In a traditional turbine-generator set, rotor shaft designers and blade designers have their own models and design process which neglects the coupled effect. Since longer blade systems have recently been employed for advanced turbine sets to get higher output and efficiency, additional consideration is required concerning rotor bending vibrations coupled with a one-nodal (k=1) blade system. Rotor-blade coupled bending conditions generally include two types so that the parallel and tilting modes of the shaft vibrations are respectively coupled with in-plane and out-of-plane modes of blade vibrations with a one-nodal diameter (k=1). In this paper, we propose a method to calculate the natural frequency of a shaft blade coupled system. According to our modeling technique, a certain blade mode is reduced to a single mass system, which is connected to the displacement and angle motions of the shaft. The former motion is modeled by the m-k system to be equivalent to the blade on the rotating coordinate. The latter motion is commonly modeled in discrete form using the beam FEM on an inertia coordinate. Eigenvalues of the hybrid system covering both coordinates provide the natural frequency of the coupled system. In order to solve the eigenfrequencies of the coupled system, we use a tracking solver based on sliding mode control. An eight-blade system attached to a cantilever bar is used for an example to calculate a coupled vibration with a one-nodal diameter between the blade and shaft.
