Vibrations of summed type under parametric excitation of a rotating shaft driven through a universal joint

In a rotating shaft driven by a universal joint (Hooke's joint), the secondary moment induces lateral vibrations while changing the joint angle. A torsional vibration then differs from that in the case of a shaft that has no deflection. Furthermore, the change of the joint angle influences the secondary moment. Therefore, we analyze equations of motion in terms of the second power of the deflection, the angle of deflection and the angle of torsion. Asymptotic analysis for the equations of motion exhibits that the universal joint must couple the lateral vibration with the torsional vibration. These vibrations become unstable and grow when the driving shaft rotates with the angular velocity nearly equal to half as large as the sum of the natural frequency of the whirling and that of the torsion. The experimental results support and confirm the analytical results.