Optimal One-plane Active Balancing of a Rigid Rotor during Acceleration

Rotating machinery, including machining spindles, industrial turbomachinery, and aircraft gas turbine engines, are very commonly used in industry. One major problem faced by these machineries is the harmful, imbalance-induced vibration. Many methods have been developed to reduce this vibration: o!-line balancing methods [1], on-line active balancing methods using mass redistribution devices [2}5], and on-line active balancing methods using magnetic bearings [6}9]. These on-line methods can be applied during the operation of the rotor if the rotating speed is a constant. In some applications, the balancing needs to be completed during speed-varying transient time in order to save time and get better performance. For example, in high-speed machining, the spindle speed could be up to 40 000 r.p.m. and the chip-to-chip time could be less than 2 s. If an active balancing scheme is used in this machine, the balancing has to be done during the acceleration period to avoid increasing the cutting cycle time. Furthermore, the maximum vibration of a rotor usually occurs when it passes through its critical speeds. To avoid this hostile vibration, balancing during acceleration is needed. Zhou and Shi [10] proposed an adaptive active balancing scheme to perform balancing during acceleration by using an innovative mass redistribution actuator. There are multiple vibration modes for a general rotor. In general, if single-plane balancing is considered, the optimal compensating imbalances are di!erent for di!erent modes. However, since the imbalance distribution of the balancer can be changed during operation, the vibration of both vibration modes can be suppressed e$ciently by only one balancer [11]. To balance multiple modes with only one balancer, a &&switching'' function for the balancer needs to be determined. In this paper, we assume the balancer is not at any nodes of the vibration modes and the optimal switching function for the balancer during acceleration is investigated. A rigid rotor model is used, but the extension to the #exible rotor with multiple vibration modes is straightforward. This paper consists of four sections. A brief review of the adaptive active balancing will be given in section 2. Section 3 presents an optimal one-plane active balancing strategy, which