Chaotic behaviour of dynamical systems, their routes to chaos, and the intermittency are interesting and investigated subjects in nonlinear dynamics. The studying of these phenomena in nonsmooth dynamical systems is of the special scientists' interest. In this paper we apply relatively young mathematical tool -continuous wavelet transform CWT -for investigating the chaotic behavior and intermittency in particular in strongly nonlinear non-smooth discontinuous 2-DOF vibroimpact system. We show that CWT applying allows to detect and determine the chaotic motion and the intermittency with great confidence and reliability, gives the possibility to demonstrate route to chaos via intermittency, to distinguish and analyze the laminar and turbulent phases.
Ensuring the reform of today's primary school provides for a certain removal of restrictions in pedagogical activity, providing teachers with freedom in interpreting educational programs, using forms and methods of working with younger students. This requires from elementary school teachers not only deep knowledge, possession of a set of relevant professional skills and abilities, but also an orientation towards pedagogical creativity, towards the needs of the student in the educational process, an understanding of their own responsibility for the results obtained, the ability to act effectively in conditions of academic freedom and decentralization of the school. In modern conditions of education reform, the relevance of the problem under study is explained by a number of factors. Therefore, we consider the solution of certain social and educational problems to be an important component of the activity of a primary school teacher.
Виконано чисельне моделювання стійкості параметричних коливань високої тонкостінної оболонки виду гіперболічного параболоїда при зовнішньому поверхневому тиску та осьовому стисканні. Редуковані матриці мас, демпфірування, жорсткості і геометричної жорсткості оболонки сформовані за допомогою процедур програмного комплексу скінченноелементного аналізу. Розв'язані задачі нелінійної статики модифікованим методом Ньютона-Рафсона та стійкості методом Ланцоша при дії статичної складової параметричного навантаження двох видів. Виконано модальний аналіз оболонки в лінійній постановці без урахування навантаження методом Ланцоша і в нелінійній постановці для визначення власних частот і форм коливань оболонки, яка навантажена статичною складовою параметричного навантаження двох видів. При формуванні редукованих моделей стійкості параметричних коливань оболонки при різних видах навантаження враховані особливості її статичної та динамічної поведінки. Ключові слова: параметричні коливання, динамічна стійкість, метод скінченних елементів, висока тонкостінна оболонка, гіперболоїд.
The nonlinear dynamic analysis of imperfect reservoir shell with a variable thickness of wall under pressure was executed. The finite-element model of reservoir in the form of a cylindrical shell in the software NASTRAN was built. The shell wall in the form of the three-cornered finite-element net was presented. Shape imperfection as a lower buckling form of perfect shell (Buckling) was modelled. Value of amplitude of imperfection was set proportionally to a minimum thickness of shell wall. The limits on the radial and tangential displacements of top edge nodes were entered, the nodes of lower edge were fastened. Excitation as external pressure, which linearly depended on time and uniform distributed on all shell elements was presented. The modal analysis of shell with modelled shape imperfections by using computational procedure of task on natural vibrations (Normal Modes) by the Lanczos method was executed. The nonlinear dynamic analysis (Nonlinear Direct Transient) of imperfect reservoir shell under pressure by N’yumark method was executed. Influence of amplitude of modelled imperfection on the shell stress-strain for different time intervals of excitation, the conditionally critical values of dynamic loading and corresponding of shell deformation forms were investigated. It was discovered that a modelled shell shape imperfection as a lower buckling form of perfect shell under static pressure in the dynamic analysis of shell under the same type of the loading was effeсtive. Influence of modelled shape imperfections amplitude on the stress-strain state of shell for different time interval of excitation, the conditionally critical values of dynamic loading and appropriate forms of shell deformation was considerable. Presented imperfection model in the modal analysis of shell was not effective. The increase of amplitude of shell imperfection led to insignificant decrease of natural frequencies and amplitudes of appropriate natural forms with the same amount of the semiwaves in the circular direction. In our opinion presented model of shell shape imperfection can be effective in the modal analysis of shell with the stress-strain state from the previous action of static pressure and for the estimation of design reliability of reservoir shell in the case of the dynamic loadings using the Bolotin probabilistic approach.
Performed analysis of the initial geometric imperfections influence on the stability of the open C-shaped bars. Test tasks were solved in MSC Nastran, which is based on the finite element method. Imperfections are given in different formulations: the general stability loss of an ideal bar, of wavy bulging of walls and shelves, of deplanation of a bar. To model imperfections, has been developed a program which for the formation of new coordinates of the nodes of the "deformed" model, the components of a vector similar to the form of stability loss are added to the corresponding coordinates of the middle surface of the bar. In this way, you can set initial imperfections in the forms of stability loss of the bar with different amplitude. Researches made with different values of the imperfection amplitude and eccentricity of applied efforts. All tasks are performed in linear and nonlinear staging. The conclusion is made regarding the influence of initial imperfections form and imperfection amplitude on the critical force in nonlinear calculations. It was found that the most affected are imperfections, which are given in the form of total loss of stability. It was revealed the influence of the imperfection amplitude on the magnitude of the critical force for such imperfections. The influence of imperfections amplitude given in the form of wavy bulging walls and in the form of deplanations is not affected on the value of the critical force.
The numeral approach to analysis of influence the real shape imperfections on natural and forced periodic vibrations of reservoir shell under pressure was presented. The approach was based on a theory and the methods of structural mechanics and calculable procedures of finite-element software complex NASTRAN. The welded steel reservoir with a variable thickness along its height was investigated. On the manufacted stage, transporting and exploitating the defects of wall shape were formed. The actual radial and angular deviations of the reservoir belts as a result measurements by theodolite were got. Geometry of shell with the real shape imperfections by spline curves and spline surfaces with adding of factual radial and angular rejections to the proper coordinates of points formative shells with an ideal surface was built. Model of reservoir in the form of a cylindrical shell with a three-cornered finite-element net was presented. The limits on the radial and tangential displacements of top edge nodes were entered, the nodes of lower edge were fastened. The periodic loading as external pressure was given. The modal analysis of shell without and with real shape imperfections by the decision of task on natural vibrations (Normal Modes) by the Lanczos method was executed. It was discovered that the natural forms of shell with real shape imperfections had local deformations in the places of maximal radial deviations from a vertical line unlike regular deformations in all natural forms of ideal shell. Natural frequencies were decreased and were slit and there wasn’t influence on its dense distribution due to the presence of shell shape imperfections. A transient and forced periodic vibrations of shell under pressure were investigated. Analysis of transient was executed by the method of direct integration (Direct Transient). The presence of shape imperfections in the shell wall influenced the increase of the transient duration and the dynamic coefficient The direct frequency analysis of shell response on periodic loading (Direct Frequency) was executed taking into account the proper ten first forms and frequencies of natural vibrations. Significant influence of the shell shape imperfections on the resonant frequencies of the forced periodic vibrations and corresponding form of deformation was discovered.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.