The first researches of V. Bolotin and F. Weidenhammer of the elastic systems parametric vibrations in nonlinear formulation led to appearance of the general nonlinear theory of elastic systems parametric vibrations [1]. The special difficulties of researches in this area consisted in the compatible study of nonlinearity and stability of parametric vibrations. The stability problem of elastic systems parametric vibrations in the most existent researches to the study of a equations system with periodic coefficients was reduced. During a long time the Mathieu-Hill equation was the basic object of dynamic stability researches of elastic systems in linear formulation. To solve of stability problem of elastic systems parametric vibrations in nonlinear formulation were used the approximate methods of nonlinear mechanics, which took beginning from the method of amplitudes which change slowly. In the asymptotic method of theory of nonlinear mechanics, which was created by N. Krylov, N. Bogolyubov, Yu. Mitropolsky, the solution of equations system was presented as a sum of elements which decomposed on the degrees of a small parameter. To research of stability of elastic systems nonlinear vibrations the methods of the equivalent linearizing and harmonic balance were also used. The Hill’s method of the generalized determinants to determine the instability areas of elastic systems periodic parametric influence was widely used. The Flock theory, which was based on the Lyapunov theorems about of motion stability in the first approaching, got wide application in the researches of decisions stability of the nonlinear ordinary differential equations system.
Nowadays the nonlinear theory of the elastic systems parametric vibrations intensively develops due to realization of numeral approaches which are based on the modern software calculation procedures. But the influence problem of the shape imperfections on the elastic systems parametric vibrations and their stability remains not enough researched. Especially, it touches the thin-walled constructions on which act the parametric loadings. The numeral approach to analysis of influence of shape imperfections on the parametric vibrations of the thin-walled constructions is presented in this article. This approach is based on the methods of nonlinear and statistical mechanics with using the calculation procedures of finite-elemets analysis software NASTRAN and the specially program to create the different form and amplitude imperfections [2-5]. The numeral approach includes: computer finite-element design of the construction model with the shape imperfections; creating a reduced model of parametric vibrations with determination of reduced mass matrix, stiffness and geometrical stiffness matrixes, damping matrix; solving the free oscillation problem of the imperfect construction taking into account the static action of the parametric loading; solving the stability problem of the periodic parametric vibrations by Hill determinants, the stochastic vibrations − in relation to moment functions.
References
1. Schmidt, G. Parametererregte Schwingungen. Under Mitarbeit von R. Schulz. Mathematik fur naturwissenschaft und technik. Band 15.Berlin: VEB Deutscher Verlag der Wissenschaften. 313 S.
URL: http://zbmath.org/0305.70022
2. Bazhenov V.A., Lukianchenko O.O., Vorona Yu.V., Kostina O.V. Stability of the Parametric Vibrations of a Shell in the Form of a Hyperbolic Paraboloid // International Applied Mechanics, 2018. − 54(3). − P. 274-286.
URL: http://link.springer.com/article/10.1007/s10778-018-0880-4
3. Lukianchenko O.O. Application of stiffness rings for improving of operating reliability of the tank with shape imperfections // Strength of Materials and Theory of Structures: Scientificand-technical collected articles. − K.: KNUBA, 2020. − Issue 104. − P. 244-256.
URL: http://doi.org/10.32347/2410-2547.2020.104.242-254
4. Okhten І.О., Lukianchenko О.О. Some aspects of consideration of initial imperfections in the calculations of stability of thin-walled elements of open profile // Strength of Materials and Theory of Structures: Scientificand-technical collected articles. − K.: KNUBA, 2021. − Issue 106. − P. 122-128.
URL: http://doi.org/10.32347/2410-2547.2021.106.122-128
5. Bazhenov V.A., Lukianchenko O.O., Vorona Yu.V., Vabyshchevych O.M. The influence of shape imperfections on the stability of thin spherical shells // Strength of Materials, 2021. − Vol.53 (6). − P.842-851.
URL: http://link.springer.com/article/10.1007/s11223-022-00351-0
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