Iteration method for equation of viscoelastic motion with fractional differential operator of damping

Computation of the Laplace inverse transform by application of the wavelet theory

“…Such a dynamic system is governed by a general constitutive relations in which the stress-strain relation is described by a linear combination of fractional derivatives of either rational or irrational order. Numerical examples show that the results are in good agreement with those in the literature [12,17,18].…”

“…These results show that the damping e ect decreases with increase of 2 when the other parameters have been chosen. We ÿnd that the predicted system response or Green function in the special case of ÿ 1 = 0:5 is in good agreement with the previous results [12,17].…”

“…Recently, fractional derivative constitutive models for describing the viscoelastic behaviour of materials have received considerable attention [11][12][13][14][15][16][17][18]. Use of such 'fractionally damped' dynamic models is motivated in large part by the fact that fractional derivative models describe the frequency dependence of dynamic properties very well and fewer parameters are usually required to represent material behaviours in comparison with the traditional Kelvin and Maxwell-based models, etc.…”

“…For example, Ingman et al [12] analysed the motion of an N -degree-of-freedom system. For fractionally damped dynamic systems, it is usually di cult to obtain analytic solutions because of the fractional derivative terms in the equations of motion.…”

Solving non-linear viscoelastic problems via a self-adaptive precise algorithm in time domain

“…Recently, the dynamic response of structures with fractional differential operator of damping was obtained by Ingman and Suzdalnitsky in an iteration method via Laplace and Fourier transformations and series expansion (see e.g. Ingman and Suzdalnitsky, 2001). Beijer provided efficient FEM based strategies for analyzing non-linear viscoelastic polymers.…”