翻译：球式平衡器是一种新型的平衡调节器材它能够通过调节球的位置和重量来实现平衡调节广泛应用于机械、仪器、电子、航空、航天等领域。目前球式平衡器的发展前景非常广阔已经成为一种重要的技术手段。随着科技的不断发展和应用的不断扩展球式平衡器在各个领域的应用也在不断增加。在机械行业中球式平衡器被广泛应用于旋转机械设备的平衡调节如风机、离心机、转子等；在电子行业中球式平衡器被用于光学设备、半导体设备、精密测试

Translation:

The sphere-type balancer is a new type of balancing adjusting equipment that can achieve balance adjustment by adjusting the position and weight of the sphere. It is widely used in mechanical, instrument, electronic, aviation, and aerospace fields. Currently, the development prospects of sphere-type balancers are very broad, and they have become an important technical means. With the continuous development of science and technology and the continuous expansion of applications, the application of sphere-type balancers in various fields is also increasing. In the mechanical industry, sphere-type balancers are widely used in the balance adjustment of rotating mechanical equipment, such as fans, centrifuges, rotors, etc.; in the electronic industry, sphere-type balancers are used in the balance adjustment of optical equipment, semiconductor equipment, precision testing instruments, etc.; in the aviation and aerospace fields, sphere-type balancers are used in the attitude control and balance adjustment of satellites, rockets, missiles, etc.

However, in the region below the critical speed, sphere-type balancers cannot effectively suppress rotor vibration and may cause a rapid increase in transient amplitude, which accelerates the fatigue wear of the rotor system and affects its service life. Currently, for the eccentricity problem, multiple flat balancers are mostly used, but the cross-plane balancing ability of the sphere in the direction of the rotor axis is ignored. This paper proposes a cross-plane sphere balancer considering the axial cross-plane balancing ability of the sphere for non-flat rotor systems, and stability analysis is performed. The specific content is as follows:

First, the concept of auto-centering and its application are introduced. Then, a planar Jeffcott rotor model is established, and the non-dimensional model of the rotor is obtained through force analysis. Then, the ratio of amplitude to eccentricity and the relationship between the angle of rotation and the speed ratio are calculated to verify the principle of auto-centering. Finally, through simulation verification, the vibration curve of the rotor is analyzed at speeds below the critical speed, equal to the critical speed, and above the critical speed to explore the vibration characteristics of the rotor.

Secondly, the anti-vibration effect of traditional sphere balancers on the Jeffcott planar rotor model is analyzed. The balancer rotor vibration model is established by deriving the Lagrangian equation, and then it is transformed into an autonomous equation. Then, the forms of the three steady-state solutions are analyzed, and bifurcation theory is used to discuss stability through bifurcation analysis and single-bifurcation analysis. Finally, numerical simulation verification is performed using Matlab to discuss the steady-state response of the rotor in balance zone I, balance zone II, and unbalanced zone.

Then, a new cross-plane sphere balancer with axial rolling freedom is proposed for the even unbalance problem caused by eccentricity in different planes. First, each part of the rotor system is disassembled and the kinematic equation is established separately. Then, they are combined and the final result is calculated using the Lagrangian equation. Then, the cross-plane balancer with double tracks is investigated, and the balance condition of the rotor mass-diameter moment is derived and analyzed. The theoretical balance position of the balance sphere and the minimum limit of the balance sphere mass under normal working conditions are obtained. The rotor motion is divided into stable zones I and II, and the effect of changes in the balance sphere mass and eccentricity mass on the rotor stability response is discussed. Then, similar research is done for the cross-plane balancer with three tracks. The rotor motion is divided into stable zones I and II and unstable zone III according to the critical speed. The stability under different speeds and changes in the balance sphere mass and eccentricity angle are verified through experiments