Analysis of Orthotropic Plates in Free Vibration Regime Using Rayleigh – Ritz Method

ABSTRACT The assumed deflection shapes used in the approximate methods such asin the Rayleigh–Ritz method were normally formulated by inspection andsometimes by trial and error, until recently, when a systematic method ofconstructing such a function in the form of characteristic orthogonal polynomials(COPs) was developed in 1985. However, vibrational analyses of plates with all edges free or clamped are much more complicated. This project aims at establishing particular expressions for the fundamental natural frequency of rectangular plates using Rayleigh Ritz Method by obtaining approximate shape functions of plate of different support conditions through characteristics orthogonal polynomials. To this end, new sets of stress – strain relations for orthotropic plates were derived. The principle of force of inertia was introduced, yielding the corresponding dynamic governing equation of orthotropic plate and hence the strain energy equation of orthotropic plate.Rayleigh – Ritz quotient was obtained by equating the strain energy equation of the plate to the kinetic energy equation of the plate. From the rayleigh’s quotient, an expression for the natural frequency of the plate was obtained in terms of the shape functions of the plate. The shape function of plate of different support conditions was obtained through characteristics orthogonal polynomials. A spreadsheet programme was developed to aid the solutions of the equations and the results show that values of fundamental natural frequencies obtained using the present studies (Rayleigh - Ritz Method) for all round clamped plates with aspect ratio, r between 0.5 to 1.2 are four times greater than that obtained from that of previous studies (Galerkin’s method). Moreso, for plates with mixed support conditions, values of fundamental natural frequencies obtained using the present study are higher than that of previous studies (Galerkin’s Method) their difference ranging from 33% for r = 0.5 to 95% for r = 1.2 while results for plate all round simply supported for both present and previous studies are the same.