Approximated gramians and balanced realization of lightly damped flexible structures

This paper presents a method of approximating the balanced realization for lightly damped flexible structures with nonmodal damping. The damped system is treated as a perturbation from the undamped system, and the controllability and observability gramians are given as a power series in the perturbation scaling factor. The approximation utilizes the special structure of second-order mechanical systems to obtain closed-form expressions for the series coefficient matrices. The series form leads also to an approximated balancing algorithm, which gives the balancing transformation explicitly