Fractional Calculus Approach to Nonlocal Three-Dimensional Vibration Analysis of Plates

Abstract

The nonlocal three-dimensional vibration analysis of rectangular plates is investigated within the framework of fractional calculus in the sense of the Caputo fractional derivative. To show the effect of the fractional derivative on nonlocality, the frequency spectra of the plates with different boundary conditions and symmetry modes are carried out for the different orders of the fractional derivative $a$ and different values of the length scale parameter $l$ . The vibration analysis is obtained by the Ritz energy method, whereas Chebyshev polynomials are used as admissible functions. The results of the frequency spectrum demonstrate that the nonlocal effect decreases and the results get closer to the values of the classical frequencies as the order of the fractional derivative approaches the classical derivative $(\alpha \to 1)$ .