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The work is devoted to methods of analysis of vibrations and stability of discrete-continuous, multi-parameter models of beams, shafts, rotors, vanes, converting to homogeneous and one-dimensional. The properties of Cauchy's influence function and the characteristic series method were used to solve the boundary problem. It has been shown that the aforementioned methods are an effective tool for solving boundary problems as described by ordinary fourth- and second-order differential equations with variable parameters.

Particular attention should be paid to the solution of the boundary problem of two-parametric spring systems with variable distribution of parameters. Universal beam-specific equations with typical support conditions including vertical support, which do not depend on beam shape and axial load type, are saved. The shape and type of load are taken into account in the form of an impact function that corresponds to any change in cross-section of the support and continuous axial load, so that the functions describing the stiffness, the mass and the continuous load are complete. As a result of the solution of the boundary vibration problem of freely bent support and any change in its cross-section, loaded with any longitudinal load, arranged on the resilient substrate, strict relations between the own frequency parameters and the load parameters were derived. Using the aforementioned methods, simple calculations were made, easy to use in engineering practice, and conditions of applicability were given. Experimental studies have confirmed the high accuracy of theoretical calculations using the proposed methods and formulas.