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
Organization (S): EDF/IMA/MN
Handbook of Référence
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.
Handbook of Référence
R5.01 booklet: Modal analysis
HI-75/97/022/A
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
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
Handbook of Référence
R5.01 booklet: Modal analysis
HI-75/97/022/A
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:
3/12
1
Definition of the problem to the eigenvalues
1.1 General
That is to say the problem with the eigenvalues according to:
To find
(,) C × Cn/(2 B + C+ A) = 0
éq 1.1-1
where A, C, B are positive symmetrical real matrices of command N.
Two cases are distinguished:
· quadratic problem: C 0,
· generalized problem: C = 0.
is called eigenvalue 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 + A) = 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.
One also shows that the clean modes are A and B orthogonal, i.e.:
iT A J
iT
I
=
ij
With
éq 1.2-3
iT B J
iT
I
= ij
B
where (I J
,
) are two clean modes.
Handbook of Référence
R5.01 booklet: Modal analysis
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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:
4/12
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
(
0 B - B 0
,) C × Cn/
+
= 0
éq 1.3-1
B
C
0
With
! 0 B!
- B 0
One poses in the continuation: B =
With
B
C
= 0 A.
Like matrices A, 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:
(
2 F)
= + I B has = -
+ I = -
+ I (
2 F)
éq 1.3-2
1 - 2
1 - 2
where
: own pulsation associated the clean mode,
F: Eigen frequency associated the clean mode,
: reduced damping.
0 B
- B 0
It is also shown that the clean modes are
orthogonal, i.e.:
B
C and
0
With
(
iT
J
iT
J
iT
I
iT
I
I + J) B + C = ij (2 I B + C)
éq 1.3-3
-
iT
J
iT
J
2
iT
I
iT
I
I J B +
With = ij
(- I B + A)
where (
I
J
I, J) are the eigenvalues respectively associated with the clean modes (,).
Note:
the clean modes are thus not A, B or C orthogonal.
Handbook of Référence
R5.01 booklet: Modal analysis
HI-75/97/022/A
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
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, command NORM_MODE [U4.06.02] makes it possible 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 of 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 =, 1m with 1 L}
N and 1 m N
K
K
.
2.2 Normalizes
Euclidean
1/2
m
2
The following standard is defined:
= (
2
lk)
K 1
=
1
1
One then obtains the normalized vector “
: “=
“J =
J J =, 1
N.
2
2
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).
Handbook of Référence
R5.01 booklet: Modal analysis
HI-75/97/022/A
Code_Aster ®
Version
4.0
Titrate:
Modal parameters and standard of the clean vectors
Date:
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Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
6/12
2.3
“Larger component with 1 normalizes”
The following standard is defined:
= max
L
K =, m
K
1
1
1
One then obtains the normalized vector “
: “=
“J =
J J =, 1
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'), is 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 grid. The name of the node and the component are specified by the user
(key words NOM_CMP and NOEUD of command NORM_MODE [U4.64.02]).
By defect the modes are normalized with standard “SANS_CMP=LAGR”.
2.4
Mass or unit generalized rigidity normalizes
1/2
That is to say a positive definite matrix of command N. The following standard is defined:
= (T
E
E)
1
1
One then obtains the normalized vector “
: “=
“J =
J J =, 1
N
.
E
E
In Code_Aster, two standards of this family are available:
· NORME=' MASSE_GENE': E = B. In a traditional problem of vibration, B is the matrix
of mass.
· NORME=' RIGI_GENE': E = A. Dans a traditional problem of vibration, A is the matrix
of rigidity.
Note:
For a rigid mode of body, one a:
=
E
To = 0
Handbook of Référence
R5.01 booklet: Modal analysis
HI-75/97/022/A
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:
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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
“traditional” become:
1/2
m
· square standard:
= (L
2
L
where is combined of,
K
K)
lk
lk
K 1
=
/
1 2
· “larger component with 1 normalizes”:
= max L =
max
L
L
(the value
K,
1 m
K
K,
1 (
K
K)
=
= m
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:
T
T
!
T
T 0
B
! =
B
=
= 2 TB + T
,
,
C,
B
()
()
B C
^
= 1
!# B,
· normalizes generalized rigidity:
T
T
!
T
T - B
0
! =
With
=
= - 2 TB
+ T
,
,
With
,
With
()
()
0
With
“
1
=
.
!With
Handbook of Référence
R5.01 booklet: Modal analysis
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Code_Aster ®
Version
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Titrate:
Modal parameters and standard of the clean vectors
Date:
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Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
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4
Modal parameters associated for the generalized problem
One in the case of places a traditional 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 M +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 RESULTAT mode_meca, 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
RESULTAT of the type mode_meca [U5.01.23] under 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:
T
K
K
= = (
2 F) =
=.
T
M 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
2
= M = where is the density of the structure.
The kinetic energy of the structure vibrating according to the mode is equal then to:
1
1
E =
2
m
2
T
C
=
Mr.
2
2
The potential energy of deformation associated with the mode is equal to:
1
1
E =
K
T
p
=
K.
2
2
Handbook of Référence
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Modal parameters and standard of the clean vectors
Date:
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Author (S):
B. QUINNEZ J.R. LEVESQUE
Key:
R5.01.03-A
Page:
9/12
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 (T) where I is the clean mode real associate with the eigenvalue I,
i= m
1,
solution of the generalized problem (in general one has m N (N is the number of degree of freedom) because one
does not take into account that part of the modal base):
(- M2
I
I + K) = 0
The generalized vector = (I)
is solution of:
i= m
1,
~
~
~
M $ ~
+ C $ + K = F (problem of command m) with:
~
M = (~M
iT
J
iT
J
ij) = (
M)
~
C = (~Cij) = (C)
~
.
K = (~K
iT
J
iT
ij) = (
K)
~
F = (~fi) = (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 direction D. In each node of the Ud vector having them
components of displacement (DX, DY, DZ) one a:
(DX = X, DY = y, DZ = Z
D
D
D) where (X, y, Z
D
D
D) are the cosine Directors of the direction D (one has
thus: x2 + y 2 + z2
D
D
D = 1).
For example, if D is 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:
(T
2
Mr. Ud)
m D, = (
.
T
M)
Handbook of Référence
R5.01 booklet: Modal analysis
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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:
10/12
4.2.2 Property
Statement:
The sum of the effective modal masses in a direction D is equal to the total mass mtotale
structure. That is written:
(iT
2
Mr. Ud)
m
=
where N is the total number of modes associated with
,
=1, (
=
iT
I
I D
I
N
M)
m
total
i=1 N
,
problem [éq 4-1]
4.2.3 Unit effective modal masses
By using the preceding property, one defines the unit effective modal masses:
2
T
MR. U
~
1
(
D)
m, D =
,
m
T
total
(M)
and one a:
~
m
= 1
I
.
, D
i=,
1 N
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 0 9
I
.
, D
i=,
1 N
This empirical relation is stated in the RCC_G (Règles of design and construction
applicable to Génie Civil).
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 grid:
·
D = direction X,
·
D = direction Y,
·
D = direction Z.
The effective modal masses and the unit effective modal masses are accessible in
concept RESULTAT of the type mode_meca [U5.01.23] under names MASS_EFFE_DX, MASS_EFFE_DY,
MASS_EFFE_DZ, MASS_EFFE_UN_DX, MASS_EFFE_UN_DY, MASS_EFFE_UN_DZ.
Handbook of Référence
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Version
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Titrate:
Modal parameters and standard of the clean vectors
Date:
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Author (S):
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Key:
R5.01.03-A
Page:
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4.3
Factors of participation
4.3.1 Definition
One defines other parameters called factor of participation:
(T
Mr. Ud)
p D, = (
.
T
M)
This parameter depends on the standardization of the mode of vibration.
As for the effective masses, one has three directions D which are those of the reference mark of
definition of the grid.
The factors of participation are accessible in concept RESULTAT of the mode_meca type
[U5.01.23] under names FACT_PARTICI_DX, FACT_PARTICI_DY, FACT_PARTICI_DZ.
4.3.2 Property
Statement:
The factors of participation associated with a direction D check the following relation:
(iT
2
iT
MR. U
2
D)
MR. U
2
m
=
M
,
where N is
=1, (
D
iT
I
=
iT
I
iT
I
I
=
D
I
N
M)
(
)
M
(p) m
total
i=1 N
,
i=1 N
,
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 Ud was considered which relates to only them
degrees of freedom of translation (DX, DY, DZ). This concept can be extended to rotations in
considering the following definition. One defines a matrix U of dimension (N ×)
6. If all nodes
grid support 3 degrees of freedom of translation and 3 others of rotation, the matrix U is
formed of the stacking of the matrices uktr (6 ×)
6 following (the index K corresponds to the node of
D
number K):
1 0 0
0
(Z
K - zc)
- (yk - teststemyç)
0 1 0 - (zk - zc)
0
(xk - teststemxç)
K
0 0 1
(yk - teststemyç) - (xk - teststemxç)
0
utr =
0 0 0
1
0
0
0 0 0
0
1
0
0 0 0
0
0
1
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Modal parameters and standard of the clean vectors
Date:
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Author (S):
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Key:
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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: (2M + C + K) = 0.
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:
TC
C
2
2 (2 F)
=
= 2
(
Re) = -
= -
,
TM m
1 - 2
1 - 2
T
2
K
K
2
2
(2 F)
=
= =
=
.
TM m
2
2
1 -
1 -
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].
Handbook of Référence
R5.01 booklet: Modal analysis
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