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Asymptotic Approaches in Nonlinear Mechanics How well is Nature simulated by the varied asymptotic models, that imaginative scientists have invented?
G. Birkhoff
This lecture deals with asymptotic methods in nonlinear dynamics. A detailed and systematic treatment of new asymptotic methods in combination with the Padé approximants is presented. The main features of the lecture are: 1) it is devoted to the basic principles of asymptotics and its applications, and 2) it deals with both traditional approaches (such as regular and singular perturbations, averaging and less widely used, new approaches such as one- and two-point Padé approximants. Many results are reported in English for the first time. The choice of topics reflects the authors' research experience and involvement in industrial applications. The narration is commonly based on examples given by applied mechanics, plates and shells. Obviously, the methods in question are really versatile in application, covering applied mathematics, physics, mechanics and other basic sciences. The author has paid special attention to examples and discussion of results rather than to burying the ideas in formalism and technical details. Progress in the applications of AM in the theory of oscillations as well as in applied mathematics on the whole is closely linked with the introduction of new small parameters and, respectively, new asymptotic procedures. This is also the field of my lecture. V.V. Bolotin proposed an effective asymptotic method for the investigation of linear continuous elastic system oscillations with complicated boundary conditions. The main idea of this approach is in the separation of the continuous elastic system into two parts. Then the matching procedure permits us to obtain a complete solution of the dynamics problem in a relatively simple form. The idea of Bolotin's asymptotic method is generelized for the nonlinear case in my lecture. A new AM for solving mixed boundary value problems is considered. The parameter s is introduced into the boundary conditions in such a way that the s=0 corresponds to the simple boundary problems and the case s=1 corresponds to the general problem under consideration. Then, the s-expansion of the solution is obtained. As a rule, the expansion of the solution is divergent just at the point s=1. The PAs are used to remove this divergence. |
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