Geometrically Non-Linear Axisymmetric Free Vibrations of Thin Isotropic Annular Plates

Authors: Boutahar Lhoucine, El Bikri Khalid, Benamar Rhali

Abstract:

The effects of large vibration amplitudes on the first axisymetric mode shape of thin isotropic annular plates having both edges clamped are examined in this paper. The theoretical model based on Hamilton’s principle and spectral analysis by using a basis of Bessel’s functions is adapted اhere to the case of annular plates. The model effectively reduces the large amplitude free vibration problem to the solution of a set of non-linear algebraic equations.

The governing non-linear eigenvalue problem has been linearised in the neighborhood of each resonance and a new one-step iterative technique has been proposed as a simple alternative method of solution to determine the basic function contributions to the non-linear mode shape considered.

Numerical results are given for the first non-linear mode shape for a wide range of vibration amplitudes. For each value of the vibration amplitude considered, the corresponding contributions of the basic functions defining the non-linear transverse displacement function and the associated non-linear frequency, the membrane and bending stress distributions are given. By comparison with the iterative method of solution, it was found that the present procedure is efficient for a wide range of vibration amplitudes, up to at least 1.8 times the plate thickness,

Keywords: Non-linear vibrations, Annular plates, Large vibration amplitudes.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1088864

References:

[1] A. W. Leissa. Vibration of plates. 1969 Office of Technology Utilization. Washington.
[2] C.L. Huang, B.E. Sandman, 1971 Large amplitude vibrations of a rigidity clamped circular plate, International Journal of non-linear Mechanics 6, 451–468.
[3] J.N. Reddy, C.L. Huang, I.R. Singh, 1981 Large deflections and large amplitude vibrations of axisymmetric circular plates, International Journal for Numerical Methods in Engineering 17, 527–541.
[4] R. Benamar, M.M.K. Bennouna and R.G. White, 1991. The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures. Part I: Simply supported and clamped –clamped beams, Journal of Sound and Vibration 149 (2), 179–195.
[5] R. Benamar, M.M.K. Bennouna and R.G. White, 1993. The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures. Part II: Fully clamped rectangular isotropic plates, Journal of Sound and Vibration 164 (2), 295–316.
[6] S. A. Vera, M. Febbo, C. A. Rosit, A. E. Dolinko. 2002. Transverse vibrations of circular annular plates with edges elastically restrained against rotation, used in acoustic underwater transducers. Ocean Engineering 29,1201-1208.
[7] M. Haterbouch, R. Benamar, 2003. The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates, part I: iterative and explicit analytical solution for non-linear transverse vibrations, Journal of Sound and Vibration 265, 123–154.