Fitted viscous damping matrix C
kj
Figure 7.4
: Fitted viscous damping matrix for the local case,
γ
= 0
.
5
, damping model 1
0
5
10
15
20
25
30
0
5
10
15
20
25
30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Fitted viscous damping matrix C
kj
Figure 7.5
: Fitted viscous damping matrix for the local case,
γ
= 2
.
0
, damping model 1
bution of the damping is revealed quite clearly and correctly. In both cases, the nonlocal nature
of the damping is hinted at by the strong negative values on either side of the main diagonal of the
matrix.
Because the symmetry preserving method uses a constrained optimization approach, numerical
accuracy of the fitting procedure might be lower compared to the procedure outlined in Chapter
5
.
132
Chapter 7. Symmetry Preserving Methods
0
5
10
15
20
25
30
0
5
10
15
20
25
30
0
1
2
3
4
5
Fitted viscous damping matrix C
kj
Figure 7.6
: Fitted viscous damping matrix for the local case,
γ
= 0
.
5
, damping model 2
0
5
10
15
20
25
30
0
5
10
15
20
25
30
0
1
2
3
4
5
Fitted viscous damping matrix C
kj
Figure 7.7
: Fitted viscous damping matrix using first 20 modes for the local case,
γ
= 0
.
5
,
damping model 2
In order to check numerical accuracy we have reconstructed the transfer functions using the com
plex modes obtained by using the fitted viscous damping matrix. Comparison between a typical
original and reconstructed transfer function
H
kj
(
ω
)
, for
k
= 11
and
j
= 24
is shown in Figure
7.11
, based on locallyreacting damping using damping model 1. It is clear that the reconstructed
7.2. Identification of Viscous Damping Matrix
133
0
5
10
15
20
25
30
0
5
10
15
20
25
30
0
1
2
3
4
5
Fitted viscous damping matrix C
kj
Figure 7.8
: Fitted viscous damping matrix using first 10 modes for the local case,
γ
= 0
.
5
,
damping model 2
0
5
10
15
20
25
30
0
5
10
15
20
25
30
0
2
4
6
8
Fitted viscous damping matrix C
kj
Figure 7.9
: Fitted viscous damping matrix for the nonlocal case,
γ
= 0
.
5
, damping model 1
transfer function agrees well with the original one. Thus the symmetry preserving viscous damp
ing matrix identification method developed here does not introduce much error due to the applied
constrains in the optimization procedure.
134
Chapter 7. Symmetry Preserving Methods
0
5
10
15
20
25
30
0
5
10
15
20
25
30
0
0.5
1
1.5
Fitted viscous damping matrix C
kj
Figure 7.10
: Fitted viscous damping matrix for the nonlocal case,
γ
= 0
.
5
, damping model 2
7.3
Identification of Nonviscous Damping
7.3.1
Theory
As has been mentioned earlier, out of several nonviscous damping models the exponential function
turns out to be the most plausible. In this section we outline a general method to fit an exponential
model to measured data such that the resulting coefficient matrix remains symmetric. We assume
that the mass matrix of the structure is known either directly from a finite element model or by
means of modal updating. Also suppose that the damping has only one relaxation parameter, so
that the matrix of the kernel functions is of the form
G
(
t
) =
μe

μt
C
(7.28)
where
μ
is the relaxation parameter and
C
is the associated coefficient matrix. In Chapter
6
a
method was proposed to obtain
μ
and
C
from measured complex modes and frequencies. This
method may yield a
C
matrix which is not symmetric. In this section we develop a method which
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 Fall '16
 Dr Nurasyikin
 Linear Algebra, Eigenvectors, Matrices, The Land, Complex number, Damping Properties