Abstract
This paper reports the analytically established conditions for the onset of auto-balancing for the case of a flat rotor model on isotropic elastic-viscous supports and an auto-balancer with a single load. The rotor is statically unbalanced, the rotation axis is vertical. The auto-balancer has a single cargo – a pendulum, a ball, or a roller. The balancing capacity of the cargo is equal to the rotor imbalance. The physical-mathematical model of the system is described. The differential equations of motion are recorded in dimensionless form relative to the coordinate system that rotates synchronously with the rotor. The so-called main movement has been found; in it, the cargo synchronously rotates with the rotor and balances it. The differential equations of motion are linearized in the neighborhood of the main movement. A characteristic equation has been constructed. It helped investigate the stability of the main movement (an auto-balancing mode) for the cases of the absence and presence of resistance forces in the system. It was established that in the absence of resistance forces in the system: – the rotor has three characteristic rotational speeds, and the first always coincides with the resonance frequency; – auto-balancing occurs when the rotor rotates at speeds between the first and second ones, and above the third characteristic speed; – the value of the second and third characteristic speeds is significantly influenced by the ratio of weight to the mass of the system; – the second and third characteristic speeds monotonously increase with an increase in the ratio of cargo weight to the mass of the system. Resistance forces significantly affect both the values of the second and third characteristic speeds and the conditions of their existence. Small resistance forces do not change the quality behavior of the system. With high resistance forces, the number of characteristic speeds decreases to one. The paper reports the results applicable to an auto-balancer with many cargoes when it balances the imbalance that equals the balancing capacity of the auto-balancer
Highlights
To apply passive auto-balancers for automatic balancing on the stroke of high-speed rotors, a deep theoretical study of the phenomenon of auto-balancing [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] is required
The results reported in [14, 15] demonstrate the feasibility of using model problems in the construction of an analytical theory of passive auto-balancers
Discussion of results obtained in the analytical study of the conditions for the onset of auto-balancing
Summary
To apply passive auto-balancers for automatic balancing on the stroke of high-speed rotors, a deep theoretical study of the phenomenon of auto-balancing [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] is required. Further research is carried out by numerical methods In this case, the results obtained are not general in nature. The results of solving model problems should make it possible to apply them to more complex rotary systems with auto-balancers. The rotor balances a ball, a roller, or a pendulum auto-balancer with a single cargo It is distinguished by the relative mathematical simplicity and by the possibility of applying the obtained results to auto-balancers with many cargoes. It is important to solve the specified model problem both in order to construct an analytical theory of passive auto-balancing and to practically implement it in order to design these devices for specific rotors
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