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Titrate:
Harmonic response by conventional dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.03-B
Page:
1/12
Manual of Reference
R4.06 booklet: Under-structuring
HP-51/98/015/A
Organization (S):
EDF/EP/AMV
Manual of Reference
R4.06 booklet: Under-structuring
Document: R4.06.03
Harmonic response by under-structuring
conventional dynamics
Summary:
After having made some recalls concerning the methods of modal synthesis and having introduced the base of
Harmonic Craig-Bampton, we present the theoretical bases of the methods of calculation of answer
harmonic by under-structuring. Initially, we establish the dynamic equations
checked by the substructures separately. Then, the taking into account of the conditions of assembly enters
substructures, enables us to determine the dynamic equations checked by the total structure. In
private individual, we stick to well highlighting the processing of the matrix of damping and of
vector of the external efforts, which intervene in the harmonic calculation of response per under-structuring.
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Code_Aster
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Version
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Titrate:
Harmonic response by conventional dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.03-B
Page:
2/12
Manual of Reference
R4.06 booklet: Under-structuring
HP-51/98/015/A
Contents
1 Introduction ............................................................................................................................................ 3
2 harmonic Response by under-structuring ......................................................................................... 4
2.1 Base of harmonic Craig-Bampton .............................................................................................. 4
2.2 Dynamic equations checked by the substructures separately ............................................ 5
2.3 Assembly of the substructures .................................................................................................... 7
2.4 Dynamic equations checked by the total structure ................................................................ 9
2.5 Implementation in Code_Aster ................................................................................................ 9
2.5.1 Study of the substructures separately ................................................................................. 9
2.5.2 Assembly and resolution ..................................................................................................... 10
2.5.3 Restitution on physical basis .............................................................................................. 10
3 Conclusion ........................................................................................................................................... 10
4 Bibliography ........................................................................................................................................ 11
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Code_Aster
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Version
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Titrate:
Harmonic response by conventional dynamic under-structuring
Date:
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Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.03-B
Page:
3/12
Manual of Reference
R4.06 booklet: Under-structuring
HP-51/98/015/A
1 Introduction
After having developed in Code_Aster of the modules of modal calculation by under-structuring of which
the theoretical bases are presented in reference [R4.06.02], the operators of calculation of answer
harmonic by modal synthesis were implemented.
The methods of modal synthesis which consist in condensing the degrees of freedom resulting from
modeling finite elements on fields of displacement particular to each substructure,
translate by important gains into calculating times and place memory.
For the harmonic problems, the studied system is subjected to a force spatially unspecified,
but sinusoidal in time. The form of the loading, the frequency of excitation and properties
modal, play each one an essential role. It is also necessary to take account of dissipation in
solid, which one can translate by the introduction of a matrix of damping. Method of calculation
harmonic by under-structuring, programmed in Code_Aster, which makes it possible to replace it
total problem by a simplified problem, proceeds in four times. First of all, of the clean modes
and of the static deformations are calculated on each substructure composing the system.
Then, the total problem is projected on these fields, and one takes account of the couplings between
substructures, on the level of their interfaces. One can then solve the problem classically
tiny room obtained. Finally, it any more but does not remain to deduce the overall solution by reconstitution from it.
General notations:
:
Pulsation (rad.s-1)
J
:
Imaginary pure unit
(
)
J
2
1
= -
NR
S
:
A number of substructures
M
:
Stamp of mass resulting from modeling finite elements
K
:
Stamp rigidity resulting from modeling finite elements
C
:
Stamp damping exit of modeling finite elements
Q
:
Vector of the degrees of freedom resulting from modeling finite elements
F
ext.
:
Vector of the forces external with the system
F
L
:
Vector of the bonding strengths applied to the system
:
Stamp vectors of the base of the substructures



:
Vector of the generalized degrees of freedom
B
:
Stamp extraction of the degrees of freedom of interface
L
:
Stamp connection
Note:
The exponent K characterizes the sizes relating to the substructure
S
K
and sizes
generalized are surmounted by a bar: for example
M
K
is the matrix of generalized mass
substructure
S
K
.
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Titrate:
Harmonic response by conventional dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.03-B
Page:
4/12
Manual of Reference
R4.06 booklet: Under-structuring
HP-51/98/015/A
2
Harmonic response by under-structuring
2.1
Base of harmonic Craig-Bampton
Methods of modal synthesis (cf [R4.06.02]), whatever their applicability
(modal, harmonic and/or transitory) the techniques of under-structuring associate those of
modal recombination. To carry out a calculation by under-structuring means that the structure is cut out
in several elements and that its displacement is calculated like the response to the bonding strengths which
connect to the other components and the external forces which are applied to him. In addition,
response of each substructure is calculated by modal recombination. Thus, all components
are defined by a base of projection made up of clean modes and modes of interface. Two
bases of projection introduced into Code_Aster were presented in the documentation of
reference [R4.06.02]:
·
base of Craig-Bampton,
·
bases of Mac Neal without/with static correction.
A third base of projection was introduced within the framework of the developments concerning
harmonic calculation of response per conventional dynamic under-structuring. It is about the base of
Harmonic Craig-Bampton.
We present this new base of projection using the following simple example:
1
2
Substructure 1
Substructure 2
Q
I
1
Q
J
1
Q
j2
Q
i2
Q
Q
Q
K
ik
jk
=
The vector of the degrees of freedom of the substructure is characterized by one
exponent who defines the number of the substructure, and an index which allows
to distinguish the degrees of freedom intern (index I), of the degrees of freedom of
border (index J).
The harmonic base of Craig-Bampton consists of modes specific to interfaces locked and of
harmonic constrained modes [bib6]. The latter are joined to the normal modes with interfaces
locked to correct the effects due to their boundary conditions. A harmonic constrained mode is
defined by the response of the substructure not deadened to a harmonic displacement, of amplitude unit
and of frequency given, imposed on a degree of freedom of connection, the other degrees of freedom of
connection being locked.
Substructure 1
Substructure 2
Modes specific to locked interfaces
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Titrate:
Harmonic response by conventional dynamic under-structuring
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Substructure 2
Substructure 1
Q
Q
=
=
E
E
J
T
J
T
Harmonic constrained modes
This modal base is more particularly appropriate to the problems of interactions fluid-structures,
for which the static loadings applicable in Code_Aster (are not not taken in
count effect of added mass). It can be used for any type of calculation (modal, harmonic
and transient).
2.2
Dynamic equations checked by the substructures separately
We will consider a structure
S
composed of
NR
S
noted substructures
S
K
. Us
let us suppose that each substructure is modelized in finite elements. We saw that in one
calculation by dynamic under-structuring, the vibratory behavior of the substructures results from
external forces which are applied to him, and of the bonding strengths which on them the others exert under
structures. Thus, on the level of the substructure
S
K
, we can write:
M Q
C Q
K Q
F
F
K K
K K
K K
ext.
K
Lk
!!
!
+
+
=
+
éq 2.2-1
where:
M
K
is the matrix of mass resulting from modeling finite elements from
S
K
C
K
is the matrix of damping resulting from modeling finite elements from
S
K
K
K
is the matrix of rigidity resulting from modeling finite elements from
S
K
F
ext.
K
is the vector of the external forces applied to
S
K
F
Lk
is the vector of the bonding strengths applied to
S
K
Q Q
Q
K
K
K
!
!!
and
are the vectors speed displacement and acceleration resulting from
modeling finite elements.
In a harmonic problem, one imposes a loading dynamic, spatially unspecified, but
sinusoidal in time. One is interested then in the stabilized answer of the system, without holding account
transitory part.
The field of the external forces is written:
F
F
ext.
K
ext.
K
J T
T
E
() {
}
=
The field of the bonding strengths is written:
F
F
Lk
Lk
J T
T
E
() {}
=
The field of displacements is written:
Q
Q
K
K
J T
T
E
() {}
=
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Titrate:
Harmonic response by conventional dynamic under-structuring
Date:
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G. ROUSSEAU, C. VARE
Key:
R4.06.03-B
Page:
6/12
Manual of Reference
R4.06 booklet: Under-structuring
HP-51/98/015/A
The fields speed and acceleration are written:
! ()
{}
!! ()
{}
Q
Q
Q
Q
K
K
J T
K
K
J T
T
J
E
T
E
=
= -
2
Finally the substructure
S
K
check the following equation:
(
) {} {
} {}
K
C
M
Q
F
F
K
K
K
K
ext.
K
Lk
J
+
-
=
+
2
éq 2.2-2
The method of modal synthesis consists in seeking the field of unknown displacement, resulting from
modeling finite elements, on an adapted space, of reduced size (transformation of Ritz).
We saw that for each substructure, this space is composed of clean modes
dynamic and of static deformations:
[
]
Q
K
K
K
ik
jk
K
K
=








=










éq 2.2-3



K
are the modal vectors associated the dynamic clean modes of
S
K
,
K
are the modal vectors associated the static deformations of
S
K
,



ik
is the vector of the generalized co-ordinates associated the clean modes of
S
K
,



jk
is the vector of the generalized co-ordinates associated the static deformations of
S
K
,



K
is the vector of the co-ordinates generalized of
S
K
.
The equation [éq 2.2-2] is projected on the basis of
S
K
by taking account of [éq 2.2-3]. This allows us
to write:
(
) {} {
} {}
K
C
M
F
F
K
K
K
K
ext.
K
Lk
J
+
-
=
+
2



éq 2.2-4
where:
M
M
K
K T
K
K
=
is the matrix of mass generalized of
S
K
,
C
C
K
K T
K
K
=
is the matrix of damping generalized of
S
K
,
K
K
K
K T
K
K
=
is the matrix of rigidity generalized of
S
K
,
{
}
{
}
F
F
ext.
K
K T
ext.
K
=
is the vector of the generalized harmonic external forces applied
with
S
K
,
{}
{}
F
F
Lk
K T
Lk
=
is the vector of the generalized bonding strengths applied to
S
K
,
{}



K O
is the vector of generalized harmonic displacements.
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Titrate:
Harmonic response by conventional dynamic under-structuring
Date:
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Key:
R4.06.03-B
Page:
7/12
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R4.06 booklet: Under-structuring
HP-51/98/015/A
By supposing that the dynamic clean modes and the static deformations are organized like
show the formula [éq 2.2-3] and by considering that the clean vectors associated the modes
dynamic are normalized compared to the unit modal mass, the matrices of mass and rigidity
generalized take the following form:
M
Id
M
M
M
K
K
K
K
K
K T
K
K
K T
K K
K T
K
K
K
K
K T
K
K
K T
K K
K T
K
K
=


=

















where:
Id
is the Identité matrix,



K
is the diagonal matrix of the squares of the own pulsations of the base.
It is shown, in the case of the method of Craig-Bampton, that the normal modes and the modes
constrained are orthogonal with respect to the matrix of rigidity whose diagonal terms are, as of
at the time, null [R4.06.02]. However, this property is not used in the algorithm programmed in
Code_Aster.
We consider, like type of dissipation, only viscous damping (it is the only one which is
supported by the tools for under-structuring in Code_Aster). Two methods are usable for
to take into account this damping:
·
the damping of Rayleigh applied at the elementary level which consists in supposing that
stamp elementary damping
C
E
associated each finite element of the model is one
linear combination of the matrices of elementary mass and rigidity
K
E
and
M
E
:
C
K
M
E
E
E
E
E
=
+
The matrix of damping is then assembled
C
K
then projected on the basis [éq 2.2-3]:
C
C
C
C
C
K
K T
K K
K T
K
K
K T
K K
K T
K
K
=














·
the damping proportional applied to the dynamic clean modes of each
substructure. The resulting matrix is thus an incomplete diagonal (one does not know
to associate damping proportional to the static deformations):
C
K
K
=



0
0
0
2.3
Assembly of the substructures
After having studied each substructure separately, one proposes to establish the equations which
govern their assembly. Let us consider two substructures
S
K
and
S
L
connected between them to the level
interface
S
S
K
L
. They are represented by their modeling finite elements and it is admitted that
their respective mesh is compatible. Thus, on the level of the interface, the nodes coincide and them
meshs in opposite are identical. Consequently, the law of action-reaction and the continuity of displacements
with the interfaces, which represent the assembly of
S
K
and
S
L
, are written:
F
F
Lk
L
S K S L
S K SSL
= -
Q
Q
S
S
K
S
S
L
K
L
K
L
=
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Titrate:
Harmonic response by conventional dynamic under-structuring
Date:
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Key:
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Page:
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R4.06 booklet: Under-structuring
HP-51/98/015/A
where:
F
Lk
S K S L
is the vector of the bonding strengths applied to the substructure
S
K
, on the level of
the interface
S
S
K
L
,
Q
S
S
K
K
L
is the vector of the degrees of freedom of the interface
S
S
K
L
resulting from modeling
finite elements of the substructure
S
K
.
Let us introduce the matrices of extraction of the degrees of freedom of the interface
S
S
K
L
:
Q
B
Q
Q
B
Q
S
S
K
S
S
K
K
S
S
L
S
S
L
L
K
L
K
L
K
L
K
L
=
=
By using the equation of projection [éq 2.2-3] and the formulation applied above to both
substructures subjected to a harmonic loading, one obtains:
B
B
S
S
K
K
K
S
S
L
L
L
K
L
K
L
=
{
}
{}
That is to say:
L
L
S
S
K
K
S
S
L
L
K
L
K
L
=
{
}
{}






éq 2.3-1
where:
L
S
S
K
K
L
is the matrix of connection of the interface
S
S
K
L
substructure
S
K
,
L
S
S
L
K
L
is the matrix of connection of the interface
S
S
K
L
substructure
S
L
.
This processing can be carried out on the level of all the interfaces of the total structure. In particular,
it is noted that the work of the bonding strengths is null on the interface
S
S
K
L
; it is thus null on
total structure:
W
S
S
Lk
S
S
K
L
S
S
L
K
L
S K SSL
K
L
S K SSL
K
L
=
+
=
F
Q
F
Q
0
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2.4
Dynamic equations checked by the total structure
The dynamic equations that checks the total structure are:
K
K
K
C
C
C
M
M
M
F
F
1
1
2
1
1
1
K
NR
K
NR
K
NR
ext.
ext.
K
S
S
S
J










+










-




























=









K
NR
S
F
F
F
F
ext.
NR
L
Lk
LN
S
S








+








1
Which, it is necessary to add the equations of connection (according to [éq 2.3-1]):
=
K L
S
S
K
K
S
S
L
L
K
L
K
L
,
{
}
{}
L
L






This system is solved by double dualisation of the boundary conditions [R3.03.01]. Its formulation
finale thus utilizes the vector of the multipliers of Lagrange



and can be written in the form
condensed:
(
) {}
{
}
{}
K
C
M
L
F
L
+
-
+
=
=
J
T
ext.
2
0









éq 2.4-1
The problem defined by the equation [éq 2.4-1] is symmetrical. In addition, its dimension is given
by the number of modes taken into account (dynamic modes and static deformations). One is
thus brought to solve a conventional harmonic problem, of reduced size, to which is associated one
linear equation of stress. Its resolution thus does not pose a problem.
2.5
Implementation in Code_Aster
2.5.1 Study of the substructures separately
Parameters
E
and
E
damping of Rayleigh are introduced, if necessary, by
the operator
DEFI_MATERIAU
[U4.23.01].
The processing of the substructures are identical to the case of modal calculation [R4.06.02]. Modes
clean dynamic are calculated with the operators:
MODE_ITER_SIMULT
[U4.52.02] or
MODE_ITER_INV
[U4.52.01]. The conditions with the interfaces of connection are applied with the operator
AFFE_CHAR_MECA
[U4.25.01]. The operator
DEFI_INTERF_DYNA
[U4.55.03] allows to define them
interfaces of connection of the substructure. The operator
DEFI_BASE_MODALE
[U4.55.04] allows
to calculate the base of complete projection of the substructure.
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The operator
MACR_ELEM_DYNA
[U4.55.05] calculates the generalized matrices of rigidity, mass and
possibly of damping of the substructure, as well as the matrices of connection.
The damping of Rayleigh is taken into account by supplementing the operand
MATR_AMOR
.
Damping proportional is introduced by the operand
AMOR_REDUIT
.
The harmonic loading is defined, on the level of the substructure, by the operators
AFFE_CHAR_MECA
[U4.25.01] (application of the force on the mesh),
CALC_VECT_ELEM
[U4.41.02]
(calculation of the associated elementary vectors) and
ASSE_VECTEUR
[U4.42.03] (assembly of the vector of
loading on the mesh of the substructure).
2.5.2 Assembly and resolution
As in the case of modal calculation [R4.06.02], the model of the complete structure is defined by
the operator
DEFI_MODELE_GENE
[U4.55.06]. Its classification is carried out by the operator
NUME_DDL_GENE
[U4.55.07]. Matrices of mass, rigidity and possibly of damping
generalized of the structure supplements are assembled according to this classification with
the operator
ASSE_MATR_GENE
[U4.55.08].
The loadings are projected on the basis of substructure to which they are applied, then
assembled starting from classification resulting from
NUME_DDL_GENE
[U4.55.07] by the operator
ASSE_VECT_GENE
[U4.55.09].
The calculation of the harmonic response of the complete structure is carried out by the operator
DYNA_LINE_HARM
[U4.54.02].
2.5.3 Restitution on physical basis
The restitution of the results on physical basis is identical to the case of modal calculation [R4.06.02]. It makes
to intervene the operator
REST_BASE_PHYS
[U4.64.01] and possibly the operator
DEFI_SQUELETTE
[U4.75.01] (creation of a mesh “skeleton”).
3 Conclusion
Method of calculation of response harmonic per modal synthesis available in Code_Aster
rest on that of modal under-structuring, also programmed. It consists in expressing
the whole of the equations in a space of reduced size, made up of modes of different
substructures, by a method of Rayleigh-Ritz. The definition of these fields is that used for
the modal under-structuring and includes/understands normal modes as well as other statics or
harmonics. The procedure employed results in a projection of the matrices and the second member
on restricted space.
In the document, we presented the bases of this method. We showed
how the projected equations were obtained starting from the problem arising in continuous space.
base of harmonic Craig-Bampton which is used for primarily to solve the problems of interaction
fluid-structures was presented. The formulation of the conditions of connection was also evoked.
We thus obtained the reduced equations, according to the generalized co-ordinates, which
allow to solve the modular problem of way, with less costs. In addition, the catch
in account of the phenomena of damping was examined. It leads to the introduction into
calculation by under-structuring of depreciation of Rayleigh or clean modal depreciation
with each substructure.
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4 Bibliography
[1]
C. VARE: “Code of Mechanics Aster - Manual of reference: Methods of analysis” [R4.06.02]
“Modal Calculation by conventional and cyclic dynamic under-structuring”
[2]
P. RICHARD: “Methods of cyclic under-structuring in finite elements” Report/ratio EDF
HP-61/91.156
[3]
P. RICHARD: “Methods of under-structuring in Code_Aster” Report/ratio EDF
HP-61/92.149
[4]
R. ROY, J. CRAIG & Mr. C. BAMPTON: “Coupling off Substructures for Dynamic Analysis” -
AIAA Newspaper, (July 1968), vol. 6, N° 7, p. 1313-1319.
[5]
R.H. Mac Neal: “A hybrid method off component mode synthesis” Computers and Structures,
(1971), vol. 1, p. 581-601.
[6]
T. KERBER: “Schedule of conditions: implementation of the harmonic under-structuring” -
Report/ratio D.E.R. HP-61/93.053
[7]
T. KERBER: “harmonic Under-structuring in Code_Aster” - Report/ratio D.E.R.
HP-61/93.104
[8]
J. PELLET: “Code of Mechanics Aster - Manual of Reference: Finite elements in
Aster " - [R3.03.01] “Dualisation of the boundary conditions”
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