Application of fractional calculus to the analysis and identification of viscoelastic systems

In the present study non-integer order or fractional derivative rheological models are applied to analysis of vibrating mechanical systems. Their effectiveness in fitting experimental data on wide intervals of frequency by means of a minimum number of parameters is first discussed in comparison with some classical integer order derivative models. A technique for evaluating an equivalent damping ratio for fractional derivative models is then introduced, and a numerical procedure for the experimental identification of the parameters of the Fractional Standard Linear Solid model is applied to a High Density Polyethylene (HDPE) beam in axial and flexural vibrations. When applying fractional derivative rheological models to vibrations the resulting equations of motion, after a fractional state-space expansion, can still be studied in terms of modal analysis, but the growth in matrix dimensions is in general so fast to make the calculations too cumbersome, especially for finite element applications. Aimed at reducing this computational effort, a discretization technique for continuous structures is also presented, based on the Rayleigh-Ritz method. The Fractional Standard Linear Solid is again the adopted model, but the same method may be applied to problems involving different rheological linear models. Finally, examples regarding two different continuous structures are proposed and discussed in detail.