Vibratory technological equipment is widely used in various industries. The vast majority of existing vibratory machines are equipped by single-or double-mass oscillatory systems and inertial or electromagnetic vibration exciters. The novelty of the present study consists in development and investigation of the three-mass oscillatory system with crank excitation mechanism. Such a system can be effectively implemented in various designs of vibratory equipment, e.g., conveyers, separators, feeders, shakers, batchers, sieves, etc. Based on the mathematical model derived in the form of differential equations of the system's motion, there are deduced the analytical expressions for determining its inertia-stiffness parameters ensuring the energy-efficient resonance operation mode. Using the solid model of the vibratory conveyer-separator designed in SolidWorks software, there is determined the input data for calculating the parameters of the oscillatory system. Based on the results of calculations, the numerical modelling of the system's motion is carried out in MathCad software. In order to verify the correctness of the theoretical investigations, the simulation of the system's motion is carried out in SolidWorks Motion software. The comparative analysis of the results of numerical modelling and computer simulation is performed, and the prospects of their implementation are considered.

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.

A continuous rod-shaped member (a body with distributed inertia and rigidity parameters), which is the object of the investigations, is considered. To ensure the optimal natural frequency concerning the oscillations of the reactive (exciting) mass of the three-mass discrete vibratory system, with the use of the Krylov-Duncan functions, the mathematical model describing forced oscillations of the continuous member considered as a disturbing body of the three-mass discrete-continuous vibratory system is established, and the corresponding frequency equation is analytically derived. The obtained theoretical results are verified using the Finite Element Method in SolidWorks software. The novelty of the present paper consists in substantiation of the possibilities of implementing the continuous rod-shaped members with distributed inertia and rigidity parameters for exciting the oscillations of the three-mass discrete-continuous inter-resonance vibratory systems.

The article considers the possibilities of developing the combined discrete-continuous vibratory systems, in which the disturbing member is designed in the form of the uniform elastic rod with distributed inertia and stiffness parameters. The forced oscillations of the continuous member of the three-mass vibratory system are analyzed. Based on the Krylov-Duncan functions (circular and hyperbolic functions), the system of equations describing the motion of the continuous rod is derived. 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 rods. The inertia-stiffness parameters of the vibratory system are determined and the analytical dependencies for calculating the reactions in supports are derived. The frequency-response curves of the considered discrete-continuous vibratory system are constructed. The deflection (bending) diagram of the continuous members is plotted for the case of forced oscillations of the combined discrete-continuous vibratory system.

An integral component in the calculation of vibration machines is the establishment of drive power necessary to set in motion an oscillating system with specified characteristics. However, difficulties often arise at this stage. The inertial characteristics of the drive directly affect the power consumption. Therefore, when calculating it, you must already know the size of the drive, which is still unknown, since it is only being installed. This question is especially relevant for low-frequency oscillatory systems with an inertial drive, in which the inertial parameters of the drive are proportional to the masses of the oscillatory systems. In such oscillatory systems, ignoring the mass of the drive when setting the consumed power to set the mechanical oscillating system in motion can lead to the fact that the vibrating machine will not be able to provide the expected (calculated) technical and technological parameters, since the massive drive oscillates with the whole system, will take on its own propulsion a significant proportion of the energy. The article justifies the analytical relation¬ships for calculating the power of vibration machines with inertial drive, taking into account the mass of vibrator motors. For this, systems of equations are solved that interconnect analytical expressions for calculating the powers necessary to bring the oscillatory systems into motion, taking into account the mass of vibrator motors. The solution to such systems is the value of the required drive power (vibrator motors) and its mass, which is already consistent with the masses of unified vibrator motors produced by manufacturers. The obtained analytical dependences make it quite easy to determine the power of the drives in one-, two- and trimass oscillation systems of vibration machines with an inertial drive. Using the formulas obtained in the article, it is possible to precisely establish the necessary drive power to set the oscillatory system in motion and uniquely select the mass of the vibrator motor.