MODERN CONSTRUCTION SOLUTIONS FOR PRESTRESSED CABLE DOMES AND WAYS TO IMPROVE THEM

To create fundamentally new innovative large-span structures of buildings and structures coverings, modern design solutions of prestressed cable domes of the Tensegrity type are considered. The service life of the first built Tensigrity domes is only 35 years. These are fairly new, effective structures that require careful study and use of modern scientific approaches for their design using software systems, since their work under load and the construction process are quite complex. The design analysis and erection of self-stressed structures is based on the invention of an equilibrium structure, the so-called tensegrity form. The search for the shape is multidimensional and consists of the stage of computational analysis of a self-stressed dome for the equilibrium position of elements and their nodes, selection of the most stable and rigid structure, as well as taking into account possible unfavorable loads during operation and the initial load in the elements from the application of prestressing. To determine the shape of cable domes, a nonlinear programming problem with given axial forces is formulated, which can be considered as the problem of minimizing the difference in the total strain energy between the elements of the cables and struts under constraints on the compatibility conditions. The first step in calculating the prestressing of a cable dome is to assess the feasibility of its geometry. The possibility of forming a cable dome of negative Gaussian curvature is considered and a method for calculating the prestressing for this new shape is investigated. The proposed method is effective and accurate in determining the allowable prestressing for a cable dome with negative Gaussian curvature and can be used for other types of prestressed structures. The new directions for the development of effective constructive solutions for large-span coatings are presented, including a suspended-dome structure, which combines the advantages of a mesh shell and a cable dome. Special attention should be paid to experimental studies on models of tensegrity domes, the results of which demonstrate the positive and negative aspects of the behavior of structures under load, the process of their erection, as well as the possibility of control and restoration during operation.

Example of Bianchi Transformation of Kuen’s Surface

The work is devoted to the study of the Bianchi transformation for surfaces of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, and the pseudosphere (Beltrami surface). Surfaces of constant negative Gaussian curvature also include Kuen’s surface and the Dini’s surface. Studying the surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. Geometric characteristics of pseudospherical surfaces are found to be related to the theory of networks, the theory of solitons, nonlinear differential equations, and sin-Gordon equations. The sin-Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface. The Bianchi transformation for the Kuen’s surface is constructed using a mathematical software package.

NORMALIZATION OF CLASSICAL HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM

В работе исследовано семейство гамильтоновых систем с двумя степенями свободы. Расчетами сечений Пуанкаре показано, что при произвольных значениях параметров функции Гамильтона система является неинтегрируемой и в ней реализуется динамический хаос. Найдено, что для трех наборов параметров рассматриваемая система является интегрируемой, однако в одном интегрируемом случае при этих же значениях параметров на поверхности потенциальной энергии имеется область с отрицательной гауссовой кривизной, в то же время в двух других случаях интегрируемости при соответствующих значениях параметров областей с отрицательной гауссовой кривизной не имеется. Таким образом, наличие областей с отрицательной гауссовой кривизной на поверхности потенциальной энергии не достаточно для развития в системе глобального хаоса. Получена классическая нормальная форма для произвольных значений параметров. The family of the Hamiltonian systems with two degrees of freedom was investigated. The calculations of the Poincaré sections show that with arbitrary values of the parameters of the Hamilton function, the system is non-integrable and dynamic chaos is realized in it. For the three parameter sets, the system in question was found to be integrable, but shows that in one integrable case on the potential energy surface (PES) there are regions with the negative Gaussian curvature. It was found that in one integrable case for the same values of the parameters, the potential energy surface has a region with the negative Gaussian curvature. At the same time, in the other two cases, the domains with negative Gaussian curvature are not integrable for the corresponding values of the parameters. Thus, the presence of regions with negative Gaussian curvature on the potential energy surface is not enough for the development of the global chaos in the system. The classical normal form for arbitrary parameter values is obtained.

Transformation of Bianchi for Minding Top

The work is devoted to the study of the Bianchi transform for surfac­es of revolution of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, the pseudosphere (Beltrami surface). The study of surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. The connection of the geometric characteristics of pseudospherical surfaces with the theory of networks, with the theory of solitons, with nonlinear differential equations and sin-Gordon equations is established. The sin-Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface. The Bianchi transform for the Minding top is constructed. Using a mathematical package, Minding's top and its Bianchi transform are constructed.

Mechanics and dynamics of translocating MreB filaments on curved membranes

MreB is an actin homolog that is essential for coordinating the cell wall synthesis required for the rod shape of many bacteria. Previously we have shown that filaments of MreB bind to the curved membranes of bacteria and translocate in directions determined by principal membrane curvatures to create and reinforce the rod shape (Hussain et al., 2018). Here, in order to understand how MreB filament dynamics affects their cellular distribution, we model how MreB filaments bind and translocate on membranes with different geometries. We find that it is both energetically favorable and robust for filaments to bind and orient along directions of largest membrane curvature. Furthermore, significant localization to different membrane regions results from processive MreB motion in various geometries. These results demonstrate that the in vivo localization of MreB observed in many different experiments, including those examining negative Gaussian curvature, can arise from translocation dynamics alone.