Ukrainian Journal of Mechanical Engineering and Materials Science

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    Nonlinear mathematical model of the five-container vibration system
    (Видавництво Львівської політехніки, 2022-02-22) Rebot, Dariia; Topilnytskyy, Volodymyr; Lviv Polytechnic National University
    The construction of a non-linear mathematical model of movement and interaction of the commanding and executive components of vibration systems is an important task. It implements vibration technologies of separation, grinding, mixing, compaction, transportation, surface product processing and technology for regulating the vibration effect on systems and mechanisms for their further research to increase the efficiency of vibrating machines, devices, and mechanisms and relevant technological processes. The article presents a generalized diagram of a five-container vibration system. On its basis, a mathematical model was developed, which in the future will make it possible to research the effectiveness of vibration machines, devices, and mechanisms. The calculation scheme of the system and the methods of nonlinear mechanics were used to build the mathematical model. The obtained mathematical model makes it possible to determine the horizontal and vertical components of the amplitude of any point of the containers of the vibration system. This will make it possible to investigate the influence of different modes of operation of the system on the amplitude and nature of vibrations of the containers, in particular, established regimes, influence reversing of the drive, the influence of the processing environment of the containers of the vibration system, influence processed parts.
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    Resonant phenomena of elastic bodies that perform bending and torsion vibrations
    (Lviv Politechnic Publishing House, 2018-01-29) Andrukhiv, Andrij; Sokil, Bohdan; Sokil, Mariia; Lviv Polytechnic National University
    The method of study of the influence of torsional oscillations of one-dimensional models of nonlinear elastic bodies, along which moves with a constant velocity continuous flow of inelastic homogeneous medium, into bending, is developed. It is believed that information on torsional oscillations is known from empirical studies. Based on the latter, the refined model of the dynamics of the process of the investigated object is constructed. The latter is a boundary value problem for nonlinear nonautonomous differential equations with partial derivatives. The imposed restrictions on power factors and the main parameters of torsional oscillations allow for the analytical study of the dynamics of the process to use the basic ideas of the asymptotic integration of equations with partial derivatives. With their help, we obtain a two-parameter set of solutions that describe the determinant parameters of bending vibrations of an elastic body. It is established that for the considered elastic body there can be resonance oscillations, which are caused not only by external factors, but also by internal – torsional oscillations. Regarding the law of the change in the basic parameters of the dynamics of the bending motion of an elastic body, its rotation around the vertical axis reduces the frequency of its own flexural oscillations of the body, and even small torsional oscillations cause an additional periodic action on the transverse. In connection with the above bending vibrations of the elastic body, which performs complex oscillations (torsion and bending), resonances are possible both at the frequency of the external periodic perturbation and at the frequencies of the torsional oscillations (internal resonances). The amplitude of the transition through the resonance: a) at the basic frequency of external perturbation takes less value for elastic bodies of greater flexural rigidity and for higher values of the relative motion of the medium; b) at the frequency of torsional oscillations for larger values of the angular velocity takes more importance; c) with “fast” transition through resonance at the frequency of external or internal perturbation is less than with “slow”. The obtained results can serve as the basis for the choice of operating parameters of elastic elements of machines that carry out complex oscillations.