New analytic method for free torsional vibration analysis of a shaft with multiple disks and elastic supports

  • Meilong Chen
  • Shuying Li
  • Hongliang Li
  • Tao Peng
  • Siyuan Liu
Original
  • 34 Downloads

Abstract

Free torsional vibration analysis of a shaft with multiple disks and elastic supports is important in mechanical engineering. As is well known, many numerical methods have been proposed to solve the problem, but exact analytic solutions are rarely reported in the literature. In this paper, a successful method is presented to solve this problem by combining the Hamilton’s principle and integral transform. The analysis results from the proposed method agree well with the results published in the studies. Compared with lumped-mass method, it shows that with lumped-mass method, the accuracy of computation of natural frequencies and modes very much depends on the numbers of simplified inertia and the structures simplified. The results demonstrate that the proposed method is superior to the lumped-parameter method in accuracy. The proposed method is used to verify the finite element method while modeling shafting. The results indicate that when using finite element modeling shaft, the principle is that the order of interpolation functions should be chosen as high as possible, the elements chosen as many as possible and the discrete finite elements of shaft divided as even as possible in a reasonable range.

Keywords

New analytic method Free torsional vibration The Hamilton principle Integral transform Lumped-mass method Finite element method 

Notation

L

Length of the shaft (m)

R

Radius of the circular section (m)

G

Shear modulus \((\hbox {N/m}^{2})\)

\(\rho \)

Mass density \((\hbox {kg/m}^{3})\)

s

Number of disk

r

Number of elastic supports

\(J_j \)

Moment of inertia with the associated disk \((\hbox {kg}\,\hbox {m}^{2})\)

\(K_i\)

Stiffness with associated elastic support (N/m)

\(\theta \)

Twist angular displacement (rad)

\(\gamma \)

Shear strain

\(\tau \)

Shear stress\((\hbox {N/m}^{2})\)

u

Shear strain energy (J)

\(U_\mathrm{s}\)

Total strain energy of the shaft (J)

\(I_\mathrm{p}\)

Polar moment of inertia of the cross section (\(\hbox {kg} \, \hbox {m}^{2})\)

\(T_\mathrm{s}\)

Kinetic energy of shafting (J)

\(T_j\)

Kinetic energy of the j-th disk (J)

\(\delta \)

Dirac Delta function

\(U_k \)

Potential energy of the k-th elastic support (J)

\(\omega \)

Constant

d

Diameter of the shaft (m)

D

Diameter of the disk (m)

L3

Width of the disk (m)

x

Non-dimensional variable

Notes

Acknowledgements

The research work is supported by the National Natural Science Foundation of China (Grant No. 51375104), Heilongjiang Province Funds for Distinguished Young Scientists (Grant No. JC 201405), China Postdoctoral Science Foundation (Grant No. 2015M581433) and Postdoctoral Science Foundation of Heilongjiang Province (Grant No. LBH-Z15038).

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

  1. 1.
    Wu, J.J.: Torsional vibration analyses of a damped shafting system using tapered shaft element. J. Sound. Vib. 306(3–5), 946–954 (2007)CrossRefGoogle Scholar
  2. 2.
    Shahgholi, M., Khadem, S.E., Bab, S.: Free vibration analysis of a nonlinear slender rotating shaft with simply support conditions. Mech. Mach. Theory. 82, 128–140 (2014)CrossRefGoogle Scholar
  3. 3.
    Boukhalfa, A., Hadjoui, A., Hamza, C.S.M.: Free vibration analysis of an embarked rotating composite shaft using the hp-version of the FEM. Lat. Am. J. Solids Struct. 7(2), 105–141 (2010)CrossRefGoogle Scholar
  4. 4.
    Bulut, G.: Dynamic stability analysis of torsional vibrations of a shaft system connected by a Hookexs joint through a continuous system model. J. Sound Vib. 333(16), 3691–3701 (2014)CrossRefGoogle Scholar
  5. 5.
    Hosseini, S.A.A., Khadem, S.E.: Free vibrations analysis of a rotating shaft with nonlinearities in curvature and inertia. Mech. Mach. Theory. 44(1), 272–288 (2009)CrossRefMATHGoogle Scholar
  6. 6.
    Salarieh, H., Ghorashi, M.: Free vibration of Timoshenko beam with finite mass rigid tip load and flexural-torsional coupling. Int. J. Mech. Sci. 48(7), 763–779 (2006)CrossRefMATHGoogle Scholar
  7. 7.
    Wu, T.X.: Analytical study on torsional vibration of circular and annular plate. Proc. Inst. Mech Eng. Part C J. Mech. Eng. Sci. 220(4), 393–401 (2006)CrossRefGoogle Scholar
  8. 8.
    Shirazi, A.A.R.H., Hematiyan, M.R.: Closed-form formulation for torsional analysis of beams with open or closed cross sections having a crack. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 227(12), 1953–1967 (2013)CrossRefGoogle Scholar
  9. 9.
    Zou, C.P., Chen, D.S., Hua, H.X.: Torsional vibration analysis of complicated multi-branched shafting systems by modal synthesis method. J. Vib. Acoust. 125(3), 317–323 (2003)CrossRefGoogle Scholar
  10. 10.
    Tang, X.L., Jin, Y.J., Zhang, J.W., Zou, L., Yu, H.S.: Torsional vibration and acoustic noise analysis of a compound planetary power-split hybrid electric vehicle. Proc. Inst. Mech Eng. Part D J. Automob. Eng. 228(1), 94–103 (2014)CrossRefGoogle Scholar
  11. 11.
    Chondros, T.G., Labeas, G.N.: Torsional vibration of a shaft with non-uniform cross section. J. Sound Vib. 301(3–5), 994–1006 (2007)CrossRefGoogle Scholar
  12. 12.
    Tang, X.L., Zhang, J.W., Zou, L., Yu, H.S., Zhang, D.J.: Study on the torsional vibration of a hybrid electric vehicle powertrain with compound planetary power-split electronic continuous variable transmission. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 228(17), 3107–3115 (2014)CrossRefGoogle Scholar
  13. 13.
    Sapountzakis, E.J.: Torsional vibrations of composite bars of variable cross-section by BEM. Comput. Methods Appl. Mech. 194(18–20), 2127–2145 (2005)CrossRefMATHGoogle Scholar
  14. 14.
    Ma, J. M, Ren, Y. S.: Vibration and stability of variable cross section thin-walled composite shafts with transverse shear. Shock Vib. 2015, 1–12 (2015)Google Scholar
  15. 15.
    Gama, A.L., De Oliveira, S.R.: An algorithm for solving torsional vibration problems based on the invariant imbedding method. Int. J. Acoust. Vib. 19(2), 107–113 (2014)MathSciNetGoogle Scholar
  16. 16.
    Wu, J.J.: Torsional vibrations of a conic shaft with opposite tapers carrying arbitrary concentrated elements. Math. Probl. Eng. 2013, 1–19 (2013)Google Scholar
  17. 17.
    Bazehhour, B.G., Mousavi, S.M., Farshidianfar, A.: Free vibration of high-speed rotating Timoshenko shaft with various boundary conditions: effect of centrifugally induced axial force. Arch. Appl. Mech. 84, 1691–1700 (2014)CrossRefGoogle Scholar
  18. 18.
    Mirtalaie, S.H., Hajabasi, M.A.: Nonlinear axial-lateral-torsional free vibrations analysis of Rayleigh rotating shaft. Arch Appl Mech (2017).  https://doi.org/10.1007/s00419-017-1265-6 Google Scholar
  19. 19.
    Bogacz, R., Noga, S.: Free transverse vibration analysis of a toothed gear. Arch. Appl. Mech. 82, 1159–1168 (2012)CrossRefMATHGoogle Scholar
  20. 20.
    Akgöz, B., Civalek, O.: A novel microstructure-dependent shear deformable beam model. Int. J. Mech. Sci. 99, 10–20 (2015)CrossRefMATHGoogle Scholar
  21. 21.
    Behzad, M., Bastami, A.R.: Effect of centrifugal force on natural frequency of lateral vibration of rotating shafts. J. Sound Vib. 274, 985–995 (2004)CrossRefGoogle Scholar
  22. 22.
    Civalek, O., Kiracioglu, O.: Free vibration analysis of Timoshenko beams by DSC method. Int. J. Numer. Methods Biomed. Eng. 26, 1890–1898 (2010)MATHGoogle Scholar
  23. 23.
    Zou, J.X., Yu, K.P.: Dynamics of Structural, p. 94. Harbin Institute of Technology Press, Harbin (2009)Google Scholar
  24. 24.
    Shang, D.Z., Li, H.L., Han, G.C.: Structural Dynamics, p. 91. Harbin Engineering University Press, Harbin (2005)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Meilong Chen
    • 1
  • Shuying Li
    • 1
  • Hongliang Li
    • 1
  • Tao Peng
    • 1
  • Siyuan Liu
    • 1
  1. 1.Harbin Engineering UniversityHarbinChina

Personalised recommendations