Based on a diophantine representation of the operational powers, a fractional derivative modelling scheme is developed to simulate frequency dependent structural damping. The diophantine set of powers is established by employing the curvature properties of the defining empirical data set. These together with a remezed least square scheme are employed to construct a Chebyschev like optimal differintegro simulation. Based on the use of the rational form resulting from the diophantine representation, a composition rule is introduced to reduce the differintegro simulation to first order form. The associated eigenvalue/vector properties are then explored. To verify the robustness-stability accuracy of the overall modelling procedure, correlation studies are also presented. Part I of this series focuses on the diophantine representation, its use in formulating a numerically more workable first order form as well as formal representations of its transient and steady state solutions. This will include investigations of the asymptotic properties of the various formulations. Part II will introduce the model fitting scheme along with a look at eigen properties and fitting effectiveness.
Padovan, J., and Sawicki, J.T. (1997). Diophantine Type Fractional Derivative Representation of Structural Hysteresis, Part I: Formulation. Computational Mechanics, Springer, 19(5), 335-341.
The final publication is available at Springer via http://dx.doi.org/10.1007/s004660050182