ON THE APPLICATION OF GALERKIN'S METHOD TO ANALYZING EIGEN VIBRATIONS OF ELASTIC BODIES

The topicality of the issues related to studying vibrations of elastic bodies and structures is discussed. The publications and results obtained in this field have been analyzed. It is noted that one of the common characteristics of all the approximate methods of analyzing boundary-value problems is certain ambiguity in formulating finite-dimensional approximations of the solution. A boundary-value problem of determining eigen frequencies of a homogeneous membrane has been formulated. The basic idea of the approaches in question is that variables used in equations of mathematical physics can always be divided into two groups, one of which consists of the so-called measureable variables, such as displacement, velocity, temperature etc., and the other one includes non-measureable ones, such as stress, pulse, heat flow etc. Issues related to various classical formulations of spectral problems arising in the theory of elasticity have been investigated. The method of integral-differential relations is described, which is an alternative to classical numerical approaches. The possibility of constructing various bilateral energy-based evaluations of the accuracy of approximate solutions, following from the method of integral-differential relations has been studied. A one-parameter family of quadratic non-negative functionals has been introduced, the stationarity condition of which together with integral-differential constraints form a complete equation set describing dynamic behavior of elastic bodies. A projection approach to analyzing spectral problems of linear plasticity theory has been considered. Using the example of the problem of free vibrations of a circular membrane, the effectiveness of the method of integral-differential relations is demonstrated. Various energy-based evaluations of an approximate solution constructed using polynomial approximations of the sought functions are proposed. Application of the standard technique of Bubnov-Galerkin's method to the problem of free vibrations is shown to lead to the appearance of complex eigen frequencies, the real part of the eigenvalue being its approximate value, whereas its imaginary part being able to serve as an evaluation of the accuracy of the solution. The proposed numerical algorithm makes it possible to uniquely evaluate the integral quality of the obtained numerical solutions.