For the purpose of analyzing the torsional vibration caused by the gravitational unbalance torque arisen in a spindle system when it is machining heavy work piece, a 10-DOF lumped parameter model was made for the machine tool spindle system with geared transmission. By using the elementary method and Runge-Kutta method in Matlab, the eigenvalue problem was solved and the pure torsional vibration responses were obtained and examined. The results show that the spindle system cannot operate in the desired constant rotating speed as far as the gravitational unbalance torque is engaged, so it may cause bad effect on machining accuracy. And the torsional vibration increases infinitely near the resonant frequencies, so the spindle system cannot operate normally during these spindle speed ranges.

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when a large scale lathe is machining a big size irregular shaped work piece like crankshaft, the spindle system of the lathe can hardly operate in the desired constant rotating speed, because of gravitational unbalance torque arisen from the eccentric unbalance weight of the work piece. Consequently, rotating speed changes and torsional vibration of the spindle can make machining accuracy worse. To analyze the torsional vibrations, 3 different lumped parameter models for the spindle system with geared transmission were derived. And the gravitational unbalance torque was also represented mathematically. Because the gravitational unbalance torque is related with the torsional vibration itself, the system with unbalanced work piece become a nonlinear self excited vibrating system. We solved the equations of motion by using Runge-Kutta method of the MATLAB.