An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives

Abstract

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.