This paper describes an analytical approach for calculating the damped critical speeds of multi-degree-of-freedom rotor-bearing systems. It is shown that to calculate the critical speeds is equivalent to finding the roots of a proposed matrix algebraic equation. The technique employes a Newton-Raphson scheme and the derivatives of eigenvalues. The system left eigenvectors are used to simplify the calculations. Based on this approach, a general-purpose computer program was developed with a finite element model of rotor-bearing systems. The program automatically generates system equations and finds the critical speeds. The program is applied to analyze a turbomachine supported by two cylindrical oil-film Journal bearings. The results are compared with reported data and the agreements are very good.