Arch Appl Mech DOI 10.1007/s00419-014-0969-0 SPECIAL John T. Katsikadelis Generalized fractional derivatives and their applications to mechanical systems Received: 19 February 2014 / Accepted: 23 October 2014 © Springer-Verlag Berlin Heidelberg 2014 Abstract New fractional derivatives, termed henceforth generalized fractional derivatives (GFDs), are intro- duced. Their definition is based on the concept that fractional derivatives (FDs) interpolate the integer- order derivatives. This idea generates infinite classes of FDs. The new FDs provide, beside the fractional order, any number of free parameters to better calibrate the response of a physical system or procedure. Their usefulness and consequences are subject of further investigation. Like the Caputo FD, the GFDs allow the application of initial conditions having direct physical significance. A numerical method is also developed for the solution of differential equations involving GFDs. Mechanical systems including fractional oscillators, viscoelastic plane bodies and plates described by such equations are analyzed. Keywords Generalized fractional derivatives · Fractional differential equations · Numerical solution · Fractional differential viscoelastic models · Viscoelastic structures 1 Introduction Fractional differential equations appear more and more frequently in different research areas and engineering applications. Various physical phenomena in the fields of viscoelasticity, diffusion procedures, relaxation vibrations, electrochemistry, etc. are successfully described by differential equations involving derivatives of fractional (non-integer) order. Very often an issue is made, which is not unjustified, of the usefulness of the FDs in mechanics and of what they accomplish in comparison with the classical derivatives. A concise answer would be that fractional calculus allows the investigation of the nonlocal response of mechanical systems. Fractional derivatives are non-local operators since they take into account the memory of the system (if time is the independent variable) as in viscoelasticity or non-local action (if space distance is the independent variable) as in non-local elasticity. In short non-locality is the main characteristic of fractional calculus when compared with integer-order calculus. It would not be excessive to say that simulating physical systems using only integer-order derivatives are similar to doing arithmetic (algebra) using only integer numbers. Thus, fractional calculus may be viewed as step from integer numbers to real ones. The related literature on the applications of fractional calculus in mechanics is extensive. The interested reader is advised to look in Ref. [1–5] and in the references therein. Several definitions of the fractional derivative of a continuously differentiable function u (t ) have been introduced [6], inter-alia, the Riemann-Liouville (R–L), the Grünwald–Letnikov (G–L) and the Erdelyi–Kober (E–K) type FDs [4]. The R–L derivative is the most popular in fractional calculus and has played an important role in the development of the theory. Modern technology uses fractional derivative models for better description of mechanical properties. This leads to fractional differential equations and thus to the necessity to apply J. T. Katsikadelis (B ) School of Civil Engineering, National Technical University of Athens, Athens, Greece E-mail: jkats@central.ntua.gr