The results of mathematic simulation of directed motion of the circulating flow of granulated working medium in the oscillating reservoir of finishing-grinding vibration machine are given. The model is based on the physical phenomenon, at which the granulated medium changes into vibro-boiling state clue to the action of vibrations. The solutions of Navier-Stokes implicated equations give the expressions of Bessel functions and their integral transformations. The model solutions ground the complex cycloid-trochoid nature of working medium kinematics on the vibration treatment operations and allow to determine the velocities of displacements of medium flows in any point of the reservoir. It is necessary for determination of the technological parameters and required equipment as well as their complex influence on the vibration treatment effectiveness. The zones of reservoir with increased impulse loading at combining the schemes of energetic action into the working medium and processed products are observed.

The multi-energy vibration processing, namely the combination of different energies or forces acting on a free abrasive medium for grinding of metal parts, is becoming more used in finishing processes, in recent years. However, the complexity that is involved in the aforementioned process requires a careful look in the particularities of the process itself in general and the movement of the abrasive media, in particular. In this paper, the nature of the collective movement of abrasive granules between the independently oscillating surfaces of the reservoir and the processed parts is described. This study presents the dissipation of the kinetic energy of the granules in a pseudo-gas from the working medium granules. The motion of the medium granules near the part surface, which is caused by pseudo-waves initiated by vibrations of the working surfaces of the vibration machine reservoir, is demonstrated. Furthermore, the nature of the motion of the granules near the oscillating part surface is described. The analysis that is presented here permits the determination of metal removal quantity from the surface of the workpiece as a result of multi-agent group action of the vibrating reservoir surface and the processed part. The optimal conditions for the finishing process can be determined based on the analysis presented.

The results of mathematical simulation have been carried out for the pattern of working medium motion providing the technological process of finishing–grinding treatment in an oscillating reservoir. With use of physics laws, it is ascertained and grounded that the flow of granules at the plane wall of reservoir is travelling oppositely to the source of vibrations, whereas the granules are drifting on the cycloid–trochoid trajectories from the wall of reservoir, where the looped displacement is maximal, to the center of reservoir in which the shift of granules is reduced to minimum because of damping and dissipation effect. The received theoretical regulations have a fundamental nature and can be used at the account of technological parameters of designed vibration machines.

The rotational motion of the medium granules under the influence of an impeller installed in the bottom of a cylindrical reservoir is considered. The dependencies of the circulation velocity of the abrasive granules, as well as the dependence of the pressure in the circulation flow of the granules on the radius of the vibrating machine cylindrical reservoir for different speeds of the impeller rotation are obtained. Furthermore, the velocities of the abrasive granules at various distances from the center of the cylindrical reservoir of the vibrating machine have been determined. The amplitudes of the tangential and radial components of the velocity of movement of pseudo-gas from abrasive granules are obtained. The total pressure on the surface of the processed part and the average velocity of the abrasive granules colliding with it are determined. The graphical dependencies of the integral metal removal on the amplitude and frequency of oscillations of the walls of the vibrating machine reservoir are given for various values of the angular velocities of the impeller rotation.

Vibration treatment in finishing processes is becoming a more complex technology, sometimes leading to weak shock waves occurring in the abrasive working medium. Here, the preconditions are discussed for the appearance of the effect of a shock wave of the vibrating working medium in a vibrating machine reservoir during multi-energy vibration processing. The peculiarities of the propagation of the force pulse of abrasive granules in the working medium, as in a pseudogas, are considered based on the kinetic theory of gases and the equations of gas dynamics. The paper also describes the motions of the parts in the working medium and the mechanism of the appearance of a weak shock wave, such as the compressive shock wave of abrasive granules. The interaction of the surface of the oscillating processed part with the granules of the working medium is studied. The characteristic of the Hugoniot adiabat for working medium granules is given. The influence of compression shock wave on the process of vibration processing is revealed. A comparison is made with experimental data.

The hydromechanics of the fluidized bed of the processing medium is presented; the conditions for its formation, stability, and the limits of existence are given. The stochasticity of a two-phase hydromechanical process in a fluidized bed has been determined. The boundary conditions on the surface of a spherical processed part and the vibrating machine reservoir surface are considered. It was found that the velocity of granules during collision with the surface of the part, in addition to the chaotic component, has a normal and tangential component. It is established that the amplitude of oscillations of one of the components of the tangential velocity of the part is twice the amplitude of its normal component. The other component of the tangential velocity of the granule at the surface of the part is ten times less than the other components of the velocity. The equations for the dynamics of a fluidized medium are compiled taking into account the nature of the flow around of the part at low Reynolds numbers. The fields of fluid flow near the part are found, and the mechanism of its flow is established. The dynamics of the movement of abrasive granules in a fluidized medium, under the action of the movement of the part, is presented. The dependences of metal removal during vibration treatment of a rotating and oscillating part in a fluidized medium of abrasive granules are obtained. It is shown that metal removal from the processed part depends on the rotation frequency of the part and vibration exciter. It is noted that the presence of granule velocity components at the surface of the part makes it possible to control the processes of work hardening and micro-cutting.