Abstract
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 methods are an effective tool for solving boundary problems described by ordinary fourth-and second-order differential equations with variable parameters. Particular attention should be paid to the solution of the border problem of two-parameter elastic 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 recorded. The shape and type of load are considered 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 methods, simple calculations were made, easy to use in engineering practice and conditions of use were given. Experimental studies have confirmed the high accuracy of theoretical calculations using the proposed methods and formulas.
Highlights
Components such as turbine blades, helicopter carrier blades, engine rotors, machine spindles, crane booms, drilling rigs, masts can be modeled with beams of variable distribution of parameters, the most important of which are: rigidity, mass, modulus of elasticity, cross section, substrate elasticity Longitudinal load
The output equation of the vibration of flexible beams with a power-varying cross-section may be in the form of the Euler equation and the general integral is expressed by the elementary functions [5,6,7]
On the basis of this, Mike has derived the characteristic equation of the cantilever beam with varying cross section, but it has a very complicated form [5]
Summary
Components such as turbine blades, helicopter carrier blades, engine rotors, machine spindles, crane booms, drilling rigs, masts can be modeled with beams of variable distribution of parameters, the most important of which are: rigidity, mass, modulus of elasticity, cross section, substrate elasticity Longitudinal load. On the basis of this, Mike has derived the characteristic equation of the cantilever beam with varying cross section, but it has a very complicated form [5]. The authors of this work [10,11] have used a series of characteristic methods to solve the boundary free-floating edge problem of a cantilever beam with a freely varying cross section. Work [5] has derived models that can be applied in engineering practice to calculate the basic vibration frequencies of the flexible beams and their conditions of use. The general form of the characteristic equation was obtained, on the basis of which it is possible to calculate the estimate of the frequencies of the own vibration and the estimates of the critical forces.
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