On the numerical evaluation of fractional derivatives in multi-degree-of-freedom systems

The numerical evaluation of fractional derivatives requires a high number of computations due to their non-local character. Particularly in the case of a multi-degree-of-freedom (mdof) system that is described by a fractional differential equation in time and is solved numerically by time integration, the numerical effort and the storage requirements explode. Therefore, a new method for the numerical evaluation of fractional derivatives is presented which reduces the numerical effort of mdof systems drastically. The algorithm is based on the Grünwald definition of fractional derivatives and, in contrast to some other concepts, maintains its benefits. The algorithm is applied to the equation of motion of a viscoelastic member, the properties of which are described by a fractional constitutive equation. Comparative calculations demonstrate the accuracy of the algorithm and the reduction in computation time and storage requirements.