مهندسی مکانیک شریف (May 2018)
FREE VIBRATION ANALYSIS OF LAMINATED COMPOSITE TRAPEZOIDAL PLATE RESTING ON THE PASTERNAK FOUNDATION
Abstract
Trapezoidal fiber reinforced composite plates are important structural elements in modern engineering industries. In this paper and for the first time, the free vibration analysis of laminated composite trapezoidal plate resting on the Pasternak type foundation has been presented. First, the kinetic and strain potential energies of the laminated composite plate based on the classical plate theory are formed. Then by using the change of variables, the trapezoidal plate is mapped into the rectangular one. Next, we have assumed the deflection of the plate as a series having orthogonal terms and unknown coefficients. It is worth mentioning that the admissible polynomials for all possible combinations of classical boundary conditions have been presented in the paper. The deflection is approximated by a set of beam characteristic orthogonal polynomials generated using the Gram-Schmidt procedure. A sufficiently large number of truncated series have been worked out to make sure the convergence criteria. These polynomials have to satisfy the essential boundary conditions of the plate. Upon substituting the above stated response in the energies terms, they have been rewritten in terms of the unknown coefficients. At the end by applying the Ritz method, a standard eigenvalues problem is obtained out of which the frequencies and corresponding mode shapes can be obtained. The obtained algorithm is very general and it is attractive regarding its versatility in handling any classic boundary conditions. Besides, it allows taking into account a great variety of anisotropic characteristics and geometric planforms. In order to establish the validity, accuracy and applicability of the described approach and self-developed computer program, numerical results have been computed for a number of plate problems for which comparison values are available in the literature. Excellent agreements are observed. Additionally, new results are also presented for plates under different conditions to investigate the influences of different parameters on the vibrational characteristics of the plate. For some plates, mode shapes of free vibration are also shown.
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