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


Comprehensive Modal Analysis for In-Plane Free Vibrations of High-Speed Annular Disks

Journal of Vibration Testing and System Dynamics 7(4) (2023) 471--507 | DOI:10.5890/JVTSD.2023.12.006

Ehsan Sarfaraz, Hamid R. Hamidzadeh

Department of Mechanical and Manufacturing Engineering, Tennessee State University, Nashville, TN 37209,


Download Full Text PDF



In view of the vast potential applications of flexible thin rotating disks, the knowledge of their vibration characteristics has been considered by many investigators. Rotating disks are the main components in various machinery applications, such as space structures, flywheels, torsional disk dampers, grinding wheels, turbine rotors, circular saw blades, computer storage devices and brake systems. Dynamic response and stability of rotating disk depend on its rotational speed. The knowledge of the in-plane vibration of rotating disks is also essential for the design of spinning disks. In most cases, to rotating a disk at a certain speed, a knowledge of modal vibrations and critical speeds of the disks are essential. An analytical solution is investigated to determine in-plane modal vibration characteristics of high speed rotating annular disks. A systematic approach based on the established governing equation for the linear in-plane free vibrations of disks is developed, and the displacements and stresses compatibilities are considered. The disk material is elastic homogeneous, thin, and isotropic and is rotating at constant angular speed. The developed analytical solution is obtained by implementing the two-dimensional plane stress theory. In this research, several possible boundary conditions for the annular disks are investigated, and natural frequencies and mode shapes of rotating disks are computed. The mode shape functions for displacements and stresses in the radial and circumferential directions are determined. In addition, variations of nondimensional modal frequencies versus a wide range of dimensionless rotational speeds for several radius ratios are presented.


  1. [1]  Love, A. (1944), A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, NY, USA.
  2. [2]  Onoe, M. (1956), Contour vibrations of isotropic circular plates, Journal of the Acoustical Society of America, 28, 1158-1162.
  3. [3]  Bhuta, P.G. and Jones, J.P. (1963), Symmetric planar vibrations of a rotating disk, The Journal of the Acoustical Society of America, 35, 982-989.
  4. [4]  Srinivasan, V. and Ramamurti, V. (1980), Dynamic response of an annular disk to a moving concentrated, in-plane edge load, Journal of Sound and Vibration, 72(2), 251-262.
  5. [5]  Burdess, J. S., Wren, T., and Fawcett, J.N. (1987), Plane stress vibrations in rotating disks, Proceedings of Institute of Mechanical Engineers, 201, 37-44.
  6. [6]  Chen, J.S. and Jhu, J.L. (1996), On the in-plane vibration and stability of a spinning annular disk, Journal of Sound and Vibration, 195(4), 585-593.
  7. [7]  Chen, J.S. and Jhu, J.L. (1997), In-plane stress and displacement distributions in a spinning annular disk under stationary edge loads, Journal of Applied Mechanics, 64, 897-904.
  8. [8]  Hamidzadeh, H.R. and Dehghani, M. (1999), Linear in-plane free vibration of rotating disks, Proceedings of the ASME 17th Biennial Conference on Mechanical Vibration and Noise, 673-682.
  9. [9]  Hamidzadeh, H.R. and Wang H. (2000), In-Plane Free Vibration of Spinning Rings an Analytical Approach, ASME International Mechanical Engineering Congress and Exposition, 108, 9-16.
  10. [10]  Hamidzadeh, H.R. (2002), In-plane free vibration and stability of rotating annular discs, Journal of Multi-body Dynamics, 216(4), 371-380.
  11. [11]  Deshpande, M. and Mote, M. (2003), In-plane Vibration of Thin disks, ASME Vibrations and Acoustic, 125(1), 68-72.
  12. [12]  Farag, N.H. and Pan, J. (2003), Modal characteristics of in-plane vibration of circular plates clamped at the outer edge, Journal of the Acoustical Society of America, 113(4), 1935-1946.
  13. [13]  Bashmal, S., Bhat, R., and Rakheja, S. (2009), In-plane free vibration of circular annular disks, Journal of Sound and Vibration, 322(1-2), 216-226.
  14. [14]  Hamidzadeh, H.R. and Sarfaraz, E. (2012), Influence of Material Damping on In-Plane Modal Parameters for Rotating Disks, Proceedings of the ASME 2012 International Mechanical Engineering Congress and Exposition, 4(Dynamics, Control and Uncertainty, Parts A and B. Houston, Texas, USA. November 9-15), 89-96.
  15. [15]  Sarfaraz, E. and Hamidzadeh, H.R. (2013), The effect of material damping on in-plane vibration characteristics of rotating disk, in: ASNT 22nd Research Symposium, 108-112.
  16. [16]  Hamidzadeh, H.R. and Sarfaraz, E. (2014), In-Plane Free Vibration and Stability of High Speed Rotating Annular Disks and Rings, in: J.A.T. Machado, D. Baleanu, A.C.J. Luo (Eds.), Discontinuity and Complexity in Nonlinear Physical Systems, Nonlinear Systems and Complexity, Springer, 389-408.
  17. [17]  Hamidzadeh, H.R. and Sarfaraz, E. (2012), Influence of embedded material at edges on natural frequencies of rotating annular disk, 2012 IEEE 4th International Conference on Nonlinear Science and Complexity (NSC), 167-176.
  18. [18]  Sarfaraz, E. and Hamidzadeh, H.R. (2013), Influence of Embedded Material on Natural Frequencies of Double Segment Rotating Disk, Journal of Applied and Nonlinear Dynamics, 2(2), 175-192.
  19. [19]  Sarfaraz, E. and Hamidzadeh, H.R. (2020), In-Plane Vibration Mode Shapes for Rotating Disks: Exact Solution, Proceedings of the ASME 2020 International Mechanical Engineering Congress and Exposition, 7B (Dynamics, Vibration, and Control. Virtual, Online. November 16-19), V07BT07A043.
  20. [20]  Bagheri, E. and Jahangiri M. (2019), Analysis of in-plane vibration and critical speeds of the functionally graded rotating disks, International Journal of Applied Mechanics, 11(2), 1950020
  21. [21]  Lyu, P., Du, J., Wang, Y. and Liu Z. (2019), Free in-plane vibration analysis of rotating annular panels with elastic boundary restraints, Journal of Sound and Vibration, 439, 434-456.
  22. [22]  Lyu, P., Du, J., and Liu, Z. (2020), A semianalytical solution for in-plane vibration analysis of annular panels with arbitrary distribution of internal point constraints, Mathematical Problems in Engineering, Article ID 7269809.