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

Nonlinear steady state vibrations of beams made of the fractional Zener material using an exponential version of the harmonic balance method

“…Because the virtual vectors and can be freely chosen from the average virtual work equation, the socalled amplitude Eqs. (32) and (33) are obtained.…”

“…Some applications of fractional calculus in mechanics can be found in [31]. Moreover, the free, steady and transient vibration of beams, frame structures and plates where the fractional models are used to describe damping properties of systems materials are, among others, presented in [1,5,25,[32][33][34][35][36][37][38]. Some useful reviews papers which presented the application of fractional calculus in mechanics are [39][40][41].…”

“…Using the well-known finite element procedure, it is possible to rewrite the integral appearing in Eq. (33) in the following matrix form:…”

Transient vibration of a fractional viscoelastic cantilever beam with an eccentric mass element at the end

“…This paper presents a study of transient vibrations of a fractional cantilever beam with a rigid mass element attached at its free end, whose center of mass does not coincide with the point of its attachment. The study is a continuation and extension of the author's earlier works (Freundlich, 2019(Freundlich, , 2021. The mentioned works have been focused on vibration of a fractional viscoelastic cantilever beam with an end mass element, whose center of gravity coincides with the point of its attachment.…”

“…The mentioned works have been focused on vibration of a fractional viscoelastic cantilever beam with an end mass element, whose center of gravity coincides with the point of its attachment. In the first mentioned work, a fractional viscoelastic Kelvin-Voigt material model was used (Freundlich, 2019), whereas the second work adopted a fractional viscoelastic Zener material model (Freundlich, 2021). Therefore, this study is dedicated to transient dynamic analysis of a cantilever beam having the end mass element, whose center of gravity is not coincident with the point of its attachment.…”