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Kim1993-MassMatrices-Eigenproblems.pdf
Con~pur~n & SW< ,WCY Vol. 46. No 6. pp 104-1048.
Printed in Great Britain.
A REVIEW
1993
0045-7949/93 S6.00 + 0.00
c’ 1993 Pergamon Press Ltd
OF MASS MATRICES
FOR EIGENPROBLEMS
KI-OOK KIM
Department
of Aerospace
Engineering,
(Receiued
Inha University,
2 January
Inchon
402-75
I, Korea
1992)
Abstract-Various
nonconsistent
mass matrices have been presented to achieve more accurate natural
frequencies
in eigenproblems
of the finite element analysis. The matrices are obtained
as a linear
combination
of lumped and consistent mass matrices. For an improved accuracy, the consistent mass
should be more weighted than the lumped mass. Instead of the mass combination,
the interpolation
functions can be combined to give nonconsistent
mass matrices, which show the same tendency. To find
a nonsingular
lumped mass matrix for the bending vibration of beams, a translational
inertia has been
proposed for rotational
degrees of freedom. The inertia effect is highly overestimated
and hence lower
natural frequencies are obtained. When combined with the consistent mass matrix, however, the modified
lumped mass matrix gives a significant improvement
for the natural frequencies of intermediate
and higher
modes. A simple corrective method was applied to get a better estimation
of the natural frequencies
through the use of the frequency dependent stiffness and mass matrices. The method shows high accuracy
without complicated
calculations.
INTRODUCTION
In eigenproblems
of
finite
element
analysis,
both
in fact, functions of
natural frequencies [I]. In the analysis, however, the
natural frequencies are not known a priori.
Hence,
the static stiffness and mass matrices are employed to
obtain the natural frequencies,
which can be good
approximations
for the lowest modes.
Usually, the highest frequencies are considered to
be less important in finite element dynamic analysis.
Firstly, it is difficult, using the finite element method,
to calculate the exact values in a continuous
structural system which has an infinite number of degrees
of freedom. Secondly, the highest frequencies
may
not have any practical meaning in large finite element
systems when the real structures do not vibrate with
those high frequencies.
In bending vibrations,
higher modes have many
points of zero displacement
which are called nodes.
As the number of nodes increases with each mode,
severe wrinkling can occur in the vibration
mode
shape. Hence, for the highest modes, the effect of
shear deformation
and rotation may not be neglected
and the simplified theory of beam bending is no
longer valid [2]. Therefore, in most cases where the
rotational
inertia of a finite element is ignored in
the formulation
of mass matrix, the finite element
analysis tries to get accurate values for the natural
frequencies of lower and intermediate
modes.
It is well known that lumped mass matrices overestimate the mass effect and hence give lower natural
frequencies than the exact ones. On the other hand,
the underestimation
of consistent mass causes higher
natural frequencies. Hence, the lumped and the consistent matrices are applicable
only for the lowest
modes.
stiffness
and
mass
matrices
are,
While consistent
mass matrices are formulated
using the same interpolation
functions
that are
assumed in the generation of stiffness matrices, other
methods are employed to get nonconsistent
mass
matrices which may give more accurate approximations for the natural frequencies of intermediate
modes as well as lower ones.
For the axial vibration of bars, NASTRAN
has
adopted the average of the lumped and the consistent
mass matrices [3]. The simple approximation
gives
excellent accuracy in lower modes, which has turned
out to be a second-order
formulation.
For intermediate modes, however, moderate
errors are still
involved.
Stavrinidis
et al. [4] have derived
analytical
expressions
to obtain nonconsistent
mass matrices
for several finite elements. Although good approximations to the exact natural frequencies are achieved
from the lowest through
to the highest modes,
the method loses accuracy in the lower modes which
are the major concern in finite element dynamic
analysis. Of course, the minor differences may be
insignificant in an engineering sense and hence can be
neglected.
The procedure
discussed here was motivated
by
work done by Stavrinidis et a/. on nonconsistent
mass
matrices. It is interesting to note that various nonconsistent
mass matrices, including the Stavrinidis
mass matrix in the axial vibration of bars, can be
obtained as a linear combination
of the lumped and
the consistent
mass matrices. In general, the consistent mass matrix should be more weighted than
the lumped mass in the combinations.
While the
Stavrinidis
mass matrix provides
good approximations to the natural frequencies for all the modes,
other matrices can give improved accuracy for modes
other than the highest ones.
1041
KI-CKIK KIM
1042
Another approach is to obtain the mass matrices
through the use of combinations of the interpolation
functions of the lumped and the consistent mass
formulations. The Stavrinidis matrices have simple
forms of interpolation functions. For most matrix
combinations, however, it is very difficult to find the
relevant interpolation
functions. The method of
matrix combination is simple and only has small
differences from the function combination.
In the present paper, various matrix combinations
have been studied to compare the accuracy in finite
element eigenproblems. It seems that nonconsistent
mass matrices can provide more accurate natural
frequencies than the lumped and the consistent mass
formulations. While some approaches show excellent
accuracy in lower modes, other methods give good
natural frequencies for intermediate or higher modes.
To get a nonsingular lumped mass matrix for the
bending vibration of beams, a translational inertia
has been adopted for rotational degrees of freedom.
As can be expected, the inertia effect is highly overestimated and hence lower natural frequencies are
obtained. When combined with the consistent mass
lumped mass
matrix, however, the nonsingular
matrix gives a significant contribution
to the
improvement of the intermediate natural frequencies.
Table
1. Mass matrices
Form
Various nonconsistent mass matrices are expressed as
a linear combination for problems of beam bending.
An iterative method has been presented to improve
the accuracy of the natural frequencies through the
use of the frequency dependent mass and stiffness
matrices. If an approximation is obtained, a simple
corrective procedure can be used in order to have
a better estimation of the natural frequencies. No
additional matrix decomposition is required when the
static stiffness matrix has been decomposed in the
first approximation as in inverse iteration methods.
It should, however, be noted that the application of
the frequency dependent matrices gives significant
improvement, especially for lower modes.
MASS MATRICES FOR AXIAL
OF A BAR
For the axial vibration of a two-node bar element
with length L, the usual lumped and consistent mass
matrices are written as
B
where m is the mass per unit length of the bar.
Linear combinations
of the lumped and the
consistent mass matrices give various forms of
nonconsistent mass matrices
Mass matrix
I1
-[
5
1
1
I I
et al.
1
0
0
I
0
3
mL
I
I
8
8
16 I
-[
7
._
7
3
2
Proposed
mL
2
?
4
Averaged
2
3
s
s
4 I
10
1 4
-[
I
3
mL
3
4
4
8
-[
13
mL
2
I
0
I
-16
1
2
mL
Table
2. Natural
I
1
1
1
frequencies
Errors
I
2
3
4
5
6
7
8
where the constraint u + fi = 1 is imposed for mass
conservation. The magnitude of the parameters a and
/I dictates relative emphasis on the lumped and the
consistent mass matrices. Further study is required to
clarify the physical meaning of the constraint and to
explain the mass effect in the case where 51+ fl # 1.
For typical values of the parameters, the corresponding mass matrices are obtained as shown in
Table 1.
As a numerical example, a uniform bar with eight
elements was used for the analysis of axial vibration
(see Fig. 1). Tables 2 and 3 show the percentage
14 61 6I
mL
-[
2
Consistent
Mode
No.
VIBRATION
for axial vibrations
a
Lumped
Stavrinidis
Fig. 1. Axial vibration of bar.
Lumped
-0.16
- 1.44
-3.97
-7.69
- 12.51
- 18.33
-25.02
- 32.42
Averaged
-0.00
-0.03
-0.20
-0.79
-2.24
-5.11
- 10.04
- 17.43
of a clamped-free
bar
(%) of each form
Proposed
+0.03
+0.26
+0.61
+0.78
+0.29
- 1.61
-5.81
- 13.03
Stavrinidis
e/ al.
+ 0.08
f0.70
+ 1.86
+3.29
f4.48
+4.47
+ 1.83
-4.88
Consistent
f0.16
+ 1.45
f4.05
+7.92
+ 12.79
+ 17.69
f20.14
+ 15.94
Review
Table
of mass matrices
3. Natural
frequencies
Errors
Mode
No.
I
2
3
4
5
6
7
Lumped
Averaged
-0.64
-2.55
-5.68
-9.97
- 15.31
-21.58
-28.64
-0.00
-0.08
-0.42
-1.38
-3.46
-1.29
- 13.42
OF INTERPOLATION
of a clamped<lamDed
functions
I-E,
Consistent
+0.32
+I.23
+2.56
+ 3.96
+4.70
+3.58
-0.95
+0.64
f2.59
+ 5.83
+ 10.27
+ 15.35
+ 19.46
+19.15
For the lumped mass matrix, the functions
defined as follows:
X
iv,(x) =;.
(3)
N2
1
0
L
1
<x<L
0
1
If nonconsistent mass matrices are to be calculated
using combinations
of the interpolation
function
[eqns (3) and (4)] instead of the mass matrices in
eqn (l), the corresponding interpolation functions
are obtained
4
N,
o<x<;.+(1-f) s(:)
<x<L
P(l-;)
a+/?(;)
?
Table 4. Number of elements and natural frequencies
Errors (%) of each form
No. of Elements
Mode
No.
Averaged
Proposed
Stavrinidis
et a/.
Consistent
1
1
- 1.38
f0.66
2
I
2
-0.08
-7.29
f0.44
-3.37
+ 1.23
+ 3.58
+2.59
+ 19.46
1
2
3
-0.02
-1.38
-11.10
f0.21
f0.66
-6.79
+0.56
+3.96
+ 1.03
+1.15
+ 10.27
+ 20.02
1
2
3
4
-0.00
-0.42
-3.46
- 13.42
f0.12
+0.74
-0.43
-9.02
+0.32
+2.56
+4.70
-0.95
+0.64
+5.83
+ 15.35
+ 19.15
I
-0.00
2
3
4
5
-0.17
-1.38
-5.50
- 14.95
+0.08
+0.57
+0.66
-1.91
- 10.53
+0.20
+ 1.72
+ 3.96
f4.35
-2.38
+0.41
+3.73
+ 10.27
+18.11
+ 18.15
3
4
5
can be
(4)
X
of
N,
oix<;
L
N,(x)=
bar
Stavrinidis et al.
+0.12
+0.44
+0.74
+0.66
+0.43
-3.37
-9.02
FUNCTIONS
As is well known, the interpolation
the consistent mass are
1043
(%) of each form
Proposed
errors of the natural frequencies for clamped-free
and clamped+zlamped bars, respectively.
The results indicate that the lumped and the
consistent matrices give significant errors and hence
are only acceptable for the first several frequencies.
It seems, in general, that the consistent mass matrix
should be more weighted than the lumped mass in
the combinations. While the Stavrinidis et al. mass
matrix provides good approximations to the natural
frequencies for all the modes, the averaged and the
proposed matrices show improved accuracy for
modes other than the highest ones.
Depending on the number of elements used in
finite element modeling, the accuracy of the calculated frequencies may vary. Table 4 shows the
errors obtained from each mass formulation for
the clamped-free bar. The same tendency can be
observed as in Tables 2 and 3.
COMBINATIONS
for eigenproblems
+3.96
+ 10.27
(5)
KI-OOK KIM
1044
Then, we get
s
L
L
~N,(x)~ dx =
0
mN2(x)’ dx
i 0
Fig. 2. Finite element
t6)
Hence, it results that
I_
m(N, + N,)‘dx
= mL(a + /?)2.
each nodal point has two degrees of freedom. Shear
deformation will be neglected. A finite element and
the corresponding nodal displacements are shown in
Fig. 2.
As is well known, the lumped and the consistent
mass matrices for bending vibration of beams are
written as
r
(7)
s0
The nonconsistent mass matrices for the values
of a and /l corresponding to those in Table 1 can be
calculated as in Table 5.
Compared with the matrix combinations,
the
nonconsistent mass matrices obtained from linear
combinations
of the interpolation
functions have
smaller diagonal and larger off-diagonal components
indicating strong coupling of the degrees of freedom.
It may be possible to get a function combination
which gives the same matrices that are shown in
Table 1. The expression is, however, complicated in
its formulation and subsequent calculations. It seems
that one of the most important properties of a mass
matrix is the ratio of the diagonal components to the
off-diagonal ones. Therefore, numerical calculations
have been omitted.
hl
= g
hl
from function
5
8
Lumped
I
B
0
-[11
mL
I
2
0
X
0
0
210
0
0
-22L
54
-22L
4L2
-13L
54
- 13L
156
13L
-3L2
22L
183
-llL
X
(8)
(9)
27
-1lL
27
6.5L
2L2
-6.5L
- 1.5L2
-6.5L
183
1lL
- 1.5L2
1lL
2L2
hl
=
s
X
163
-25.5L
47
9.5L
-25.5L
7.5L2
-9.5L
-3L?
47
-9SL
163
25.5L
9.5L
-3L2
25.51,
7.5LZ
z
2
I
4
0
1
(1
For numerical tests, five identical elements are used
in the finite element modeling as shown in Fig. 3.
5
et al.
;lO,
Stavrinidis et al. [4] have obtained a mass matrix as
I
Proposed
Consistent
0
Hence, the averaged mass matrix is obtained as
0
7
Stavrinidis
0
combinations
Mass matrix
4
Averaged
0
156
6.5L
‘I
0
=g
In this section, a uniform beam will be considered
illustrate the numerical results for bending
vibration. The motion is in the plane and hence
5. Mass matrices
0
0
to
Table
210
1
MASS MATRICES FOR BENDING VIBRATION
OF BEAMS
Form
for beam bending.
I
1
mL
2
1
-[6
1
2
1
2
3
4
5
[email protected]@[email protected]@
Fig. 3. Bending
vibration
of beam.
6
Review of mass matrices for eigenproblems
1045
Table 6. Natural frequencies of a simply supported beam
Errors (%) in natural frequencies
Mode
Lumped
Averaged
Stavrinidis et al.
I
2
3
4
5
6
7
8
9
IO
-0.01
-0.25
- I .86
-9.09
N/A
N/A
N/A
N/A
N/A
N/A
-0.00
-0.02
-0.31
- 1.38
-3.74
-9.38
-11.42
- 17.84
-23.27
-29.98
- 40.06
-0.02
-0.30
-2.53
+ 56.91
+ 34.09
+40.36
+ 57.58
+81.85
+ 79.83
It should be noted that the element displacement
represents the transverse deflection or translation and
hence the rotational inertia is not included in the mass
matrix.
Concerning the boundary conditions, two types
were considered: simply supported and cantilever
beams. Hence, the problem has 10 unconstrained
degrees of freedom. All of the IO modes have been
calculated in order to check the natural frequencies of
higher modes.
Tables 6 and 7 show the percentage errors of the
natural frequencies. Since the lumped mass matrix is
singular, only four and five frequencies are obtained
for each boundary condition. The several lowest
natural frequencies are good enough in both cases.
On the other hand, the averaged matrix gives
improved accuracy for lower modes due to the
contribution
of the consistent mass matrix. For
higher modes, however, excessive errors make the
calculations meaningless.
The Stavrinidis et al. and the consistent mass
matrices show accurate natural frequencies for lower
modes. Large errors are still observed in intermediate
and higher modes in which the translational inertia of
the rotational degrees of freedom plays significant
roles. Only half of the calculated natural frequencies
are acceptable in an engineering sense. Hence, twice
the number of degrees of freedom of the desired
natural frequencies are required in the calculations.
Therefore, the inertia of the rotational degrees of
freedom needs further study.
Consistent
+0.01
+0.17
+0.79
+2.30
+ 10.99
+ll.26
+ 19.02
+ 29.42
+ 37.63
+27.16
LUMPED MASS WITH INERTIA FOR
ROTATIONAL DOF
As is shown in eqn (8), the translational inertia
of the rotational
degrees of freedom in the
lumped mass matrix is set to zero. Hence, the
matrix becomes singular, which causes difficulties
in many cases. The singularity of the mass matrix
can be removed by providing the corresponding
translational inertia.
Hence, we get
~N,(x)~ dx = mL
1
mN,(x)' dx = !$
L
I
mN,(x)N,(x) dx = 0.
0
unit rotation
unit tnrnslation
II
L
0
++
y(x)=-x+$
N,(x)=1
Fig. 4. Interpolation
for finite element.
Table 7. Natural frequencies of a cantilever beam
Errors (%) in natural frequencies
Mode
Lumped
Averaged
IO
- I .80
- 5.90
-9.31
- 13.62
-23.44
N/A
N/A
N/A
N/A
N/A
-0.91
-3.07
-5.03
-7.69
- 14.26
+43.12
+ 36.82
+47.60
+70.13
+91.77
Stavrinidis ef al.
-0.00
-0.10
-0.70
-2.30
-4.75
- II.69
- 16.08
-20.82
-25.87
-33.32
Consistent
+o.oo
+0.05
+0.36
+1.17
+ 1.58
+ 12.97
+ 18.29
+28.85
+42.51
+ 67.83
(12)
KI-OOK KIM
1046
Table 8. Interpolation functions for lumped mass matrix
X
N,
N,
N,
N,
04x<;
1
2(-x+$)
0
0
L
? <x<L
0
0
1
Errors (%) of each form
Mode
2(-X +;L)
Distributing the mass properties evenly onto the
nodal points, we obtain a nonsingular lumped mass
matrix as
M=z
210
0
0
0
0
17SL2
0
0
0
0
210
0
0
0
0
17.5Li
Table 10. Natural frequencies of a simply supported beam
1
. (13)
In finite element formulations, the element displacement w(x) is expressed in terms of the nodal
displacements W, , 0,) W,, and OZ.
Nonsingular
- 1.62
-6.15
- 12.88
- 20.99
- 29.80
- 38.79
-47.54
-55.71
-63.08
-69.62
1
2
3
4
5
6
7
8
9
10
B[Mcl.
Form
c(
9. Nonconsistent
B
~P-‘ol- ~2W21 - KDIP’)
1
(15)
[&]2-
2
3
s
s
mass matrix
1
8
9
1
Mass matrix
-6.5L
21
6.5L
-6.5L
- 1.5L’
183
IIL
IIL
10.75L’ I
- 13.2L
32.4
9.4L’
-7.8L
- 1.8L’
177.6
l3.2L
-7.8L
13.2L
- 19.56L
9
quadratic
E {O), (16)
5.5L’
-11.56L
-2.67L=
48
-I
l.56L
162
19.56L
7.8L
9.4L’
1
I l.56L
-2.67L’
19.56L
5.5L’
1
7’8
for beam bending
- 1.8L’
Proposed
’
45AE [ 718
- l.5L’
Previous
FOR
where I = w2. [K,] and [M,,] are the usual static
stiffness and consistent mass matrices. The matrices
[M,] and [&] represent the second-order dynamic
correction. It should be noted that the formula gives
good approximations for lower modes only. For the
natural frequencies of higher modes higher-order
terms are required.
For the axial vibration of bars, the consistent
formulation gives
10.75L’
1
-0.17
-0.60
- 1.05
-1.33
+2.74
-0.81
- 1.60
-4.58
-11.78
-24.41
Przemieniecki [ 1] has proposed the
approximation for natural frequencies
-11L
Averaged
Proposed
HIGHER-ORDER
APPROXIMATION
EIGENVALUES
The percentage errors of the natural frequencies for
the typical nonconsistent mass matrices in Table 9
are illustrated in Tables 10 and 11. The nonsingular
lumped mass highly overestimates the mass effect.
The third combination
gives excellent approxi-
Table
-0.65
-2.51
-5.40
-9.22
- 12.25
- 19.82
-27.08
- 35.47
-44.58
-53.88
(14)
The corresponding interpolation functions can be
defined as in Table 8.
As in the axial vibration of bars, nonconsistent
mass matrices for bending can be obtained as a linear
combination of the consistent and the nonsingular
lumped matrices
[MNCI = aPfJ+
Previous
-0.81
-3.15
-6.78
-11.54
- 16.10
- 24.23
-32.17
-40.78
- 49.60
-58.19
mations for the natural frequencies of intermediate
and higher modes.
NK,l-
w(x) = N, W, + N,0, + N3W, + N4Q2.
Averaged
I
(17)
Review of mass matrices for eigenproblems
Table 11.Natural frequencies of a cantilever beam
Errors (%) of each form
Mode
Nonsingular
1
-2.53
- 10.09
- 17.12
-23.77
- 30.26
-37.14
- 44.42
-51.87
- 59.20
- 64.66
2
3
4
5
6
7
8
9
IO
Averaged
Previous
Proposed
- 1.29
-5.44
-9.94
- 14.85
- 19.97
-24.14
- 30.50
- 37.39
-45.18
-53.30
- I .03
-4.41
-8.17
- 12.41
- 17.02
-20.10
-26.01
- 32.45
-40.10
-48.83
-0.29
- I .25
-2.27
-3.34
-5.08
-0.81
-2.77
-4.26
-8.22
- 17.33
When an approximation
W,PP
or
A+,,
is known,
for the natural frequency
the expression can be
Even when the exact frequencies are used for
the approximation, the quadratic correction cannot
get the exact natural frequencies, as shown in the
third column. On the contrary, the approximate
values from [K,,] and [M,] give more accurate
natural frequencies in the fourth column. This is
because the higher values of o0 magnify the mass
effect in the dynamic correction and hence lower the
approximation close to the exact natural frequencies.
The corrective method can be applied to the
problem of beam bending. Note that good accuracy
can be obtained for natural frequencies in the
neighborhood of an approximation. Hence, vector
iteration methods which calculate the eigenvalues
one by one, are preferred in the eigensolution.
rewritten as
]&I {Xl = 4W,l+
1047
CONCLUSIONS
&ppW,l
- [WI 1x1
(18)
which, in fact, represents a dynamic correction for
the mass matrix.
For the appropriate value Apppof each mode, an
initial analysis with [K,] and [M,,] can be performed
using iterative or transformation methods. As is well
known, transformations
are preferred when all the
eigenvalues are desired.
Table 12 shows the comparison of the natural
frequencies obtained using the consistent mass and
the quadratic correction. The bar was modeled with
eight elements as shown in Fig. 1. The first column
represents the exact theoretical values. The natural
frequencies from [K,] and [MO] are shown in the
second column and are used as the approximate
values in the quadratic formulation.
Significant
improvement can be observed, especially for lower
modes. For higher modes, however, moderate errors
still remain. Therefore, it seems that more emphasis
should be put on minimizing the error of the lower
modes rather than reducing the average error of all
the modes including the highest ones.
It has been proved that the proper forms of
nonconsistent mass matrices give accurate natural
frequencies for higher modes as well as lower ones.
Various forms can be obtained simply as a linear
combination of the lumped and the consistent mass
matrices. Depending on the modes desired for the
natural frequencies, a proper form may be selected.
The procedure is simple and straightforward.
It
seems, in general, that the consistent mass matrix
should be more weighted than the lumped matrix.
Another method is to use a combination of the
interpolation functions of the lumped and consistent
mass matrices. The function combination
shows
a small difference from those obtained through
the matrix combination with the same parameters.
In practice, however, the same tendency can be
observed. Hence, the matrix combination has been
tested in numerical calculations.
It is believed that the constraint Q + fl = 1 assigned
on the combination
implies the conservation of
element mass. The physical meaning needs to be
clarified. Further study will be required to determine
the inertia effect in the case where c( + jI f 1.
Table 12. Consistent mass and auadratic correction
Errors (%) in natural frequencies
Modes
n
I .5708
4.7124
7.8540
10.9956
14.1372
17.2788
20.4204
23.5619
[&I?&1
1.5733
(+0.16)
4.7808
(+ 1.45)
8.1719
( + 4.05)
1I .8662
( + 7.92)
15.9455
(+ 12.79)
20.3358
( + 17.69)
24.5327
( + 20.14)
27.3182
(+ 15.94)
Quadratic form with w.++,
o&Xl
WO
1.5708
1.5708
( + 0.00)
( + 0.00)
4.7147
4.7128
( + 0.05)
(+ 0.01)
7.8827
7.8601
( + 0.37)
( + 0.08)
II.1415
11.0344
(+ 1.33)
( + 0.35)
14.6092
14.3000
(+ 3.34)
(+ 1.15)
17.7912
18.4015
( + 2.97)
( + 6.50)
22.3713
21.5801
( + 9.55)
( + 5.68)
24.7700
25.3541
(+ 7.61)
(+ 5.13)
1048
KI-WOK KIM
In the problem of beam bending, a translational
inertia has been proposed for rotational degrees of
freedom in order to get a nonsingular
lumped mass
matrix. The modified matrix highly overestimates
the
inertia effect and hence gives lower natural frequencies than the exact ones. If combined with the consistent mass matrix, however, the lumped mass matrix
shows significant improvement
for intermediate
and
higher modes. Although
it might be too early to
generalize the results, it is obvious that proper combination can give very accurate natural frequencies.
The procedure of dynamic correction through the
use of frequency dependent
structural matrices can
be utilized to get natural frequencies of better accuracy. Since the natural frequencies
obtained
from
the static stiffness and consistent mass matrices are
larger than the exact ones, the corrective mass term
is overestimated
resulting in approximation
closer to
the exact natural frequencies.
The method shows
significant improvement
especially for lower modes.
REFERENCES
I.
Them-J, of Murrix
Structurul
Ancdysis, pp. 273-339.
Dover, New York (1985).
2. L. Meirovitch,
Elements
of
Vihralion
Analysis,
pp. 220- 227. McGraw-Hill (1986).
J. S. Przemieniecki.
3. NASTRAN Theoretical Manual. MacNeal-Schwendler
Corp., Los Angeles, CA (1972).
4. C. Stavrinidis, J. Clinckemaillie and J. Dubois, New
concepts for finite-element mass matrix formulations.
AIAA
Jnl 27, 1249-1255
(1989).
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