Bifurcations and Stability of Nonlinear Vibrations of a Three-Layer Composite Shell with Moderate Amplitudes

Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher Anatolii Pidhornyi Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 26, no. 2, 2023 (June)
Pages 6-15
Cited by J. of Mech. Eng., 2023, vol. 26, no. 2, pp. 6-15



Kostiantyn V. Avramov, Anatolii Pidhornyi Institute of Mechanical Engineering Problems of NAS of Ukraine (2/10, Pozharskyi str., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0002-8740-693X

Borys V. Uspenskyi, Anatolii Pidhornyi Institute of Mechanical Engineering Problems of NAS of Ukraine (2/10, Pozharskyi str., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0001-6360-7430

Inna A. Urniaieva, Kharkiv National University of Radio Electronics (14, Nauky ave., Kharkiv, 61166, Ukraine), e-mail:, ORCID: 0000-0001-9795-6954

Ivan D. Breslavskyi, McGill University (817 Rue Sherbrooke O #270, Montréal, QC H3A 0C3, Canada), ORCID: 0000-0002-9666-9731



The authors derived a mathematical model of geometrically nonlinear vibrations of three-layer shells, which describes the vibrations of the structure with amplitudes comparable to its thickness. The high-order shear theory is used in the derivation of this model. Rotational inertia is also taken into account. At the same time, the middle layer is a honeycomb structure made thanks to additive FDM technologies. In addition, each shell layer is described by five variables (three displacement projections and two rotation angles of the normal to the middle surface). The total number of unknown variables is fifteen. To obtain a model of nonlinear vibrations of the structure, the method of given forms is used. The potential energy, which takes into account the quadratic, cubic, and fourth powers of the generalized displacements of the structure, is derived. All generalized displacements are decomposed by generalized coordinates and eigenforms, which are recognized as basic functions. It is proved that the mathematical model of shell vibrations is a system of nonlinear non-autonomous ordinary differential equations. A numerical procedure is used to study nonlinear periodic vibrations and their bifurcations, which is a combination of the continuation method and the shooting method. The shooting method takes into account periodicity conditions expressed by a system of nonlinear algebraic equations with respect to the initial conditions of periodic vibrations. These equations are solved using Newton’s method. The properties of nonlinear periodic vibrations and their bifurcations in the area of subharmonic resonances are numerically studied. Stable subharmonic vibrations of the second order, which undergo a saddle-node bifurcation, are revealed. An infinite sequence of bifurcations leading to chaotic vibrations is not detected.


Keywords: shell of double curvature, additive technologies, honeycomb structure, bifurcation behavior.


Full text: Download in PDF



  1. Derevianko, I., Uspensky, B., Avramov, K., Salenko, A., & Maksymenko-Sheiko, K. (2023). Experimental and numerical analysis of mechanical characteristics of fused deposition processed honeycomb fabricated from PLA or ULTEM 9085. Journal of Sandwich Structures and Materials, vol. 25, iss. 2, pp. 264–283.
  2. Uspensky, B., Derevianko, I., Avramov, K., Polishchuk, O., & Salenko, A. (2022). Experimental and numerical study on fatigue of sandwich plates with honeycomb core manufactured by fused deposition modeling. Applied Composite Materials, vol. 29 (5), pp. 2033–2061.
  3. Matthews, N. Additive metal technologies for aerospace sustainment. (2018). Aircraft Sustainment and Repair, pp. 845–862.
  4. Boparai, K. S. & Singh, R. (2017). Advances in fused deposition modeling. Reference Module in Materials Science and Materials Engineering.
  5. Wilkins, D. J., Bert, C. W., & Egle, D. M. (1970). Free vibrations of orthotropic sandwich conical shells with various boundary conditions. Journal of Sound and Vibration, vol. 13, iss. 2, pp. 211–228.
  6. Kanematsu, H. H. & Hirano, Y. (1988). Bending and vibration of CFRP – faced rectangular sandwich plates. Computers and Structures, vol. 10, iss. 2, pp. 145–163.
  7. Frostig, Y., Baruch, M., Vilnay, O., & Sheinman, I. (2000). High-order theory for sandwich beam with transversely flexible core. ASCE Journal of Engineering Mechanics, vol. 118, iss. 5, pp. 1026–1043.
  8. Malekzadeh, K., Khalili, M. R., & Mittal, R. K. (2005). Local and global damped vibrations of plates with a viscoelastic soft flexible core: An improved high-order approach. Journal of Sandwich Structures and Materials, vol. 7, iss. 5, pp. 431–456.
  9. Frostig, Y. & Thomsen, O. T. (2004). High-order free vibration of sandwich panels with a flexible core. International Journal of Solids and Structures, vol. 41, iss. 5–6, pp. 1697–1724.
  10. Hohe, J., Librescu, L., & Oh, S. Y. (2006). Dynamic buckling of flat and curved sandwich panels with transversely compressible core. Computers and Structures, vol. 74, iss. 1, pp. 10–24.
  11. Catapano, A. & Montemurro, M. (2014). A multi-scale approach for the optimum design of sandwich plates with honeycomb core. Part I: homogenisation of core properties. Composite Structures, vol. 118, pp. 664–676.
  12. Reddy, J. N. (1984). A simple higher-order theory for laminated composite plates. ASME Journal of Applied Mechanics, vol. 51, iss. 4, pp. 745–752.
  13. Amabili, M. (2018). Nonlinear mechanics of shells and plates in composite, soft and biological materials. Cambridge: Cambridge University Press, 568 p.
  14. Meirovitch, L. (1970). Fundamentals of vibrations. New York: McGraw Hill Higher Education, 826 p.
  15. Parker, T. S. & Chua, L. O. (1989). Practical numerical algorithms for chaotic systems. New York: Springer, 348 p.
  16. Seydel, R. (1991). Tutorial on continuation. International Journal of Bifurcation and Chaos, vol. 1, no. 1, pp. 3–11.
  17. Seydel, R. (1997). Nonlinear computation. International Journal of Bifurcation and Chaos, vol. 7, no. 9, pp. 2105–2126.
  18. Doedel, E., Keller, H. B., & Kernevez, J. P. (1991). Numerical analysis and control of bifurcation problems (I) Bifurcation in finite dimensions. International Journal of Bifurcation and Chaos, vol. 1, no. 3, pp. 493–520.
  19. Avramov, K. (2016). Bifurcation behavior of steady vibrations of cantilever plates with geometrical nonlinearities interacting with three-dimensional inviscid potential flow. Journal of Vibration and Control, vol. 22, iss. 5, pp. 1198–1216.
  20. Avramov, К. & Raimberdiyev, T. (2017). Bifurcations behavior of bending vibrations of beams with two breathing cracks. Engineering Fracture Mechanics, vol. 178, pp. 22–38.


Received 27 March 2023

Published 30 June 2023