A method for solving eigenvalue problems for complex linear systems

The natural vibrations of a complex system whose subsystems interact at finitely many points are considered. We assume that the system and its subsystems are described by ordinary self-adjoint linear differential operators with discrete spectrum. The couplings between subsystems are specified by linear homogeneous differential relations. We solve the problem of synthesizing the natural frequencies and vibration modes of the system on the basis of given eigenfunctions of the subsystems. An efficient numerical-analytical method is suggested in which the solution is sought in the form of series in the eigenfunctions of the subsystems. To take into account the influence of local couplings between the subsystems, we introduce correction functions; this permits one to obtain the solution in the form of rapidly converging series. The method is intended for determining the natural dynamic characteristics of spacecraft structures. A sample analysis of a model of such a structure is given.