Abstract
In this paper we consider parametric oscillations of flexible plates within the model of von Kármán equations. First we propose the general iterational method to find solutions to even more general problem governed by the von Kármán–Vlasov–Mushtari equations. In the language of physics the found solutions define stress–strain state of flexible shallow shell with a bounded convex space ΩεR 2 and with sufficiently smooth boundary Γ. The new variational formulation of the problem has been proposed and his validity and application has been discussed using precise mathematical treatment. Then, using the earlier introduced theoretical results, an effective algorithm has been applied to convert problem of finding solutions to hybrid type partial differential equations of von Kármán form to that of the ordinary differential (ODEs) and algebraic (AEs) equations. Mechanisms of transition to chaos of deterministic systems with infinite number of degrees of freedom are presented. Comparison of mechanisms of transition to chaos with known ones is performed. The following cases of longitudinal loads of different sign are investigated: parametric load acting along X direction only, and parametric load acting in both directions X and Y with the same amplitude and frequency.
Similar content being viewed by others
References
Vlasov, V.Z., General Theory of Shells and Its Applications in Technology, Gostekhizdat, Moskva, 1949.
Volmir, A.S., Flexible Plate and Shells, Gostekhizdat, Moskva, 1956.
Mushtari, Kh.M. and Galimov, K.Z., Nonlinear Theory of Elastic Shells, Tatknigoizdat, Kazan, 1957.
Awrejcewicz, J. and Krysko, V.A., Dynamics of Continuous Systems, WNT, Warsaw, 1999 (in Polish).
Awrejcewicz, J. and Krysko, V.A., Dynamics and Stability of Shells with Thermal Excitations, WNT, Warsaw, 1999 (in Polish).
Awrejcewicz, J. and Krysko, V.A., ‘3D theory versus 2D approximate theory of the free orthotropic (isotropic) plates and shells vibrations. Part 1. Derivation of governing equations and Part 2, Numerical algorithms and analysis’, J. Sound Vib. 226(5) (1999) 807–829, 831–871.
Awrejcewicz, J., Krysko, V.A. and Kutsemako, N., ‘Free vibrations of doubly curved in-plane non-homogeneous shells’, J. Sound Vib. 225(4) (1999) 701–722.
Awrejcewicz, J. and Krysko, V.A., ‘Vibration analysis of the plates and shells of moderate thickness’, J. Tech. Phys. 40(3) (1999) 277–305.
Awrejcewicz, J. and Krysko, V.A., Techniques and Methods of Plates and Shells Analysis, Łódź Technical University Press, Łódź, 1996.
Kirchhoff, G., Vorlesungen über Mathematische Physik. 1. Mechanik, 1976.
Love, A.E., A Treatise on Mathematical Theory of Elasticity, Cambridge University Press, Cambridge, 1927.
Pietraszkiewicz, W., ‘Geometrically nonlinear theories of thin elastic shells’, Adv. Mech. 12(1) (1989) 51–130.
Awrejcewicz, J., Asymptotic Methods — A New Perspective of Knowledge, Łódź Technical University Press, Łódź, 1995 (in Polish).
Awrejcewicz, J., Andrianov, I.V. and Manevich, L.I., Asymptotic Approach in Nonlinear Dynamics: New Trends and Applications, Springer-Verlag, Berlin, 1998.
Andrianov, I.V. and Awrejcewicz, J., ‘New trends in asymptotic approaches. Part 1. Summation and interpolation methods’, Appl. Mech. Rev. 54(1) (2001) 69–92.
Chang, S.I., Bajaj, A.K. and Davies, P., ‘Bifurcation and chaotic motions in resonantly excited structures’, in: Awrejcewicz, J. (ed.), Bifurcation and Chaos — Theory and Applications, Springer-Verlag, Berlin, 1995.
Awrejcewicz, J., ‘Strange nonlinear behaviour governed by a set of four averaged amplitude equations’, Meccanica 31 (1996) 347–361.
Chueshov, I.D., ‘Strong solutions and the attractors for von Kármán equations’, Math. USSR Sbornik 69 (1991) 25–36 (in Russian).
Chueshov, I.D., ‘Finite dimensionality of an attractor in some problems of nonlinear shell theory’, Math. USSR Sbornik 61 (1988) 419–428 (in Russian).
Favini, A., Horn, M.A., Lasiecka, I. and Tataru, D., ‘Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation’, Differ. Integral Eq. 9 (1996) 267–294.
Lasiecka, I. and Heyman, W.S., ‘Asymptotic behaviour of solutions in nonlinear dynamic elasticity’, Discret. Contin. Dyn. Syst. 1 (1995) 237–252.
Lasiecka, I., ‘Finite dimensionality of attractors associated with von Kármán plate equations and boundary damping’, J. Differ. Eq. 117 (1995) 357–389.
Lasiecka, I., ‘Local and global compact attractors arising in nonlinear elasticity’, J. Math. Anal. Appl. 196 (1995) 332–360.
Lasiecka, I. and Tataru, D., ‘Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping’, Differ. Integral Eq. 6 (1993) 507–533.
Lasiecka, I., ‘Finite dimensionality and compactness of attractors for von Kármán equations with nonlinear dissipation’, Nonlinear Differ. Eq. Appl. 6 (1999) 437–472.
Krysko, V.A., Awrejcewicz, J. and Bruk, V.M., ‘The existence and uniqueness of solution of one coupled thermomechanics problem’, J. Appl. Anal. 8(1) (2002) 129–139.
Krysko, V.A., Awrejcewicz, J. and Bruk, V., ‘On the solution of a coupled thermo-mechanical problem for non-homogeneous Timoshenko-type shells’, J. Math. Anal. Appl. 273(2) (2002) 409–416.
Lions, J., Quelques Méthods de Resolution des Problèmes aux Limite Non-linéaires, Dunod Gauthier-Villars, Paris, 1969.
Lions, J. and Magenes, E., Non-homogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972.
Kantorovich, L.V., ‘On one of direct methods of an approximate solution of the problem of minimum of a doubled integral’, Izv. AN SSSR, Math. Real Sci. 5 (1933) 647–652.
Vlasov, V.Z., ‘A new practical method of computations of sandwich coverings and shells’, Ind. Design 11, 12 (1932) 33–37, 21–26.
Kantorovich, L.V., ‘On convergence of the reduction method to ordinary differential equations’, Dokl. AN SSSR 30(7) (1941) 579–582.
Vlasovoy, Z.A., ‘On the method of reduction to ordinary differential equations’, Bull. Math. Inst. AN SSSR 53 (1959) 16–36.
Kirichenko, V.F. and Krysko, V.A., ‘On the problem of solution of non-linear boundary value problems using the Kantorovich—Vlasov method’, Differ. Eq. 16(12) (1980) 2186–2189.
Vorovich, I.I., ‘On direct method in theory of nonlinear shallow shells’, Appl. Math. Mech. 20(4) (1956) 449–474.
Michlin, S.G., Variational Methods in Mathematical Physics, Nauka, Moscow, 1970.
Kirichenko, V.E. and Krysko, V.A., ‘Method of variational iterations in the plates theory and its background’, Appl. Mech. 17(4) (1981) 71–76.
Ramberg, W., Pherson, A.E. and Levy, S., ‘Normal pressure tests of rectangular plates’, NACAT 849 (1942).
Krysko, V.A., Nonlinear Status and Dynamics of Non-homogeneous Shells, Saratov University Press, Saratov, 1976.
Krysko, V.A. and Krysko, A.V., ‘Bifurcation problems and elastic stability loss in nonlinear theory of plates’, in: Mechanics of Plates and Shells in XXI Century, Saratov, 1999 (in Russian).
Krysko, V.A., Vakhalaeva, T.V. and Krysko, A.V., ‘Complex symmetric oscillations and bifurcations of plates under longitudinal loads of different sign’, Trans. Nizhenogorod University Series Mechanics 2 (2000) 153–160 (in Russian).
Krysko, V.A., Vakhalaeva, T.V. and Krysko, A.V., ‘Dynamics and bifurcations of flexible elasto-plastic systems with nonuniform boundary conditions along edge under longitudinal loads of different sigh’, in: Current Problems of Mechanics of Shells. Proceedings of the International Conference, 2000, Kazan, pp. 284–289 (in Russian).
Awrejcewicz, J., Krysko, V.A. and Krysko, A.V., ‘Spatial-temporal chaos and solitons exhibited by von Kármán model’, Int. J. Bif. Chaos 12(7) (2002) 1465–1513.
Awrejcewicz, J. and Krysko, V.A., Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells, Springer-Verlag, Berlin, 2003.
Volmir, A.S., Stability of Deformable Systems, Nauka, Moscow, 1967.
Stoliarov, N.N. and Riabov, A.A., ‘Stability and post-critical behaviour of rectangular plates with variable thickness’, in: Investigation of Shells, Kazan, 1981, pp. 135–145.
Feigenbaum, M.J., ‘Universal behaviour in nonlinear systems’, Los Alamos Sci. 1 (1980) 4–27.
Pomeau, Y. and Manneville, P., ‘Intermittent transition to turbulence in dissipative dynamical systems’, Commun. Math. Phys. 74 (1980) 189–197.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Awrejcewicz, J., Krysko, V. & Krysko, A. Complex Parametric Vibrations of Flexible Rectangular Plates. Meccanica 39, 221–244 (2004). https://doi.org/10.1023/B:MECC.0000022845.52667.b0
Issue Date:
DOI: https://doi.org/10.1023/B:MECC.0000022845.52667.b0