This note 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.