# Review of Rotor Balancing Methods

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of the ICM

#### 2.1. Development of the ICM

#### 2.2. Operational Steps of the ICM

**Choose appropriate balance planes and experimental speeds.**In general, the positions of balance planes are determined by engineers when designing rotors. The number of the balance planes should not be less than the modal modes within the maximum working speed. The experimental speeds should be chosen in different working conditions and the sensors should be set up on the specific axial positions (e.g., on the bearing support) as measurement points.**Measure the initial vibration.**Start the rotor and measure the vibrations (including the amplitude and the phase angle) at different measurement points and at different speeds.**Measure the vibration after installing trial weights on the rotor.**Install the trial weight whose mass is known on the first balance plane and record its position (including the radius and the phase angle). Start the rotor again and measure the vibrations at different measurement points and at different speeds. Repeat these operations in other balance planes.**Calculate the influence coefficients and the balance corrections.**When the rotor rotates at ${\Omega}_{h}$, the influence coefficients can be calculated using:$${a}_{ij}^{h}=\frac{{A}_{ij}^{h}-{A}_{i0}^{h}}{{m}_{tj}{r}_{j}\left(\mathrm{cos}{\phi}_{j}+\mathrm{i}\mathrm{sin}{\phi}_{j}\right)}$$

- 5.
**Balancing the rotor according to the calculated results.**Due to the inevitable errors in the principle, measurement and operation, it is necessary to repeat the balancing process until the required precision is reached.

#### 2.3. Advantages and Disadvantages of the ICM

- As each element is measured by experiment, these coefficients can reflect the influences of the vibration mode, the support stiffness, and other factors.
- There is no need to know the dynamic response in advance. Enough sensitive information can be achieved at all critical speeds if the rotor operates safely within the normal speed range.
- The ICM is readily computerized and automated.
- Data manipulation techniques can be used to compensate for measurement errors.
- The ICM is an entirely empirical procedure, which requires minimal foreknowledge of rotor dynamics.

- As each influence coefficient is measured by experiment, some measurement error is inevitable.
- A significant number of revolutions are required to obtain sensitivity data at the highest balancing speed.
- The use of the least square method can affect the previously balanced modes, unless these modes are heavily weighted, and the other modes may receive insufficient emphasis.
- The misuse of the non-independent balance planes often results in impractical (and generally inappropriate) results, and the balance process thus becomes invalid.

## 3. Review of the MBM

#### 3.1. Development of the MBM

#### 3.2. Operational Steps of the MBM

**The initial vibration should be measured firstly.**Install the rotor on the dynamic balancing machine or the bearing support. Drive the rotor close to its first critical speed, usually at 90%, and then the unbalanced vibration of the rotor or the bearing support can be measured and recorded.

**Measure the vibration after installing the trial weights.**Install a trial weight on the balance plane near the middle of the rotor (near the peak of the vibration mode, as shown in Figure 2). In order to simplify the calculation, the position of the trial weight is similar to the radius of the balance correction, and the phase angle can be chosen freely. Then drive the rotor to the same speed as in step (a), and measure and record the unbalance vibration of the rotor or the bearing support.

**Calculate the balance correction and balance the first mode shape.**The balance correction ${m}_{j}$ is given by:

**Repeat the aforementioned operations and balance other vibration modes within the working speed range.**

#### 3.3. Advantages and Disadvantages of the MBM

- The number of start–stop times required at the highest balancing speed is minimized.
- Good sensitivity at the highest balancing speed can always be achieved.
- Balancing a specific mode is permitted, and this balancing does not affect the previously balanced (usually, lower) modes.
- The MBM can be an entirely empirical procedure that requires only an understanding of the modal character of the unbalance. Nevertheless, it is in practice most often used in conjunction with analytically determined mode shapes rather than in an empirical manner.

- The planar mode assumption can be invalid for systems with substantial damping or bearing cross-coupling effects.
- In theory, balance planes can be set at an arbitrary axial location, but for sensitivity purposes, they are usually set at peaks and troughs of the rotor’s vibration mode. Therefore, in order to determine the test speeds and balance planes, the critical speed and its vibration mode within the operating speed range should be understood before experimenting.
- Balancing results are generally based on the vibration measured by only one or two sensors for a specific mode, which cannot result in a uniformly well-balanced rotor.
- Modal balancing is not usually automated and it does not easily lend itself to production applications.
- The trimming of lower modes while balancing higher modes is liable to affect the higher modes (although, in theory, this can be avoided with the addition of extra balancing planes).

## 4. New Balancing Methods

#### 4.1. Nonlinear Rotor-Balancing Methods

#### 4.2. Transient Rotor-Balancing Methods

#### 4.3. Balancing Methods Using Homologous Information Fusion Technology

#### 4.4. Balancing Methods for Specific Rotors

#### 4.4.1. Balancing Methods for Asymmetric Rotors and Overhung Rotors

#### 4.4.2. Balancing Methods for a Dual-Rotor System

#### 4.4.3. Balancing Methods for Bending Rotors

#### 4.5. Other Rotor Balancing Methods

**Transfer function method.**Cao et al. [67] examined the selection of a balance plane in high-speed balancing methods and proposed the transfer function method for high-speed flexible rotors. Tiwari et al. [68] applied a similar ideology, identifying the unbalance using a numerical simulation. Khulief et al. [69] generalized this method to field balancing.**The balancing method without trial weights.**For the cases of installing trial weights and multiple start–stop actions of rotors, Wang et al. [70] proposed a balancing method without trial weights for high speed flexible rotors. By using the finite element method to simulate the dynamic characteristics of the rotor, in this way, the vibration mode function can be obtained. Li et al. [71] carried out a series of tests without the trial weights on a high-speed flexible simulated rotor tester. They found that the vibration amplitude was reduced by more than 70% after one balance. The results verified that this method could obtain the unbalance magnitude and direction accurately and quickly. However, because of the limitation of the rotor tester, this method can only balance the first mode shape. Liu et al. [72] used a similar method, but it had the disadvantage of the dependence of the balance accuracy on the finite element calculation.**Online balancing method.**Zhang [78] reviewed the online balancing research achievement, proposed a new type of pure mechanical balancing head and achieved online balancing. Wang et al. [79] presented the electromagnetic online automatic balancing system and discussed its principle, structure and balancing method.**Balancing method by using a specific device.**For aeroengines, engineers often used the Multiplane/Multispeed Balancing Method or the Exact Point-Speed ICM. This method is based on the ICM and it is generally carried out on a special balancing device. For this method, multiple planes and multiple speeds are needed to obtain enough influence coefficients. For a slender shaft, it is usually necessary to attach a precision collar. By using this method, NASA had successfully balanced the engines of a T700 [80,81], T53/T55 [82,83], and so on. In China, Deng et al. [84] adopted a similar method to balance engines.**Using slow-speed data to balance high-speed rotors.**Tresser et al. [85] proposed that by using slow-speed data via parametric excitation can achieve balance for super-critical rotating structures. Analytical, numerical and experimental results were shown to validate this method.**New methods that combine emerging technologies with traditional balancing methods.**Untaroiu et al. [86] combined the convex optimization technology with the ICM to balance flexible rotors. Mohammadi et al. [87] combined the Imperialist Competitive Algorithm with the ICM to balance flexible rotors. Saldarriaga et al. [88] used an inverse problem approach to balance flexible rotating machines.

## 5. Conclusions

- The nonlinearity cannot be ignored for the system containing non-smooth factors such as collision, shock and dry friction. Therefore, the conventional balancing method based on linear assumption should be modified using nonlinear theory.
- For high-speed rotor systems, such as aeroengine rotor systems and large thermal power-generation unit rotor systems, it is harmful if starting and stopping the rotor multiple times. Therefore, further research needs to be done in order to improve the balancing efficiency. The transient rotor balancing method and the method of using slow-speed data to balance the high-speed rotor system may solve this problem.
- Balancing methods should be improved by use of the multisource information fusion technologies. The signals of the unbalanced vibration of the rotor are not only determined by the dynamics of the rotor, but also closely related to test and analysis technology. Multi-information fusion and integration technologies will play a certain role in improving the accuracy and speed of rotor dynamic balance.
- For special rotor systems, different kinds of methods should be applied. For example, when balancing the dual-rotor system of an aeroengine, if balancing the inner and outer rotors separately, the system unbalance would occur even if a single rotor has been balanced. Therefore, a specific balancing method should be developed for dual-rotor systems.
- More and more methods such as transfer function method, no trial weights methods, online balancing methods will be developed to make rotor dynamic balance more convenient, more accurate, more energy saving and faster.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Xu, M.; Marangoni, R.D. Vibration analysis of a motor-flexible coupling-rotor system subject to misalignment and unbalance, part i: Theoretical model and analysis. J. Sound Vib.
**1994**, 176, 663–679. [Google Scholar] [CrossRef] - Wang, H.Y.; Zhang, Z.S.; Xu, X.L. Rotor Balancing Technology and Balancing Machine, 1st ed.; China Machine Press: Beijing, China, 1988. (In Chinese) [Google Scholar]
- Deng, W.Q.; Tang, G.; Gao, D.P. Research summary of rotor dynamic characteristics and dynamic balance. Gas Turbine Exp. Res.
**2008**, 21, 57–62. (In Chinese) [Google Scholar] - Zhou, S.; Shi, J. Active Balancing and Vibration Control of Rotating Machinery: A Survey. Shock Vib. Dig.
**2001**, 33, 361–371. [Google Scholar] [CrossRef] [Green Version] - Parkinson, A.G. Balancing of Rotating Machinery. Proc. Inst. Mech. Eng. Part C Mech. Eng. Sci.
**1991**, 205, 53–66. [Google Scholar] [CrossRef] - Foiles, W.C.; Allaire, P.E.; Gunter, E.J. Rotor balancing. Shock Vib.
**1998**, 5, 325–336. [Google Scholar] [CrossRef] - Thearle, E. Dynamic balancing of rotating machinery in the field. Trans. ASME
**1934**, 56, 745–753. [Google Scholar] - Baker, J. Methods of rotor-unbalance determination. ASME J. Appl. Mech.
**1939**, 61, A1–A6. [Google Scholar] [CrossRef] - Goodman, T.P. A least-squares method for computing balance corrections. J. Manuf. Sci. Eng.
**1964**, 83, 273–277. [Google Scholar] [CrossRef] - Tessarzik, J.M.; Badgley, R.H.; Anderson, W.J. Flexible rotor balancing by the exact point- speed influence coefficient method. J. Eng. Ind.
**1972**, 94, 148–158. [Google Scholar] [CrossRef] - Tessarzik, J.M. Flexible Rotor Balancing by the Influence Coefficient Method: Multiple Critical Speeds with Rigid or Flexible Supports; National Aeronautics and Space Administration: Washinghton, DC, USA, 1975. [Google Scholar]
- Gu, J.L. Rotor Dynamics; National Defense Industry Press: Beijing, China, 1985. (In Chinese) [Google Scholar]
- Zhong, Y.E.; He, Y.Z.; Wang, Z. Rotor Dynamics; Tsinghua University Press: Beijing, China, 1987. (In Chinese) [Google Scholar]
- Deng, W.Q. Experiment Investigation of Dynamic Characteristics and High Speed Dynamic Balance of a Aeroengine Flexible rotor. Ph.D. Thesis, Nanjing University of Aeronautics and Astronautics, Nanjing, China, 2006. (In Chinese). [Google Scholar]
- Bin, G.; Yao, J.; Jiang, Z.; Gao, J. Solving method of influence coefficient for rotor dynamic balance based on finite element model. J. Vib. Meas. Diagn.
**2013**, 33, 998–1002. (In Chinese) [Google Scholar] - Urbikain, G.; Olvera, D.; López de Lacalle, L.N.; Elías-Zúñiga, A. Stability and vibrational behaviour in turning processes with low rotational speeds. Int. J. Adv. Manuf. Technol.
**2015**, 80, 871–885. [Google Scholar] [CrossRef] - Urbikain, G.; Alvarez, A.; López de Lacalle, L.N.; Arsuaga, M.; Alonso, M.A.; Veiga, F. A Reliable Turning Process by the Early Use of a Deep Simulation Model at Several Manufacturing Stages. Machines
**2017**, 5, 15. [Google Scholar] [CrossRef] - Grobel, L.P. Balancing turbine-generator rotors. Nav. Eng. J.
**2010**, 65, 868–874. [Google Scholar] - Kellenberger, W. Balancing flexible rotors on two generally flexible bearings. Brown Boveri Rev.
**1967**, 54, 603–617. [Google Scholar] - Kellenberger, W. Should a Flexible Rotor Be Balanced in N or (N + 2) Planes? Trans. ASME
**1972**, 548–588. [Google Scholar] [CrossRef] - Palazzolo, A.B.; Gunter, E.J. Modal balancing of a multi-mass flexible rotor without trial weights. In Proceedings of the ASME 1982 International Gas Turbine Conference and Exhibit 1982, London, UK, 18–22 April 1982; ASME: New York, NY, USA, 1982; pp. 1–11. [Google Scholar]
- Meacham, W.L.; Talbert, P.B.; Nelson, H.D.; Cooperrider, N.K. Complex modal balancing of flexible rotors including residual bow. J. Propuls. Power
**1988**, 4, 245–251. [Google Scholar] [CrossRef] - Bucher, I.; Ewins, D.J. Modal analysis and testing of rotating structures. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.
**2001**, 359, 61–96. [Google Scholar] [CrossRef] - Sharp, R.S. Flexible rotor balancing: A review of principles and practices. Tribol. Int.
**1980**, 13, 211–217. [Google Scholar] [CrossRef] - Darlow, M.S.; Smalley, A.J.; Parkinson, A.G. Demonstration of a unified approach to the balancing of flexible rotors. J. Eng. Power
**1981**, 103, 101–107. [Google Scholar] [CrossRef] - Darlow, M.S. Balancing of high-speed machinery: Theory, methods and experimental results. Mech. Syst. Signal Process.
**1987**, 1, 105–134. [Google Scholar] [CrossRef] - Huang, W.H.; Wu, X.H.; Jiao, Y.H.; Xia, S.B.; Chen, Z.B. Review of nonlinear rotor dynamics. J. Vib. Eng.
**2000**, 13, 5–17. (In Chinese) [Google Scholar] - Luo, Y.G.; Bao, W.B.; Jin, Z.H.; Wen, B.C. Study on dynamic behavior of nonlinear rigid unbalanced rotor system. J. Vib. Shock
**2002**, 21, 84–86. (In Chinese) [Google Scholar] - Guskov, M.; Sinou, J.J.; Thouverez, F. Multi-dimensional harmonic balance applied to rotor dynamics. Mech. Res. Commun.
**2007**, 35, 537–545. [Google Scholar] [CrossRef] [Green Version] - Turpin, A.; Sharan, A.M. Balancing of rotors supported on bearings having nonlinear stiffness characteristics. J. Eng. Gas Turbines Power
**1994**, 116, 718–726. [Google Scholar] [CrossRef] - Cao, S.Q.; Chen, Y.S.; Ding, Q.; Lang, Z.G.; Zhang, Y.A. Nonlinear transfer function method of high-speed dynamic balance of rotor. In Proceedings of the 2003 Large Generator Set Vibration and Rotor Dynamics Academic Conference, Jiaozuo, China; 2003; pp. 300–304. (In Chinese). [Google Scholar]
- Nauclér, P.; Söderström, T. Unbalance estimation using linear and nonlinear regression. Automatica
**2010**, 46, 1752–1761. [Google Scholar] [CrossRef] - Green, K.; Champneys, A.R.; Lieven, N.J. Bifurcation analysis of an automatic dynamic balancing mechanism for eccentric rotors. J. Sound Vib.
**2006**, 291, 861–881. [Google Scholar] [CrossRef] [Green Version] - Yang, Y.F.; Ren, X.M.; Xu, B. Review of international researches on rotor dynamics. Mech. Sci. Technol. Aerosp. Eng.
**2011**, 30, 1775–1780. (In Chinese) [Google Scholar] - Chen, P.; Liao, M.F. Research on transient equilibrium method of flexible rotor. In Proceedings of the Chinese Society of Aeronautics and Astronautics Aeronautical Society Aviation Engine Structural Strength Vibration Academic Conference, Wuyishan, China; 1998; pp. 280–284. (In Chinese). [Google Scholar]
- Zheng, L.X.; Gao, X.G.; Li, X.F. Transient field balancing technique for a micro turbo-jet engine. Meas. Diagn.
**2008**, 28, 282–285. (In Chinese) [Google Scholar] - Fu, C.; Ren, X.M.; Yang, Y.F.; Deng, W.Q. Transient dynamic balancing of rotor system with parameter uncertainties. J. Dyn. Control
**2017**, 15, 453–458. (In Chinese) [Google Scholar] - Huang, X.; Zhou, J.P.; Wen, G.R.; Jiang, H.; Tan, Y. Balancing under all working conditions of rotor based on parameterized time-frequency analysis. J. Vib. Meas. Diagn.
**2017**, 37, 134–139. (In Chinese) [Google Scholar] - Huang, J.P. Study on the Transient Balancing Method of Rotors. Ph.D. Thesis, Northwestern Polytechnical University, Xi’an, China, 2006. (In Chinese). [Google Scholar]
- Goldman, P.; Muszynska, A. Application of full spectrum to rotating machinery diagnostics. Orbit
**1999**, 20, 17–21. [Google Scholar] - Qu, L.S.; Liu, X.; Peyronne, G.; Chen, Y.D. The holospectrum: A new method for rotor surveillance and diagnosis. Mech. Syst. Signal Process.
**2003**, 3, 255–267. [Google Scholar] [CrossRef] - Qu, L.S.; Qiu, H. Rotor balancing based on holospectrum analysis: Principle and practice. Chin. Mech. Eng.
**1998**, 9, 60–63. (In Chinese) [Google Scholar] - Han, J.; Guan, H.L.; Liang, C. Vector spectrum: A practical analysis method of the rotary machine fault diagnosis. J. Mech. Strength
**1998**, 20, 212–215. [Google Scholar] - Han, J. Research on the Application of Full Vector Spectrum Technology and Equipment Fault Diagnosis. Ph.D. Thesis, Tongji University, Shanghai, China, 2005. (In Chinese). [Google Scholar]
- Xu, B.G.; Qu, L.S. Holobalancing of asymmetric rotors. J. Xi’an Jiaotong Univ.
**2000**, 34, 60–65. (In Chinese) [Google Scholar] - Lang, G.F.; Lin, J.; Liao, Y.H. Phase compensation method of holobalancing. J. Mech. Eng.
**2014**, 50, 16–21. (In Chinese) [Google Scholar] [CrossRef] - Xu, B.; Qu, L.; Sun, R. The optimization technique-based balancing of flexible rotors without test runs. J. Sound Vib.
**2000**, 238, 877–892. [Google Scholar] [CrossRef] - Han, D.J. Generalized modal balancing for non-isotropic rotor systems. Mech. Syst. Signal Process.
**2007**, 21, 2137–2160. [Google Scholar] [CrossRef] - Kang, Y.; Liu, C.P.; Sheen, G.J. A modified influence coefficient method for balancing unsymmetrical rotor-bearing systems. J. Sound Vib.
**1996**, 194, 199–218. [Google Scholar] [CrossRef] - Kang, Y.; Sheen, G.J.; Wang, S.M. Development and modification of a unified balancing method for unsymmetrical rotor-bearing systems. J. Sound Vib.
**1997**, 199, 349–369. [Google Scholar] [CrossRef] - Yu, T.; Han, Q.K.; Li, S.D.; Wen, B.; Zhang, Z.S. Study on dynamic characteristics and imbalance response of double-over-hung rotor system. J. Vib. Meas. Diagn.
**2007**, 27, 186–189. (In Chinese) [Google Scholar] - Zhang, J.Q. Investigation of Pure Mechanical On-Line Balancing System. Master’s Thesis, Zhejiang University, Hangzhou, China, 2006. (In Chinese). [Google Scholar]
- Liao, M.F.; Liu, Y.Q.; Wang, S.J.; Wang, Y.; Pin, L. The vibration features of a twin spool rotor system with an inter-bearing. Mech. Sci. Technol. Aerosp. Eng.
**2013**, 32, 641–646. (In Chinese) [Google Scholar] - Han, J.; Gao, D.P.; Hu, X.; Chen, G. Research on beat vibration of dual-rotor for aero-engine. Acta Aeronaut. Et Astronaut. Sin.
**2007**, 28, 1369–1373. (In Chinese) [Google Scholar] - Zhang, Z.X.; Jin, Z.J.; He, S.Z. Study on whole machine balancing method for coaxial dual-rotor system with little rotating speed difference. Chin. J. Mech. Eng.
**2004**, 40, 40–44. (In Chinese) [Google Scholar] [CrossRef] - Yang, J.; He, S.Z.; Wang, L.Q. Dynamic balancing of a centrifuge: Application to a dual-rotor system with very little speed difference. J. Vib. Contro
**2004**, 10, 1029–1040. [Google Scholar] [CrossRef] - Deepthikumar, M.B.; Sekhar, A.S.; Srikanthan, M.R. Modal balancing of flexible rotors with bow and distributed unbalance. J. Sound Vib.
**2013**, 332, 6216–6233. [Google Scholar] [CrossRef] - Wu, X.H.; Liu, Z.S.; Zhang, X.Y. Characteristic analysis of bending faults of shaft. J. Vib. Eng.
**1990**, 3, 95–102. (In Chinese) [Google Scholar] - Xia, Y.L. Study on Vibration Characteristics and Reduction Technology of Rotating Machinery with Bend Rotor. Master’s Thesis, Southeast University, Nanjing, China, 2017. (In Chinese). [Google Scholar]
- Nicholas, J.C.; Gunter, E.J.; Allaire, P.E. Effect of residual shaft bow on unbalance response and balancing of a single mass flexible rotor: Part l: Balancing. J. Eng. Gas Turbines Power
**1976**, 98, 182–187. [Google Scholar] [CrossRef] - Nicholas, J.C.; Gunter, E.J.; Allaire, P.E. Effect of residual shaft bow on unbalance response and balancing of a single mass flexible rotor—part II: Balancing. J. Eng. Gas Turbines Power
**1976**, 98, 171–181. [Google Scholar] [CrossRef] - Deepthikumar, M.B.; Sekhar, A.S.; Srikanthan, M.R. Balancing of flexible rotor with bow using transfer matrix method. J. Vib. Control
**2014**, 20, 225–240. [Google Scholar] [CrossRef] - Liu, J.Y.; Ren, P.Z.; Liao, M.F.; Zhao, M.Y.; Yang, H.Y. A study on the vibration of flexible rotor due to its initial bending and unbalance. J. Vib.
**1998**, 18, 282–286. (In Chinese) [Google Scholar] - Parkinson, A.G.; Darlow, M.S.; Smalley, A.J. Balancing flexible rotating shafts with an initial bend. Aiaa J.
**1984**, 22, 683–689. [Google Scholar] [CrossRef] - Rao, J.S. A note on jeffcott warped rotor. Mech. Mach. Theory
**2001**, 36, 563–575. [Google Scholar] [CrossRef] - He, G.A.; Zhang, S.J.; Zhang, X.Y. Dynamic balance research on progressive bending rotor faults of large turbo-generator units. Turbine Technol.
**2014**, 56, 439–442. (In Chinese) [Google Scholar] - Cao, S.Q.; Chen, Y.S.; Ding, Q.; Zhang, Y.A.; Lang, Z.G. Transfer function technique of dynamic balancing for high-speed rotors. J. Mech. Strength
**2002**, 24, 500–504. (In Chinese) [Google Scholar] - Tiwari, R.; Chakravarthy, V. Simultaneous identification of residual unbalances and bearing dynamic parameters from impulse responses of rotor–bearing systems. Mech. Syst. Signal Process.
**2006**, 20, 1590–1614. [Google Scholar] [CrossRef] - Khulief, Y.A.; Mohiuddin, M.A.; El-Gebeily, M. A new method for field-balancing of high-speed flexible rotors without trial weights. Int. J. Rotating Mach.
**2014**, 2014, 1–11. [Google Scholar] [CrossRef] [Green Version] - Wang, W.M.; Gao, J.J.; Jiang, Z.N.; Yan, L.I. Principle and application of no trial weight field balancing for a rotating machinery. Zhendong Yu Chongji/J. Vib. Shock
**2010**, 29, 212–215. (In Chinese) [Google Scholar] - Xi, X.F.; Zheng, L.X.; Liu, Z.X. Theoretical and experimental research on balancing of flexible rotors without trial weights. J. Vib. Meas. Diagn.
**2013**, 33, 565–570. (In Chinese) [Google Scholar] - Liu, G.Q.; Zheng, L.X.; Mei, Q.; Huang, J.J. Balancing method of flexible rotor across second order without trial weights. Acta Aeronaut. Et Astronaut. Sin.
**2014**, 35, 1019–1025. (In Chinese) [Google Scholar] - Yan, L.T.; Xiao, G. CFM56-3 Aeroengine local balancing technology. Int. Aviat.
**1989**, 32–34. (In Chinese) [Google Scholar] - Jiang, G.Y.; Wang, D.Y.; Jiao, Y.Q. Experimentd investigation of local balanced technique for large aeroengine rotor. Aeroengine
**2008**, 34, 19–22. (In Chinese) [Google Scholar] - Chen, X.; Liao, M.F.; Zhang, X.M.; Wang, S.J. Field balancing technology for low pressure rotors of high bypass ratio turbofan engines. Hangkong Dongli Xuebao/J. Aerosp. Power
**2017**, 32, 808–819. (In Chinese) [Google Scholar] - Chen, B.Y. Balance technology for aeroengine. Aeronaut. Sci. Technol.
**1996**, 7–11. (In Chinese) [Google Scholar] - Chen, B.Y. Development of balance technology for aeroengines. J. Propuls. Technol.
**1998**, 19, 105–109. (In Chinese) [Google Scholar] - Zhang, M. Study on Transient Dynamic Balance Method and its Influencing Factors of Single-Disc Overhung Rotor. Master’s Thesis, Northwestern Polytechnical University, Xi’an, China, 2017. (In Chinese). [Google Scholar]
- Wang, X.X.; Zeng, S. Study of an on-line automatic dynamic balancing system and its dynamic balancing method when used on a flexible rotor. Therm. Energy Power Eng.
**2003**, 18, 53–57. (In Chinese) [Google Scholar] - Burgess, G.; Rio, R. T700 Power Turbine Rotor Multiplane/Multispeed Balancing Demonstration; Technical Information Service: Springfield, VA, USA, 1979. [Google Scholar]
- Walton, J.; Lee, C.; Martin, M. High Speed Balancing Applied to the t700 Engine; Mechanical Technology Inc.: Latham, NY, USA, 1989. [Google Scholar]
- Martin, M.A. T55 Power Turbine Rotor Multiplane/Multispeed Balancing Study; Mechanical Technology Inc.: Latham, NY, USA, 1982. [Google Scholar]
- Martin, M.R. Development of a Multiplane/Multispeed Balancing System for Turbine Engines; Mechanical Technology Inc.: Latham, NY, USA, 1984. [Google Scholar]
- Deng, W.Q.; Gao, D.Q. High speed dynamic balance technique applied to flexible rotors of a small-sized engine. Gas Turbine Exp. Res.
**2003**, 16, 30–33. (In Chinese) [Google Scholar] - Tresser, S.; Dolev, A.; Bucher, I. Dynamic balancing of super-critical rotating structures using slow-speed data via parametric excitation. J. Sound Vib.
**2018**, 415, 59–77. [Google Scholar] [CrossRef] [Green Version] - Untaroiu, C.D.; Allaire, P.E.; Foiles, W.C. Balancing of flexible rotors using convex optimization techniques: Optimum min-max lmi influence coefficient balancing. J. Vib. Acoust.
**2008**, 130, 111–120. [Google Scholar] [CrossRef] - Mohammadi, N.; Mohammadzadeh, A. Balancing of the flexible rotors with ica methods. Int. J. Res. Rev. Appl. Sci.
**2015**, 23, 54–64. [Google Scholar] - Saldarriaga, M.V.; Steffen, V.; Der Hagopian, J.; Mahfoud, J. On the balancing of flexible rotating machines by using an inverse problem approach. J. Vib. Control
**2011**, 17, 1021–1033. [Google Scholar] [CrossRef]

**Figure 1.**A rotating rotor whose axis is not coincident with the geometric axis. (x-the rotor’s rotation axis, o-the disk’s rotation center, o’-the disk’s geometric center, e-the disk’s eccentric distance, φ-the disk’s rotation phase angle.)

**Figure 4.**Configuration of the numerical example [47]. (A,B,C,D-four disks on the shaft, Kx- stiffness of bearing, Cx- damping of bearing, S,T- measuring plane locations.)

**Figure 5.**Theoretical model of the numerical example [47]. (1-10-ten lumped masses, Kx-stiffness of bearing, Cx-damping of bearing.)

**Figure 6.**Flow chart of the balancing methodology [57].

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, L.; Cao, S.; Li, J.; Nie, R.; Hou, L.
Review of Rotor Balancing Methods. *Machines* **2021**, *9*, 89.
https://doi.org/10.3390/machines9050089

**AMA Style**

Li L, Cao S, Li J, Nie R, Hou L.
Review of Rotor Balancing Methods. *Machines*. 2021; 9(5):89.
https://doi.org/10.3390/machines9050089

**Chicago/Turabian Style**

Li, Liqing, Shuqian Cao, Jing Li, Rimin Nie, and Lanlan Hou.
2021. "Review of Rotor Balancing Methods" *Machines* 9, no. 5: 89.
https://doi.org/10.3390/machines9050089