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.
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