The use of fractional derivatives has proved to be very successful in describing the behavior of damping materials, in particular, the frequency dependence of their parameters. In this article the three-parameter model with fractional derivatives of order 1/2 is applied to single-degree-of-freedom systems. This model leads to second-order semidifferential equations of motion for which previously there were no closed-form solutions available. A new procedure that permits to obtain simple closed-form solutions of these equations is introduced. The method is based on the transformation of the equations of motions into a set of first-order semidifferential equations. The closed-form expression of he eigenvalues and eigenvectors of an associated eigenproblem are used to uncouple the equations. Using the Laplace transform method, closed-form expressions to calculate the impulse response function, the step response function and the response to initial conditions are derived.
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.
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