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The equation of motion of a structure installed with dampers modeled by fractional derivatives and
excited by earthquake induced ground motion x g (r) will also have fractional derivative terms:
M{3c} + CW + CD ff (W)+K f U} = -M{r}^(r) (3.4)
where M and C, respectively, are the mass and inherent damping matrix of the structure; [x] is
relative displacement vector; K, is the total stiffness matrix including the effect of the damper, and C
is the damping matrix due to the contribution of the damper. The solution of these equations for an
arbitrarily defined input such as earthquake acceleration time history (that are defined at discrete time
points) is a complicated task. Only a few analyses with earthquake induced ground motions have been
reported in the literature, and they have been primarily limited to the response calculation of single
degree of freedom systems. Makris and Constantinou (1991) used the Fourier transform methods to
obtain the numerical results. Koh and Kelly (1990) used the LI algorithm to solve the problem of a
single degree of freedom base-isolated system in which the damping characteristics of the base
isolation element was represented by a fractional derivative model. L-l signifies the use of the
Luoiville-Riemann definition for the fractional derivative. The Gl algorithms using the Grundwald
representation have also been used. These algorithms can be extended for direct integration of the
equations of motion of the multi-degree of freedom (MDOF) structures installed with fractional
derivative modeled dampers. In these schemes, the regular derivatives in the equations of motion are
represented either by some finite difference schemes or by Newmark-/ approach. The development
of these numerical integration schemes for MDOF systems can be found in Chang (2002).
These equations can also be solved by a special eigenvector expansion approach, herein called as the
modal approach similar to that used for linear systems with regular derivatives, if the exponent of the
fractional derivative can be expressed as a rational number, that is a = l/m, i