solutions clean (8 in this case) associated 2i and vectors I gathered in
2
spectral matrix =
[]
[]
I and the modal matrix = I.
The modal transformation consists in writing: u0 = Q what leads to:
[- 2 I+ J] Q = T F0
I is the identity,
here is diagonal =
[]
(
)
II bus damping is proportional C = K.
N
T
The answer is written: U
I
I
0 =
F
2
0
i=1 I - 2 + J II
One obtains the exact solution by taking all the clean modes.
One deduces some: &u = J U
and
&u
2
= - U
0
0
0
0
2.2
Results of reference
Displacement according to X of the P4 point for certain frequencies.
2.3
Uncertainty on the solution
Semi-analytical solution.
2.4 Reference
bibliographical
[1]
J. PIRANDA: Note of use of the software of modal analysis MODAN - Version 0.2 (1990).
Laboratory of Mécanique Appliquée - Université de Franche County - Besancon (France).
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.0
Titrate:
SDLD21 - Système mass-arises to 8 ddl with viscous damping device
Date:
23/06/03
Author (S):
O. NICOLAS
Key: V2.01.021-C Page:
4/10
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
3.2
Characteristics of the grid
A number of nodes: 10
A number of meshs and types: 9 SEG2
3.3 Functionalities
tested
Commands
DISCRETE AFFE_CARA_ELEM GROUP_MA “K_T_D_L'
GROUP_MA
“A_T_D_L'
GROUP_MA
“M_T_D_N'
“MECHANICAL” AFFE_MODELE VERY “DIS_T'
GROUP_NO
“DIS_T'
AFFE_CHAR_MECA DDL_IMPO GROUP_NO
FORCE_NODALE
NOEUD
DYNA_LINE_HARM MATR_AMOR
DEFI_LIST_REEL BEGINNING
INTERVALLE
RECU_FONCTION LIST_FREQ
TEST_FONCTION
TEST_RESU
IMPR_RESU
LIRE_RESU
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.0
Titrate:
SDLD21 - Système mass-arises to 8 ddl with viscous damping device
Date:
23/06/03
Author (S):
O. NICOLAS
Key: V2.01.021-C Page:
5/10
3.4
Results of modeling A
Parts real and imaginary of component DX of the displacement of the P4 point.
Frequency Reference
Aster %
Difference
5.00
1.0237 E4
1.02369 E4
0.0004
8.5187 E6
8.51874 E6
5.50
4.5066 E4
4.50662 E4
0.0004
7.7914 E4
7.79143 E4
6.00
9.4101 E5
9.41096 E5
0.0002
1.0585 E5
1.05851 E5
10.00
8.4143 E7
8.41427 E7
0.0024
1.0335 E6
1.03346 E6
15.00
1.2656 E5
1.26556 E5
0.0032
5.6652 E6
5.66517 E6
20.00
2.9784 E6
2.97844 E6
0.0003
6.6970 E6
6.69700 E6
25.00
1.2536 E6
1.25362 E6
0.0008
5.2703 E6
5.27033 E6
30.00
2.0904 E6
2.09042 E6
0.0009
5.4821 E6
5.48215 E6
35.00
4.5447 E6
4.54473 E6
0.0011
1.1190 E6
1.11903 E6
39.50
2.6895 E6
2.68949 E6
0.0003
3.0505 E7
3.05048 E7
Parts real and imaginary of component DX the speed of the P4 point.
Frequency Reference
Aster %
Difference
5.00
2.6762 E4
2.6762 E4
0.000
3.2160 E3
3.21603 E3
5.50
2.6925 E2
2.69252 E2
0.001
1.5574 E2
1.55737 E2
6.00
3.9904 E4
3.99052 E4
0.000
3.5475 E3
3.54752 E3
10.00
6.4937 E5
6.49347 E5
0.002
5.2869 E5
5.28685 E5
15.00
5.3393 E4
5.33929 E4
0.003
1.1928 E3
1.19276 E3
20.00
8.4157 E4
8.41570 E4
0.001
3.7428 E4
3.74282 E4
25.00
8.2786 E4
8.27862 E4
0.001
1.9691 E4
1.96919 E4
30.00
1.0333 E3
1.03334 E3
0.001
3.9403 E4
3.94035 E4
35.00
2.4608 E4
2.46089 E4
0.001
9.9943 E4
9.99439 E4
39.50
7.5709 E5
7.57086 E5
0.000
6.6749 E4
6.67494 E4
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.0
Titrate:
SDLD21 - Système mass-arises to 8 ddl with viscous damping device
Date:
23/06/03
Author (S):
O. NICOLAS
Key: V2.01.021-C Page:
6/10
Parts real and imaginary of component DX of the acceleration of the P4 point.
Frequency Reference
Aster %
Difference
5.00
1.0103 E1
1.01035 E1
0.000
8.4076 E3
8.40766 E3
5.50
5.3819 E1
5.38190 E1
0.000
9.3047 E1
9.30470 E1
6.00
1.3374 E1
1.33738 E1
0.000
1.5044 E2
1.50439 E2
10.00
3.3218 E3
3.32182 E3
0.002
4.0801 E3
4.07996 E3
15.00
1.1242 E1
1.12415 E1
0.003
5.0322 E2
5.03217 E2
20.00
4.7033 E2
4.70337 E2
0.001
1.0575 E1
1.05755 E1
25.00
3.0931 E2
3.09320 E2
0.001
1.3004 E1
1.30040 E1
30.00
7.4273 E2
7.42739 E2
0.001
1.9478 E1
1.94780 E1
35.00
2.1979 E1
2.19788 E1
0.001
5.4116 E2
5.41178 E2
39.50
1.6566 E1
1.65662 E1
0.000
1.8789 E2
1.87898 E2
3.5 Remarks
Contents of the file results:
Values of the displacement of component DX of the P4 point for all the frequencies from 5 to 40 Hz
by step of 0.5 (Cas initial test of VPCS).
Values the speed and the acceleration of component DX of the P4 point for some
frequencies of vibration.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.0
Titrate:
SDLD21 - Système mass-arises to 8 ddl with viscous damping device
Date:
23/06/03
Author (S):
O. NICOLAS
Key: V2.01.021-C Page:
7/10
4 Modeling
B
4.1
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
4.2
Characteristics of the grid
A number of nodes: 10
A number of meshs and types: 9 SEG2
4.3 Functionalities
tested
Commands
DISCRETE AFFE_CARA_ELEM GROUP_MA “K_T_D_L'
GROUP_MA
“A_T_D_L'
GROUP_MA
“M_T_D_N'
“MECHANICAL” AFFE_MODELE VERY “DIS_T'
GROUP_NO
“DIS_T'
AFFE_CHAR_MECA DDL_IMPO GROUP_NO
FORCE_NODALE
NOEUD
MODE_ITER_SIMULT
MACRO_PROJ_BASE
DYNA_LINE_HARM MATR_AMOR
REST_BASE_PHY
DEFI_LIST_REEL BEGINNING
INTERVALLE
RECU_FONCTION LIST_FREQ
TEST_FONCTION
TEST_RESU
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.0
Titrate:
SDLD21 - Système mass-arises to 8 ddl with viscous damping device
Date:
23/06/03
Author (S):
O. NICOLAS
Key: V2.01.021-C Page:
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4.4
Results of modeling B
Parts real and imaginary of component DX of the displacement of the P4 point.
Frequency Reference
Aster %
Difference
5.00
1.0237 E4
1.02369 E4
0.0004
8.5187 E6
8.51874 E6
5.50
4.5066 E4
4.50662 E4
0.0004
7.7914 E4
7.79143 E4
6.00
9.4101 E5
9.41096 E5
0.0002
1.0585 E5
1.05851 E5
10.00
8.4143 E7
8.41427 E7
0.0024
1.0335 E6
1.03346 E6
15.00
1.2656 E5
1.26556 E5
0.0032
5.6652 E6
5.66517 E6
20.00
2.9784 E6
2.97844 E6
0.0003
6.6970 E6
6.69700 E6
25.00
1.2536 E6
1.25362 E6
0.0008
5.2703 E6
5.27033 E6
30.00
2.0904 E6
2.09042 E6
0.0009
5.4821 E6
5.48215 E6
35.00
4.5447 E6
4.54473 E6
0.0011
1.1190 E6
1.11903 E6
39.50
2.6895 E6
2.68949 E6
0.0003
3.0505 E7
3.05048 E7
4.5 Remarks
Contents of the file results:
Values of the displacement of component DX of the P4 point for all the frequencies from 5 to 40 Hz
by step of 0.5 (Cas initial test of VPCS).
Values the speed and the acceleration of component DX of the P4 point for some
frequencies of vibration.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.0
Titrate:
SDLD21 - Système mass-arises to 8 ddl with viscous damping device
Date:
23/06/03
Author (S):
O. NICOLAS
Key: V2.01.021-C Page:
9/10
5 Modeling
C
5.1
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
5.2
Characteristics of the grid
A number of nodes: 10
A number of meshs and types: 9 SEG2
5.3 Functionalities
tested
Commands
DISCRETE AFFE_CARA_ELEM GROUP_MA “K_T_D_L'
GROUP_MA
“A_T_D_L'
GROUP_MA
“M_T_D_N'
“MECHANICAL” AFFE_MODELE VERY “DIS_T'
GROUP_NO
“DIS_T'
AFFE_CHAR_MECA DDL_IMPO GROUP_NO
FORCE_NODALE
NOEUD
MODE_ITER_SIMULT MATR_AMOR
DYNA_LINE_HARM AMOR_REDUIT
DEFI_LIST_REEL BEGINNING
INTERVALLE
RECU_FONCTION LIST_FREQ
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.0
Titrate:
SDLD21 - Système mass-arises to 8 ddl with viscous damping device
Date:
23/06/03
Author (S):
O. NICOLAS
Key: V2.01.021-C Page:
10/10
5.4
Results of modeling C
Eigen frequencies of the structure for the sequence numbers from 1 to 5.
Sequence number
Reference
Aster %
Difference
1
5.5271 5.5271848238694
0.002
2
10.8868 1.088524727521
0.014
3
15.9155 1.5910519939851
0.031
4
20.4606 20.449995091940
0.052
5
24.384 24.366059022201
0.074
Damping reduce structure for the sequence numbers from 1 to 5.
Sequence number
Reference
Aster %
Difference
1
0.00868241 8.6824088833463D-03
1.29E-05
2
0.017101 1.7101007166284D-02
4.19E-05
3
0.025 2.5000000000002D-02
9.19E-12
4
0.0321394 3.2139380484326D-02
6.07E-05
5
0.0383022 3.8302222155950D-02
5.78E-05
6
Summary of the results
The results obtained are excellent, which is normal for a direct integration.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
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