[01] Ramzy M. Abumandour, Basic Engineering Science Department, Faculty of Engineering, Menofia University, Shebin El-Kom, Menofia, Egypt. [02] M. H. Kamel, Engineering Mathematics and Physics Department, Faculty of Engineering, Cairo University, Giza, Egypt. [03] M. M. Nassar, Engineering Mathematics and Physics Department, Faculty of Engineering, Cairo University, Giza, Egypt.

This paper study the vibration analysis using the differential quadrature method (DQM) which has very wide applications in the field of structural vibration of various elements such as beams, plates, cylindrical shells and tanks. One of the most advantages of the DQM is its simple forms for nonlinear formulations. In this paper, the free vibration of uniform and non-uniform beams resting on fluid layer under axial force under three sets of boundary conditions, that is, simply–simply supported (S–S), clamped–clamped supported (C–C) and clamped–simply supported (C–S) were studied using the generalized differential quadrature (GDQ). The proposed approach directly substitutes the boundary conditions into the governing equations (SBCGE). The approach of directly SBCGE is presented to overcome the drawbacks of previous approaches in treating the boundary conditions. The non-dimensional natural frequency and the normalized mode shapes of uniform and non-uniform beams were obtained. Results show good agreement with the previous analytical solutions. The effect of the varying cross section area on the vibration was studied. This work reflects the power of the DQM in solving non-uniform problems.

Uniform and Non-Uniform Beam, Vibration, Differential Quadrature Method

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