Bifurcation of a Shaft With Hysteretic-Type Internal Friction Force of Material
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摘要: 应用平均法研究迟滞型材料阻尼转轴的分岔.首先用Hamilton原理推导出复数形式的转轴运动微分方程,然后用平均法求出各阶模态主共振时的平均方程,并分析定常解的稳定性,最后用奇异性理论分析正常运动和失稳运动响应(异步涡动)的分岔.研究表明,一定参数条件下,转轴在通过各阶临界转速(主共振)时,可能会因受到冲击而失稳(Hopf分岔).正常运动响应在不平衡量较大时有滞后和跳跃现象,而失稳运动响应是一类余维数较高的非对称分岔.由于内阻尼的非线性,响应随转速增加时还可能产生二次Hopf分岔,对应原系统的双调幅运动.做好动平衡及提高外阻尼水平是避免这种大幅值自激振动的有效措施.Abstract: 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.
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