Abstract
Wave theory of movement is used to study the mathematical model of a physical system which describes oscillations of a one-dimensional elastic body under the impact of a moving continuous flow of a homogeneous environment. This model accounts for nonlinear elastic properties of the body at transverse oscillations, as well as environment density and movement velocity. Oscillation amplitude and frequency variation laws in nonresonant modes and under the impact of harmonic perturbation are obtained. Variation laws of the aforesaid parameters are defined by geometrical characteristics of the elastic body, physical and mechanical properties of the material, the velocity of the moving environment, the angular velocity of elastic body rotation, and external factors.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Magrab, E.B.: An Engineer’s Guide to Mathematica. Wiley, Hoboken (2014)
Jones, D.I.G.: Handbook of Viscoelastic Vibration Damping. Wiley, Hoboken (2001)
Sobotka, Z.: Theory of Plasticity and Limit Design of Plates. Elsevier, Amsterdam (1989)
Chen, L.-Q., Yang, X.-D., Cheng, C.-J.: Dynamic stability of an axially moving viscoelastic beam. Eur. J. Mech. A/Solids 23, 659–666 (2004)
Hatami, S., Azhari, M., Saadatpour, M.M.: Free vibration of moving laminated composite plates. Compos. Struct. 80, 609–620 (2007)
Banichuk, N., Jeronen, J., Neittaanmaki, P., Tuovinen, T.: Static instability analysis for traveling membranes and plates interacting with axially moving ideal fluid. J. Fluids Struct. 26, 274–291 (2010)
Czaban, A., Szafraniec, A., Levoniuk, V.: Mathematical modelling of transient processes in power systems considering effect of high-voltage circuit breakers. Przeglad Elektro-techniczny 95(1), 49–52 (2019)
Mockersturm, E.M., Guo, J.: Nonlinear vibration of parametrically excited, visco-elastic, axially moving strings. J. Appl. Mech. ASME 72, 374–380 (2005)
Kuttler, K.L., Renard, Y., Shillor, M.: Models and simulations of dynamic frictional contact. Comput. Methods Appl. Mech. Engrg. 177, 259–272 (1999)
Lim, C.W., Li, C., Yu, J.-L.: Dynamic behaviour of axially moving nanobeams based on non-local elasticity approach. Acta Mech. Sinica 26, 755–765 (2010)
Wickert, J.A., Mote Jr., C.D.: Classical vibration analysis of axially-moving continua. J. Appl. Mech. ASME 57, 738–744 (1990)
Pukach, P.Ya., Kuzio, I.V.: Nonlinear transverse vibrations of semiinfinite cable with consideration paid to resistance. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu, no. 3, pp. 82–86 (2013)
Pukach, P., Il’kiv, V., Nytrebych, Z., Vovk, M., Pukach, P.: On the asymptotic methods of the mathematical models of strongly nonlinear physical systems. In: Advances in Intelligent Systems and Computing, vol. 689, pp. 421–433 (2018)
Lavrenyuk, S.P., Pukach, P.Ya.: Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables. Ukrainian Math. J. 59, no. 11, pp. 1708–1718 (2007)
Buhrii, O.M.: Visco-plastic, newtonian, and dilatant fluids: stokes equations with variable exponent of nonlinearity. Matematychni Studii, vol. 49, no. 2, pp. 165–180 (2018)
Nytrebych, Z., Malanchuk, O., Il’kiv, V., Pukach, P.: On the solvability of two-point in time problem for PDE. Italian J. Pure Appl. Math. 38, 715–726 (2017)
Pukach, P.: Investigation of bending vibrations in Voigt-Kelvin bars with regard for non-linear resistance forces. J. Math. Sci. 215(1), 71–78 (2016)
Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)
Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)
Papargyri-Beskou, S., Tsepoura, K.G., Polyzos, D., Beskos, D.E.: Bending and stability analysis of gradient elastic beams. Int. J. Solids Struct. 40, 385–400 (2003)
Vardoulakis, I., Sulem, J.: Bifurcation Analysis in Geomechanics. Blackie/ Chapman and Hall, London (1995)
Gao, X.-L., Park, S.K.: Variational formulation of a modified couple stress theory and its application to a simple shear problem. Zeitschrift furangewandte Mathematik und Physik 59, 904–917 (2008)
Ma, H.M., Gao, X.-L., Reddy, J.N.: A non-classical Reddy-Levinson beam model based on a modified couple stress theory. Int. J. Multiscale Comput. Eng. 8, 167–180 (2010)
Belmas, I.V., Kolosov, D.L., Kolosov, A.L., Onyshchenko, S.V.: Stress-strain state of rubber-cable tractive element of tubular shape. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu, vol. 2, pp. 60–69 (2018)
Mahmoodi, S.N., Jalili, N.: Non-linear vibrations and frequency response analysis of piezoelectrically driven microcantilevers. Int. J. Non-Linear Mech. 42, 577–587 (2007)
Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)
Nayfeh, A.H., Mook, D.T.: Non-Linear Oscillations. Wiley, New York (1979)
Pain, H.J.: The Physics of Vibration and Waves, 6th edn. Wiley, New York (2005)
Chen, L.-Q., Chen, H.: Asymptotic analysis of nonlinear vibration of axially accelerating visco-elastic strings with the standard linear solid model. J. Eng. Math. 67, 205–218 (2010)
Bayat, M., Barari, A., Shahidi, M.: Dynamic response of axially loaded Euler-Bernoulli beams. Mechanika 17(2), 172–177 (2011)
Teslyuk, V.M.: Models and Information Technologies of Micro-electromechanical Systems Synthesis. Vezha and Кo, Lviv (2008)
Nytrebych, Z., Il’kiv, V., Pukach, P., Malanchuk, O.: On nontrivial solutions of homogeneous Dirichlet problem for partial differential equations in a layer. Kragujevac J. Mathem. 42(2), 193–207 (2018)
Pukach, P.Ya., Kuzio, I.V., Nytrebych, Z.M., Ilkiv, V.S.: Analytical methods for determining the effect of the dynamic process on the nonlinear flexural vibrations and the strength of compressed shaft. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu 5, 69–76 (2017)
Pukach, P.Ya., Kuzio, I.V., Nytrebych, Z.M., Ilkiv, V.S.: Asymptotic method for investigating resonant regimes of non–linear bending vibrations of elastic shaft. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu 1, 68–73 (2018)
Kauderer, H.: Nonlinear Mechanics. Izdatelstvo Inostrannoy Literatury, Moscow (1961). (in Russian)
Pukach, P., Nytrebych, Z., Ilkiv, V., Vovk, M., Pukach, Yu.: On the mathematical model of nonlinear oscillations under the impact of a moving environment. In: Proceedings of International scientific conference Computer sciences and information technologies (CSIT-2019), vol. 1, pp. 71–74 (2019)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Pukach, P., Il’kiv, V., Nytrebych, Z., Vovk, M., Pukach, P. (2020). Modified Asymptotic Method of Studying the Mathematical Model of Nonlinear Oscillations Under the Impact of a Moving Environment. In: Shakhovska, N., Medykovskyy, M.O. (eds) Advances in Intelligent Systems and Computing IV. CSIT 2019. Advances in Intelligent Systems and Computing, vol 1080. Springer, Cham. https://doi.org/10.1007/978-3-030-33695-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-33695-0_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-33694-3
Online ISBN: 978-3-030-33695-0
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)