Example 6.1 Obtain the torsional natural frequency of a cantilever rotor system as shown in Fig. 6.3. The end B1 of the shaft has fixed end conditions. The shaft diameter is 10 mm and the length of the span is 0.2 m. The disc D1 is thin, and has mass of 10 kg and the polar mass moment of inertia equal to 0.02 kg-m2. Neglect the mass of the shaft. Compare the torsional natural frequency so obtained with the transverse natural frequency of the same rotor system.
Figure 6.3 A cantilever rotor system
Answer: For the present problem the torsional stiffness of the shaft can be obtained as
Hence, the torsional frequency is given as
Hence, if the rotor has cyclic torque variation with a period of 1/22.3 sec then the rotor might undergo to the resonance under torsional vibrations. To have a comparison with the transverse natural frequency, the bending stiffness is given as
Hence, the transverse natural frequency of the same rotor system is given as
If the same rotor has small amount of unbalance and if rotor is spinning around 9.9 Hz speed, then the rotor might undergo to the resonance under transverse vibrations. For the present case, the transverse natural frequency is much lower than the torsional natural frequency.
Answer.