Algorithms for seismic analysis of MDOF systems with fractional derivatives

Viscoelastic dampers are often considered for use in structural systems to reduce their dynamic response. The frequency dependent storage and loss moduli of the viscoelastic material are sometimes modeled using the fractional derivatives. This introduces fractional derivatives in the equations of motion. Herein, several schemes, specialized for arbitrary inputs such as earthquake induced ground motions, are presented for direct numerical integration of such equations to obtain the dynamic response of multi-degrees-of-freedom damper–structure systems. Relative numerical accuracies of the proposed schemes are examined. The numerical analyses with fractional derivatives require that all previous time response values be used, but this is invariably time consuming. The possibility of discarding some early time response values without compromising the accuracy of the calculations is, thus, of natural interest. It is shown that such truncations are, indeed, possible but a straightforward omission of preceding time values can introduce unacceptable errors in the calculated responses. To reduce such errors a new algorithm is proposed.