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The unbalance-induced vibration of a high-speed rotor directly affects the manufacturing accuracy. To effectively suppress this undesired vibration and avoid the complicated process of using trial weights during the balancing adjustment, a modal-based balancing method for a high-speed rotor without trial weights is proposed. First, the matrix sweep operation is employed to acquire the unbalance equivalent plane (EP). Next, the equivalent concentration methods, using the vector feedback principle (VFP) and modal equivalent principle (MEP), respectively, are studied and compared, while the equivalent transfer of the continuous unbalance vector to the EP is realized. Then, through modal analysis and the MEP, a balancing method, without trial weights required, is proposed for the high-speed rotor, which only needs to collect vibration data below critical speed. Finally, the rotor model and the presented method are validated on a rotor test platform, where the simulation and experiment results show that the unbalance-induced vibration has been effectively suppressed, ensuring smooth and safe operation of the rotor at high speed.


Introduction
High-speed rotating systems, as the driving source of cutting tools or actuators, are widely used in manufacturing equipment such as CNC machine tools (Brecher et al., 2007), and their dynamic performance directly affects the manufacturing accuracy. A high-speed rotor is the core component of a typical rotating system. Due to material anisotropy, installation eccentricity and other factors, the actual rotation axis of a rotor is usually not aligned with its geometric axis, which induces unbalance and vibration when working at high speed (Knowles et al., 2017). If the vibration exceeds a certain limit, the quality and efficiency of the rotation system will decrease, resulting in the failure of the manufactured parts. Therefore, how to suppress the unbalance-induced vibration becomes a key problem of the high-speed rotor system's application.
There are two classic methods for balancing flexible rotors (Kellenberger, 1972;Kang et al., 1996;Foiles et al., 1998;Xu et al., 2001): the influence coefficient (IC) method and the modal balancing (MB) method. Regarding the former, Zhou et al. (2004) presented an active balancing method of the rotor system during acceleration, which relies on the ICs at different speeds. Xu and Fan (2013) analysed the vibration form of a rigid rotor under unbalance inertia force: the amplitude and phase of the unbalance-induced vibration signal are obtained based on the principle of the cross-correlation method. Chu et al. (2018) introduced the system-level balancing method, where the IC approach is applied to solve the problem of excessive floor panel vibration. In the case of the MB method, Han (2007) aimed at the new generalized modal balancing for non-isotropic rotor systems, which uses the derived unbalance modal responses from the complete modal analysis. Deepthikumar et al. (2013) described the eccentricity distribution of the finite-element model by a polynomial curve and estimated the unbalance distributions based on the MB principle and the measured vibration responses; then, the rotor is balanced at its first bending critical speed. Since the above two methods have their own advantages and disadvantages, scholars try to integrate the advantages of the two methods and put forward a unified balancing method (Darlow et al., 1981;Kang et al., 1997). Khulief et al. (2014a) developed a low-speed unbalance identification method for high-speed rotors, which relies on the knowledge of the ICs and the modal characteristics of the rotor. However, these methods still need to use trial weights during the balancing process, which complicates the operation of balancing and increases the cost.
If the unbalance distribution of the rotor can be accurately identified, without the need for trial weights, the balancing process will become simple and safe. Thus, the balancing method without trial weights attracted research focus. Scholars have done a lot of work on such methods (Moton, 1985;Zhu and Wang, 1995;Xu et al., 2000;Delgado, 2002;El-Shafei et al., 2004). Ramlau and Niebsch (2009) described a new method for the detection and reconstruction of imbalances in the rotor of a wind turbine, avoiding test-weight measurements, while the wind turbine model can be used to predict the vibrations, given a specific imbalance distribution. Li et al. (2013) provided a new balancing method without trial weights for flexible rotors, which is based on the traditional MB method and combines the dynamic characteristics of the rotor. Khulief et al. (2014b) developed a hybrid experimental and analytical technique for balancing high-speed flexible rotors: this approach does not rely on trial runs and does not require one to operate the supercritical rotor in a high-speed balancing facility. Bin et al. (2014) developed a virtual dynamic balancing methodology without trial weights and have validated it by solving a shafting dynamic balancing example with no trial-weight requirements. Zhang et al. (2016) proposed a balancing method based on a virtual IC matrix for flexible rotors without trial weights, and the validity of the ICs is verified by an experiment on a rotor test rig. Wang et al. (2017) provided an efficient way, which is based on a novel measurement point vector method, for rotor balancing without test runs and external excitations, and only the measured unbalance response of the rotor shaft under steady-state operating conditions is needed. Ye et al. (2018) proposed a balancing method without trial weights by using the dynamic similitude scale model; this method could be used to directly obtain the required coefficients for the balancing problem of the prototype system through a similarity model test.
Although remarkable balancing effects are obtained by the aforementioned approaches, there are still some issues worth further study, for example the optimization of unbalance correcting planes, the balancing realization based on the lowspeed data collection, and efficiency improvement of the balancing process. Moreover, in some methods, the constructed model is only used to obtain the virtual vibration IC, while the characteristics of rotor dynamics are not taken into account. Aiming to solve the problem at hand, the authors have devised a modal-based balancing method for a high-speed rotor, which makes full use of modal information of the rotor to solve and correct the continuous unbalance vector. The scientific contribution of this paper is the proposal of a nontrial-weight balancing method, which adapts to optimal se-lection of the equivalent correcting plane and requires one only to collect vibration data below critical speed. The paper is structured into five sections. Following the introduction, Sect. 2 includes the selection of the equivalent plane for correcting unbalance, and the equivalent transfer principle of the continuous unbalance vector to the equivalent plane is presented. Based on this, a modal-based unbalance identification method is proposed. Section 3 analyses the effectiveness of the modal equivalent principle and the proposed balancing method by numerical simulation. The experimental platform is presented in Sect. 4 along with the experimental results and discussion. Lastly, the conclusions are given in Sect. 5.

Selection of equivalent plane
In theory, the higher the number of unbalance correcting planes, the better the desired effect will be. However, when there are too many correcting planes, the condition number of the corresponding IC matrix will get higher, which eventually leads to unreasonable counterweight results. Therefore, there is always at least one group of planes, called equivalent planes (EPs), which are able to produce an equivalent excitation effect to the one derived by all the correcting planes. The matrix sweep operation (MSO) can be used to obtain EPs in the unbalance problem. Suppose a ii (i = 1, 2,..., N ) is a diagonal element in A ∈ C N×N, a ii = 0, the MSO process from matrix A to B with a ii as the pivot element can be expressed as B = MSO(A, i), and each element in matrix B can be expressed as If the maximum value of a ii is not less than the preset threshold value µ, the correcting planes described in matrix A can be analysed successively through the MSO with the maximum value as the pivot element. In this process, the correcting plane with the smallest multicol