A direct integration formulation for exponentially damped structural systems

This paper presents a direct integration formulation for damped structural systems in which damping is characterised by exponential law models. In effect, for non-viscously damped systems the damping forces appearing in the equation of motion are represented by convolution integrals, which take into account the complete history of the load. The proposed formulation employs the Laplace transformation of the motion equation to transform it into a differential equation with time derivative orders higher than two, in contrast to the classical M–C–K–F equation of structural dynamics. Due to the difficulty to manipulate these high order derivatives, the particular case of a damping model with only two kernel functions is taken into account. Then, an implicit direct integration scheme is proposed to discretize the motion equation, which becomes an equivalent second-order Meq–Ceq–Keq–Feq equation. In contrast with other methods revised in the literature, the proposed formulation does not employ internal variables, which normally enlarge the size of the system. Besides, standard direct integration schemes such as those of Newmark’s family can be employed. In the last part of the paper two numerical examples are presented: one for a three degrees-of-freedom system, and another one for a finite element application.