Code_Aster
®
Version
6.4
Titrate:
Note of modeling of the mechanical cushioning
Date:
08/11/02
Author (S):
J. PELLET
Key
:
U2.06.03-B
Page
:
1/6
Instruction manual
U2.06 booklet: Dynamics
HT-66/02/003/A
Organization (S):
EDF-R & D/AMA
Instruction manual
U2.06 booklet: Dynamics
Document: U2.06.03
Note of modeling of damping
mechanics
Summary
Linear and non-linear dynamic analyzes, for the study of the vibratory response with an excitation in
force or moving imposed or for the modal analysis complexes, require to add characteristics
of mechanical cushioning to the characteristics of rigidity and mass.
One has several conventional modelings, applicable to all the types of finite elements available:
·
the model of viscous damping,
·
the model of damping hysteretic (known as also “structural damping”) for the analysis
harmonic of viscoelastic materials.
For the analyzes using a modal base of real clean modes, it is possible to introduce coefficients
of damping modal.
Code_Aster
®
Version
6.4
Titrate:
Note of modeling of the mechanical cushioning
Date:
08/11/02
Author (S):
J. PELLET
Key
:
U2.06.03-B
Page
:
2/6
Instruction manual
U2.06 booklet: Dynamics
HT-66/02/003/A
1
Model of viscous damping
The model of viscous damping is most usually used. It corresponds to modeling
of a dissipated energy proportional to the vibratory speed:
Cu
U
Cv
v
T
T
2
1
2
1
=
=
D
E
éq 1-1
where
C
is the matrix of viscous damping, with real coefficients.
It leads to the conventional equations of the dynamics of the structures:
()
T
F
Ku
Cu
Driven
=
+
+
éq 1-2
with
K
stamp rigidity and
M
stamp of mass.
1.1
Viscous damping proportional “total”
This modeling, easy to implement, corresponds to:
M
K
C
+
=
éq 1.1-1
It is currently available, by using the operator
COMB_MATR_ASSE
[U4.72.01], after having
assembled the matrices of rigidity and mass with real coefficients, but it is of a low utility:
·
validation of algorithms of resolution,
·
useless for the industrial studies, because it does not make it possible to represent the heterogeneity of
the structure compared to damping (dissipation with the supports or the assemblies). Of
more total identification of the coefficients
and
is not possible, in modal analysis
experimental, that for two Eigen frequencies
[
]
2
1
F
F
distinct; it gives, for
Eigen frequencies
[
]
2
, F
F
F
I
I
with
I
I
F
2
=
, a law of evolution of damping
tiny room of the form:
I
I
I
+
=
1.2
Viscous damping proportional of the elements of the model
1.2.1 Characteristics
of damping
It is possible to build a matrix of damping starting from each element of the model,
as for rigidity and the mass.
Two functionalities are usable:
·
the assignment of discrete elements, on meshs
POI1
or
SEG2
, by the operator
AFFE_CARA_ELEM
[U4.42.01]. The aforementioned makes it possible to define, with several possible modes of description, a matrix
of damping for each degree of freedom.
·
the definition of a characteristic of damping for any elastic material by the operator
DEFI_MATERIAU
[U4.43.01] by:
AMOR_ALPHA
:
[R]
AMOR_BETA
:
this material being then affected with the meshs concerned.
Code_Aster
®
Version
6.4
Titrate:
Note of modeling of the mechanical cushioning
Date:
08/11/02
Author (S):
J. PELLET
Key
:
U2.06.03-B
Page
:
3/6
Instruction manual
U2.06 booklet: Dynamics
HT-66/02/003/A
1.2.2 Calculation of the matrices of damping
For all the types of finite elements (of continuous, structural or discrete mediums), it is possible of
to calculate the real elementary matrices corresponding to the option of calculation
“AMOR_MECA”
, afterwards
to have calculated the elementary matrices corresponding to the options of calculation
“RIGI_MECA”
and
“MASS_MECA”
or
“MASS_MECA_DIAG”
. Each elementary matrix is then of the form:
·
when the material
I
, of characteristics of viscous damping proportional (
I
I
), is
affected with the element
elem
C
elem
=
I K
elem
+
I m
elem
·
for a discrete element
C
elem
=
has
discrete
This operation is possible with:
mel
[
matr_elem_DEPL_R] = CALC_MATR_ELEM
(
/
OPTION:
“AMOR_MECA”
MODEL:
Mo
[model]
CHAM_MATER:
chmat
[cham_mater]
CARA_ELEM:
will cara
[cara_elem]
);
The assembly of all the elementary matrices of damping is obtained with the operator
ASSE_MATRICE
usual [U4.61.22]. It will be noted that one must use same classifications and it
even mode of storage that for the matrices of rigidity and mass (operator
NUME_DDL
[U4.61.11]).
It is noticed that the matrix of damping obtained is, in general, nonproportional:
M
K
C
+
=
°
1.2.3 Use of the matrix of viscous damping
The matrix
C
is usable for the direct linear dynamic analysis (key word
MATR_AMOR
) with
operators of linear dynamic response:
·
transient
DYNA_LINE_TRAN
[U4.53.02]
·
harmonic
DYNA_LINE_HARM
[U4.53.11]
It is essential for the modal analysis complexes with the operators of search of the values
clean:
·
by iterations opposite
MODE_ITER_INV
[U4.52.04]
·
by simultaneous iterations
MODE_ITER_SIMULT
[U4.52.03]
For the analyzes in modal base, one must project this matrix in the subspace defined by one
together
real clean modes. This operation is possible with the operator
PROJ_MATR_BASE
[U4.63.12]. Let us note that in the general case (
C
nonproportional), the matrix
projected is not diagonal. It remains nevertheless usable (key word
AMOR_GENE
) for the calculation of
dynamic response in force or imposed in modal space, with the operator of
linear dynamic response:
·
transient
DYNA_TRAN_MODAL
[U4.53.21]
Code_Aster
®
Version
6.4
Titrate:
Note of modeling of the mechanical cushioning
Date:
08/11/02
Author (S):
J. PELLET
Key
:
U2.06.03-B
Page
:
4/6
Instruction manual
U2.06 booklet: Dynamics
HT-66/02/003/A
1.2.4 Use of viscous modal damping
For the analyzes in modal base of real clean modes, the dynamic differential equation in
generalized co-ordinates:
()
T
F
Q
Q
Q
I
T
I
I
I
I
I
I
I
µ
=
+
+
2
2
&
&&
éq
1.2.4-1
fact of appearing a modal damping coefficient
I
expressed like a fraction of
critical damping and generalized mass of the mode
I
µ
, which depends on the mode of standardization
clean mode.
In the case of a matrix of damping
C
strictly proportional, coefficients
I
deduce from the diagonal terms of the matrix of damping generalized
C
T
by:
I
T
I
I
T
I
I
I
M
C
=
2
and, in the case of clean modes normalized with the unit modal mass,
I
Ti
I
I
C
=
2
One can use this relation in the case of a matrix of damping
C
nonproportional, in
applying the assumption of BASILE, who is acceptable for weak depreciation (in particular if it
does not have there damping localized dominating) and of the real clean modes sufficiently uncoupled.
The modal damping coefficients can be provided by control (key word
AMOR_REDUIT
)
with two operators for:
·
transitory analysis in modal space
DYNA_TRAN_MODAL
[U4.53.21]
·
seismic analysis by spectrum of oscillator
COMB_SISM_MODAL
[U4.84.01]
Let us note that there is not any tool for automatic extraction of these coefficients, starting from the matrix
of damping generalized
C
T
)
, concept produced by the operator
PROJ_MATR_BASE
[U4.63.12].
2
Model of damping hysteretic
The model of damping hysteretic is usable to treat the harmonic answers of
structures with viscoelastic materials. The damping coefficient hysteretic
is
determined starting from a test under harmonic cyclic loading with the pulsation
for which one
a relation stress-strain obtains which makes it possible to define:
·
the energy dissipated by cycle in the form:
=
cycle
D
D
E
·
the YOUNG modulus complexes E * starting from the relation stress-strains:
(
)
(
)
(
)
J
E
E
J
E
E
J
E
E *
E
E
J
T
J
T
J
+
=
+
=
+
=
=
=
=
=
-
1
*
sin
cos
,
1
2
1
0
0
0
0
0
0
0
0
where
phase
amplitudes
and
with
and
Code_Aster
®
Version
6.4
Titrate:
Note of modeling of the mechanical cushioning
Date:
08/11/02
Author (S):
J. PELLET
Key
:
U2.06.03-B
Page
:
5/6
Instruction manual
U2.06 booklet: Dynamics
HT-66/02/003/A
with
(
)
cos
0
0
1
=
E
= real part and
(
)
sin
0
0
2
=
E
= imaginary part
tg
E
E
=
=
2
1
= factor of dissipation
This led to the equations of the dynamics of the structures:
(
)
()
=
+
+
F
U
J
K
U
M
1
*
&&
éq 2-1
with
K
stamp real elastic rigidity,
M
stamp of mass and
the damping coefficient
hysteretic. Let us note that one often speaks about complex matrix of rigidity.
2.1
“Total” damping hysteretic
This modeling, easy to implement, corresponds to:
(
)
()
=
+
+
-
F
U
K
K
J
M
2
éq
2.1-1
It is currently available, by using the operator
COMB_MATR_ASSE
[U4.72.01], after having
assembled the matrix of rigidity to real coefficients, but it is of a low utility:
·
validation of algorithms of resolution,
·
useless for the industrial studies, because it does not make it possible to represent the heterogeneity of
the structure compared to damping (dissipation located in particular areas
structure treated with viscoelastic materials).
2.2
Damping hysteretic of the elements of the model
2.2.1 Characteristics
of damping
It is possible to build a complex matrix of rigidity starting from each element of the model,
as for real rigidity and the mass.
Two functionalities are usable:
·
the assignment of discrete elements, on meshs
POI1
or
SEG2
, by the operator
AFFE_CARA_ELEM
[U4.42.01]. The aforementioned makes it possible to define, with several modes of
description possible, a matrix of real rigidity for each degree of freedom and one
damping coefficient hysteretic to apply to this matrix.
AMOR_HYST:
éta
[R]
·
the definition of a characteristic of damping for any elastic material by the operator
DEFI_MATERIAU
[U4.43.01] by the key word:
AMOR_HYST:
éta
[R]
this material being then affected with the meshs concerned.
Code_Aster
®
Version
6.4
Titrate:
Note of modeling of the mechanical cushioning
Date:
08/11/02
Author (S):
J. PELLET
Key
:
U2.06.03-B
Page
:
6/6
Instruction manual
U2.06 booklet: Dynamics
HT-66/02/003/A
2.2.2 Calculation of the matrices of damping
For all the types of finite elements (of continuous, structural or discrete mediums), it is possible of
to calculate the complex elementary matrices corresponding to the option of calculation
“RIGI_MECA_HYST”
,
after having calculated the elementary matrices corresponding to the options of calculation
“RIGI_MECA”
.
Each elementary matrix is then of the form:
·
when the material
I
, of characteristics of damping hysteretic
I
, is affected with
the element
elem
K
*
elem = kelem (1 + J
I)
·
for a discrete element defined by a matrix of rigidity
kdiscret
and a coefficient
of damping hysteretic
K
*
elem = kdiscret (1 + J
)
This operation is possible with:
mel
[
matr_elem_DEPL_C] = CALC_MATR_ELEM
(
/
OPTION:
“RIGI_MECA_HYST”
MODEL:
Mo
[model]
CHAM_MATER:
chmat
[cham_mater]
CARA_ELEM:
will cara
[cara_elem]
RIGI_MECA:
rigi
[matr_elem_ *]
CHARGE
:
l_char
[l_char_meca]
);
Assembly of the complex matrix of rigidity
*
K
, starting from the elementary matrices is obtained
with the operator
ASSE_MATRICE
usual [U4.61.22]. It will be noted that one must use the same one
classification and same mode of storage as for the matrix of mass (operator
NUME_DDL
[U4.61.11]).
The loading used for the calculation of the matrix of real rigidity (
OPTION “RIGI_MECA”
) must be
informed by the key word
“LOAD”
for the calculation of the matrix of elementary rigidity complex.
2.2.3 Use of the complex matrix of rigidity
The complex matrix of rigidity
*
K
is usable for the direct linear dynamic analysis (key word
MATR_RIGI
) with the operator of dynamic response linear:
·
harmonic
DYNA_LINE_HARM
[U4.53.11]
For the search for eigenvalues, no functionality is currently available for
the use of the model of hysterical damping.
For the analyzes in modal base, no functionality is currently available for the use
model of damping hysteretic.