Steady-state nonlinear vibrations of a fractional cantilever beam with an eccentric tip mass element

The paper presents nonlinear steady-state vibrations of a cantilever beam with a eccentric mass element attached to its free end. The Euler-Bernoulli beam theory and the fractional Kelvin-Voigt model of the beam material are assumed. A fractional Caputo derivative of an order of \((0 < γ ≤ 0)\) is used to formulate a viscoelastic material law. The beam nonlinear equation of motion is derived using Hamilton's principle. The characteristic equation, modal frequencies, eigenfunction and orthogonality condition are obtained for the linear beam vibrations. The multiple scales method is used to solve the equation of motion of the beam in the vicinity of the first resonance. The first eigenfunction of linear vibrations is used as an approximate solution for the nonlinear vibrations. The impact of order of the fractional derivative on the beam vibrations is studied. The performed analysis shows that the order of the fractional derivative affects the nonlinear oscillation. An increase of the order of a fractional derivative decreases amplitude of vibrations.