Large Amplitude Free Vibration Analysis of Thin Rectangular Plates: Simple Closed-form Solutions

Abstract

Large amplitude free vibration behavior of thin, isotropic rectangular plate configurations are expressed in the form of simple closed-form solutions by using an application of the Ritz method based on coupled displacement fields. Influence of plate aspect ratio a b and Poisson ratio . / on the behavior of back-bone curves is briefly discussed for various boundary configurations of the rectangular plate. Proposed closed-form solutions are corrected for the simple harmonic motion (SHM) assumption using the well established harmonic balance method which is applicable for the homogeneous form of cubic non-linear Duffing equation.Keywords. Large amplitude free vibration, Thin rectangular plates, Ritz method, Closed-form solutions, Coupled displacement fields, von-Kármán type of geometric nonlinearity, Non-linear frequency. Nomenclature a = length of the rectangular plate b = width of the rectangular plate C = axial rigidity of the plate E h .1 2 / Á D = flexural rigidity of the plate E h 3 12.1 2 / Á E = Young's modulus h= thickness of the plate N x , N y and N xy = in-plane stress resultants M x , M y and M xy = moment resultants u = displacement in x-direction v = displacement in y-direction U = strain energy of the plate w = transverse displacement T = kinetic energy of the plate x, y = in-plane co-ordinates z 1 = maximum reference amplitude of the rectangular plate z 1 h = maximum reference amplitude to thickness ratio x , y and xy = in-plane strain terms L = linear frequency parameter h! L 2 a 4 4 D Á NL H = non-linear frequency parameter obtained on the basis of SHM assumption  h! NL H 2 a 4 4 D à NL = non-linear frequency parameter h! NL 2 a 4 4 D Á = Poisson ratiox , y and xy = curvature terms Subscripts L = linear NL = non-linear NL H = non-linear based on harmonic motion assumption x = partial derivative with respect to x xx = second partial derivative with respect to x y = partial derivative with respect to y yy = second partial derivative with respect to y xy = partial derivative with respect to x and y 1 Introduction