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Among the characteristics of the transfer matrix method that can cause numerical diffi-
culties is the build-up of roundoff and truncation errors by the progressive multiplication
of the form
U j
U 2 U 1 z a
z j
=
ยทยทยท
=
Uz a
(5.13)
Rather than converging to z j , this can converge to
z , the eigenvector of the first eigenvalue
of U . This is the result of the vectors in U becoming linearly dependent and the determinant
of the system of equations approaching the value zero. Even when z a is known exactly, the
solution can converge to
z .
Numerical difficulties also occur when the transfer matrix manipulations involve dif-
ferences of large numbers, which can lead to inaccuracies in the computations. This can
occur, for example, if a very stiff spring is included in the model. Also, this difficulty can be
expected if the effect of occurrences on one boundary is small on the other boundary, i.e.,
the solution appears to die out rapidly. In such cases, the calculation of the initial conditions
(
z a
)
can involve differences between large numbers, a hazardous numerical operation at
best.
Conclusions
There are two principal reasons for implementing a transfer matrix solution in combi-
nation with a displacement method. First, in many cases, the switch to the combination
approach yields a procedure that is more efficient computationally than the pure pro-
gressive transfer matrix multiplication. Second, the transfer matrix method tends to be
numerically unstable for many practical problems, whereas the displacement method may
eliminate such difficulties. There are many circumstances under which it is desirable to
use a combination of mixed (e.g., transfer matrix) and displacement methods. For example,
the displacement method can be used to compute the responses at the nodes, followed
by the use of the transfer matrix method to print out the responses along the element
between the nodes.
5.2
General Structural Systems
Structural systems of arbitrary geometry are usually analyzed with force or displacement
methods, whereas the transfer matrix method is appropriate only for structural systems
with a line-like configuration. However, transfer matrices are also useful for the develop-
ment of stiffness matrices or for computing displacements and forces along a member if
the nodal responses have been calculated with another method. In contrast to the trans-
fer matrix method, wherein the system matrix resulting from progressive multiplication
of element matrices remains small regardless of the system complexity, the force and dis-
placement methods develop system matrices whose size depends on the complexity of the
system model. Before considering details of the force and displacement methods, we will
define what may appear to be cumbersome notation. It is essential, however, to have generic
notation so that arbitrary configurations can be handled. Table 5.3 gives a summary of the
most frequently occurring notation in the matrix analysis of general structural systems. The
structural analysis methods to be developed here also provide the foundation for the finite
element method which can be applied to very general systems.
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