Extension of Linear Systems by Fractional Derivatives

The model description by linear, time-invariant systems (LTIs) has been established for many, many years. All modal analysis identification methods were based on these systems. The algorithms for OMA and classical, FRF-based EMA are fully developed and have a high maturity state. However the description of real structures is not sufficient in some cases. The demand is also to improve the mathematical model, which is base for the parameter estimation. The paper shows how fractional calculus (fractional derivatives) extends the linear model in a genuine way. This helps to describe especially highly damped systems. It is shown how material behavior can varied by fractional derivatives. On this basis a complete system of fractional differential equations can be established and solved. Especially the systems with excitation can be solved in frequency domain by related frequency response functions. These FRFs can be fitted to measured FRFs of real structures. The extended model parameters are identified. Practical examples are shown after a base explication for a simple generic oscillator. Currently the method starts with a classical linear parameter estimation and uses optimization techniques to find the fractional parameters. An outlook to the application within OMA is given.