Modified Asymptotic Method of Studying the Mathematical Model of Nonlinear Oscillations Under the Impact of a Moving Environment
- 177 Downloads
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.
Keywords
Mathematical model Physical system Dynamical process Nonlinear oscillations Wave theoryReferences
- 1.Magrab, E.B.: An Engineer’s Guide to Mathematica. Wiley, Hoboken (2014)Google Scholar
- 2.Jones, D.I.G.: Handbook of Viscoelastic Vibration Damping. Wiley, Hoboken (2001)Google Scholar
- 3.Sobotka, Z.: Theory of Plasticity and Limit Design of Plates. Elsevier, Amsterdam (1989)zbMATHGoogle Scholar
- 4.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)CrossRefGoogle Scholar
- 5.Hatami, S., Azhari, M., Saadatpour, M.M.: Free vibration of moving laminated composite plates. Compos. Struct. 80, 609–620 (2007)CrossRefGoogle Scholar
- 6.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)CrossRefGoogle Scholar
- 7.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)Google Scholar
- 8.Mockersturm, E.M., Guo, J.: Nonlinear vibration of parametrically excited, visco-elastic, axially moving strings. J. Appl. Mech. ASME 72, 374–380 (2005)CrossRefGoogle Scholar
- 9.Kuttler, K.L., Renard, Y., Shillor, M.: Models and simulations of dynamic frictional contact. Comput. Methods Appl. Mech. Engrg. 177, 259–272 (1999)MathSciNetCrossRefGoogle Scholar
- 10.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)MathSciNetCrossRefGoogle Scholar
- 11.Wickert, J.A., Mote Jr., C.D.: Classical vibration analysis of axially-moving continua. J. Appl. Mech. ASME 57, 738–744 (1990)CrossRefGoogle Scholar
- 12.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)Google Scholar
- 13.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)Google Scholar
- 14.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)Google Scholar
- 15.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)Google Scholar
- 16.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)Google Scholar
- 17.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)MathSciNetCrossRefGoogle Scholar
- 18.Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)MathSciNetCrossRefGoogle Scholar
- 19.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)CrossRefGoogle Scholar
- 20.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)CrossRefGoogle Scholar
- 21.Vardoulakis, I., Sulem, J.: Bifurcation Analysis in Geomechanics. Blackie/ Chapman and Hall, London (1995)Google Scholar
- 22.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)MathSciNetCrossRefGoogle Scholar
- 23.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)CrossRefGoogle Scholar
- 24.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)Google Scholar
- 25.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)CrossRefGoogle Scholar
- 26.Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)zbMATHGoogle Scholar
- 27.Nayfeh, A.H., Mook, D.T.: Non-Linear Oscillations. Wiley, New York (1979)zbMATHGoogle Scholar
- 28.Pain, H.J.: The Physics of Vibration and Waves, 6th edn. Wiley, New York (2005)CrossRefGoogle Scholar
- 29.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)CrossRefGoogle Scholar
- 30.Bayat, M., Barari, A., Shahidi, M.: Dynamic response of axially loaded Euler-Bernoulli beams. Mechanika 17(2), 172–177 (2011)CrossRefGoogle Scholar
- 31.Teslyuk, V.M.: Models and Information Technologies of Micro-electromechanical Systems Synthesis. Vezha and Кo, Lviv (2008)Google Scholar
- 32.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)Google Scholar
- 33.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)Google Scholar
- 34.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)CrossRefGoogle Scholar
- 35.Kauderer, H.: Nonlinear Mechanics. Izdatelstvo Inostrannoy Literatury, Moscow (1961). (in Russian)Google Scholar
- 36.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)Google Scholar