A Novel Axial Vibration Model of Multistage Pump Rotor System with Dynamic Force of Balance Disc

  • Wenjie ZhouEmail author
  • Yuhua Cao
  • Ning Zhang
  • Bo Gao
  • Ning Qiu
  • Weibin Zhang
Original Paper



In this paper, a novel axial dynamic model including the transient force of balance disc is proposed to predict the vibration characteristics of multistage pump rotor system.


To obtain the dynamic force, the rotating effect of the rotor system is considered and the Navier–Stokes equations are further simplified on the basis of the geometric structure and inner flow characteristic of the balance disc. In addition, based on finite element method and matrix operation, a novel axial motion model of rotor system is established. The efficient Newmark method is applied to describe the dynamic response of the coupled rotor system.


The pressure distribution in axial clearance and the corresponding dynamic force present obvious nonlinear reduction as the axial gap increases from 0.2 mm to 1 mm. The inner chamber pressure is more sensitive to the inlet pressure than the rotating speed, especially when the axial gap is 0.2 mm. Moreover, the axial steady amplitude of the rotor system is directly proportional to the rotating speed and initial axial gap, while it is inversely proportional to the outer radius of the balance disc. In addition, the vibration frequencies for axial vibration are multiple even when the motion of the rotor system is regularly reciprocating.


The transient force of the balance disc needs to be considered in the calculation of axial rotor dynamics for the multistage pump. The calculated results can provide references for the design of the balance disc and an axial vibration model of the multistage pump rotor system.


Balance disc Axial dynamic characteristics Transient force model Multistage pump rotor Finite element method (FEM) 

List of Symbols


Cross-sectional area

b1, b2

Radial and axial clearance


Young’s modulus

fr, fz, fθ

Body force components in the radial, axial and circumferential direction


Length of shaft element

l1, l2

Length of radial and axial clearance



p1, p2, p3

Inlet pressure, inner pressure and outlet pressure of balance disc


Total axial force of balance disc

q1, q2

Flux in radial and axial clearance

r1, r2

Inner radius and outer radius of radial clearance

r3, r4

Inner radius and outer radius of axial clearance


Kinetic energy of rotor system


Kinetic energy of elastic shaft element



ur, uz, uθ

Velocity in the radial, axial and circumferential direction


Potential energy of rotor system


Potential energy of elastic shaft element


Empirical coefficient


Time step




Dynamic viscosity


Shaft mass for length unit


Kinematic viscosity


Fluid density




Empirical coefficient


Rotating speed



This work was supported by the National Natural Science Foundation of China (Grant No. 51706087), the Project funded by China Postdoctoral Science Foundation (Grant No. 2018M642177), the Zhejiang Postdoctoral Preferential Foundation (Grant No. zj20180009), the Open Research Subject of Key Laboratory of Fluid and Power Machinery (Xihua University), Ministry of Education (Grant No. szjj2019-009), the Key Research and Development Program of Zhenjiang (Grant No. GY2018023) and Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). The authors also thank Anthony Akurugo Alubokin and Longjie Yu for their help.

Compliance with Ethical Standards

Conflict of Interest

There are no conflicts of interest as declared by the authors.


  1. 1.
    Zhang N, Liu X, Gao B et al (2019) DDES analysis of the unsteady wake flow and its evolution of a centrifugal pump. Renew Energy 141:570–582CrossRefGoogle Scholar
  2. 2.
    Ni D, Yang M, Gao B et al (2018) Experimental and numerical investigation on the pressure pulsation and instantaneous flow structure in a nuclear reactor coolant pump. Nucl Eng Des 337:261–270CrossRefGoogle Scholar
  3. 3.
    Wang K, Lu X, He X (2018) Experimental investigation of vibration characteristics in a centrifugal pump with vaned diffuser. Shock Vib 2018:486536Google Scholar
  4. 4.
    Wang C, Chen X, Qiu N et al (2018) Numerical and experimental study on the pressure fluctuation, vibration, and noise of multistage pump with radial diffuser. J Braz Soc Mech Sci Eng 40(10):481CrossRefGoogle Scholar
  5. 5.
    Zhang N, Gao B, Li Z et al (2018) Unsteady flow structure and its evolution in a low specific speed centrifugal pump measured by PIV. Exp Therm Fluid Sci 97:133–144CrossRefGoogle Scholar
  6. 6.
    Zhou W, Wei X, Wei X et al (2014) Numerical analysis of a nonlinear double disc rotor-seal system. J Zhejiang Univ Sci A 15(1):39–52MathSciNetCrossRefGoogle Scholar
  7. 7.
    Varney P, Green I (2017) Steady-state response of a flexibly mounted stator mechanical face seal subject to dynamic forcing of a flexible rotor. J Tribol 139(6):062201CrossRefGoogle Scholar
  8. 8.
    Zhou W, Qiu N, Wang L et al (2018) Dynamic analysis of a planar multistage centrifugal pump rotor system based on a novel coupled model. J Sound Vib 434:237–260CrossRefGoogle Scholar
  9. 9.
    Xu Q, Niu J, Yao H et al (2018) Fluid-induced vibration elimination of a rotor/seal system with the dynamic vibration absorber. Shock Vib 2018:1738941Google Scholar
  10. 10.
    Pan G (2009) Distribution of velocity and pressure within balance disk of multistage-section pump. Xihua University, Chengdu (in Chinese) Google Scholar
  11. 11.
    Will BC, Benra FK, Dohmen HJ (2012) Investigation of the flow in the impeller side clearances of a centrifugal pump with volute casing. J Therm Sci 21(3):197–208CrossRefGoogle Scholar
  12. 12.
    Li W (2013) Model of flow in the side chambers of an industrial centrifugal pump for delivering viscous oil. J Fluids Eng 135(5):051201CrossRefGoogle Scholar
  13. 13.
    Zhao W, Wang Z, Wang L et al (2013) Study and numerical simulation on dynamic equilibrium of balance disc in multistage pump. J Mech Eng 49(4):163–167 (in Chinese) CrossRefGoogle Scholar
  14. 14.
    Zhao W, Wei L, Yan S (2017) Effect of structural parameters of balance disc on dynamic equilibrium rule of rotor system in multistage pump. J Lanzhou Univ Technol 43(4):53–58 (in Chinese) Google Scholar
  15. 15.
    Guha A, Sengupta S (2013) The fluid dynamics of the rotating flow in a Tesla disc turbine. Eur J Mech B/Fluids 37:112–123MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Turkyilmazoglu M (2014) Nanofluid flow and heat transfer due to a rotating disk. Comput Fluids 94:139–146MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wu W, Xiao B, Hu J et al (2018) Experimental investigation on the air-liquid two-phase flow inside a grooved rotating-disk system: flow pattern maps. Appl Therm Eng 133:33–38CrossRefGoogle Scholar
  18. 18.
    Wang CC, Yao YD, Liang KY et al (2010) Axial vibration study of a mobile fan motor. IEEE Trans Magn 46(6):1397–1400CrossRefGoogle Scholar
  19. 19.
    Evrensel CA, Finley CD (2002) Verification of axial rotor oscillations in cryogenic turbine generators. The 9th international symposium on transport phenomena and dynamics of rotating machinery, Honolulu, HawaiiGoogle Scholar
  20. 20.
    Patel TH, Darpe AK (2009) Vibration response of misaligned rotors. J Sound Vib 325(3):609–628CrossRefGoogle Scholar
  21. 21.
    Luo R, Hu G, Wang Q (2011) Axial vibration analysis and validity of high-speed rotor of cantilever SGCMG based on a triangle-star equivalent transform action. Aerosp Control Appl 37(02):14–20Google Scholar
  22. 22.
    Zhou Y, Peng FY, Cao XH (2011) Parameter sensitivity analysis of axial vibration for lead-screw feed drives with time-varying framework. Mechanika 17(5):523–528CrossRefGoogle Scholar
  23. 23.
    Ye J, Zheng C, Yao X (2013) Analysis of coupled bending-axial vibration of a rotor. Adv Mater Res 662:608–611CrossRefGoogle Scholar
  24. 24.
    Kashani MT, Hashemi SM (2018) A finite element formulation for bending-torsion coupled vibration analysis of delaminated beams under combined axial load and end moment. Shock Vib 2018:1348970Google Scholar
  25. 25.
    Kashani MT, Jayasinghe S, Hashemi SM (2015) Dynamic finite element analysis of bending-torsion coupled beams subjected to combined axial load and end moment. Shock Vib 2015:471270Google Scholar
  26. 26.
    Sukkar R, Yigit AS (2008) Analysis of fully coupled torsional and lateral vibrations of unbalanced rotors subject to axial loads. Kuwait J Sci Eng 35(2B):143–170Google Scholar
  27. 27.
    Liu Y, Shu DW (2014) Coupled bending-torsion vibration of a homogeneous beam with a single delamination subjected to axial loads and static end moments. Acta Mech Sin 30(4):607–614MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tchomeni BX, Alugongo AA, Masu LM (2014) In situ modelling of lateral-torsional vibration of a rotor-stator with multiple parametric excitations. World Acad Sci Eng Technol Int J Mech Aerosp Ind Mechatron Eng 8(11):1855–1861Google Scholar
  29. 29.
    Towliat Kashani MT, Jayasinghe S, Hashemi SM (2014) On the flexural-torsional vibration and stability of beams subjected to axial load and end moment. Shock Vib 153532Google Scholar
  30. 30.
  31. 31.
    Liu Z, Xu L, Jia X et al (2013) Analysis of liquid flow and axial force calculation in axial clearance for floating impeller of centrifugal pump. Trans Chin Soc Agric Eng 29(12):79–85 (in Chinese) Google Scholar
  32. 32.
    Zhou W (2016) The dynamic characteristics study on multi-stage centrifugal pump rotor coupled system. Zhejiang university, Hangzhou (in Chinese) Google Scholar
  33. 33.
    Zhou WJ, Wei XS, Wu GK et al (2015) Numerical research on dynamic lateral vibration of a pump-turbine’s shaft system. J Eng Res 3(4):127–148CrossRefGoogle Scholar
  34. 34.
    Gong RZ, Wang H, Zhao JL et al (2014) Influence of clearance parameters on the rotor dynamic character of hydraulic turbine shaft system. Proc Inst Mech Eng C J Mech Eng Sci 228(2):262–270CrossRefGoogle Scholar
  35. 35.
    Qiang H, Yuan S, Fengzhang R et al (2017) Numerical simulation and experimental study of the air-cooled motorized spindle. Proc Inst Mech Eng C J Mech Eng Sci 231(12):2357–2369CrossRefGoogle Scholar
  36. 36.
    Childs DW, Norrbin CS, Phillips S (2014) A lateral rotor dynamics primer on electric submersible pumps (esps) for deep subsea applications. Proceedings of the 30th international pump users symposium. Turbomachinery Laboratories, Texas A&M Engineering Experiment StationGoogle Scholar

Copyright information

© Krishtel eMaging Solutions Private Limited 2019

Authors and Affiliations

  • Wenjie Zhou
    • 1
    Email author
  • Yuhua Cao
    • 1
  • Ning Zhang
    • 1
    • 2
  • Bo Gao
    • 1
  • Ning Qiu
    • 3
  • Weibin Zhang
    • 4
  1. 1.School of Energy and Power EngineeringJiangsu UniversityZhenjiangChina
  2. 2.Mechanical Engineering DepartmentUniversity College LondonLondonUK
  3. 3.Research Center of Fluid Machinery Engineering and TechnologyJiangsu UniversityZhenjiangChina
  4. 4.Key Laboratory of Fluid and Power Machinery, Ministry of EducationXihua UniversityChengduChina

Personalised recommendations