Transient vibrations of a fractional Kelvin-Voigt viscoelastic cantilever beam with a tip mass and subjected to a base excitation

In the paper, transient vibrations of a Bernoulli-Euler cantilever beam with a rigid mass attached at the end and subjected to base motion are presented. The tip mass centre of gravity coincides with the endpoint of the beam. The viscoelastic properties of the beam material are described using a fractional Kelvin-Voigt model. The Riemann-Liouville fractional derivative of an order of 0 < γ ≤ 1 is used. Exact relationships for the natural frequencies and mode shapes of the beam are derived. Moreover, a method of calculating the damped natural frequencies of the analyzed beam is presented. The forced-vibration solution of the beam is derived using the mode superposition method. Transient movement of the base is described by an oscillating function with a linearly time-varying frequency. A convolution integral of the fractional Green's and forcing functions is used to achieve the beam response. The Green's function is formulated by two terms. The first term describes damped vibrations around the drifting equilibrium position, while the second describes the drift of the equilibrium position. The dynamic responses are calculated numerically. The impact of a fractional order derivative on transient responses when passing through resonance regions of the analyzed beam is shown. The beam responses obtained using a fractional viscoelastic material model are compared with those obtained using an integer viscoelastic material model. In the system analyzed, the influence of the term describing the drift of the equilibrium position on the beam deflection is relatively low and may be neglected. The results obtained indicate that an increase in the derivative order causes a decrease in the vibration amplitudes of the beam. The calculations reveal that the vibration amplitudes in the region of the second resonance are larger than in the region of the first resonance for some parameters of the system under consideration, which is not the case when using integer order derivatives in the viscoelastic material model.