Archive of Applied Mechanics

, Volume 88, Issue 6, pp 955–979 | Cite as

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


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.


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



Length of the shaft (m)


Radius of the circular section (m)


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

\(\rho \)

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


Number of disk


Number of elastic supports

\(J_j \)

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


Stiffness with associated elastic support (N/m)

\(\theta \)

Twist angular displacement (rad)

\(\gamma \)

Shear strain

\(\tau \)

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


Shear strain energy (J)


Total strain energy of the shaft (J)


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


Kinetic energy of shafting (J)


Kinetic energy of the j-th disk (J)

\(\delta \)

Dirac Delta function

\(U_k \)

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

\(\omega \)



Diameter of the shaft (m)


Diameter of the disk (m)


Width of the disk (m)


Non-dimensional variable



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.


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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

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