The problems of synthesis and substantiation of elasticity parameters of the resonant vibratory device with electromagnetic drive and one flat spring are considered. At first, the harmonic systems with oscillation frequencies of 50 Hz and 100 Hz were investigated. Then, various asymmetric piecewise linear characteristics of elasticity were carried into effect on one flat spring using auxiliary intermediary fixed cylindrical supports. Due to this, the corresponding vibro-impact operation modes were obtained. The resonant systems characterized by improved functioning efficiency were carried into effect using the new technique of optimization synthesis of elasticity parameters. The resonant systems being investigated were implemented in practice. The basic experimental investigations of their kinematic, dynamic and energetic parameters were carried out. The fundamental result of the investigation consists in confirmation of the improved dynamic efficiency of vibro-impact systems with pulsed electromagnetic excitation designed according to the new technique. The proposed systems may be used in technological processes of materials compaction and screening, of surface treatment of machine parts and in processes associated with nanotechnology.

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Асимптотичний метод дослідження резонАнсних режимів неліній них згинАльних коливАнь пружного вАлА Purpose. To develop a method for determining resonant modes of industrial equipment of elastic shaft type, which is widely used in the mining industry, through the study of mathematical model of nonlinear oscillations. Mathematical models of oscillatory systems previously were studied in the literature mainly based on the numerical and experimental approaches. This paper proposes using a combination of the wave theory of motion and asymptotic methods of nonlinear mechanics using special apparatus of periodic functions to investigate the vibrational dynamics of the system and conditions of resonance phenomena in it, as well as to describe the method for determining the resonance curves to increase the margin of safety of industrial equipment. Methodology. Methods for studying resonance amplitudes and frequencies, determining the strength characteristics of equipment are based on the use of asymptotic methods of nonlinear mechanics, wave motion theory and theory of special Ateb-functions. Findings. In this work the conditions of resonance amplitude and frequency depending on the system parameters were obtained analytically for these nonlinear vibrational systems of elastic shaft and the overall method for determining the resonance curves was described. Originality. For the first time a complete analysis of the impact of physical, mechanical and geometrical factors of the dynamic pricess on the resonant frequency and amplitude in systems such as elastic shaft was conducted on the UDC 534.1

Автоматизація виробничих процесів у машинобудуванні та приладобудуванні. Вип. 52. 2018 32 УДК 621.923.7 \ I. V. Kuzio, V. M. Zakharov, V. M. Korendiy doi The purpose of the paper. Substantiation of structure (design), parameters and operation modes of the improved vibratory finishing machine. Analysis of dynamical processes which occur during "lap over lap" dressing. Investigation methodology. Mathematical model of motion of the mechanical system of vibratory finishing machine was developed on the basis of Lagrange differential equations of the second order. For the purpose of describing friction between the working surfaces of the laps, the Coulomb friction model was used. Stiffness parameters of all elastic elements were modeled in accordance with the Hooke's law. Energy losses in elastic elements during their tension-compression were taken into account by corresponding coefficients of dissipation (damping). Simulation modelling of motion of the machine's mechanical oscillatory system was carried out in MathCAD software by means of solving the derived differential equations of the system's motion using the numerical methods of Runge-Kutta. Obtained results. Structural and functional peculiarities of the improved vibratory finishing machine for lapping flat surfaces of cylindric and prismatic parts were considered. Design (calculation) diagram of its mechanical oscillatory system was substantiated and differential equations of motion of oscillating masses were derived. Simulation modelling of the laps' motion during their dressing was carried out and the correspondence of the obtained results to the input modelling parameters (operation in near-resonance mode with the given oscillation amplitudes) was analyzed. Scientific novelty. For the first time we obtained following results: spatial design (calculation) diagram of mechanical oscillatory system of the improved vibratory finishing machine was proposed; mathematical model of plane-parallel motion of oscillating masses (with circular trajectories of oscillations) was developed; possibilities of performing the laps dressing using "lap over lap" method were substantiated by means of ensuring their circular oscillations. Practical value. The results of the performed investigations can be used during designing new and improving existing structures of vibratory finishing machines for finishing treatment (lapping) of flat surfaces of cylindric and prismatic parts.

Problem statement. Designing and manufacturing of efficient resonant vibratory lapping machines with linear oscillations of laps demand an accurate and detailed calculation of parameters of their elastic systems and electromagnetic drives. Purpose. The main objective of this research consists in derivation of analytical dependencies for calculating the stiffness and excitation parameters of mechanical oscillatory system of vibratory finishing machine in order to ensure its resonance operation mode. Methodology. The technique of the research is based on fundamental concepts of engineering mechanics, strength of materials and theory of mechanical vibrations. Findings (results). The design diagram of mechanical oscillatory system of vibratory finishing machine with linear oscillations of laps is considered and corresponding equations of motion are presented. Analytical dependencies for calculating stiffness and excitation parameters of the system are deduced. The example of parameters calculation is given and time dependencies of the system's motion are constructed. Originality (novelty). The mathematical model of linear oscillations of the three-mass mechanical system of vibratory finishing machine was developed. The possibilities of performing the laps dressing using "lap over lap" method were substantiated. Practical value. The results of the performed investigations can be used during designing new designs and improving existing structures of vibratory finishing machines for lapping flat surfaces of cylindric and prismatic parts. Scopes of further investigations. In further investigations, it is necessary to analyse the influence of the viscous damping on the system's motion. In order to substantiate (justify) the obtained theoretical results, the experimental investigations should be carried out.