Establishment of the original frequency of the continual section of the interreson research machine Rayleigh–Ritz method

2020;
: 5-15
1
Lviv Polytechnic National University
2
Lviv Polytechnic National University
3
Lviv Polytechnic National University
4
Lviv Polytechnic National University
5
Lviv Polytechnic National University
6
Hetman Petro Sahaidachny National Army Academy

Aim. Is to establish the first natural frequency of the continuum in the form of a rod (body with distributed parameters) of the interresonant vibrating machine. Method. The work is based on fundamental methods of mechanics, in particular the theory of mechanical oscillations. The natural frequencies and reactions in the supports of the continuum were found using the method of initial parameters, using the Krylov functions and the Rayleigh-Ritz method. Results. The paper considers a discrete model of an interresonant vibrating machine and establishes the partial frequency of its reactive mass, which is the initial parameter that the continuous section must provide. The frequency equation of the continuum section is formed using analytical methods. The value of its first natural frequency was found, which practically coincided with the partial frequency of the discrete model. The continuous section in the form of a rod was modeled in the software product SOLIDWORKS Simulation. Scientific novelty. For the first time, the most optimal scheme of fastening the continuous section in the form of an elastic rod mounted on two hinged supports, one of the ends of which is cantilevered, and the other is driven by an eccentric. For the first time using the Rayleigh-Ritz method, the natural frequency of the continuum section was analytically found, which is a determining parameter for the synthesis of interresonant discrete-continuum vibrating machines in which the partial frequency of the discrete model was matched with the value of the first natural frequency of the rod. Practical significance. The established analytical expressions will allow to form the effective concept of construction of an engineering technique of calculation of discrete-continuous vibrating technological equipment of various technological purpose, capable to realize effectively interresonant operating modes. Sufficient accuracy of the proposed analytical expressions contributes to their widespread use in practice.

1. A. Buchacz, Calculation of flexibility of vibrating beam as the subsystem of mechatronic system by means the exact and approximate methods, Proceedings in Applied Mathematics and Mechanics 9/1 2009, 373-374.
https://doi.org/10.1002/pamm.200910160
2. A. Buchacz, The supply of formal notions to synthesis of the vibrating discrete-continuous mechatronic systems, Journal of Achievements in Materials and Manufacturing Engineering, International OCOSCO World Press 44/2 2011, 168-178.
3. Albert R. Probert, "Linear vibratory conveyor," GB Patent 2,238,841А, February 06.1991.
4. Arthur W. Forman, "Improvements in Vibratory Feeders," GB Patent 1,256,225, July12.1971.
5. Franklin C. Pereny, "Vibrator," US Patent 4,117,381, September 26, 1978.
6. Gulertan Vural, Albert Linz, "Vibratory Compacting Machine," US Patent 3909148A, Sep. 30, 1975.
7. I. M. Babakov, Teoriya kolebaniy [Theory of oscillations] / I. M. Babakov. - L.: Nauka 1968. [in Russian].
8. Jakhin B. Popper, Kyriat Motzkin, "Vibrating surface apparatus," US Patent 4,315,817A Feb. 16, 1982.
9. Ya. G. Panovko, Osnovy prikladnoy teorii kolebaniy i udara [Fundamentals of applied theory of vibrations and impact] / Ya. G. Panovko. - M.: Mashinostroyeniye, 1976. [in Russian].
10. John C. O'Connor, "Vibration Producing Mechanism" US Patent 2,353,492, November 07.1944.
11. John M. Morris, "Wibratory drive mechanism," US Patent 3,180,158 April 27, 1965.
12. Joseph W. Sherwen (The General Electric Company Limited), "Improvements in or relating to electro-magnetic vibrating equipment," GB Patent 664720A , January 09,1952.
13. Joseph W. Sherwen (The General Electric Company Limited), "Improvements in or relating to Electro-Magnetic Vibrating Equipment," GB Patent 745518A, February 29, 1956.
14. Mikata Yoshitaka, Нiguchi Sunao, "Electromagnetic Vibration Conveyance Feeder for Combination Balance or the Like," JP Patent S63282615A, November 18, 1988.
15. Mikata Yoshitaka, Higuchi Tadashi, Harada Shogo, "Two trough, electromagnetically vibratory feeder," EP Patent 0349693A3, September 26, 1990.
16. Moulsdale Timothy, John Blackwall, Wright Machnery Limited (GB), Mendelewicz Ran (IL), "Vibratory Conveyor" WO Patent 2006/125998A1, November 30,2006.
17. Lanets Oleksii. Osnovy rozrakhunku ta konstruyuvannya vibratsiynykh mashyn. Knyha 1. Teoriya ta prak-tyka stvorennya vibratsiynykh mashyn z harmoniynym rukhom robochoho orhana: navch. Posibnyk [Fundamentals of analysis and design of vibrating machines. Book 1. Theory and Practice of Development of Vibratory Machines with Harmonic Motion of the Working Element body: textbook. manual]/ O. Lanets'. - L'viv: Vydavnytstvo L'vivs'koyi politekhniky, 2018. [in Ukrainian].
18. Peter T. Jones, Kennewick, "Conveying Apparatus," US Patent 6,868,960 B2, March 22, 2005.
19. Richard B. Kraus, "Linear Drive for Vibratory Apparatus," CA Patent 2,518,736C, Sep. 01, 2009.
20. S. D. Ponomarev i dr., Raschety na prochnost v mashinostroyenii [Strength calculations in mechanical engineering]: v 3 t. / Ponomarev S. D. i dr. - M.: Mashgiz 1959. [in Russian].
21. Thomas H. Falconer (Eriez Manufacturing Company), "Three Mass Vibratory Feeder" US Patent 4,961,491, October 09, 1990.
22. Wang Yun, "Three-mass asymmetric vibrating mill," CN Patent 104941764A, September 30, 2015.
23. William R. Brown, "Three Mass Electromagnetic Feeder," US Patent 4,378,064, March 29, 1983.
24. William W. Triggs (GB) (The Traylor Vibrator Company (US)), "Improvements in and relating to Method of and Apparatus for Conveying and Conditioning Materials," GB Patent 494,206A, October 21, 1938.
25. William W. Triggs (GB) (The Traylor Vibrator Company (US)), "Improvements in and relating to Method of and Apparatus for Conveying and Conditioning Materials," GB Patent 504,782A, April 24, 1939.