Transient vibrations of a fractional Zener viscoelastic cantilever beam with a tip mass

Abstract The presented work concerns the kinematically excited transient vibrations of a cantilever beam with mass element fixed to its free end. Euler–Bernoulli theory and fractional Zener model material are assumed. A Caputo derivative is used formulate viscoelastic law. characteristic equation, modal frequencies, eigenfunction orthogonality conditions achieved for considered. equations motion system solved numerically. numerical solution multi-term differential equation obtained by means conversion mixed ordinary equations, each order $$0 < \gamma \le 1$$ 0 < ? ? 1 . time histories during passage through resonance calculated. comparison between responses an integer presented. calculations performed reveal that use damping affects on system. calculated show some values $$\gamma$$ , amplitudes occurring in area second greater than those first resonance, which does not occur case derivative. Moreover, evaluation made difference results using Kelvin model. It shown physical parameters, calculation both models virtually same models, simpler, Kelvin–Voigt can be instead This simplifies decreases needed make calculations.