{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:44:55Z","timestamp":1760060695651,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T00:00:00Z","timestamp":1757635200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The second approximation of the averaging method is constructed in the problem of motion along a horizontal rough plane of a vibration-driven mechanical system consisting of a carrying body (robot housing) in contact with the plane and two internal masses oscillating vertically and horizontally according to a harmonic law with the same frequency. In the absence of a phase shift in the first approximation, the directed motion of the system as a single whole (locomotion) cannot be detected. In the second approximation, an expression for the average velocity of motion is obtained. The values of velocity calculated using the obtained formula are in good agreement with the results of numerical calculations. The described technique for constructing the second approximation can be used to solve other problems with dry friction.<\/jats:p>","DOI":"10.3390\/axioms14090694","type":"journal-article","created":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T13:43:02Z","timestamp":1757684582000},"page":"694","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Second Approximation of the Averaging Method in the Dynamics of Controlled Motions of Vibration Systems with Dry Friction"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9481-689X","authenticated-orcid":false,"given":"Igor","family":"Zeidis","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Technische Universit\u00e4t Ilmenau, Max-Planck-Ring 12, 98693 Ilmenau, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1939-0099","authenticated-orcid":false,"given":"Nikolay","family":"Bolotnik","sequence":"additional","affiliation":[{"name":"Laboratory of Robotics and Mechatronics, Institute for Problems in Mechanics of the Russian Academy of Sciences, 101-1 Vernadsky Ave, 119526 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Klaus","family":"Zimmermann","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Technische Universit\u00e4t Ilmenau, Max-Planck-Ring 12, 98693 Ilmenau, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,12]]},"reference":[{"key":"ref_1","unstructured":"Blekhman, I.I. (2004). Vibrational Mechanics, World Scientific."},{"key":"ref_2","unstructured":"Fidlin, A. (2005). Nonlinear Oscillations in Mechanical Engineering, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Andrianov, I., and Awrejcewicz, J. (2024). Asymptotic Methods for Engineers, CRC Press.","DOI":"10.1201\/9781003467465"},{"key":"ref_4","first-page":"287","article-title":"The homotopy perturbation method for nonlinear oscillators with discontinuities","volume":"151","author":"He","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_5","unstructured":"Bogolubov, N.N., and Mitropolskii, Y.A. (1961). Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Sanders, J.A., Verhulst, F., and Murdock, J. (1985). Averaging Methods in Nonlinear Dynamical Systems, Springer.","DOI":"10.1007\/978-1-4757-4575-7"},{"key":"ref_7","unstructured":"Zimmermann, K., Zeidis, I., and Behn, C. 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Iberoam."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1007\/BF01062050","article-title":"Method of partial averaging in the systems of standard form with discontinuous right-hand sides","volume":"18","author":"Plotnikov","year":"1993","journal-title":"Ukr. Math. J."},{"key":"ref_16","unstructured":"Appel, P. (1893). Trait\u00e9 de M\u00e9canique Rationnelle, Gauthier-Villars."},{"key":"ref_17","unstructured":"Whittaker, E.T. (1917). A Treatise on the Analytical Dynamics of Particles and Rigid Dodies, Cambridge University Press."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Anh, L.X. (2003). Dynamics of Mechanical Systems with Coulomb Friction, Springer.","DOI":"10.1007\/978-3-540-36516-7"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/9\/694\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:44:47Z","timestamp":1760035487000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/9\/694"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,12]]},"references-count":18,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["axioms14090694"],"URL":"https:\/\/doi.org\/10.3390\/axioms14090694","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,9,12]]}}}