Variational approaches to solving initial-boundary-value problems in the dynamics of linear elastic systems

Problems of the controlled motion of an elastic body are considered in the linear theory. Using the method of integrodifferential relations, a family of quadratic functionals is introduced, which define the state of the elastic body, and variational formulations of the initial-boundary-value problem of dynamics are given. Euler's equations and boundary and terminal relations corresponding to them are obtained from the condition for the functionals to be stationary. It is shown that there is a relation between the proposed formulations and the Hamilton variational principle in the case of boundary-value and time-periodic problems of dynamics. A numerical algorithm is developed for finding the motions of an elastic body, based on piecewise-polynomial approximations and a criterion is proposed for estimating the quality of the approximate solutions. An example of the calculation and analysis of the forced transverse motions of a rectilinear beam with a square cross section is given for the three-dimensional model. (C) 2010 Elsevier Ltd. All rights reserved.