Abstract
In this paper we consider in general form the compilation of a dynamic model of a cantilever rod with a variable cross section that performs bending vibrations. A real mechanical system is a system with distributed parameters. The task of dynamic modeling is the creation of a more convenient calculation scheme. The dynamic model of a rod of an alternating cross section obtained in this case is a discrete system with a concentrated mass at the end of the reduced mass associated with fixing a weightless elastic rod. The problem of determining the parameters of a discrete system is solved from the condition that the kinetic energies of the initial (real) and reduced (discrete) systems are equal. As a result of the work, formulas were obtained that make it possible to determine the reduced mass of the beam and its bending stiffness concentrated at the end of the console. The major goal of this paper is to address the derivation of the frequency equation of flexural vibrating cantilever beam considering the bending moment generated by an additional mass at the free end of beam, not just the shear force. It is a transcendental equation with two unambiguous physical meaning parameters. And the influence of the two parameters on the characteristics of frequency and shape mode was made. The results show that the inertial moment of the mass has the significant effect on the natural frequency and the shape mode. And it is more reasonable using this frequency equation to analyze vibration and measure modulus.
Highlights
When dynamically studying a mechanical system, it is necessary to compile a dynamic model of mechanics [1]
The dynamic model of a rod of an alternating cross section obtained in this case is a discrete system with a concentrated mass at the end of the reduced mass associated with fixing a weightless elastic rod
The major goal of this paper is to address the derivation of the frequency equation of flexural vibrating cantilever beam considering the bending moment generated by an additional mass at the free end of beam, not just the shear force
Summary
When dynamically studying a mechanical system, it is necessary to compile a dynamic model of mechanics [1]. To determine the true loads in the contours of the links of the actuating mechanisms of operating machines, it is convenient to use differential equations describing the processes occurring during the operation of the machines under study This applies to various types of hoisting and transport machines [6, 7]. The main reason for this is flexural oscillations of the cantilever beam with parametric excitation caused by a change in the mass of the material being discharged [9, 10] The study of such devices with the help of differential equations will reveal the causes of overloads of the bearing parts of the drive and suggest measures for their improvement
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