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Code_Aster
®
Version
4.0
Titrate:
Modal parameters and standard of the clean vectors
Date:
10/09/97
Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
1/12
Manual of Reference
R5.01 booklet: Modal analysis
HI-75/97/022/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R5.01 booklet: Modal analysis
R5.01.03 document:
Modal parameters and standard of the clean vectors
Summary:
In this document, one describes:
·
various possibilities in Code_Aster to normalize the clean modes,
·
important modal parameters associated the clean modes.
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Code_Aster
®
Version
4.0
Titrate:
Modal parameters and standard of the clean vectors
Date:
10/09/97
Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
2/12
Manual of Reference
R5.01 booklet: Modal analysis
HI-75/97/022/A
Contents
1 Definition of the problem to the eigenvalues .......................................................................................... 3
1.1 General information ...................................................................................................................................... 3
1.2 Generalized problem ........................................................................................................................ 3
1.3 Quadratic problem ..................................................................................................................... 4
2 Standard of the clean modes of the generalized problem .............................................................................. 5
2.1 Components of a clean mode ..................................................................................................... 5
2.2 Euclidian norm ........................................................................................................................... 5
2.3 “Larger component with 1 normalizes” ............................................................................................. 6
2.4 Mass or unit generalized rigidity ................................................................................. 6 normalizes
3 Standard of the clean modes of the quadratic problem ........................................................................... 7
3.1 Euclidian norms and “larger component with 1” .................................................................... 7
3.2 Mass or unit generalized rigidity ................................................................................. 7 normalizes
4 modal Parameters associated for the generalized problem .................................................................. 8
4.1 Generalized sizes .................................................................................................................. 8
4.1.1 Definition ................................................................................................................................ 8
4.1.2 ................................................................................................................................ Use 9
4.2 Effective modal masses and unit effective modal masses .............................................. 9
4.2.1 Effective modal masses .................................................................................................... 9
4.2.2 Property ............................................................................................................................... 10
4.2.3 Unit effective modal masses ................................................................................... 10
4.2.4 .............................................................................................................................. Use 10
4.2.5 Directions privileged in Code_Aster .......................................................................... 10
4.3 Factors of participation ............................................................................................................... 11
4.3.1 Definition .............................................................................................................................. 11
4.3.2 Property ............................................................................................................................... 11
4.3.3 .............................................................................................................................. Use 11
4.4 Unit vector displacement ........................................................................................................ 11
5 modal Parameters associated for the quadratic problem ............................................................. 12
6 Bibliography ........................................................................................................................................ 12
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Code_Aster
®
Version
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Titrate:
Modal parameters and standard of the clean vectors
Date:
10/09/97
Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
3/12
Manual of Reference
R5.01 booklet: Modal analysis
HI-75/97/022/A
1
Definition of the problem to the eigenvalues
1.1 General
That is to say the problem with the eigenvalues according to:
To find
(
)
(
)
,
/
0000
×
+
+
=
C C
N
2
B
A.C.
éq 1.1-1
where
WITH C B
,
are positive symmetrical real matrices of command
N
.
Two cases are distinguished:
·
quadratic problem:
C
0
,
·
generalized problem:
C
=
0
.
eigenvalue is called and
clean vector. In the continuation, one will speak about clean mode for
and
one will introduce the concept of Eigen frequency.
To solve this problem, several methods are available in Code_Aster and one returns it
reader with the documents [R5.01.01] and [R5.01.02].
1.2 Problem
generalized
The generalized problem can be written in the form:
To find
(
)
(
)
,
/
×
-
+
=
N
2
B WITH
0
éq 1.2-1
One introduces two other sizes which make it possible to characterize the clean mode:
(
)
= =
2 F
éq 1.2-2
where
: own pulsation associated the clean mode
,
F
: Eigen frequency associated the clean mode
.
It is also shown that the clean modes are
With
and
B
orthogonal, i.e.:
iT
J
ij
iT
I
iT
J
ij
iT
I
With
With
B
B
=
=



éq 1.2-3
where
(
)
I
J
,
are two clean modes.
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Code_Aster
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Version
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Titrate:
Modal parameters and standard of the clean vectors
Date:
10/09/97
Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
4/12
Manual of Reference
R5.01 booklet: Modal analysis
HI-75/97/022/A
1.3 Problem
quadratic
The quadratic problem [éq 1.1-1] can be put in another form of size doubles (one speaks about
linear reduction [R5.01.02]):
To find
()
,
/
×




+ -








=
C C
N
0 B
B C
B 0
0
With
0
éq 1.3-1
One poses in the continuation:
!
!
B
0 B
B C
With
B 0
0
With
=


= -




.
Like the matrices
WITH C B
,
are real, the values and clean modes imaginary are combined
two to two.
One introduces three other sizes which make it possible to characterize the clean mode:
(
) ()
= +
= -
-
+
= -
-
+
ad interim B
I
F
I
F
1
2
1
2
2
2
éq 1.3-2
where
: own pulsation associated the clean mode
,
F
: Eigen frequency associated the clean mode
,
: reduced damping.
It is also shown that the clean modes are
0 B
B C




and
-




B 0
0
With
orthogonal, i.e.:
(
)
(
)
(
)
I
J
iT
J
iT
J
ij
I
iT
I
iT
I
I
J
iT
J
iT
J
ij
I
iT
I
iT
I
+
+
=
+
-
+
=
-
+



B
C
B
C
B
With
B
With
2
2
éq 1.3-3
where
()
I
J
,
are the eigenvalues respectively associated with the clean modes
(
)
I
J
,
.
Note:
the clean modes are thus not
WITH B
C
,
or
orthogonal.
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Code_Aster
®
Version
4.0
Titrate:
Modal parameters and standard of the clean vectors
Date:
10/09/97
Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
5/12
Manual of Reference
R5.01 booklet: Modal analysis
HI-75/97/022/A
2
Normalizes clean modes of the generalized problem
One supposes to have calculated a couple
(
)
,
solution of the problem [éq 1.2-1]:
is the eigenvalue
associated the clean mode
. One considers for the moment only the case of the generalized problem.
In Code_Aster, the control
NORM_MODE
[U4.06.02] allows to impose a type of standardization
for the whole of the modes.
2.1
Components of a clean mode
That is to say a clean mode
components
()
J J N
=
1,
.
Among these components, one distinguishes:
·
the components or degrees of freedom called “physics” (they are for example the degrees
of freedom of displacement (
DX, DY, DZ
), degrees of freedom of rotation (
DRX, DRY, DRZ
), it
potential characterizing an irrotational fluid (
PHI
),…),
·
the components of Lagrange (the parameters of Lagrange are unknown factors
additional which is added with the “physical” problem initial so that the conditions with
limits are checked [R3.03.01]).
In Code_Aster, one has three families of standards:
·
normalizes
Euclidean,
·
normalizes: “larger component with 1” among a group of degrees of freedom defined,
·
mass or unit generalized rigidity normalizes.
They successively are described.
Previously, one defines
L
a family of indices which contains
m
terms:
{
}
L
L K
m
L
N
m N
K
K
=
=
,
,
.
1
1
1
with
and
2.2 Normalizes
Euclidean
The following standard is defined:
()
2
2
1
1 2
=


=
L
K
m
K
/
The normalized vector is then obtained
:
.
=
=
=




1
1
1
2
2
J
J
J
N
In Code_Aster, two standards of this family are available:
·
NORME=' EUCL'
:
L
corresponds to the whole of the indices which characterize a degree of
physical freedom,
·
NORME=' EUCL_TRAN'
:
L
corresponds to the whole of the indices which characterize a degree
of physical freedom of displacement in translation (
DX, DY, DZ
).
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Code_Aster
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Titrate:
Modal parameters and standard of the clean vectors
Date:
10/09/97
Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
6/12
Manual of Reference
R5.01 booklet: Modal analysis
HI-75/97/022/A
2.3
“Larger component with 1 normalizes”
The following standard is defined:
=
=
max
,
K
m
L
K
1
The normalized vector is then obtained
:
.
=
=
=




1
1
1
J
J
J
N
In Code_Aster, five standards of this family are available:
·
NORME=' SANS_CMP=LAGR'
:
L
corresponds to the whole of the indices which characterize one
physical degree of freedom,
·
NORME=' TRAN'
:
L
corresponds to the whole of the indices which characterize a degree of
physical freedom of displacement in translation (
DX, DY, DZ
),
·
NORME=' TRAN_ROTA'
:
L
corresponds to the whole of the indices which characterize a degree
of physical freedom of displacement in translation and rotation (
DX, DY, DZ, DRX, DRY, DRZ
),
·
NORME=' AVEC_CMP'
or
“SANS_CMP”
:
L
is built either by taking all the indices which
correspond to types of components stipulated by the user (for example the standard
displacement following axis X: '
DX
') (
NORME=' AVEC_CMP'
), that is to say by taking the complementary one
of all the indices which correspond to types of components stipulated by the user
(
NORME=' SANS_CMP'
),
·
NORME=' NOEUD_CMP'
:
L
corresponds to only one index which characterizes a component of one
node of the mesh. The name of the node and the component are specified by the user
(key words
NOM_CMP
and
NODE
control
NORM_MODE
[U4.64.02]).
By defect the modes are normalized with the standard
“SANS_CMP=LAGR”.
2.4
Mass or unit generalized rigidity normalizes
That is to say a positive definite matrix of command
N
. The following standard is defined:
(
)
E
E
=
T
1 2
/
The normalized vector is then obtained
:
,
=
=
=




1
1
1
E
E
J
J
J
N
.
In Code_Aster, two standards of this family are available:
·
NORME=' MASSE_GENE'
:
E B
=
. In a conventional problem of vibration,
B
is the matrix
of mass.
·
NORME=' RIGI_GENE'
:
E WITH
=
. In a conventional problem of vibration,
With
is the matrix
of rigidity.
Note:
For a mode
rigid body, one a:
E
With
=
=
0
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Code_Aster
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Titrate:
Modal parameters and standard of the clean vectors
Date:
10/09/97
Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
7/12
Manual of Reference
R5.01 booklet: Modal analysis
HI-75/97/022/A
3
Normalizes clean modes of the quadratic problem
3.1
Euclidian norms and “larger component with 1”
For the quadratic problem, one has the same standards as for the generalized problem.
clean modes being complex, one works with the square product. Various standards
“conventional” become:
·
square standard:
(
)
2
1
1 2
=


=
L
L
K
m
K
K
/
where
L
K
is combined of
L
K
,
·
“larger component with 1 normalizes”:
(
)
=
=
=
=




max
max
,
,
/
K
m
L
K
m
L
L
K
K
K
1
1
1 2
(the value
absolute in the real field becomes the module in the complex field).
3.2
Mass or unit generalized rigidity normalizes
With regard to the standard “masses or generalized rigidity”, denomination by analogy with
generalized problem, one uses as matrix associated with the standard, that which intervenes in the writing
quadratic problem put in the reduced form [éq 1.3-1].
One has then:
·
normalizes generalized mass:
(
)
(
)
!
!
B
B
0 B
B C
B
C
=




=




=
+
T
T
T
T
T
T
,
,
,
2
^
=
1
!#
B
,
·
normalizes generalized rigidity:
(
)
(
)
!
!
With
With
B 0
0
With
B
With
=




=
-




= -
+
T
T
T
T
T
T
,
,
,
2
=
1
!With
.
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Code_Aster
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Version
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Titrate:
Modal parameters and standard of the clean vectors
Date:
10/09/97
Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
8/12
Manual of Reference
R5.01 booklet: Modal analysis
HI-75/97/022/A
4
Modal parameters associated for the generalized problem
One in the case of places a conventional generalized problem of vibration. One a:
·
With
=
K
is the matrix of rigidity,
·
B
=
M
is the matrix of mass.
That is to say a couple
(
)
,
solution of the problem:
(
)
-
+
=
2
MR. K
0
éq 4-1
In the continuation, one defines successively the following sizes:
·
sizes
generalized,
·
effective modal mass and unit effective modal mass,
·
factor of participation.
To know the names of the parameters associated with the clean modes and how y to reach in
structure of data
Mode_meca RESULT
, one returns the reader to the document [U5.01.23].
4.1 Sizes
generalized
4.1.1 Definition
Two generalized sizes are defined:
·
Mass generalized of the mode
:
m
T
=
M
,
·
Generalized rigidity of the mode
:
K
T
=
K
.
These quantities depend on standardization on
. These sizes are accessible in the concept
RESULT
of type
mode_meca
[U5.01.23] under the names
MASS_GENE, RIGI_GENE
.
Notice 1:
One with the following relation between the pulsation (or the frequency) of the mode and the mass and rigidity
generalized of the mode:
(
)
= =
=
=
2 F
K
m
T
T
K
M
.
Notice 2:
From the physical point of view, the generalized mass (which is a positive value) can be interpreted
like the mass moving:
m
T
=
=
M
2
where
is the density of the structure.
Kinetic energy of the structure vibrating according to the mode
is equal then to:
E
m
C
T
=
=
1
2
1
2
2
2
M
.
Potential energy of deformation associated with the mode
is equal to:
E
K
p
T
=
=
1
2
1
2
K
.
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Code_Aster
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Titrate:
Modal parameters and standard of the clean vectors
Date:
10/09/97
Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
9/12
Manual of Reference
R5.01 booklet: Modal analysis
HI-75/97/022/A
4.1.2 Use
During a calculation by modal recombination [R5.06.01], one seeks a solution of the equation of
dynamics:
()
MX Cx Kx
$$
$
+
+
=
F T
,
in the form
()
X
=
=



I
I
I
m
T
1,
where
I
real associate with the eigenvalue is the clean mode
I
,
solution of the generalized problem (in general one has
m N
(
N
of degree of freedom) because one is the number
does not take into account that part of the modal base):
(
)
-
+
=
M
K
I
I
2
0
The generalized vector
()






=
=
I I m
1,
is solution of:
~ $$ ~ $ ~
~
M
C
K









+
+
=
F
(problem of command
m
) with:
()
(
)
()
(
)
()
(
)
()
(
)
~
~
~
~
~
~
~
~
M
M
M
C
C
C
K
K
K
=
=
=
=
=
=
=
=
ij
iT
J
ij
iT
J
ij
iT
J
I
iT
F
F
F
.
The modes of vibration of the generalized problem are
K
and
M
orthogonal [R5.01.01]. Matrices
~
M
and
~
K
are then diagonal and are consisted of the rigidities and masses generalized of each
mode. The matrix
~
C
is usually full if one does not make additional assumptions on
C
[R5.05.04].
4.2
Effective modal masses and unit effective modal masses
4.2.1 Effective modal masses
That is to say
U
D
an unit vector in the direction
D
. In each node of the vector
U
D
having them
components of displacement (
DX, DY, DZ
) one a:
(
)
DX
DY
DZ
=
=
=
X
y
Z
D
D
D
,
,
where
(
)
X
y Z
D
D
D
,
,
are the cosine Directors of the direction
D
(one has
thus:
X
y
Z
D
D
D
2
2
2
1
+
+
=
).
For example, if
D
is the direction
X
, the vector
U
D
has all its components
DX
equal to 1 and its
other components equal to 0.
One defines the effective modal masses in the direction
D
by:
(
)
(
)
m
D
T
D
T
,
=
MR. U
M
2
.
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Code_Aster
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Version
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Titrate:
Modal parameters and standard of the clean vectors
Date:
10/09/97
Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
10/12
Manual of Reference
R5.01 booklet: Modal analysis
HI-75/97/022/A
4.2.2 Property
Statement:
The sum of the effective modal masses in a direction
D
is equal to the total mass
m
total
structure. That is written:
(
)
(
)
m
m
total
iT
D
iT
I
I
N
D
I
N
I
=
=
=
=
MR. U
M
2
1
1
,
,
,
where
N
is the total number of modes associated with
problem [éq 4-1]
4.2.3 Unit effective modal masses
By using the preceding property, one defines the unit effective modal masses:
(
)
(
)
~
,
m
m
D
total
T
D
T
=
1
2
MR. U
M
,
and one a:
~
,
,
m
I
D
I
N
=
=
1
1
.
Modal masses
~
,
m
D
and
m
D
,
are independent of the standardization of the mode
of vibration.
4.2.4 Use
“Empirical” relation:
At the time of a study “seismic stress of a structure in a direction
D
“by a method of
modal recombination, one must preserve the modes of vibration which have a unit effective mass
important and it is considered that one has a good modal representation if for the unit of the modes
preserved one a:
~
,
,
,
m
I
D
I
N
=
1
0 9
.
This empirical relation is stated in the RCC_G (Rules of design and construction
applicable to the Civil Engineering).
4.2.5 Directions privileged in Code_Aster
In Code_Aster, one has three directions which are those of the reference mark of definition of the mesh:
·
D
= direction
X
,
·
D
= direction
Y
,
·
D
= direction
Z
.
The effective modal masses and the unit effective modal masses are accessible in
concept
RESULT
of type
mode_meca
[U5.01.23] under the names
MASS_EFFE_DX,
MASS_EFFE_DY,
MASS_EFFE_DZ,
MASS_EFFE_UN_DX,
MASS_EFFE_UN_DY,
MASS_EFFE_UN_DZ
.
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Code_Aster
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Titrate:
Modal parameters and standard of the clean vectors
Date:
10/09/97
Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
11/12
Manual of Reference
R5.01 booklet: Modal analysis
HI-75/97/022/A
4.3
Factors of participation
4.3.1 Definition
One defines other parameters called factor of participation:
(
)
(
)
p
D
T
D
T
,
=
MR. U
M
.
This parameter depends on the standardization of the mode of vibration
.
As for the effective masses, one has three directions
D
who are those of the reference mark of
definition of the mesh.
The factors of participation are accessible in the concept
RESULT
of type
mode_meca
[U5.01.23] under the names
FACT_PARTICI_DX, FACT_PARTICI_DY, FACT_PARTICI_DZ.
4.3.2 Property
Statement:
Factors of participation associated with a direction
D
check the following relation:
(
)
(
)
(
)
()
m
p
m
total
iT
D
iT
I
I
N
iT
D
iT
I
iT
I
I
N
D
I
N
I
=
=






=
=
=
=
MR. U
M
MR. U
M
M
2
1
2
1
2
1
,
,
,
,
where
N
is
the total number of modes associated with the problem [éq 4-1].
This result is obtained easily by expressing the factor of participation according to the modal mass
effective and by using the result stated with [§ 4.2.3].
4.3.3 Use
These parameters are used in particular to calculate the response of a structure subjected to one
seism by spectral method. One returns the reader to the document [R4.05.03].
4.4
Unit vector displacement
In what precedes, a unit vector of displacement was considered
U
D
who relates to only them
degrees of freedom of translation (
DX, DY, DZ
). This concept can be extended to rotations in
considering the following definition. A matrix is defined
U
of dimension
(
)
N
×
6
. If all nodes
mesh support 3 degrees of freedom of translation and 3 others of rotation, the matrix
U
is
formed of the stacking of the matrices
(
)
U
tr
K
D
6.6
×
following (the index
K
corresponds to the node of
number
K
):
(
)
(
)
(
)
(
)
(
) (
)
U
tr
K
K
C
K
C
K
C
K
C
K
C
K
C
Z
Z
y
y
Z
Z
X
X
y
y
X
X
=
-
-
-
-
-
-
-
-
-














1 0 0
0
0 1 0
0
0 0 1
0
0 0 0
1
0
0
0 0 0
0
1
0
0 0 0
0
0
1
background image
Code_Aster
®
Version
4.0
Titrate:
Modal parameters and standard of the clean vectors
Date:
10/09/97
Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
12/12
Manual of Reference
R5.01 booklet: Modal analysis
HI-75/97/022/A
where
(
)
X y Z
K
K
K
,
,
are the co-ordinates of the node and
(
)
X y Z
C
C
C
,
,
are the co-ordinates of the center
instantaneous of rotation.
One can thus define effective modal masses, factors of participation associated with
degrees of freedom of rotation.
For the moment, the calculation of these parameters is not available in Code_Aster.
5
Modal parameters associated for the quadratic problem
One writes the quadratic problem in the form:
(
)
2
0
M
C K
+
+
=
.
For the quadratic problem, one calculates only three parameters which correspond to the sizes
generalized following:
·
mass generalized (real quantity):
m
T
=
M
,
·
generalized rigidity (real quantity):
K
T
=
K
,
·
generalized damping (real quantity):
C
T
=
C
.
Attention, if one normalizes the clean mode with the standard “masses generalized”, one does not have in the case
quadratic:
m
=
1
. One can make the same remark concerning generalized rigidity.
By using the relations of orthogonality and the fact that the clean elements appear per pairs
combined, one can write the following relations:
()
(
)
(
)
T
T
T
T
C
m
F
K
m
F
C
M
K
M
=
=
= -
-
= -
-
=
=
= - = -
2
2
1
2
2
1
1
2
1
2
2
2
2
2
2
2
Re
,
.
6 Bibliography
[1]
J.R. LEVESQUE, L. VIVAN, Fe WAECKEL: Seismic response by spectral method
[R4.05.03].
[2]
D. SELIGMANN, B. QUINNEZ: Algorithms of resolution for the generalized problem
[R5.01.01].
[3]
D. SELIGMANN, R. MICHEL: Algorithms of resolution for the quadratic problem
[R5.01.02].
[4] Operator
NORM_MODE
[U4.06.02].