Abstract: A transfer matrix-direct integration method is proposed,which emplo7S the transfer matrix method to derive the equations of motion of a "characteristic disk" and the direct integration method to determine critical speed,mode and unbalance response and analyze stability.The modified transfer matrix（see equation（12））for an uniform shaft is derived so as to consider the effects of its distributed mass,moment of inertia and shear deformation.A numerical example（see fig.2）is given to explain the procedure of applying transfer matrix method to deriving the characteristic equations（16）.The unbalance response,critical speed and mode of a model rotor-bearing system are calculated and experimentally determined.The analytical results are basically consistent with the experimental data as shown in the table.The transfer matrix method is straightforward in use and capable of considering the effect of support impedance.In principle,the establishment of the characteristic equations by using the transfer matrix is similar to the Ricatti transfer matrix method,therefore it is superior to the classic transfer matrix method.The number of governing equations deduced in this way is below eight,much less than that of the mode synthesis method.The direct integration method is suitable for both linear and nonlinear systems.It can obtain the unbalance response,critical speed and whirl orbit and analyze stability with enhanced accuracy in the meantime.All the advantages mentioned above are concentrated in the transfer matrix-direct integration method.The procedure proposed should be modified properly while the rotor-bearing system is markedly asymmetric or it is necessary to analyze the transient response or general precession.