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Orthogonal Tapered Beam Functions in the Study of Free Vibrations for Non-uniform Isotropic Rectangular Plates

M.F. Liu1

Department of Applied Mathematics, I-Shou University, No.1, Sec. 1, Syuecheng Rd., DashuTownship, Kaohsiung County. 840, Taiwan R.O.C. E-mail:

* Corresponding Author:Email:email

Computers, Materials & Continua 2011, 22(2), 97-128.


A new invented Orthogonal Tapered Beam Functions (OTBFs) have been introduced in this paper and used in accordance with the Rayleigh-Ritz method to determine the natural frequencies and mode shapes of the non-uniform rectangular isotropic plates with varying thickness in one or two directions. The generation of the OTBFs is based on the static solution of a one-dimensional beam problem subjected to constant applied load, and then extends to an orthogonal or orthonomal infinite set of admissible functions by performing the three-term recurrence scheme. A wide range of non-uniform rectangular plate whose domain is referenced by a so-called truncation factor and thickness variation is presented by the so-called taper factor is investigated in the present study in order to practice the validity and efficiency of the proposed methodology. Following the Levy approach, the fourth order differential equation governing the free vibration of non-uniform isotropic plate can be transferred into an 8th order eigenfrequency equation by inserting the OTBFs as shape functions. Three different combinations of clamped, simply-supported and free boundary conditions are imposed around the non-uniform plate and the aspect ratio which indicates the width over the length of the rectangular plate is ranging from half to two. The effects of taper factor together with the truncation factor, boundary condition and aspect ratio on the natural frequencies of free vibration are demonstrated for the first few modes by using the finite terms approximation of the proposed admissible functions. A general computer program has been implemented to solve the eigenfrequency equation, some numerical results are tabulated and comparisons of the results with those available in literature are also presented.


Cite This Article

M. Liu, "Orthogonal tapered beam functions in the study of free vibrations for non-uniform isotropic rectangular plates," Computers, Materials & Continua, vol. 22, no.2, pp. 97–128, 2011.

cc This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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