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Version
6.4
Titrate:
SDLD320 - Transitory Réponse of a free system of 3 masses and 2 springs
Date:
01/03/04
Author (S):
E. BOYERE, T. QUESNEL Key
:
V2.01.320-A Page:
1/8
Organization (S): EDF-R & D/AMA, IRCN
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
V2.01.320 document
SDLD320 - Transitory Réponse of a free system
of 3 masses and 2 springs under excitation
harmonic
Summary:
One considers the transitory analysis of a discrete system masses/arises linear with three degrees of freedom
completely free. This system has a non-proportional damping. A sinewave excitation is
applied at an end of the system.
In this problem, one tests, through a discrete model, the calculation of the transitory response of a system
whose rigid modes are not fixed. One is interested only in the transient state. For that, one will seek
the solution by an integration on the complete modal basis (DYNA_TRAN_MODAL [U4.53.21]).
The results obtained (displacement, speed and acceleration) are compared with an average of results
coming from industrial codes and a method of integration numerical of type - Newmark improved.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A
Code_Aster ®
Version
6.4
Titrate:
SDLD320 - Transitory Réponse of a free system of 3 masses and 2 springs
Date:
01/03/04
Author (S):
E. BOYERE, T. QUESNEL Key
:
V2.01.320-A Page:
2/8
1
Problem of reference
1.1 Geometry
k1
k2
m1
m
m
2
3
F0 sin (T)
P1
P
P
2
3
C1
C2
1.2
Properties of materials
Stiffnesses of connection: k1 = 4. 109 N.m1, k2 = 5.33 108 N.m1
Specific masses: m1 = 106 kg, m2 = m3 = 12.106 kg
One-way viscous damping: C1 = 1.2566 106 kg.s1, C2 = 9.0478 106 kg.s1
1.3
Boundary conditions and loadings
Completely free system.
Loading at the P3 point following axis X: F (T) 0=F0 sin (T) for T 0 with F0= 5.104 NR and =19 rad.s
1.
1.4 Conditions
initial
The system is at rest with t=0: (
U 0) = 0 and
()
0 = 0.
dt
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A
Code_Aster ®
Version
6.4
Titrate:
SDLD320 - Transitory Réponse of a free system of 3 masses and 2 springs
Date:
01/03/04
Author (S):
E. BOYERE, T. QUESNEL Key
:
V2.01.320-A Page:
3/8
2
Reference solution
2.1
Method of calculation used for the reference solution
The search of the transitory response of this problem to damping nonproportional, and where them
rigid modes are not fixed, can be carried out by numerical integration in real space:
[M] {U &} + [C] {U &} + [K] {U} = {F
N
N
N
}.
For that, the answer was calculated with two industrial codes:
·
PERMAS: Diagram of integration of Newmark (=0,25, =0,5), t=104s,
Diagram of integration with cubic interpolation of Hermite [bib1], t=104s,
·
ABAQUS: Diagram of integration of Hilber-Hughes-Taylor [bib2] (=-0,05), t=104s,
and method of integration of - Newmark improved [bib3]:
[M] [C] [K]
+
+
+
+
+
+
{
F
F
F
2 M
K
U
N2
N 1
N
N + 2}
{
} {
} {}
[] []
=
+
-
{un+}
t2
2t
3
3
t2
1
3
[M] [C] [K]
+ -
+
-
{a}
t2
2t
3
where N, n+1, n+2 respectively indicate the calculations carried out at times tn, tn+1=tn+t and tn+2=tn+2t where
T is the increment of appointed time.
To start, one takes:
·
u0 and U 1 = U
-
0 - T
u&0
·
F
2F
F
- =
-
1
0
1
The step of adopted time is t=10-5s.
2.2
Results of reference
Displacement, speed and acceleration of the P3 point.
Differential of displacement enters the points P3 and P1.
Relative displacement of the P3 point compared to the P1 point
1,00E-05
5,00E-06
(m) 1 0,00E+00
- U
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
U 3
- 5,00E-06
- 1,00E-05
time (S)
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A
Code_Aster ®
Version
6.4
Titrate:
SDLD320 - Transitory Réponse of a free system of 3 masses and 2 springs
Date:
01/03/04
Author (S):
E. BOYERE, T. QUESNEL Key
:
V2.01.320-A Page:
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2.3
Uncertainty on the solution
Average of numerical solutions.
2.4 References
bibliographical
[1]
J.H. ARGYRIS, PC DUNNE and T. ANGELOPOULOS “Non-linear oscillations using the
finite technical element “Comp. Meth. Appl. Mech. Engng., Vol.2, 1972, pp. 203-254
[2]
H. Mr. HILBER, T.J.R. HUGHES and R.L. TAYLOR “Improved numerical dissipation for time
integration algorithms in structural dynamics “Earthquake Engineering and Structural
Dynamics, Vol.5, 1977, pp. 283-292
[3]
Structural N.M. NEWMARK “A method off computation for dynamics” Proceeding ASCE
J.Eng. Mech. Div E-3, July 1959, pp. 67-94
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A
Code_Aster ®
Version
6.4
Titrate:
SDLD320 - Transitory Réponse of a free system of 3 masses and 2 springs
Date:
01/03/04
Author (S):
E. BOYERE, T. QUESNEL Key
:
V2.01.320-A Page:
5/8
3 Modeling
With
3.1
Characteristics of modeling
Discrete elements of rigidity, damping and mass.
y
P
P2
P
1
3
.
.
.
X
N1
N2
N3
Z
Characteristics of the elements:
DISCRET:
nodal masses
M_TR_D_N
rigidities
linear K_TR_D_L
depreciation
linear
A_TR_D_L
No boundary conditions, in all the nodes: DX, DY, DZ, DRX, DRY, DRZ free.
Names of the nodes: P1 = N1, P2 = N2, P3 = N3.
Method of calculation:
Integration on the modal basis supplements with Newmark (=0,25, =0,5),
No time: t=10-4s then modal recombination.
Duration of observation: 5s.
3.2
Characteristics of the grid
A number of nodes: 3
A number of meshs and type: 2 meshs SEG2
3.3
Functionalities tested
Commands
DISCRETE AFFE_CARA_ELEM
MAILLE
“K_TR_D_L'
MAILLE
“A_TR_D_L'
NOEUD
“M_TR_D_N'
MODE_ITER_SIMULT
CALC_FREQ
(Option: “CENTER”)
DYNA_TRAN_MODAL NEWMARK
REST_BASE_PHYS NOM_CHAM:“DEPL”
CALC_FONCTION COMB
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A
Code_Aster ®
Version
6.4
Titrate:
SDLD320 - Transitory Réponse of a free system of 3 masses and 2 springs
Date:
01/03/04
Author (S):
E. BOYERE, T. QUESNEL Key
:
V2.01.320-A Page:
6/8
4
Results of modeling A
4.1 Values
tested
·
Displacement of the P3 point
Time Displacement
Displacement
Difference
(S)
Reference (m)
Aster (m)
(%)
0,09
6,7395 E-6
6,73326 E-6
- 0,093
0,32
1,1019 E-5
1,10002 E-6
- 0,171
1,18
3,6683 E-5
3,66122 E-5
- 0,193
4,92
1,6615 E-4
1,65849 E-4
- 0,181
·
Speed of the P3 point
Time Speed Speed
Difference
(S)
Reference
(Mr. S 1)
Aster (Mr. S 1)
(%)
0,05
1,3425 E-4
1,34131 E-4
- 0,088
0,32
- 6,4111 E-5
- 6,41097 E-4
- 0,002
1,18
1,6104 E-5
1,60598 E-5
- 0,274
3,55
4,4262 E-5
4,41720 E-5
- 0,203
·
Acceleration of the P3 point
Time Acceleration
Acceleration Difference
(S)
Reference
(Mr. S 2)
Aster (Mr. S 2)
(%)
0,09
- 3,5694 E-3
- 3,56634 E-3
- 0,086
0,18
- 4,3924 E-3
- 4,38933 E-3
- 0,070
0,55
4,3766 E-3
4,37283 E-3
- 0,086
1,18
4,2459 E-3
4,24264 E-3
- 0,077
4,92
- 4,2233 E-3
- 4,21962 E-3
- 0,087
·
Relative displacement of the P3 point compared to the P1 point
Time u3-u1
u3-u1 Différence
(S)
Reference (m)
Aster (m)
(%)
0,18
8,0987 E-6
8,04800 E-6
- 0,626
0,55
- 6,2246 E-6
- 6,21194 E-6
- 0,203
0,82
5,3064 E-6
5,34121 E-6
0,656
1,18
- 4,5552 E-6
- 4,52071 E-6
- 0,757
1,92
- 3,0416 E-6
- 3,04417 E-6
0,085
3,55
1,8448 E-6
1,82742 E-6
- 0,942
4,92
1,4832 E-6
1,47526 E-6
- 0,535
4.2 Remarks
In addition to the comparison for the values tested, one checks that the variables different kinematics
that those related to the translation according to X remain null.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A
Code_Aster ®
Version
6.4
Titrate:
SDLD320 - Transitory Réponse of a free system of 3 masses and 2 springs
Date:
01/03/04
Author (S):
E. BOYERE, T. QUESNEL Key
:
V2.01.320-A Page:
7/8
5
Summary of the results
·
To obtain a good precision of the results, it is initially necessary to obtain a base
modal precise and perfectly orthogonal (MODE_ITER_SIMULT):
by avoiding the multiple modes (different rigidity on the nonexcited ddl),
by calculating the rigid modes of body correctly (to prefer the option “Center” in
MODE_ITER_SIMULT with the other options),
by specifying method “JACOBI” for a modal complete extraction.
·
The precision of the results is good as well for displacements for speeds and
accelerations.
For the elastic response of the system (relative displacements u3-u1), the numerical precision is one
little worse because of the numerical office plurality of the errors on the absolute values.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A
Code_Aster ®
Version
6.4
Titrate:
SDLD320 - Transitory Réponse of a free system of 3 masses and 2 springs
Date:
01/03/04
Author (S):
E. BOYERE, T. QUESNEL Key
:
V2.01.320-A Page:
8/8
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Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A
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