Vietnam Journal of Mechanics, Vietnam Academy of Science and Technology, pp. 259–273 DOI:10.15625/0866-7136/9772 VIBRATIONS OF FRACTIONAL HALF- AND SINGLE-DEGREE OF FREEDOM SYSTEMS Valentina Ciaschetti 1 , Isaac Elishakoff 2,∗ , Alessandro Marzani 1 1 Universit` a degli Studi di Bologna – DICAM, Italy 2 Florida Atlantic University, Boca Raton, USA ∗ E-mail: elishako@fau.edu Received June 05, 2017 Abstract. In this paper we study vibrations of fractional oscillators by two methods: the triangular strip matrix approach, based on the Gr ¨ unwald-Letnikov discretization of the fractional term, and the state variable analysis, which is suitable for systems with frac- tional derivatives of rational order. Some examples are solved in order to compare the two approaches and to conduct comparison with benchmark problems. Keywords: Fractional oscillator, fractional differential equation, numerical solution. 1. INTRODUCTION Oscillators play an important role in scientific and engineering fields, since they represent the simplest model adopted to observe the dynamic behavior of complex struc- tures. In particular, the so called fractional oscillator, that is a generalization of the clas- sical harmonic oscillator in the fractional calculus framework, has started to attract in- creasing attention in the last decade. It has been found that it can describe many systems by equations consisting of derivatives with fractional order, allowing to obtain more ac- curate and detailed results. A relevant issue is that the derivative of fractional order at any point of the domain has a local property only when the order is an integer number. For non-integer cases, the fractional derivative is a nonlocal operator and depends on the past values of the function (left derivative) or future ones (right derivative). Various methods has been developed for the solution of fractional oscillator: Gaul and Schmidt [1] presented a method based on the Gr ¨ unwald-Letnikov definition of frac- tional derivatives for numerically evaluating the fractional time-derivatives in conjunc- tion with the time –integration of fractional differential equations is particularly useful for problems which comprise multiple degrees of freedom systems with a high num- ber of time-integration steps and where the immense amount of computational efforts can be drastically reduced without losing the properties and the benefits of fractional c  2017 Vietnam Academy of Science and Technology