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We note that the above uncoupling procedure (i.e. damping matrix [C] is also diagonal) will
work only in the presence of proportional damping (i.e. if , where a , a 2
1
are constants. In fact, [C] can be expanded in terms of powers of [M] and [K] and still
uncouple eqn (2.66) into N nodal equations (Bathe, 1982).

2.3.3 Damping in MDOF systems

In analogy with the SDOF oscillator, a damping coefficient βor a damping ratio are defined
(rather arbitrarily, given the coupling inherent in MDOF systems) for each modal equation as
βor n , respectively. From a practical viewpoint, the first modal equation corresponding to
n
the lowest eigenfrequency (or highest modal period) and which approximates the response of
the system to quasi-static application of the load, is the dominant one. Thus, it is essential that
correct values of damping are prescribed to this mode and also to a few more of the lower
ones. Furthermore, it is customary to assign rather large values of damping to the higher
modes so as to dampen out unwanted high frequency oscillations in the system.

2.3.4 Time integration methods

As previously mentioned, the equations of motion of an MDOF system need to be solved for
the displacement vector {U} as a function of time t. An alternative to modal analysis described
in the previous section is the use of time marching algorithms, which essentially integrate
over time the matrix differential equation (i.e. eqn (2.66)). There are many time marching
algorithms in use today, but they all fall into two basic groups: (i) direct methods and (ii)
predictor—corrector methods. The accuracy achieved through time integration is a key issue
t
and primarily depends on the size of the time step ∆ used, which is obviously judged with
respect to the magnitude of the natural periods of the system. Time stepping algorithms can
also be subdivided into unconditionally and conditionally stable ones. We note in passing that
algorithm stability does not necessarily imply accuracy. Obviously, time marching can be
used in conjunction with SDOF systems as well. Also, among the best known algorithms used
in structural dynamics are those by Houbolt, Newmark and Wilson (Bathe, 1982). Finally,
Table 2.1 presents Newmark’s method, while Figure 2.20 compares the results obtained by
various algorithms for the second storey displacement of a two-storey frame under lateral
loads which vary as sine functions in time. The exact results were obtained through modal
analysis in conjunction with the closed form solution given by eqn (2.23) for each of the two
modes.

2.3.5 Numerical example

We examine here the three storey plane frame with rigid girders shown in Figure 2.21. In
addition to the vertical static loads, the frame is subjected to dynamically