An approximate analytical procedure for natural vibration analysis of free rectangular plates

• Extraordinary mode shapes for free plate vibration are identified, as a novelty.
• Natural modes are approximated by simple mathematical modes.
• Analytical formulae are derived for natural frequencies.
• Accuracy of predicted natural response is checked by general FE software.

Natural vibrations of free rectangular plates are usually analysed by numerical methods since it is not possible to obtain the closed form analytical solution. In this paper a simple analytical procedure for estimation of natural frequencies of free thin rectangular plates, based on the Rayleigh's quotient, is presented. First, natural modes are assumed in the usual form as products of beam natural modes in longitudinal and transverse direction, satisfying the grillage boundary conditions. Based on a detailed FEM analysis some additional natural modes are recognized, which are defined as sum and difference of the cross products of beam modes. Their frequency spectra are very close and identical in some special cases manifesting in such a way double frequency phenomenon. These three families of natural mode shapes form a complete natural frequency spectrum of a free rectangular plate as a novelty. The reliable approximation of natural modes enables application of the Rayleigh's quotient for estimation of higher natural frequencies. Application of the developed procedure is illustrated in the case of a free thin square and rectangular plate. The obtained results are compared with those determined by FEM and also with more rigorous ones from the relevant literature based on the Rayleigh–Ritz method. The achieved accuracy is acceptable from the engineering point of view, and the procedure can be applied to improve the hydroelastic analysis of VLFS.