Spectral analysis of vibrating plates with general shape Giuseppe Catania, Silvio Sorrentino DIEM, Department of Mechanical Engineering, University of Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy ABSTRACT Lightweight plate structures are widely used in many engineering and practical applications. The analysis and design of such structures call for efficient computational tools, since exact analytical solutions for vibrating plates are currently known only for some standard shapes in conjunction with a few basic boundary conditions. The present paper deals with the adoption of a set of eigenfunctions evaluated from a simple structure as a basis for the analysis of plates with both general shape and general boundary conditions in the Rayleigh-Ritz condensation method. General boundary conditions are introduced in the functional of the potential energy by additional terms, and both trigonometric and polynomial interpolation functions are implemented for mapping the shape of the plate in Cartesian coordinates into natural coordinates. Flexural free vibration analysis of different shaped plates is then performed: skew, trapezoid and triangular plates, plates with parabolic edges, elliptic sector and annular plates. The proposed method can also be directly applied to variable thickness plates and non-homogeneous plates, with variable density and stiffness. Purely elastic plates are considered; however the method may also be applied to the analysis of viscoelastic plates. The results are compared to those available in the literature and using standard finite element analysis. Keywords Vibrating Plates, Rayleigh-Ritz method, Coordinate mapping. Nomenclature a reference-length parameter w displacement D flexural stiffness of the plate x Cartesian coordinate E Young’s modulus y Cartesian coordinate h thickness of the plate I beam eigenfunction J Jacobian matrix K natural coordinate K stiffness matrix N distributed stiffness M mass matrix O dimensionless frequency n modal index Ȟ Poisson’s ratio P interpolation function 3 total potential energy q modal coordinate Uh mass per unit area U potential of the strain energy Ȧn modal natural frequency V potential of applied loads [ natural coordinate 1 Introduction In recent decades, lightweight plate structures have been widely used in many engineering and practical applications. The analysis and design of such structures call for efficient computational tools, since exact analytical solutions for vibrating plates are currently known only for some standard shapes in conjunction with a few basic boundary conditions [1]. For the analysis of plates of arbitrary shape, or with general boundary conditions, numerical methods such as the finite difference method, the finite element method or the finite strip method are usually applied to the problem. Vibration analyses of plates of different shapes have been carried out extensively by several researches. Durvasula [2] computed the natural frequencies of clamped skew plates, Babu and Reddy [3] studied skew orthotropic plates by means of T. Proulx (ed.), Linking Models and Experiments, Volume 2, Conference Proceedings of the Society for Experimental Mechanics Series 5, 77 DOI 10.1007/978-1-4419-9305-2_6, © The Society for Experimental Mechanics, Inc. 2011
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