Response spectra generation using a fractional differential model

Abstract

Over the past three decades, fractional calculus has been of increasingly interest to geotechnical researchers for studying time-dependent problems such as those that arise in Rheology. From previous investigations on Rheology (i.e., [1,2]) it has been shown that the accuracy requirements and the merits of hereditary phenomena with long memory make the fractional calculus an attractive mathematical tool for developing constitutive models. Although the advantages of this mathematical tool were mainly brought out by applying it to rheological problems, the authors are certain that fractional calculus can be used as alternative to integer models for evaluating design response spectra to carry out aseismic structural designs. Accordingly, this paper presents an alternative to generating response spectra of recorded or synthetic time histories using fractional differential equations (FDEs). As it is well known an acceleration response spectrum is the locus of the absolute value of maxima response accelerations of a single degree of freedom system (SDFS) for different natural periods and at a constant damping. A loci of response spectra is developed repeating the above mentioned computations for a set of different damping values. As broadly known, the equation of motion of the simple oscillator is obtained from the equilibrium of inertia, damping, elastic forces and dynamic excitation. This force equilibrium equation yields to a second order differential equation. In this paper only the first order derivative (velocity) of the second order differential equation of motion for the single degree of freedom system is considered fractional. Varying the order of the first order derivative (velocity) influences directly the damping capacity and tangentially the stiffness of the simple oscillator. Accordingly, its natural frequency and spectral ordinates of its response will be modified. As will be shown in the paper these modifications lead to more severe seismic conditions raising the question on the appropriateness of current design spectra specified in construction building codes and used in practice. Pondering this issue, the authors considered that it would be of great interest to use a fractional differential equation to model the simple damped oscillator and develop an alternative way to compute response spectra, particularly if these are more critical than those computed by currently used procedures. Thus, in this work a fractional differential equation (FDE) is used to generate response spectra of a number of recorded acceleration time histories. Furthermore, it is shown that the exponent α of the velocity term (dx/dt into dαx/dtα) of the FDE can be expressed in terms of the SDFS stiffness and damping parameters providing the exponent α physical meaning. Also, it is shown that the SDFS damping ratio is frequency dependent.