Abstract
The free vibration of annular thick plates with linearly varying thickness along the radial direction is studied, based on the linear, small strain, three-dimensional (3-D) elasticity theory. Various boundary conditions, symmetrically and asymmetrically linear variations of upper and lower surfaces are considered in the analysis. The well-known Ritz method is used to derive the eigen-value equation. The trigonometric functions in the circumferential direction, the Chebyshev polynomials in the thickness direction, and the Chebyshev polynomials multiplied by the boundary functions in the radial direction are chosen as the trial functions. The present analysis includes full vibration modes, e.g., flexural, thickness-shear, extensive, and torsional. The first eight frequency parameters accurate to at least four significant figures for five vibration categories are obtained. Comparisons of present results for plates having symmetrically linearly varying thickness are made with others based on 2-D classical thin plate theory, 2-D moderate thickness plate theory, and 3-D elasticity theory. The first 35 natural frequencies for plates with asymmetrically linearly varying thickness are compared to the finite element solutions; excellent agreement has been achieved. The asymmetry effect of upper and lower surface variations on the frequency parameters of annular plates is discussed in detail. The first four modes of axisymmetric vibration for completely free circular plates with symmetrically and asymmetrically linearly varying thickness are plotted. The present results for 3-D vibration of annular plates with linearly varying thickness can be taken as benchmark data for validating results from various plate theories and numerical methods.
Similar content being viewed by others
References
Leissa, A.W.: Vibrations of Plates. NASA SP-160, USA (1969)
Leissa A.W.: Recent research in plate vibrations: classical theory. Shock Vib. Dig. 9, 13–24 (1977)
Leissa A.W.: Recent research in plate vibrations, 1973–1976: complicating effects. Shock Vib. Dig. 10, 21–35 (1978)
Leissa A.W.: Plate vibration research, 1976–1980: classical theory. Shock Vib. Dig. 13, 11–22 (1981)
Leissa A.W.: Plate vibration research, 1976–1980: complicating effects. Shock Vib. Dig. 13, 19–36 (1981)
Leissa A.W.: Recent studies in plate vibrations: 1981–1985, Part I: classical theory. Shock Vib. Dig. 19, 11–18 (1987)
Leissa A.W.: Recent studies in plate vibrations: 1981–1985, Part II: complicating effects. Shock Vib. Dig. 19, 19–24 (1987)
Mindlin R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. J. Appl. Mech. ASME 18, 31–38 (1951)
Civalek Ö.: Use of eight-node curvilinear domains in discrete singular convolution method for free vibration analysis of annular sector plates with simply supported radial edges. J. Vib. Control 16, 303–320 (2010)
Civalek Ö., Ersoy H.: Free vibration and bending analysis of circular Mindlin plates using singular convolution method. Commun. Numer. Methods Eng. 25, 907–922 (2009)
Zhong H., Yu T.: Flexural vibration analysis of an eccentric annular Mindlin plate. Arch. Appl. Mech. 77, 185–195 (2007)
Whitney J.M., Sun C.T.: A higher-order theory for extensional motion of laminated composites. J. Sound Vib. 30, 85–97 (1973)
Levinson M.: An accurate simple theory of the statics and dynamics of elastic plates. Mech. Res. Commun. 7, 343–350 (1980)
Reddy J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech. ASME 5, 745–752 (1984)
Leung A.Y.T.: An unconstrained third order plate theory. Comput. Struct. 40, 871–875 (1991)
Liew K.M., Xiang Y., Kitipornchai S.: Research on thick plate vibration: a literature survey. J. Sound Vib. 180, 163–176 (1995)
Zhou, D.: Three-dimensional Vibration Analysis of Structural Elements Using Chebyshev-Ritz Method. PhD Thesis, The University of Hong Kong (2003)
Conway H.D.: An analogy between the flexural vibrations of a cone and a disc of linearly varying thickness. ZAMM 37, 406–407 (1957)
Conway H.D.: Some special solutions for the flexural vibration of discs of varying thickness. Ingenieur Arch. 26, 408–410 (1958)
Lenox T.A., Conway H.D.: An exact, closed form, solution for the flexural vibration of a thin annular plate having a parabolic thickness variation. J. Sound Vib. 68, 231–239 (1980)
Jain R.K., Prasad C., Soni S.R.: Axisymmetric vibrations of circular plates of linearly varying thickness. ZAMP 23, 941–947 (1972)
Yang J.S.: The vibration of circular plate with variable thickness. J. Sound Vib. 165, 178–185 (1993)
Wang J.: General power series solutions of the vibration of classical circular plates with variable thickness. J. Sound Vib. 202, 593–599 (1997)
Duan W.H., Quek S.T., Wang Q.: Generalized hypergeometric function solutions for transverse vibration of a class of non-uniform annular plates. J. Sound Vib. 287, 785–807 (2005)
Singh B., Saxena V.: Axisymmetric vibration of a circular plate with double linear variable thickness. J. Sound Vib. 179, 879–897 (1995)
Selmane A., Lakis A.A.: Natural frequencies of transverse vibrations of non-uniform circular and annular plates. J. Sound Vib. 220, 225–249 (1999)
Wu T.Y., Liu G.R.: Free vibration analysis of circular plates with variable thickness by the generalized differential quadrature rule. Int. J. Solids Struct. 38, 7967–7980 (2001)
Soni S.R., Rao C.L.A.: On radially symmetric vibration of orthotropic non-uniform discs including shear deformation. J. Sound Vib. 42, 57–63 (1975)
Gupta U.S., Lal R.: Axisymmetric vibrations of polar orthotropic Mindlin annular plates of variable thickness. J. Sound Vib. 98, 565–573 (1985)
Gupta U.S., Lal R., Sharma S.: Vibration of non-homogeneous circular Mindlin plates with variable thickness. J. Sound Vib. 320, 1–17 (2007)
Xiang Y., Zhang L.: Free vibration analysis of stepped circular Mindlin plates. J. Sound Vib. 280, 633–655 (2005)
Efraim E., Eisenberger M.: Exact vibration analysis of variable thickness thick annular isotropic and FGM plates. J. Sound Vib. 299, 720–738 (2007)
Malekzadeha P., Shahpari S.A.: Free vibration analysis of variable thickness thin and moderately thick plates with elastically restrained edges by DQM. Thin Walled Struct. 43, 1037–1050 (2005)
Irie T., Yamada G., Aomura S.: Free vibrations of a Mindlin annular plate of varying thickness. J. Sound Vib. 66, 187–197 (1979)
Quintana M.V., Nallim L.G.: A variational approach to free vibration analysis of shear deformable polygonal plates with variable thickness. Appl. Acoust. 71, 393–401 (2010)
So J., Leissa A.W.: Three-dimensional vibrations of thick circular and annular plates. J. Sound Vib. 209, 15–41 (1998)
Liew K.M., Yang B.: Three-dimensional elasticity solutions for free vibrations of circular plates: a polynomials-Ritz analysis. Comput. Methods Appl. Mech. Eng. 175, 189–201 (1999)
Liew K.M., Yang B.: Elasticity solutions for free vibrations of annular plates from three-dimensional analysis. Int. J. Solids Struct. 37, 7689–7702 (2000)
Zhou D., Au F.T.K., Cheung Y.K., Lo S.H.: Three-dimensional vibration analysis of circular and annular plates via the Chebyshev-Ritz method. Int. J. Solids Struct. 40, 7689–7700 (2003)
Xu R.Q.: Three-dimensional exact solutions for the free vibration of laminated transversely isotropic circular, annular and sectorial plates with unusual boundary conditions. Arch. Appl. Mech. 78, 543–558 (2008)
Kang J.H., Leissa A.W.: Three-dimensional vibrations of thick, linearly tapered, annular plates. J. Sound Vib. 271, 927–944 (1998)
Taher H.R.D., Omidi M., Zadpoor A.A., Nikooyan A.A.: Free vibration of circular and annular plates with variable thickness and different combinations of boundary conditions. J. Sound Vib. 296, 1084–1092 (2006)
Kang J.H.: Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation. Comput. Struct. 81, 1663–1675 (2003)
Fox L., Parker I.B.: Chebyshev Polynomials in Numerical Analysis. Oxford University Press, London (1968)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhou, D., Lo, S.H. Three-dimensional vibrations of annular thick plates with linearly varying thickness. Arch Appl Mech 82, 111–135 (2012). https://doi.org/10.1007/s00419-011-0543-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-011-0543-y