A variational approach to modeling and optimization of the dynamics for an elastic beam with variable cross section

The paper studies the modeling and optimization of the dynamic behavior of heterogeneous beam structures. Usually, a boundary-value problem is formulated as a general differential equation with variable coefficients. One of the common characteristic features inherent in boundary-value problems of mathematical physics is a certain ambiguity in their formulation. By introducing new variables that characterize the behavior of the system, the boundary-value problem reduces to three ordinary differential equations with variable coefficients. New variables have a clear physical meaning. One function is the linear momentum density, and the other a bending moment in the beam cross section. Such a formulation of the problem of free vibrations of a beam of variable cross section allows us to reduce the system of differential equations to one fourth-order equation, but written in terms of the momentum or moments functions. This equations are equivalent to the original one formulated in displacements, but have different forms. The state equations are taken into account integrally in accordance with the ideas of the method of integrodifferential relations. A numerical algorithm is developed for solving direct and inverse problems of the beam dynamics with variable cross section based on the Ritz method and the technique of semidiscrete polynomial approximations of the desired functions. The effectiveness of the approach is demonstrated by the example of controlled movements of a thin rectilinear elastic inhomogeneous rod. The control problem is to optimally transfer the rod from the initial to the given final state. The results of numerical analysis are presented.