Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)13400Research ArticleStability of Non-Linear Vibrations of Doubly Curved Shallow ShellsMukharlyamovR Grobgar@mail.ruAmabiliMmarco.amabili@mcgill.caGarzieraRrinaldo.garziera@gmail.comRiabovaK Sergeevnakseniia.riabova@studenti.unipr.itPeoples Friendship University of RussiaMcGill UniversityUniversity of Parma150220162536317092016Copyright © 2016,2016Large amplitude (geometrically non-linear) vibrations of doubly curved shallow shells with rectangular boundary, simply supported at the four edges and subjected to harmonicexcitation normal to the surface in the spectral neighbourhood of the fundamental mode are subject of investigation in this paper. The ﬁrst part of the study was presented by the authors in [M. Amabili et al. Nonlinear Vibrations of Doubly Curved Shallow Shells. Herald of Kazan Technological University, 2015, 18(6), 158-163, in Russian]. Two diﬀerent non-linear strain-displacement relationships, from the Donnell’s and Novozhilov’s shell theories, are used to calculate the elastic strain energy. In-plane inertia and geometricimperfections are taken into account. The solution is obtained by Lagrangian approach. The non-linear equations of motion are studied by using (i) a code based on arclengthcontinuation method that allows bifurcation analysis and (ii) direct time integration. Numerical results are compared to those available in the literature and convergence of the solution is shown. Interaction of modes having integer ratio between their natural frequencies, giving rise to internal resonances, is discussed. Shell stability under dynamic load is also investigated by using continuation method, bifurcation diagram from direct time integration and calculation of the Lyapunov exponents and Lyapunov dimension. Interesting phenomena such as (i) snap-through instability, (ii) subharmonic response, (iii) period doubling bifurcations and (iv) chaotic behavior have been observed.nonlinear vibrationsshallow shellsequationmotionstabilityнелинейные колебанияпологие оболочкиуравнениедвижениеустойчивость[Нелинейные колебания пологих оболочек двоякой кривизны / М. Амабили, Р. Гарзиера, Р. Мухарлямов, К. Рябова // Вестник КГТУ. - 2015. - Т. 18, № 6. - С. 158-163.][Amabili M. Comparison of Shell Theories for Large-Amplitude Vibrations of Circular Cylindrical Shells: Lagrangian Approach // Journal of Sound and Vibration. 2003. No 264. Pp. 1091-1125.][Amabili M. Nonlinear Vibrations of Circular Cylindrical Panels // Journal of Sound and Vibration. 2005. No 281. Pp. 509-535.][Amabili M. Nonlinear Vibrations of Rectangular Plates with Different Boundary Conditions: Theory and Experiments // Computers and Structures. 2004. No 82. Pp. 2587-2605.][Wolfram S. The Mathematica Book, 4th edition. Cambridge, UK: Cambridge University Press, 1999.][AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont) / E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, X. Wang. Montreal, Canada: Concordia University, 1998.][Argyris J., Faust G., Haase M. An Exploration of Chaos. Amsterdam: North-Holland, 1994.][Kobayashi Y., Leissa A. W. Large Amplitude Free Vibration of Thick Shallow Shells Supported by Shear Diaphragms // International Journal of Non-Linear Mechanics. 1995. No 30. Pp. 57-66.]