ENOC 2008, Saint Petersburg, Russia, June, 30-July, 4 2008 APPLICATION OF A HYBRID WKB-GALERKIN METHOD TO A NONLINEAR PLATE DYNAMIC PROBLEM WITH TIME DEPENDENT DAMPING COEFFICIENT Victor Z. Gristchak Applied Mathematics Department Faculty of Mathematics Zaporizhzhya National University Ukraine grk@zsu.zp.ua Olga A. Ganilova Applied Mathematics Department Faculty of Mathematics Zaporizhzhya National University Ukraine lionly@rambler.ru Abstract The objective of the analysis is to obtain a closed form approximate analytical solution for the nonlinear differential equation of a loaded plate considering a time variant damping coefficient. The solution of the problem is obtained by using perturbation and a hybrid WKB-Galerkin method. Results are presented of comparison of the solutions based on different approaches. Keywords Nonlinear dynamic problem, perturbation method, a hybrid WKB-Galerkin method, time dependent damping coefficient. 1 Introduction Many physical phenomena involving oscillation cannot be represented in terms of linear theory. Thus non-linear theory potentially enables a description of phenomena which are otherwise hidden for the problem in the context of linear theory. A special branch of oscillation theory is devoted to non-linear oscillations with specific properties. Such kinds of motion can be observed in plates and shells with large displacements if the strains and displacement are non-linearly related. This class of problem is an integral part of non-linear deformable solid mechanics. As a rule, investigation in terms of non-linear theory results in nonlinear differential equations. Analytical solution of these equations, especially nonlinear differential equations with variable coefficients, as in this paper, causes many mathematical problems. Therefore, a wide variety of approximate methods such as perturbation techniques, the method of multiple scales and the averaging method, have been proposed for nonlinear differential equations in mechanics. However, it should be noticed that most nonlinear differential equations which are solved by applying such approximate methods are differential equations with constant coefficients. In this work a problem is presented of plate oscillation which is described by a more general nonlinear differential equation with variable coefficients. The perturbation technique and a hybrid (Wentzel-Kramer-Brillouin) WKB-Galerkin method are used to solve this problem. The hybrid WKB-Galerkin method enables especially good results to be obtained for approximate solution of a singular differential equation which contains a parameter multiplying the highest order derivative. According to the specific procedure, presented in [Gristchak, Dmitrieva, 1995; Gristchak, Ganilova, 2006], solution of the linear differential equations is conducted in two stages: initially by obtaining the WKB-solution of the problem and then by application of the Bubnov-Galerkin orthogonality procedure, taking asymptotic coefficients into consideration. However, typically, the algebra for the WKB method becomes more tedious as higher order terms are computed, and frequently the work required rises so fast from term to term that even with computational assistance very few terms can be computed. Thus for cases where higher order terms may have significant effect, it is important to get as much use of the information contained in the lower order terms as possible. The hybrid WKB-Galerkin method seems to extend greatly the power and usefulness of the WKB method [Steele, 1971; Steele, 1989] without significant computational effort. Hybrid methods have proved to be useful in a wide variety of applications such as structural mechanics problems, applications to slender-bodies and thermal problems [Geer, Andersen, 1989; Geer, Andersen, 1990; Geer, Andersen, 1991; Gristchak, Dmitrieva, 1995; Gristchak, Ganilova, 2006]. Significantly, according to results obtained in different branches of mechanics the hybrid WKB-Galerkin method shows a higher accuracy of solution compared to the perturbation and WKB methods.