The chaotic synchronization of the vibrations of contacting multilayer beam–plate–shell structures under external loading is studied. There are gaps between their layers. Such systems are called structurally nonlinear. Their behavior is studied comprehensively, and the influence parameters that characterize the safe and critical modes are determined. A method to study the chaotic phase synchronization of various nature is developed. Vibration synchronization for the following systems is analyzed: (i) a stack of two-layer plates (each equation is linear; however, there is structural nonlinearity if the plates contact), (ii) a plate reinforced with a beam, (iii) a stack of two-layer beams (each component is both physically and geometrically nonlinear; their contact interaction causes structural nonlinearity), (iv) a stack of two-layer shells (each component is both physically and geometrically nonlinear; their contact interaction causes structural nonlinearity)
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References
I. A. Birger, ”Some general methods to solve problems of plasticity,” Prikl. Mat. Mekh., 15, No. 6, 765–770 (1951).
B. Ya. Kantor, Contact Problems in the Nonlinear Theory of Shells of Revolution [in Russian], Naukova Dumka, Kyiv (1990).
A. S. Desyatova, M. V. Zhigalov, V. A. Krys’ko, and O. A. Saltykova, “Dissipative dynamics of geometrically nonlinear Bernoulli–Euler beams,” Mech. Solids, 43, No. 6, 948–956 (2008).
V. A. Krys’ko, M. V. Zhigalov, O. A. Saltykova, and A. V. Krys’ko, “Effect of transverse shears on complex nonlinear vibrations of elastic beams,” J. Appl. Math. Techn. Phys., 52, No. 5, 834–840 (2011).
V. A. Krys’ko and I. V. Kravtsova, ”Control of the chaotic vibrations of flexible spherical shells,” Izv. RAN, Mekh. Tverd. Tela, 41, No. 1, 161–172 (2006).
I. V. Papkova, V. A. Krys’ko, and V. V. Soldatov, “Analysis of nonlinear chaotic vibrations of shallow shells of revolution by using the wavelet transform,” Mech. Solids, 45, No. 1, 85–93 (2010).
A. V. Altukhov and M. V. Fomenko, “Elastic vibrations of sandwich plates with diaphragms at the edges,” Int. Appl. Mech., 50, No. 2, 179–186 (2014).
E. Ya. Antonyuk and A. T. Zabuga, “Dynamics of a two-link vehicle in an L-shaped corridor revisited,” Int. Appl. Mech., 50, No. 2, 222–230 (2014).
J. Awrejcewicz and B. A. Krysko, Chaos in Structural Mechanics, Springer, Berlin–London (2008).
J. Awrejcewicz and V. A. Krysko, Introduction to Asymptotic Methods, Chapman&Hall/CRC, London–New York (2006).
J. Awrejcewicz, B. A. Krysko, and A. V. Krysko, Thermodynamics of Plates and Shells, Springer, Berlin–London (2007).
J. Awrejcewicz, A. B. Krysko, T. B. Yakovleva, D. S. Zelenchuk, and B. A. Krysko, “Chaotic synchronization of vibrations of a coupled mechanical system consisting of a plate and beams,” Latin Amer. J. Solids Struct., 10, 161–172 (2013).
S. V. Bosakov, “Contact problems for a plate as an inclusion in an elastic half-space,” Int. Appl. Mech., 50, No. 2, 187–195 (2014).
Ya. M. Grigorenko and L. S. Rozhok, “Applying discrete Fourier series to solve problems of the stress state of hollow noncircular cylinders,” Int. Appl. Mech., 50, No. 2, 105–127 (2014).
S. Gutschmidt and O. Gottlieb, “Bifurcations and loss of orbital stability in nonlinear viscoelastic beam arrays subject to parametric actuation,” J. Sound Vibr., April, 3835–3855 (2010).
L. P. Khoroshun and O. I. Levchuk, “Stress distribution around cracks in linear hardening materials subject to tension: Plane problem,” Int. Appl. Mech., 50, No. 2, 128–140 (2014).
L. P. Khoroshun and E. N. Shikula, “Deformation and long-term damage of physically nonlinear fibrous materials,” Int. Appl. Mech., 50, No. 1, 58–67 (2014).
A. V. Krysko, M. I. Koch, Y. V. Yakovleva, U. Nackenhorst, and V. A. Krysko, “Chaotic nonlinear dynamics of cantilever beams under the action of signs-variables loads,” PAMM, Special Issue: 82nd Annual Meeting of the Int. Assoc. Appl. Math. Mech. (GAMM) (Graz 2011), 11, No. 1, 327–328 (2011).
V. P. Legeza and D. V. Legeza, “Vibration of a string with moving end,” Int. Appl. Mech., 50, No. 1, 87–91 (2014).
A. I. Manevich and Z. Kolakowski, “Revisiting the theory of transverse vibrations of plates with shear deformation,” Int. Appl. Mech., 50, No. 2, 196–205 (2014).
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Translated from Prikladnaya Mekhanika, Vol. 50, No. 6, pp. 117–132, November–December 2014.
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Krys’ko, V.A., Yakovleva, T.V., Dobriyan, V.V. et al. Wavelet-Analysis-Based Chaotic Synchronization of Vibrations of Multilayer Mechanical Structures. Int Appl Mech 50, 706–720 (2014). https://doi.org/10.1007/s10778-014-0669-z
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DOI: https://doi.org/10.1007/s10778-014-0669-z