An analytical technique for determining the damped or undamped critical speeds and generating a critical speed map for a general flexible rotor in isotropic bearings has been developed. Rotordynamics theory related to this research work is studied including synchronous unbalance response, critical speed, gyroscopic effect, critical speed map, and stability. The method of solution is based on the Transfer Matrix approach containing both real and complex variable notations for deriving the overall system matrices. Muller's Method is applied to search for real or complex eigenvalues of the system. The corresponding mode shapes are found by back substitution. An accurate and efficient computer program, BEAM VII, has been designed to perform the analysis. Through the discussion of a series of numerical examples published by other authors, the confirmation of the rotor dynamics theory and precision of the computer code are achieved. / Master of Science