This paper addresses nonlinear viscoelastic behaviour of fractional systems with variable time-dependent fractional order. In this case, the main challenge is that the Boltzmann linear superposition principle, i.e. the theoretical basis on which linear viscoelastic fractional operators are formulated, does not apply in standard form because the fractional order is not constant with time. Moving from this consideration, the paper proposes a novel approach where the system response is derived by a consistent application of the Boltzmann principle to an equivalent system, built at every time instant based on the fractional order at that instant and the response at all the previous ones. The approach is readily implementable in numerical form, to calculate either stress or strain responses of any fractional system where fractional order may change with time. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

No abstract

No abstract

No abstract

The dynamics of beams equipped with tuned mass dampers is of considerable interest in engineering applications. Here, the purpose is to introduce a comprehensive framework to address the stochastic response of the system under stationary and nonstationary loads, considering inertia effects along the spring of every tuned mass damper applied to the beam. For this, the key step is to show that a tuned mass damper with spring inertia effects can be reverted to an equivalent external support, whose reaction force on the beam depends only on the deflection of the attachment point. On this basis, a generalized function approach provides closed analytical expressions for frequency and impulse response functions of the system. The expressions can be used for a straightforward calculation of the stochastic response, for any number of tuned mass dampers. Numerical results show that spring inertia effects may play an important role in applications, affecting considerably the system response.