Comparative analysis of dynamic stability of cylindrical and conical shells under periodic axial compression
DOI:
https://doi.org/10.32347/2410-2547.2023.110.344-352Keywords:
cylindrical and conical shell, periodic axial compresson, dynamic stability, method of curvilinear grids, parameter continuation method, Newton–Kantorovich methodAbstract
A comparative analysis of the dynamic stability of cylindrical and conical shells with the same geometric and mechanical characteristics under periodic uniformly distributed axial compression was presented. The study of the stability of steady periodic vibrations of thin elastic shells was based on the joint use of the method of curvilinear grids, the projection method and the parameter continuation method combined with the Newton–Kantorovich method. Geometrically nonlinear relations of the thin elastic shells theory are formulated on the basis of the vector approximation of the displacements function in the general curvilinear coordinate system in tensor form and satisfy the Kirchhoff-Love hypothesis. The discretization of the differential equations of the steady forced vibrations in the direction of the generating shells using the method of curvilinear grids was carried out. The components of the elements displacement vectors of the shells middle surface in the circular directionare approximated by trigonometric series. Reduction of the number of generalized coordinates of the discrete dynamic model of shells steady forced vibrations was performed by the Bubnov-Galerkin basis reduction method. A transition from vector ordinary differential equations to a nonlinear system of algebraic equations was made. The construction of a mathematical model of the dynamic stability of steady forced nonlinear vibrations of thin elastic shells was performed according to Floquet's theory using the projection method. The criterion for the loss of stability was the equality to zero of the determinant of the matrix of linearized equations of steady forced nonlinear vibrations of shells according to the Lyapunov theorem. A comparative analysis of frequencies and modes of natural vibrations of cylindrical and conical shells with the same geometric and mechanical characteristics and boundary conditions was performed. Nonlinear steady vibrations of the shells due to periodic axial compression were studied. The critical values of the dynamic load and the corresponding forms of loss of shell stability in the range of lower frequencies of their natural vibrations were obtained.
References
Volmir A.S. Ustojchivost deformirovanykh sistem [Stability of deformable systems]. –М.: Nauka, 1967, 984 s.
Grigolyuk Е.І., Kabanov V.V. Ustoichivost obolochek [Shell stability]. – М.: Nauka, 1978, 359 s.
Guliaev V.I., Bazhenov V.А., Gotsulyak Е.А., Dekhtyaruk Е.S., Lizunov P.P. Ustojchivost periodicheskih proczesov v nelinejnyh mekhanicheskih sistemah [Stability of periodic processes in the nonlinear mechanical systems]. Lviv, Vyschia shkola, 1983, 287 s.(rus).
Grigorenko Ya.M., Guliaev V.I. Nelyneinye zadachy teoryy obolochek y metody ykh reshenyia (obzor) [Nonlinear tasks of theory of shells and methods of their decision (review)] // Prykladnaia mekhanyka, 1991. − T. 27, №10, S. 3-23 s.(rus).
Gotsulyak Е.А., Zablotsky S.V., Kondakov G.S. and other. Kompleks programm dlya rascheta ustoychivosti i kolebaniy obolochek slozhnoy formy (REDBAZ) [Software for calculating the stability and vibrations of shells of complex shape ] (REDBAZ). – Kyiv: KNUBA, 1988.(rus).
Gaydaychuk V.V., Kirichuk А.А., Paliy О.М. Dynamika povzdovzhnikh kolyvan’ tonkoyi tsylindrychnoyi obolonky [Dynamics of longitudinal vibrations of a thin cylindrical shell] // Opir materialiv i teoriia sporud: nauk.-tekh. zbirn. – K.: KNUBA, 2007. – Vyp. 81, S. 51-56 (ukr).
Kirichuk А.А., Paliy О.М. Vplyv heometrychnykh kharakterystyk na stiykist’ ustalenykh kolyvan’ tsylindrychnykh obolonok [Influence of geometric characteristics on the stability of steady vibrations of cylindrical shells ]// Opir materialiv i teoriia sporud: nauk.-tekh. zbirn. – K.: KNUBA, 2008. – Vyp. 82, S. 102-110 (ukr).
Kirichuk А.А., Paliy О.М. Matematychna model’ parametrychnykh neliniynykh kolyvan’ tonkykh obolonok [Mathematical model of parametric nonlinear vibrations of thin shells] // Vistnyk HNTU. – Herson: HNTU, 2008. – Vyp. 2(31), S. 230-234. (ukr).
Lukianchenko O.О., Paliy О.М. Chyselne modeliuvannia stiikosti parametrychnykh kolyvan tonkostinnoi obolonky vidiemnoi hausovoi kryvyzny [Numeral design of vibrations stability of the thin-walled shell with negative гаусової curvature] // Opir materialiv i teoriia sporud: nauk.-tekh. zbirn., K.: KNUBA, 2018. – Vyp. 101, S. 45-59 (ukr).
Paliy О.М., Lukianchenko O.О. Chastotnyi analiz vidhuku hiperbolichnoho paraboloida na periodychne povzdovzhnie navantazhennia [Frequency analysis of response of hyperbolical paraboloid on the periodic longitudinal loading] // Opir materialiv i teoriia sporud: nauk.-tekh. zbirn. – K.: KNUBA, 2019. – Vyp. 102, S. 199-206 (ukr).
Lukianchenko O.O., Kostina O.V., Paliy O.M. Periodichni kolyvania obolonky rezervuaru z realnymy nedoskonalostiamy formy vid dii poverhnevogo tysku [Periodic vibrations of reservoir shell with the real shape imperfections under pressure] // Opir materialiv i teoriia sporud: nauk.-tekh. zbirn. – K.: KNUBA, 2022. – Vyp. 108, S. 255-266.(ukr).
Downloads
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
