Fractional Derivatives and Complex Modes of Vibration

As described herein, we develop a method to obtain a fractional derivative response of a vibratory system with multiple degrees of freedom (DOF). To obtain fractional-order derivatives/integrals of dynamic response at a certain point on a structure presents technical difficulties because measurements of fractional-order derivative/integral responses in structural dynamics yield some implementation techniques. However, our method obviates special sensors with additional signal-conversion functions. Therefore, existing displacement and velocity sensors can work. Obtaining fractional derivative responses can be accomplished using three methods. Using any of the three methods, fractional states can be expressed with complex vibration modes, in which each point of the system oscillates with a phase that is different from “in-phase” or “out-of-phase.”