Classical approach to determining the natural frequency of continual subsystem of three-mass inter-resonant vibratory machine

Abstract

Problem statement. The three-mass vibratory system can be defined by five basic parameters: inertial parameters of the masses and stiffness parameters of two spring sets. Unlike the classical discrete system, the discrete-and-continual one consists of two rigid bodies connected by one spring set that form the discrete subsystem, and of the reactive mass considered as deformable (elastic) body characterized by certain stiffness and inertial parameters, which are related with one another. Purpose. The main objective of the paper consists in determining the first natural frequency of the continual subsystem of the three-mass discrete-and-continual vibratory machine. Methodology. While carrying out the investigations, it is used the classical theory of oscillations of straight elastic rods. Findings (results). The engineering technique of determining the first natural frequency of the continual subsystem of the three-mass vibratory machine is developed and approved by means of analytical calculations and numerical simulation. Originality (novelty). The optimal diagram of supporting the continual subsystem (elastic rod) is substantiated. The possibilities of exciting the vibrations of the three-mass discrete-and-continual mechanical system using the eccentric drive are considered. Practical value. The obtained research results and the developed calculation techniques can be used be engineers and designers dealing with various technological and manufacturing equipment that use vibratory drive. Scopes of further investigations. While carrying out further investigations, it is necessary to develop the model of combined discrete-and-continual system of three-mass vibratory machine, and to carry out the numerical simulation of the system’s motion under different operational conditions.

Mathematical modelling of forced oscillations of continuous members of resonance vibratory system

“…In Fig. 1(a), there is presented the three-mass discrete vibratory system whose bodies performs straight-line (linear) oscillations [10]. It consists of the active mass connected to the intermediate mass with the help of the spring unit of stiffness .…”

“…The intermediate mass of the vibratory system is supported by the vibration isolators of stiffness . The mathematical model of the three-mass discrete oscillatory system can be derived in the amplitude form as follows [10]:…”

“…In accordance with the input data [10], the forced circular frequency = 99.5 rad/s, the amplitude displacement of the active mass = 5•10 -3 m, the inertial parameters = 83.7 kg, = 62.1 kg, the stiffness of the spring unit…”

“…Despite the fact that numerous scientific publications are dedicated to discrete-continuous oscillatory systems, the problems of synthesizing the three-mass structures with rod-shaped elastic members excited by crank mechanisms are not thoroughly investigated. In the previous publications of the authors, there were considered the technique of calculation of the inertia-stiffness parameters of the three-mass discrete oscillatory system ensuring its operation in energy-efficient resonance mode [9], and the forced and free oscillations of an elastic rod considered as a disturbing body of the three-mass discrete-continuous oscillatory system [10,11]. The novelty of the present paper consists in deriving the mathematical model of the discrete-continuous vibratory system, in which the model of the discrete subsystem is combined with the model of the continuous subsystem by applying the reactions in the supports holding the uniform elastic rod.…”

Determination of the first natural frequency of an elastic rod of a discrete-continuous vibratory system

“…The analysis of numerous publications dedicated to oscillatory systems of various vibratory technological machines has shown that the problems of implementation of combined discrete-continuous three-mass systems are not thoroughly investigated. Therefore, in the previous publications of the authors, the optimal inertia and stiffness parameters of the three-mass discrete vibratory systems were substantiated [9], and the stress-strain state of the double-span elastic beam was analyzed [10]. The novelty of the present paper consists in determination of natural frequencies and substantiation of the possibilities of implementing the continuous rod-shaped members with distributed inertia and rigidity parameters for exciting the oscillations of the threemass discrete-continuous inter-resonance vibratory systems.…”