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 aii(i=1,2,…,N) is a diagonal element in A∈CN×N, aii≠0, the MSO process from matrix A to B with aii as the pivot element can be expressed as B=MSO(A,i), and each element in matrix B can be expressed as 1bii=1/aii,bij=aij/aii,bji=aji/aii,j≠i(i,j=1,2,…,N),bkl=akl-akiail/aii,k≠i,l≠ii,j=1,2,…,N. If the maximum value of aii 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 multicollinearity corresponding to the maximum value will always be selected first, while the plane with strong multicollinearity will be excluded so as to achieve obtaining of the EPs. Based on this principle, suppose H is the subscript set of diagonal elements after the MSO, while P is the set of other subscript elements. A represents the matrix of ICs between different correcting planes and unbalance vectors. The threshold value is considered to be μ=Nσ/W, where σ is the standard deviation and W is the maximum correction mass that can be applied. The basic steps to derive the EPs of unbalance-correcting planes based on the MSO are as follows.

The values are initiated: H(0)=∅,P(0)={1,2,…,N}, A(0)=ATA; find i1∈P(0) so that ai1i1(0)=aii(0); if ai1i1(0)≥μ, continue to step 1; otherwise, the algorithm ends, while the EP does not exist.

Based on the obtained EPs, the continuous unbalance vector can be corrected using the traditional IC balancing method, which can be expressed as 2W=-A-1δ, where W∈CK×1 is the counterweight vector on K correcting planes, δ∈CJ×1 is the vibration response on J measuring planes, and A is the IC matrix between the vibration response and the unbalance vector.

If K≤J, Eq. (2) has a unique solution or an optimized solution, and the continuous unbalance vector is directly corrected with a finite number of correcting planes, which can be called the vector feedback principle (VFP). Based on this, the continuous unbalance vector is equivalently transferred to the EP, producing the continuous unbalance correction without solving the unbalance forces on all nodes. However, the VFP does not consider the modal characteristics of the rotor, while its dynamic balance effect will be difficult to maintain near the critical speed. Therefore, it is necessary to analyse the equivalent principle of the unbalance vector from the perspective of the modal shape.

As shown in Fig. 1, the dynamics model of the rotor-bearing system can be divided into two parts, the finite-element model of the rotor and the one of the journal-bearing restraint. It is assumed that the FE model has N nodes, while the N-1 shaft segments can be assembled by using the common finite-element procedure (Cao and Altintas, 2005; Zou et al., 2019). The radial displacement and rotation of a single element, in the generalized coordinate system, can be expressed as δn=yn,θzn,zn,-θynT, (n=1,2,…,N), where yn and zn are the displacements of the y and z axes, respectively, while θyn and θzn are the rotation angles of the y and z axes, respectively.

The dynamic differential equation of the rotor is described as 3Mδ¨+Cδ˙+Kδ=Fu+Fb, where M is the system mass matrix, M∈R4N×4N, K is the system stiffness matrix, K∈R4N×4N, and C is the system damping and gyroscopic matrix, C∈R4N×4N. Fu represents the unbalance force and is defined as Fnu=Fynu,0,Fznu,0T, Fu∈R4N×1. Fb represents the oil film force and is defined as Fnb=Fyb,0,Fzb,0T, Fb∈R4N×1, where Fyb and Fzb are the components of the oil film force (Capone, 1991; Xie et al., 2021) in the y and z directions, respectively.

The main vibration modal shape of the ith order is expressed as Φi=φix1,φix2,…,φixNT∈R4N×1, i=1,2,…,n, where φixj=φixj,0,0,0T, i=1,2,…,n, while the vibration modal shape of the rotor can be expressed as Φ=[Φ1,Φ2,…,Φn]∈R4N×n; the unbalance response δ on the measuring planes can be converted to ξ in modal coordinates: 4δ=Φξ, where δ∈C4N×1, ξ=ζ+iη, ξ and η are the components of the deflection on the ζ-x and η-x planes, respectively.

By substituting Eq. (4) into Eq. (3) and multiplying both sides of the equation by ΦT, the dynamic differential equation of the rotor becomes 5ΦTMΦξ¨+ΦTCΦξ˙+ΦTKΦξ=ΦTFu+Fb. Due to the orthogonality of each vibration modal shape, ΦTMΦ=M∗, ΦTCΦ=C∗ and ΦTKΦ=K∗ are diagonal matrices, while the elements on the diagonal are mi∗=ΦiTMΦi, ci∗=ΦiTCΦi and ki∗=ΦiTKΦi. Similarly, the N order modal unbalance force can be expressed as F∗=ΦTF, where F=Fu+Fb. Therefore, Eq. (5) can be written as 6M∗ξ¨+C∗ξ˙+K∗ξ=F∗, where M∗∈Rn×n, C∗∈Rn×n, K∗∈Rn×n, and F∗∈Rn×1.

Assuming that the particular solution of the vibration response is ξ=ξ0ejΩt and that F∗=F0∗ejΩt is the simple harmonic excitation, then Eq. (6) can be written as 7(K∗-M∗Ω2+jΩC∗)ξ0=F0∗. The deflection of the unbalanced rotor can be expressed as 8δx=Φξ=∑r=1∞ηr2Ψr∫0lfxϕr(x)dxϕr(x)1-ηr22+4ζ2ηr2, where cr represents the component of the rth modal shape in εu(x), αr represents the phase of this component, ϕr(x) is the modal shape, while the subscript r represents the order; Ψr=∫0lmxϕr2xdx is the correcting coefficient.

To make the concentrated force loaded on the EP equivalent to the distributed unbalanced force on the rotor, the vibrations generated by the two should be equal. Assuming that the equivalent concentrated force on EP is p(xk) (k=1,2,…,K), the following equation needs to be established: 9δx=Φξ=∑r=1∞ηr2Ψr∫0lfxϕrxdx-Ψr∑k=1Kpxkϕrxkϕr(x)1-ηr22+4ζ2ηr2=0, where ηr=Ω/ωr, κ=arctan[2ζηr/(1-ηr2)] represents the mechanical lag angle and ζ represents the modal damping coefficient, which is derived by experiment or obtained by a proportional relation to stiffness and mass.

Then, the equivalent balancing equation can be derived from Eq. (9): 10Ψr∑k=1Kpxkϕrxk=Ψr∫0lfxϕrxdx. According to Eq. (10), the continuous unbalance vector distribution on the rotor can be equivalently concentrated, without changing the vibration characteristics of the rotor, as long as the modal unbalance force on the rotor remains constant, which can be called the modal equivalent principle (MEP). Since a small unbalance force at low speed cannot excite the vibration modal shape of the rotor, the MEP is more applicable to high-speed rotor operation.

In the above analysis, the continuous distributed unbalance vector can be equivalently concentrated to EP, according to the MEP. Based on this fact, the modal-based unbalance identification method is presented in this section. Given the unbalance response analysis in Eq. (3), it can be concluded that the maximum relativity between the vibration response and unbalance vector takes place along the radial direction. Hence, the radial motion alone is used as the balancing target in the following approach. Assuming that the unbalance masses are distributed on U nodes, while only J(J≤N) measuring planes and K(K≤N) correcting planes are used to eliminate the unbalance force. The vibration response and the unbalance force are transformed between physical and modal coordinates according to the following: 11δ=ΦJξ,F∗=ΦUTF, where ΦJ=Φ1J,Φ2J,…,ΦnJ∈RJ×n, ΦiJ=φix1,φix2,…,φixJT∈RJ×1 is a sub-vector of the vibration modal shape with J measuring planes, ξ∈Cn×1 is the response in modal coordinates, ΦUT=Φ1U,Φ2U,…,ΦnUT∈Rn×U, ΦiU=φix1,φix2,…,φixUT∈RU×1 is a sub-vector of the vibration modal shape with U unbalanced nodes, F={f(x1),f(x2),…,f(xU)}T∈CU×1 is the unbalance force distributed on U nodes, and F∗∈Cn×1 is the unbalance force of the first n order modes in modal coordinates.

Due to the limitation of the measurement planes and the uncertainty of the unbalance nodes, the ΦJ and ΦUT may not necessarily be of full rank. Therefore, Eq. (11) should be transformed into the following form: 12ξ=ΦJTΦJ-1ΦJTδ,13F=ΦUΦUT-1ΦUF∗. Substituting Eq. (12) into the Fourier-transformed form of Eq. (6) leads to the following: 14F∗=K∗-Ω2M∗+jΩC∗ΦJTΦJ-1ΦJTδ. Next, substituting Eq. (14) into Eq. (13) provides the following: 15F=ΦUΦUT-1ΦUK∗-Ω2M∗+jΩC∗×ΦJTΦJ-1ΦJTδ. According to the analysis of the MEP, the unbalance vector distribution on the rotor can be corrected by the counterweights W added on the EP, as long as the following modal equivalence relation is satisfied. 16ΦKTW=-ΦUTF, where ΦKT=Φ1K,Φ2K,…,ΦnKT∈Rn×K, ΦiK=φix1,φix2,…,φixUT∈RK×1 is a sub-vector of the vibration modal shape with K correcting planes, and W={w(x1),w(x2),…,w(xk)}T∈CK×1 is the counterweight vector applied on the correcting planes.

By substituting Eq. (13) into Eq. (14), the counterweight vector can be obtained as 17W=-ΦKΦKT-1ΦKΦUTΦUΦUT-1ΦU×K∗-Ω2M∗+jΩC∗ΦJTΦJ-1ΦJTδ.

To verify the equivalent method of the unbalance force, the finite-element model of the rotor system, consisting of 50 elements and 51 nodes, is established, corresponding to Fig. 2. The mass disk units, which can be used as correcting planes, are located at the 11th, 18th, 23rd, 29th, 34th and 41st nodes, the measuring planes are located at the 15th and 37th nodes, while the journal-bearing units are located at the 5th and 47th nodes.

The MSO analysis of the model, as illustrated in Fig. 2, shows that the third and fifth disks are EPs, while the rest of the planes are redundant ones. The relationship between EPs and other planes is as follows: 18a1=0.8315a3-0.4720a5,a2=1.2200a3-0.4471a5,a4=0.4504a3+0.6520a5,a6=-0.2062×a3+0.7589a5, where α represents the IC and the subscript represents the respective correcting plane.

In order to prove the validity of EPs, the relationship between the left- and right-hand sides, in Eq. (18), is determined at 2400, 3200 and 4000 r/min, respectively, while comparison of the findings follows. Since there are two measuring planes used to collect vibration information, each disk can obtain six ICs, which are then verified. Figure 3 shows that the results obtained by both sides of Eq. (18) are identical; that is to say, the effect of the first, second, fourth and sixth disks on the vibration characteristics of the rotor can be completely replaced by that of the third and fifth disks, proving the existence of EPs.

To verify the effectiveness of the equivalent principle on the EP for dynamic balancing, the distributed unbalance vectors in redundant planes are equivalent to EPs, according to the VFP and MEP, respectively, while the unbalance response characteristics, before and after equivalent operation, are compared and analysed. Here, the vibration response is obtained based on the rotor model, as shown in Fig. 2 at 3400 r/min, while the unbalance vectors on the first, second, fourth and sixth disks are, respectively, equivalent to the ones on the third and fifth disks. The details of equivalent results are listed in Table 1, while the comparison of unbalance responses after equivalence of disks, based on the VFP and MEP, is shown in Fig. 4.

As shown in Fig. 4, if the unbalance vector on a redundant disk is transferred to the third and fifth disks using the MEP, the rotor unbalance response is consistent with the initial state, within the full speed range. However, if the VFP is employed to implement the equivalent transfer operation, the unbalance response is only consistent with the initial state, at around 3400 r/min, while the variation of the two response curves becomes obvious, at a different speed value. This phenomenon is due to the fact that the VFP does not consider the influence of the rotor modal shape, while rotor vibration is essentially the superposition of the multiple-modal vibrations excited by the unbalance. If the distributed unbalanced vector is directly transferred to the EP through the VFP, the obtained equivalent result does not consider the modal characteristics. When the rotor speed changes, the current balance effect cannot be maintained. As a result, the MEP is more suitable than the VFP for dynamic balancing of high-speed rotors.

In order to verify the effectiveness of the proposed unbalance identification method, based on the EP and MEP, numerical simulation of dynamic balancing is performed on the rotor model (Fig. 2). The virtual initial unbalance, as shown in Table 2, is artificially applied on the six disks of the rotor.

The balancing speed is set at 3400 r/min, and the counterweights on the third and fifth disks can be calculated by Eq. (17), as 24.33 g mm at 244.51∘ and 7.22 g mm at 297.39∘. Next, in order to suppress the unbalance response at the measuring planes, the identified counterweights are substituted into the rotor model. The unbalance response before and after the balancing procedure is illustrated in Fig. 5.

As illustrated in Fig. 5, the comparative results show that the proposed method can effectively identify the unbalance vector. After the balancing procedure, the residual vibration below and beyond the critical speed is very small, which demonstrates the effectiveness of the method.

In modal test, an impact excitation, generated by an impulse force hammer with a model of LC-01A, is applied on the rotor, while the response signal is acquired by an acceleration sensor with a model of ULT2001. Both the excitation and response signals are collected by the data acquisition device NI-9220 at a sampling frequency of 15 kHz. The measured excitation and response signal are illustrated in Fig. 7, while the simulated and measured frequency response functions (FRFs) of the rotor are shown in Fig. 8.

According to Figs. 7 and 8, the simulated and experimentally measured FRFs are in reasonable agreement. In the simulated FRFs, the first three natural frequencies have been found to be 79.5, 286.5 and 544.5 Hz. Meanwhile, the first three natural frequencies, as obtained by the measured FRFs, are 81, 291 and 559.5 Hz. The deviation between simulation and experiment of the first three order modes is 1.9 %, 1.5 % and 2.6 %, respectively, indicating the high accuracy of the model.

To verify the effectiveness of the presented unbalance method, the balancing experiments are performed on the experimental platform shown in Fig. 6. The eddy current displacement sensor WD502A and the photoelectric sensor E3X-HD41 are used to collect the amplitude and phase of unbalanced vibration, respectively. In order to be consistent with the simulation conditions as much as possible, the balancing speed is set to 3400 r/min, and the sampling frequency is set to 10 kHz. According to Eq. (17), the identified counterweights on the third and fifth disks are 15.92 g mm at 250.70∘ and 7.22 g mm at 69.47∘, respectively.

After completing the counterweight vector identification, the identified results need to be added on the corresponding mass disks to suppress the unbalance vibration. Since there are 10 evenly distributed holes on the third and fifth disks for adding counterweight screws, and the weight of the counterweight screws is limited to certain values, the identified counterweight vector needs to be decomposed into different holes of the disks. In this specific case, in order to realize the correction process, 0.5 and 0.8 g screws are added to the fifth and eighth holes of the third disk, respectively, while 0.9 g and 0.5 g screws are added to the first and fifth holes of the fifth disk, respectively.

In order to observe the change in the axis orbit along with radial direction before and after the balancing procedure, the three-dimensional holospectrum vibration in the full speed range on the measuring plane is illustrated in Fig. 9. The outer (red) curve represents the initial vibration and the inner (green) curve is the residual vibration after the balancing procedure.

As shown in Fig. 9 and Table 3, the system can operate smoothly through the first critical speed, while the unbalance vibration is significantly reduced even in the full speed range. The comparison between the initial vibration and the residual vibration shows that the proposed unbalance identification procedure is effective, and the counterweight vector, as identified by the dynamics model and the proposed method, is accurate.

It can be seen that the residual unbalance response is not completely suppressed after the balancing operation. The possible error sources are as follows: first, the rotor platform manufacturing and assembly accuracy limit the dynamic balancing effect. In the presented case, there is a specific initial bending of the experimental rotor: since both bending-induced vibration and unbalanced vibration present the same rotating-frequency characteristics, the bending-induced vibration will be regarded as unbalanced vibration in the unbalance identification process, which leads to some fake balancing operations during the experiment. Second, the accuracy of the dynamic model needs to be further improved. There are still differences between the model-based vibration modal data and the actual rotor state. Finally, the balancing quality is affected by other factors, such as the accuracy of data collection and the robustness of the rotor system.

In addition, when the speed is lower than 3800 r/min, the residual unbalance response becomes larger as the speed decreases. This phenomenon can be explained as follows: the rotor vibration is the result of the combined action of fault factors such as bending, unbalance, electromagnetic interference, and both the mechanical and electrical run-out, among others. The unbalance vibration only accounts for a lesser share of the overall vibration at the lower speed, and the vibration response is mainly caused by non-unbalance factors. This explains why the residual response at low speed in Fig. 9 is higher than that at 3800 r/min after dynamic balancing.