Reduction of the unbalance excited vibrations during the resonance passage of a rotor

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Many rotor dynamic systems are faced with large oscillation amplitudes while passing through the resonance. The coupled differential equations for the position of the center of the shaft and the rotation angle describe the dynamic behaviour of the rotordynamic system. With an appropriate external excitation the amplitude of the oscillations can be reduced. An unbalance excited oscillator is used to study the effects of a modulated angular velocity, where the angular velocity is prescribed as linear increasing with a superimposed harmonic function. The resulting excitation force is expressed as a series of Bessel functions of the first kind. In order to achieve small amplitudes near the resonance frequency of the system, special conditions for the ratio of the modulation frequency and the angular acceleration and the argument of the Bessel function of the first kind with integer order zero are derived. These requirements are first developed for a linear oscillator with an excitation force. Due to the analogy of the solution of the equation of motion these solutions are then applied directly to an unbalance excited oscillator. The results show smaller displacement amplitudes compared to the case with only linear increasing angular velocity. For the resulting motion the required torque is computed.