Nonlinear dynamic analysis in the frequency domain of structural systems

It is well known that the solution of the forced vibration of a N-DOF dynamical problem is very cumbersome when conditions which allow the equation uncoupling do not exist. In the literature several techniques were proposed to overcome the problem, but they were mainly focused on a particular problem in turn. So, we deemed useful to search for a unifying procedure able to deal with different sources of non-linear behavior introducing only minor changes in the operation flow. In this respect, attention is paid to the Alternating Frequency/Time domain method (AFT) which draws its robustness from the speed and switching capabilities of the Fast Fourier Transform; moreover, taking advantage of the pseudo-force concept, we can arrive at a solution method featuring greater generality and able to solve different non-linear dynamical problems by means of specialization of the same conceptual framework (G-AFT or generalized AFT). In the first part of the paper the theoretical background is discussed in detail and the proposed algorithms are presented. In the second one, several problems of technical relevance are documented and solved, highlighting the efficiency, convergence and accuracy of the presented algorithm. With the aim to show that the G-AFT procedure can handle various dissipative models, in the examples we focus on different kind of damped dynamic systems widely adopted in the seismic protection of structures. For cases such as an eleven storey building or a block simulated power plant we introduce the soil-structure interaction effect by means of non-proportional damping; the responses computed either by direct frequency analysis or by iteration are compared with existing solutions or with time domain solutions determined through the Newmark b method. An original example prepared by the authors and fully referenced is then worked out in order to show the capability of the method when the Coulomb damping is taken into account; this effect covers a significant practical relevance in the base isolation field. Finally, the problem of a simple oscillator with added viscoelastic dampers is considered, taking into account the temperature dependence of the behavior, due to the variability of the material constitutive mechanical parameters