## 1 Introduction  
Mass imbalance is a common fault of rotating machinery. To achieve higher production efficiency, the working speed of these machines has been increased. Higher rotational speed demands higher dynamic balance accuracy, and this demand has led scholars to study better dynamic balancing methods. In general, there are two categories of research on dynamic balance technology: offline balance technology and online automatic balance technology. Owing to the difference in balancing principles, offline balance research can take several directions, such as the influence coefficient method [1; 2; 3], modal analysis method [4], field balancing [5; 6], and the virtual dynamic balance method [7; 8]. These methods can reduce the unbalance vibration of the machinery in the field; however, usingthis procedure, the rotating machinery needs to stop working and even stop several times, which results in time and economic losses. Online automatic balance technology can automatically decrease the unbalance vibration using a balance actuator that is rotating with the rotor when the vibration amplitude exceeds the limit value in the rotor operation process. This avoids the start and stop process in the offline balance. Based on the structural features of the balance actuator, there are several types of online automatic balance technology, such as ball-type [9; 10], hydraulic type [11], pendulum type [12], magnetic-bearing type [13], electromechanical type [14], electromagnetic-ring type [15; 16], and liquid-type [17; 18; 19; 20]. The first three can redistribute mass to automatically compensate for the unbalance based on the self-alignment phenomenon occurring in the planar rotor, and these are called passive balancing devices. The latter four need a control unit to send commands during the balancing operation and are called active balancing devices, which have better balancing accuracy than the passive devices. The magnetic-bearing type actuator reduces the unbalance vibration by means of synchronous electromagnetic force which cannot be removed after balancing, thus its scope of application is limited by the large power consumption [13]. The electromechanical type and the electromagnetic-ring type actuators use the eccentric mass blocks to compensate for the initial unbalance of the rotors [14; 15; 16], and, to change the mass distribution of the mass blocks, several movable components are needed in the structure design, such as motors, transmission chains, and ball bearings, which are easily deformed and locked under centrifugal force in high-speed applications. The liquid-type actuator compensates for the initial unbalance through the liquid mass distribution, and its commercial product is called liquid-injection type or Hydrokompenser [17]. Given that the construction of the liquid-injection type is simpler and it does not have movable components in the rotating part, it is more suitable than the other three active balancing actuators. Unfortunately, the liquid injecting procedure can cause environmental pollution, restricts the application of the products, and results in the balance capacity decreasing with the balancing number. Therefore, many studies have been done to solve these disadvantages. For example, the liquid injection and dripping type actuator was proposed to solve the problem of losing balancing ability, by possessing a continuous balancing ability [18]; and the liquid transfer type actuator for the grinding machine was proposed to solve the liquid injection problem during balancing by redistributing the liquid in the balancing disc through compressed air [19; 20]. However, these studies were conducted in low or medium speed situations, i.e., below 5500 rpm; therefore, the issue of how to solve the disadvantages in high-speed applications over 10,000 rpm has not been studied. Furthermore, at present, these liquid actuators are installed on the shaft or on the end, and there is no solution to the requirement for mounting in the hollow rotor.