Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 612

This dynamic stiffness matrix has yielded the exact natural frequencies. This
is because no approximate, e.g., consistent or lumped mass, modeling was em-
ployed. Use of exact mass modeling permits a coarser mesh, i.e., fewer elements,
to be employed in the model.

A determinant search, as would be required by Eq. (10.42) or Eq. (9) in Example 10.8, is
often a numerically cumbersome, inefficient process that perhaps should be avoided. The
lack of efficiency is due to the need to compute the value of the determinant for each trial
value of the frequency. An in-depth review of the problems associated with the use of the
dynamic stiffness matrix is given in Kim (1993). The determinant search can be avoided
by establishing an eigenvalue problem in which the structural matrices are not functions
of the frequency parameter

. This was the case for the eigenvalue problem utilizing the
static stiffness matrix and the consistent mass matrix, or the lumped mass matrix which
was presented earlier in this section.
Further study of the accuracy of structural matrices along with the corresponding fre-
quencies is merited. Begin with a frequency dependent mass matrix obtained in a fashion
similar to the formation of the dynamic stiffness matrix k i dyn . Place the exact (frequency
dependent) shape function N in (Eq. 10.5).

ω

m i N T N dV

=

V γ

(10.43)

or, in the case of a beam element,

b
a ρ

m i N T N dx

= (10.44)

The same exact mass matrix is obtained by differentiating the element dynamic stiffness
matrix by the frequency parameter ω [Fergusson and Pilkey, 1992 and Richards and Leung,

1977]

k i dyn
∂ω

∂

m i

=− (10.45)

2

It is possible to economize somewhat in the calculation of a frequency-dependent mass
matrix by constructing a “quasi-static” mass matrix m i defined as

m i N 0 γ = N dV (10.46)

V

where N 0 is an element static shape function such as the N given in Eq. (10.8). In this notation
with the static shape function denoted by N 0 , the consistent mass matrix of Section 10.1.1
is given by

N 0 γ N 0 dV (10.47)

V

Define a frequency-dependent stiffness matrix k i (ω) in terms of the dynamic stiffness ma-

trix k i dyn (ω) and a frequency-dependent mass matrix m i

(ω)

as

k i dyn (ω) = k i

2 m i

(ω) − ω (ω) (10.48)

Symbolic manipulation software is frequently helpful in implementing the operations re-
quired in forming the frequency-dependent structural matrices. Assemble the global ma-
trices M using the element matrices m i and k i

(ω) and K (ω) (ω) (ω). The eigenvalue problem

then is embodied in

2 M

(ω) − ω (ω)

] φ

=

[ K 0 (10.49)