A Cardinal‐Based Approximate Method for Time Fractional Forced Vibration Analysis of Euler–Bernoulli Viscoelastic Beam

ABSTRACT In this work, fractional derivative in the Caputo form is employed to express time fractional forced vibration of Euler–Bernoulli viscoelastic beam equation. The viscoelastic behavior of the beam is described using a fractional Kelvin‐Voigt model, which implements a parallel combination of spring and spring‐pot elements. A hybrid approach based on the piecewise Chebyshev cardinal functions (PCCFs) and Chebyshev cardinal functions (CCFs) is utilized for solving this equation. In this way, some operational matrices of these basis functions are obtained. By considering a hybrid expansion of the problem's solution via the PCCFs for the temporal variable and CCFs for the spatial variable and employing the interpolation property of these basis functions, together with the collocation scheme, the solution of the problem is obtained via solving a related algebraic system of equations. In addition, we provide an analytical solution for the problem in classical mode. The results indicate that the proposed numerical approach exhibits excellent capability in solving the fractional differential equation. Furthermore, the study investigates the effects of different parameters such as the thickness‐to‐length ratio, type of applied loading, and fractional derivative order on the obtained results.