Skip Navigation Links
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland


C. Steve Suh (editor)

Texas A&M University, USA


Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China


Generalized Formulation for Free Vibration of Elastic Solids with Static Loads and Application to Rotating Tapered Cantilever Beam Vibration

Journal of Vibration Testing and System Dynamics 5(2) (2021) 149--168 | DOI:10.5890/JVTSD.2021.06.004

Renfan Luo

Finmere, Rugby, Warwickshire, UK

Download Full Text PDF



By solving the three one-dimensional (1D) nonlinear dynamic differential equations analytically, it has been proved that unless the nonlinear terms are in first order, a nonlinear dynamic system never has a vibration natural frequency. For an elastic solid with nonlinear deformation and with static loads including a rotational angular velocity, a virtual small factor has been introduced to ensure a small deformation, a generalized formulation to predict vibration frequencies has been developed. Tapered rotating cantilever beams have been used to validate the formulation against FE analysis, and the analytical and FE results are in a good agreement.


  1. [1]  Lin, W.S. and Donald T. (1992), Greenwood, General dynamic equations of motion for elastic body systems, Journal of Guidance, Control, and Dynamics, 15(6).
  2. [2]  Cao, D.Q. and Tucker, R.W. (2008), Nonlinear dynamics of elastic rods using the Cosserat theory: Modelling and simulation, International Journal of Solids and Structures, 45, 460--477.
  3. [3]  Luo, R. (2014), Formulating frequency of uniform beams with tip mass under various axial loads, Proc ImechE Part C, J Mechanical Engineering Science, 228(I), 67-76.
  4. [4]  Luo, R. (2012), Free transverse vibration of rotating blades in a bladed disk assembly, J Acta Mech, 223, 1385-1396.
  5. [5]  Luo, R. (2019), Transverse vibration of long offshore pipes, International Journal of Pressure Vessels and Pipes, 175, 103928.
  6. [6]  Leissa, A.W. (2005), The historical bases of the Rayleigh and Ritz methods, \underline {Journal of Sound and Vibration}, 287(4--5), Pages 961-97.
  7. [7]  Naguleswaran, S. (2004), Transverse vibration of a uniform Euler-Bernoulli beam under linearly varying axial force, J Sound Vib, 275, 47-57.
  8. [8]  Wu, J., Shao, M., Wang, Y., Wu, Q., and Nie, Z. (2017), Nonlinear vibration characteristics and stability of the printing moving membrane, Journal of Low Frequency Noise, Vibration and Active Control, 36(3), 306--316.
  9. [9]  Marynowski, K. (2008), Nonlinear vibration of axial moving paper web, Journal of Theoretical and Applied Mechanics, 46(3), pp 565-580, Warsaw.
  10. [10]  Tian, W., et al, (2019), Analysis of Nonlinear Vibration and Dynamic Responses in a Trapezoidal Cantilever Plate Using the Rayleigh-ritz Approach Combined with Affine Transformation, Mathemtical Problems in Engineering, Vol. 9278069.
  11. [11]  Bakhtiari-Nejad, F. and Nazari, M. (2009), Nonlinear vibration analysis of isotropic cantilever plate with viscoelastic laminate, Nonlinear Dynamics, 56(4), 325-356.
  12. [12]  Golpayegani, I.F. (2018), Calculation of Natural Frequencies of Bi-Layered Rotating Functionally Graded Cylindrical Shells, Journal of Solid Mechanics, 10(1), pp 216-231.
  13. [13]  Lang, Z.Q. and Billings, S.A. (1996), Output frequency characteristics of nonlinear systems, Int. J Control, 64(6), 1049-1067.
  14. [14]  Lang, Z.Q. and Billings, S.A. (2004), Energy Transfer Properties of Nonlinear Systems in Frequency Domain, Research Report, No 862, The University of Sheffield.
  15. [15]  Lang, Z.Q. et al., (2009), Theoretical study of the effects of nonlinear viscous damping on vibration isolation of sdof systems, Journal of Sound and Vibration, 323, 352-365.
  16. [16]  Link, M. et al, (2010), An Approach to Non-linear Experimental Modal Analysis, Proceedings of the Int. Modal Analysis Conference, IMAC-XXVII, Jacksonville, FL.
  17. [17]  Kuether, R.J. et al, (2015), Nonlinear normal modes, modal interactions and isolated resonance curves, Journal of Sound and Vibration, 351, pp 299-310.
  18. [18]  Thomas, O., Senechal, A., and Deu, J.F. (2016), Hardening/softening behaviour and reduced order modelling of nonlinear vibration of rotating cantilever beams, Nonlinear Dynamics, 86(2), pp 1293-1318.
  19. [19]  Bokain, A. (1990), Natural frequencies of beams under tensile axial loads, J Sound Vib, 142, 481-498.
  20. [20]  Banerjee, J.R., Su, H., and Jackson, D.R. (2006), Free vibration pf rotating tapered beams using the dynamic stiffness method, Journal of Sound and Vibration, 298, 1034-1054.
  21. [21]  Banerjee, J.R. and Kennedy, D. (2014), Dynamic Stiffness Method for Inplane Vibration of Rotating Beams Including Coriolis Effects, Journal of sound and Vibration, 333, 7299-7312.
  22. [22]  Wang, Z. and Li, R. (2018), Transverse vibration of rotating tapered cantilever beam with hollow circular-cross-section, Shock and Vibration, (5), 1-14.