Analytic Theory of Seven Classes of Fractional Vibrations Based on Elementary Functions: A Tutorial Review

This paper conducts a tutorial review of the analytic theory of seven classes of fractional vibrations based on elementary functions. We discuss the classification of seven classes of fractional vibrations and introduce the problem statements. Then, the analytic theory of class VI fractional vibrators is given. The analytic theories of fractional vibrators from class I to class V and class VII are, respectively, represented. Furthermore, seven analytic expressions of frequency bandwidth of seven classes of fractional vibrators are newly introduced in this paper. Four analytic expressions of sinusoidal responses to fractional vibrators from class IV to VII by using elementary functions are also newly reported in this paper. The analytical expressions of responses (free, impulse, step, and sinusoidal) are first reported in this research. We dissert three applications of the analytic theory of fractional vibrations: (1) analytical expression of the forced response to a damped multi-fractional Euler–Bernoulli beam; (2) analytical expressions of power spectrum density (PSD) and cross-PSD responses to seven classes of fractional vibrators under the excitation with the Pierson and Moskowitz spectrum, which are newly introduced in this paper; and (3) a mathematical explanation of the Rayleigh damping assumption.