The bifurcation of a shaft with hysteretic internal friction of material was analysed. Firstly, the differential motion equation in complex form was deduced using Hamilton principle. Then averaged equations in primary...The bifurcation of a shaft with hysteretic internal friction of material was analysed. Firstly, the differential motion equation in complex form was deduced using Hamilton principle. Then averaged equations in primary resonances were obtained using the averaging method. The stability of steady_state responses was also determined.Lastly, the bifurcations of both normal motion (synchronous whirl) and self_excited motion (non_synchronous whirl) were investigated using the method of singularity. The study shows that by a rather large disturbance, the stability of the shaft can be lost through Hopf bifurcation in case the stability condition is not satisfied. The averaged self_excited response appears as a type of unsymmetrical bifurcation with high orders of co_dimension. The second Hopf bifurcation, which corresponds to double amplitude_modulated response, can occur as the speed of the shaft increases. Balancing the shaft carefully to decrease its unbalance level and increasing the external damping are two effective methods to avoid the appearance of the self_sustained whirl induced by the hysteretic internal friction of material.展开更多
文摘The bifurcation of a shaft with hysteretic internal friction of material was analysed. Firstly, the differential motion equation in complex form was deduced using Hamilton principle. Then averaged equations in primary resonances were obtained using the averaging method. The stability of steady_state responses was also determined.Lastly, the bifurcations of both normal motion (synchronous whirl) and self_excited motion (non_synchronous whirl) were investigated using the method of singularity. The study shows that by a rather large disturbance, the stability of the shaft can be lost through Hopf bifurcation in case the stability condition is not satisfied. The averaged self_excited response appears as a type of unsymmetrical bifurcation with high orders of co_dimension. The second Hopf bifurcation, which corresponds to double amplitude_modulated response, can occur as the speed of the shaft increases. Balancing the shaft carefully to decrease its unbalance level and increasing the external damping are two effective methods to avoid the appearance of the self_sustained whirl induced by the hysteretic internal friction of material.
