Nonlinear forced vibration of orthotropic rectangular plates using the method of multiple scales

Abstract

The method of multiple scales (MMS) in conjunction with the Galerkin method is used to analyze the nonlinear forced and damped response of a rectangular Orthotropic plate subjected to a uniformly distributed harmonic transverse loading. The effects of damping and in-plane loads are considered. The analysis considers simply supported as well as clamped plates. For each case, both movable and immovable edge conditions are considered. By using MMS, all possible resonances such as primary, subharmonic, and superharmonic reso- nances are studied. For the undamped response without in-plane loading, comparisons of the MMS results with those obtained by the finite-element method show excellent agreement. EVELOPMENT of composite materials comprising lam- inates of Orthotropic or multilayered anisotropic materi- als Recently has been receiving substantially growing research efforts. Due to the increasing demands for energy-efficient, high strength, minimum weight aircraft designs, many re- searchers believe that the use of composite materials offers promising alternatives for aircraft designs. Thin, laminated, composite plates subjected to transverse periodic loadings could encounter deflections of the order of plate thickness or even higher. Responses of this kind cannot be predicted by using the linear theory. Consequently, the need to study large- amplitude-deflection vibrations of composite structures is of paramount importance. The literature survey shows that the equations of motion for the large deflection analysis of heterogeneous anisotropic plates using the von Karman geometrical nonlinearity were first considered by Whitney and Leissa.1 Based on these equa- tions, different methods of analysis have been developed by several researchers. An excellent number of collections on nonlinear free and forced vibrations of composite plates cov- ering the work through 1979 can be found in the comprehen- sive book by Chia. 2 Bert3 has conducted a survey on the dynamics of composite plates for the period 1979-81. A re- view of the literature on nonlinear vibrations of plates can be found in the review paper by Sathyamoorthy4 and the book by Nayfeh and Mook.5 Large deflection analysis of symmetrically laminated rec- equations of motion are presented in terms of the lateral displacement and stress function. The equations are nondi- mensionalized following the transformation introduced by Brunelle and Oyibo.9 Though multimode analysis can be treated, the present study is focused on single-mode analysis. A deflection function representing the first mode and satisfy- ing the boundary conditions is assumed, and subsequently the stress function is found. Next, the Galerkin method is applied to obtain the modal equation, which is solved analytically by using the method of multiple scales (MMS).10 The MMS also provides solutions for subharmonic and superharmonic reso- nances. The effects of damping ratio, plate aspect ratio, and in-plane loading then are studied.