Abstract
This study has two aims. The first aim is to describe a number of wonderful phenomena caused by the action of vibration1 on nonlinear mechanical systems along with important applications of these effects in technology. The second aim is to present a general mechanical-mathematical approach to the description and investigation of this class of phenomena. We call the approach to be described herein “Vibrational Mechanics”. While it is a new approach, it is based on the classical idea of averaging in the theory of nonlinear oscillations and in the theory of the stability of motion.
By vibration we mean the mechanical oscillations whose period is much shorter than the characteristic interval in which the motion of the system is considered and whose swing is far smaller than the characteristic size of the system.
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Bibliography
I. I. Blekhman. Vibrational Mechanics. World Scientific, Singapore, 2000, 509 pp.
I. I. Blekhman, editor. Selected Topics in Vibrational Mechanics. World Scientific, Singapore, 2004, 410 pp.
I. I. Blekhman, A. D. Myshkis, and Ya. G. Panovko. Mechanics and applied mathematics: Logic and peculiarities of applications of mathematics. Nauka, Moscow, 1990, 2nd ed. (in Russian), 356 pp; Germany translation: Angewandte Mathematik. Gegenstand, Logik, Besonderheiten. Veb Deutscher Verlag der Wissenschaften, Berlin, 1989, 350 S.
I. I. Blekhman. On two resonant effects under the action of a high frequency vibration on nonlinear systems. Khimicheskaya promyshlennost, 81(7):329–331, 2004 (in Russian).
I. I. Blekhman and L. Sperling. Behavior of the pendulum od Stephenson-Kapitsa with inner degrees of freedom. In Proc. of the XXII Summer School “Advanced Problems in Mechanics” (APM’2004), June 24–July 1, St. Petersburg (Repino), Russia, p. 59–67, 2004.
I. I. Blekhman. Synchronization in Science and Technology. New York: ASME Press, 1998, 255 pp.
W. B. Fraser and A. R. Champneys. The “Indian rope trick” for a parametrically excited flexible rod: nonlinear and subharmonic analysis. Proc. Roy. Soc., London, A458, p. 1353–1373, 2002.
E. Kremer. Vibrational liquid-bubble interaction and acoustic induced flow in gas suspension. In Proc. of the IUTAM Symposium “Liquid-Particle in Suspension Flows”, Grenoble, France, April, 1994.
J. J. Thomsen. Vibration and Stability: Advanced Theory, Analysis and Tools. Springer-Verlag, Berlin-Heidelberg, 2003, 403 pp.
K. Zimmerman, I. Zeidis, J. Steigenberger and M. Pivovarov. An approach to worm-like motion. In Proc. of the XXI ICTAM, Warsaw, Aug., 2004.
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© 2007 CISM, Udine
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Blekhman, I.I. (2007). Vibrational Mechanics - a General Approach to Solving Nonlinear Problems. In: Elishakoff, I. (eds) Mechanical Vibration: Where do we Stand?. International Centre for Mechanical Sciences, vol 488. Springer, Vienna. https://doi.org/10.1007/978-3-211-70963-4_12
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DOI: https://doi.org/10.1007/978-3-211-70963-4_12
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-68586-0
Online ISBN: 978-3-211-70963-4
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