A comparison of numerical methods applied to a fractional model of dampingmaterials

The use of fractional derivatives in the constitutive equations of systems with damping materials provides a powerful tool for modeling these systems because the model does not exhibit many of the shortcomings of those based on integer-order derivatives. The resulting equations of motion possess clo sed-form solutions only for single-degree-of-freedom systems and only for a small number of loadings. For practical applications, therefore, the equat ions of motion must be solved using numerical methods. This paper presents two numerical schemes to solve single-degree- and multi-degree-of-freedom s ystems with fractional damping subjected to a number of commonly used loadi ng conditions. The techniques employed are based on the central difference method and the average acceleration method. Whenever possible, the numerica l results are compared with the analytical solutions. The results of the tw o numerical methods are essentially identical, with the exact solutions for zero initial conditions, but differ for nonzero condition!; and large damp ing. For small damping, the average method has the advantage of its simpler formulation, I:specially with regard to the starting values. For arbitrary damping, however, the central difference method, :in view of its robustnes s, is the preferred method.