A Comparison of Numerical Methods Applied to a Fractional Model of Damping Materials

Abstract

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 short comings of those based on integer-order derivatives. The resulting equations of motion possess closed-form solutions only for single-degree-of-freedom systems and only for a small number of loadings. For practical applications, therefore, the equations of motion must be solved using numerical methods. This paper presents two numerical schemes to solve single-degree- and multi-degree-of-freedom systems with fractional damp ing subjected to a number of commonly used loading conditions. The techniques employed are based on the central difference method and the average acceleration method. Whenever possible, the numerical results are compared with the analytical solutions. The results of the two numerical methods are essentially identical, with the exact solutions for zero initial conditions, but differ for nonzero conditions and large damping. For small damping, the average method has the advantage of its simpler formulation, especially with regard to the starting values. For arbitrary damping, however, the central difference method, in view of its robustness, is the preferred method.