Two-mode nonlinear vibration of orthotropic plates using method of multiple scales

Abstract

Nonlinear forced oscillation of a rectangular orthotropic plate subjected to uniform harmonic excitation is solved using the method of multiple scales. The governing equations are based on the von Karman type geometrical nonlinearity, and the effect of damping is included. The general multimode solution is developed for simply supported boundary conditions, and the solution is specialized for two-symmetric modes analysis. The primary resonances and the subharmonic and superharmonic secondary resonances are studied in detail. HIN laminated composite panels subjected to transverse periodic loadings can encounter deflections of the order of panel thickness or even higher. The effect of these periodic excitations on the panel can be very severe. Responses of this kind cannot be predicted by linear theory. Consequently, the need to study large deflections using nonlinear methods of analysis is of paramount importance. The formulation of the equations governing the fundamen- tal kinematic behavior of the laminated composite plates in the presence of the von Karman geometrical nonlinearity is attributed to Whitney and Leissa.1 Based on these equations, various methods have been developed to solve nonlinear free and forced vibrations of composite panels. A good survey on mainly nonlinear free and forced vibrations of isotropic plates is given in a book by Nayfeh and Mook.2 The most compre- hensive work on geometrically nonlinear analysis of both static and dynamic behavior of the laminated panels through 1972 is collected in a book by Chia. 3 Bert4 has conducted a survey on the dynamics of composite panels for the period of 1979-81. A review of literature on linear vibrations of plates can be found in a review paper by Sathyamoorthy.5 Relatively few investigations have been reported on the nonlinear forced vibration of isotropic or composite panels under harmonic excitations. Yamaki6 presented a one-term solution for free and forced vibrations of the rectangular plates, using Galerkin's method. Lin7 studied the response of a nonlinear flat panel to periodic and randomly varying load- ings. Nonlinear forced vibrations of beams and rectangular plates were studied by Eisely8 using a single-mode Galerkin's method in conjunction with the Linstedt-Duffing perturbation technique. Free and forced response of beams and plates undergoing large-amplitude oscillations using the Ritz averag- ing method were studied by Srinivasan.9 Bennett10 studied the nonlinear vibration of simply supported angle-ply laminated plates by considering the instability regions of the response of such plates subjected to harmonic excitations. Nonlinear free and forced vibration of a circular plate with clamped bound-