Bifurcations and Stability of Nonlinear Vibrations of a Three-Layer Composite Shell with Moderate Amplitudes
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 int...
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| Format: | Article |
| Language: | English |
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NAS of Ukraine, A. Pidhornyi Institute of Mechanical Engineering Problems
2023-06-01
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| Series: | Journal of Mechanical Engineering |
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| _version_ | 1827820991826362368 |
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| author | Kostiantyn V. Avramov Borys V. Uspenskyi Inna A. Urniaieva Ivan D. Breslavskyi |
| author_facet | Kostiantyn V. Avramov Borys V. Uspenskyi Inna A. Urniaieva Ivan D. Breslavskyi |
| author_sort | Kostiantyn V. Avramov |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-12T01:36:10Z |
| format | Article |
| id | doaj.art-5e10771ea18d4535adc20f6a7958a221 |
| institution | Directory Open Access Journal |
| issn | 2709-2984 2709-2992 |
| language | English |
| last_indexed | 2024-03-12T01:36:10Z |
| publishDate | 2023-06-01 |
| publisher | NAS of Ukraine, A. Pidhornyi Institute of Mechanical Engineering Problems |
| record_format | Article |
| series | Journal of Mechanical Engineering |
| spelling | doaj.art-5e10771ea18d4535adc20f6a7958a2212023-09-11T06:34:07ZengNAS of Ukraine, A. Pidhornyi Institute of Mechanical Engineering ProblemsJournal of Mechanical Engineering2709-29842709-29922023-06-0126261510.15407/pmach2023.02.006Bifurcations and Stability of Nonlinear Vibrations of a Three-Layer Composite Shell with Moderate AmplitudesKostiantyn V. Avramov0https://orcid.org/0000-0002-8740-693XBorys V. Uspenskyi1https://orcid.org/0000-0001-6360-7430Inna A. Urniaieva2https://orcid.org/0000-0001-9795-6954Ivan D. Breslavskyi3https://orcid.org/0000-0002-9666-9731Anatolii Pidhornyi Institute of Mechanical Engineering Problems of NAS of UkraineAnatolii Pidhornyi Institute of Mechanical Engineering Problems of NAS of UkraineKharkiv National University of Radio ElectronicsMcGill UniversityThe 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.shell of double curvatureadditive technologieshoneycomb structurebifurcation behavior |
| spellingShingle | Kostiantyn V. Avramov Borys V. Uspenskyi Inna A. Urniaieva Ivan D. Breslavskyi Bifurcations and Stability of Nonlinear Vibrations of a Three-Layer Composite Shell with Moderate Amplitudes Journal of Mechanical Engineering shell of double curvature additive technologies honeycomb structure bifurcation behavior |
| title | Bifurcations and Stability of Nonlinear Vibrations of a Three-Layer Composite Shell with Moderate Amplitudes |
| title_full | Bifurcations and Stability of Nonlinear Vibrations of a Three-Layer Composite Shell with Moderate Amplitudes |
| title_fullStr | Bifurcations and Stability of Nonlinear Vibrations of a Three-Layer Composite Shell with Moderate Amplitudes |
| title_full_unstemmed | Bifurcations and Stability of Nonlinear Vibrations of a Three-Layer Composite Shell with Moderate Amplitudes |
| title_short | Bifurcations and Stability of Nonlinear Vibrations of a Three-Layer Composite Shell with Moderate Amplitudes |
| title_sort | bifurcations and stability of nonlinear vibrations of a three layer composite shell with moderate amplitudes |
| topic | shell of double curvature additive technologies honeycomb structure bifurcation behavior |
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