Mathematical models of~nonlinear longitudinal-cross oscillations of~object with~moving borders

V. N. Anisimov ; V. L. Litvinov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 19 (2015) no. 2, pp. 382-397.

Résumé

The nonlinear formulation of problems for describing longitudinal-cross oscillations of objects with moving borders is noted. These mathematical models consist of a system of two nonlinear partial differential equations with the higher time derivative of the second order and the fourth-order by the spatial variable. The nonlinear boundary conditions on moving boundary have a higher time derivative of the second order and the third-order by the spatial variable. The geometric nonlinearity, visco-elasticity, the flexural stiffness of the oscillating object and the elasticity of the substrate of object are taken into account. Boundary conditions in the case of energy exchange between the parts of the object on the left and right of the moving boundary are given. The moving boundary has got a joined mass. The elastic nature of borders joining is considered. The longitudinal-cross oscillations of objects with moving borders of high intensity can be described by the resulting differential model. The Hamilton's variational principle is used in the formulation of the problem.

Détail

Keywords: longitudinal-cross oscillations, moving borders, boundary value problems, mathematical models, boundary conditions, nonlinear system of partial differential equations
Mots-clés : variational principles.

Bibliographie

[1] Goroshko O. A., Savin G. N., Vvedenie v mekhaniku deformiruemykh odnomernykh tel peremennoi dliny [Introduction to the Mechanics of Deformable One-Dimensional Bodies of Variable Length], Naukova dumka, Kiev, 1971, 270 pp. (In Russian)

[2] Zhu W. D., Chen Y., “Theoretical and experimental investigation of elevator cable dynamics and control”, J. Vib. Acoust., 128:1 (2006), 66–78 | DOI

[3] Samarin Yu. P., Anisimov V. N., “Forced transverse vibrations of the flexible link at dispersal”, Izv. vuzov. Mashinostroenie, 1986, no. 12, 17–21 (In Russian)

[4] Boyle (Jr) J. M., Bhushan B., “Vibration modeling of magnetic tape with vibro-impact of tape-guide contact”, J. Sound Vibr., 289:3 (2006), 632–655 | DOI

[5] Lezhneva A. A., “Bending vibration of beam of variable length”, Izv. Akad. Nauk USSR. MTT, 1970, no. 1, 159–161 (In Russian)

[6] Ding Hu, Chen Li-Qun, “Galerkin methods for natural frequencies of high-speed axially moving beams”, J. Sound Vibr., 329:17 (2010), 3484–3494 | DOI

[7] Guo Y., Yang S., Guo W., “Analysis of dynamic characteristics of steel spring supported floating track bed”, Zhendong Ceshi Yu Zhenduan = Journal of Vibration, Measurement and Diagnosis, 26:2 (2006), 146–150 (In Chinese)

[8] Lei X.-Y, “Effects of abrupt changes in track foundation stiffness on track vibration under moving loads”, Zhendong Gongcheng Xuebao = Journal of Vibration Engineering, 19:2 (2006), 195–199 (In Chinese)

[9] Sahebkar S. M., Ghazavi M. R., Khadem S. E.,Ghayesh M. H., “Nonlinear vibration analysis of an axially moving drillstring system with time dependent axial load and axial velocity in inclined well”, Mechanism and Machine Theory, 46:5 (2011), 743–760 | DOI | Zbl

[10] Inácio O., Antunes J., Wright M. C. M., “Computational modelling of string-body interaction for the violin family and simulation of wolf notes”, J. Sound Vibr., 310:1–2 (2008), 260–286 | DOI

[11] Tikhonov V. S., Abramov A. A., “Transverse Vibrations of a Flexible String with Time-Varying Length in Flow”, Vest. Mosk. Univ. Ser 1. Matematika, Mekhanika, 1993, no. 5, 45–48 (In Russian) | MR | Zbl

[12] Vesnitskii A. I., Volny v sistemakh s dvizhushchimisia granitsami [Waves in systems with moving boundaries and loads], Fizmatlit, Moscow, 2001, 320 pp. (In Russian)

[13] Anisimov V. N., Litvinov V. L., “Investigation of Resonance Characteristics of Mechanical Objects with Moving Borders by Application of the Kantorovich–Galyorkin Method”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2009, no. 1(18), 149–158 (In Russian) | DOI

[14] Anisimov V. N., Litvinov V. L., Korpen I. V., “On a method of analytical solution of wave equation describing the oscillations sistem with moving boundaries”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2012, no. 3(28), 145–151 (In Russian) | DOI | Zbl

[15] Kotera T., “Vibration of a string with time-varying length”, Memoirs of the Faculty of Engineering, Kobe University, 24, 1978, 45–54 (in Japanese)

[16] Zhu W. D., Zheng N. A., “Exact response of a translating string with arbitrarily varying length under general excitation”, ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (September 4–7, 2007), v. 1, 21st Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C, Las Vegas, Nevada, USA, 2008, DETC2007-34590, 1995–2013 | DOI

[17] Zhu W. D., Zheng N. A., “Exact response of a translating string with arbitrarily varying length under general excitation”, J. Appl. Mech., 75:3 (2008), 031003, 14 pp. | DOI

[18] Brake M. R., Wickert J. A., “Frictional vibration transmission from a laterally moving surface to a traveling beam”, J. Sound Vibr., 310:3 (2008), 663–675 | DOI

[19] Myshkis A. D., Matematika dlia tekhnicheskikh vuzov [Mathematics for Technical Colleges], Lan', St. Petersburg, 2002, 640 pp. (In Russian)

[20] Anisimov V. N., Litvinov V. L., Korpen I. V., “The formulation of the problem of the beam fluctuations with moving spring-loaded support”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Tekhn. Nauki, 2013, no. 1(37), 93–98 (In Russian)