Geodesic
Limit equations of motion for mechanical systems with vibrating elements
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2015), pp. 23-33

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This paper studies mechanical systems with rapidly oscillating elements. We suggested a method to obtain limiting equations of such systems when the oscillation frequency tends to infinity. We considered a mechanical system with an arbitrary number of degrees of freedom subject to rapid periodic vibrations with zero mean. Its motion can be described by second-order equations. Also we obtained an explicit form of the limiting equations of motion written in the form of a system of first-order differential equations. 
@article{VMUMM_2015_5_a4,
     author = {A. A. Markeeva and M. A. Levin},
     title = {Limit equations of motion for mechanical systems with vibrating elements},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {23--33},
     publisher = {mathdoc},
     number = {5},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2015_5_a4/}
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A. A. Markeeva; M. A. Levin. Limit equations of motion for mechanical systems with vibrating elements. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2015), pp. 23-33. http://geodesic.mathdoc.fr/item/VMUMM_2015_5_a4/