The Effects of Large Vibration Amplitudes on the Mode Shapes and Natural Frequencies of Thin Elastic Structures, Part II: Fully Clamped Rectangular Isotropic Plates

Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution

“…This method does not require a very good initial estimate of the solution or a step procedure, similar to other methods described for beams and rectangular plates. [20][21][22][23] It was adopted here to ensure rapid convergence when varying the amplitude, which allowed solutions to be obtained with a quite reasonable number of iterations. The fundamental nonlinear mode shape was calculated in the neighborhood of the linear solution corresponding to a small numerical value of the coefficient a r0 (r 0 = 1) of the basic function w * r0 .…”

“…Such an assumption has been made when calculating the first two nonlinear mode shapes of circular plates and fully clamped rectangular plates. 16,20,21,23 For the first nonlinear mode shape, the range of validity of this assumption has been discussed in the light of the experimental and numerical results obtained for the nonlinear frequency-amplitude dependence and the nonlinear bending stress estimates obtained at large vibration amplitude. 20,22 In order to examine the effects of large vibration amplitudes on the membrane stress patterns for clamped circular plates, the contribution of the in-plane displacement U should be taken into account in the membrane strain expression.…”

Non-linear forced vibration analysis of rectangular plates including the coupling between transverse and in-plane displacements.

“…The expressions for the mass tensor m ij , and the linear rigidity tensor k ij presented in [4,5] are unchanged, but the expression for the non-linear rigidity tensor b ijkl is affected in the present theory. Thus, the general features of the model remain the same and the mathematical treatment of the set of non-linear algebraic equations giving arise to solution of the problem of large vibration amplitudes of rectangular plates is carried out using an explicit analytical solution procedure proposed previously [6].…”

“…Geometrically non-linear vibrations of plates have been analysed by numerous researchers using analytical, numerical, or combined approximate methods [1][2][3]. In a previous series of papers, a semi-analytical model for geometrically non-linear free and forced vibrations of elastic thin straight structures, such as beams, homogeneous and symmetrically laminated rectangular plates has been developed by Benamar et al [4]. The model was based on Hamilton's principle and spectral analysis, and was of the same spirit as the well-known Rayleigh-Ritz method, used for numerical approximate solution of linear problems of vibration.…”

A family of conservative finite difference schemes for the dynamical von Karman plate equations

“…Various models have been proposed; among the most frequently encountered are those related to that of von Karman [1,2], which may be written in an especially elegant form, and which often serve as the point of departure for studies of bucking and nonlinear modal vibration [3,4]. Because of the particularly simple form of the nonlinearity, it is also an excellent candidate for the application of energy-conserving numerical methods, which are the subject of this article.…”