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Note that the consistent mass matrix leads to higher frequencies than the “exact” values.
Also, notice that the error in the approximate frequency grows for the higher modes. In
order to make the consistent mass-based frequencies more accurate, more elements should
be included in the model.
The above example shows that in an eigenvalue analysis using consistent mass matrices,
errors may develop. The errors may fall into two categories: roundoff error and discretiza-
tion error. We will examine the discretization error here. It can be shown [Hughes, 1987] that
when the consistent mass and stiffness matrices are derived from the principle of virtual
work, the approximate eigenvalues satisfy
(
k
+
1
)/(
r
+
1
)
h
ch
2
(
k
−
r
)
λ
λ
≤
λ
≤
λ
+
(10.39)
h
where
from the finite
element solution,
h
is the mesh parameter which is the diameter of the smallest circle that
contains the largest element of the mesh,
c
is a constant independent of
h
and
λ
is the
th exact eigenvalue,
λ
is the approximate value for
λ
1is
the highest order derivative appearing in the functional of the principle of virtual work
of Eq. (10.4), and
k
is the degree of the complete polynomial appearing in the element
shape functions. It can be seen from Eq. (10.39) that
,r
+
h
→
λ
λ
when
h
→
0
,
and the rate of
convergence depends on
h, k,
and
r
For a certain
h
, the accuracy of the approximation of
the eigenvalue deteriorates for higher modes. This is because 0
.
<λ
≤
λ
≤
λ
...
,
and
1
2
3
becomes larger,
ch
2
(
k
−
r
)
λ
(
k
+
1
)/(
r
+
1
)
h
when
has a larger interval in
which to vary. Thus, in order to make the higher frequencies more accurate, more elements,
i.e., a finer mesh, should be included in the model.
also becomes larger and
λ
EXAMPLE 10.6 Eigenvalues for a Beam Using a Lumped Mass Model
Use the displacement method to find the natural frequencies of the beam of Fig. 10.6, using
the lumped parameter model discussed in Section 10.1.2.
Begin by assembling the global stiffness and mass matrices. For a lumped mass model of
a beam element of length
, with the mass concentrated equally at the two ends, without tak-
ing into account the mass associated with the rotational DOF,
1000
0000
0010
0000
=
ρ
2
m
i
i
=
1
,
2
(1)
where
v
is the mass per unit length. Assemble the system mass
matrix by following the procedure outlined in Example 10.5. This leads to the assembled
mass matrix
=
[
w
θ
w
θ
b
]
T
,
and
ρ
a
a
b
10
0
0
00
00
0
0
00
=
ρ
2
001
+
10
+
000
M
(2)
000
+
00
+
000
00
0
0
10
00
0
0
00
The global stiffness matrix is still given by Eq. (3) of Example 10.5.
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