A time integration algorithm for time-varying systems with non-classical damping based on modal methods

Frequently the dynamic behaviour of systems with time varying parameters and non-classical damping have to be analysed. Hence efficient numerical routines are needed to treat the case of transient response of such systems in the time domain as their computational solution is very time consuming. In this paper an algorithm is presented, which efficiently computes the transient response of time-varying systems with non-classical damping. The transient solution is described in the symmetric modes of the system matrices and is split into a part of the solution, which is driven by the excitation load and a second part, which considers the induced load due to the time dependent parameters. Green's functions can be applied for both parts of the motion as the modal equations are solved by means of a Duhamel-type integral formulation. The time evolution of the total response can be calculated without any iteration for linear systems with non-classical damping and time-varying mass, if a certain variation of the solution within the time step is assumed. Depending on the based variation of the solution one gets a family of algorithms, which can be analysed with respect to the numerical characteristics, like the stability, the numerical dissipation and dispersion and the accuracy. The presented semi-analytic algorithm allows the consideration of more general systems with unsymmetric matrices like that of rotordynamic systems without any changes in the formulation. A modal reduction can easily be introduced within the algorithm in order to increase its efficiency. The time integration algorithm is applied to systems with time-varying mass and nonclassical damping, where the effectiveness of the presented solution procedure is demonstrated.