Civil Engineering Reference
In-Depth Information
For comparison purposes, we note that the exact solution [2] for the natural circular
frequencies
of a b
ar in axial vibration yields the fundam
enta
l natural circular frequency
as
1
.
571
/
L
√
E
/
and the second frequency as
4
.
712
/
L
√
E
/
. Therefore, the error for the
first computed frequency is about 2.5 percent, while the error in the second frequency is
about 19 percent.
It is also informative to note (see Problem 10.12) that, if the lumped mass matrix
approach is used for this example, we obtain
E
E
1
.
531
L
3
.
696
L
1
=
2
=
rad/sec
The solution for Example 10.4 yielded two natural circular frequencies for
free axial vibration of a bar fixed at one end. Such a bar has an infinite number of
natural frequencies, like any element or structure having continuously distributed
mass. In finite element modeling, the partial differential equations governing
motion of continuous systems are discretized into a finite number of algebraic
equations for approximate solutions. Hence, the number of frequencies obtain-
able via a finite element approach is limited by the discretization inherent to the
finite element model.
The inertia characteristics of a bar element can also be represented by a
lumped mass matrix, similar to the approach used in the spring-mass examples
earlier in this chapter. In the lumped matrix approach, half the total mass of the
element is assumed to be concentrated at each node and the connecting material
is treated as a massless spring with axial stiffness. The lumped mass matrix for a
bar element is then
10
01
=
AL
2
(10.65)
[
m
]
Use of lumped mass matrices offers computational advantages. Since the ele-
ment mass matrix is diagonal, assembled global mass matrices also are diagonal.
On the other hand, although more computationally difficult in use, consistent
mass matrices can be proven to provide upper bounds for the natural circular fre-
quencies [3]. No such proof exists for lumped matrices. Nevertheless, lumped
mass matrices are often used, particularly with bar and beam elements, to obtain
reasonably accurate predictions of dynamic response.
10.5 BEAM ELEMENTS
We now develop the mass matrix for a beam element in flexural vibration. First,
the consistent mass matrix is obtained using an approach analogous to that for the
bar element in the previous section. Figure 10.9 depicts a differential element of
a beam in flexure under the assumption that the applied loads are time dependent.
As the situation is otherwise the same as that of Figure 5.3 except for the use of
