The objective of this study is to minimize the weight of a damped anisotropic roto-bearing system considering whirl natural frequency and stability. The system is modeled as an assemblage of rigid disks, flexible shafts and discrete bearings. The system design variables are the crosssectional areas of shaft elements and the properties of bearings. To analyze the system, the polynomial method which is derived by rearranging the calculations performed by a transfer matrix method is adopted. For the optimization, the optimization software IDOL (Integrated Design Optimization Library) which is based on the Augmented Lagrange Multiplier (ALM) method is employed. Also, an analytical design sensitivity analysis of the system is used for high accuracy and efficiency. To demonstrate the usefulness of the proposed optimal design program incorporating analysis, design sensitivity analysis, and optimization modules, a damped anisotropic rotor-bearing system is optimized to obtain 34$ weight reduction.