Dynamics of flexible shells and Sharkovskiy's periodicity

Complex vibration of flexible elastic shells subjected to transversal and sign-changeable local load in the frame of nonlinear classical theory is studied. A transition from partial to ordinary differential equations is carried out using the higher-order Bubnov-Galerkin approach. Numerical analysis is performed applying theoretical background of nonlinear dynamics and qualitative theory of differential equations. Mainly the so-called Sharkovskiy periodicity is studied

An iterative algorithm for solution of contact problems of beams, plates and shells

An iterative algorithm to solve efficiently one-sided interaction between two rectangular plates within the Kirchhoff hypothesis is proposed. Then a proof of convergence of this algorithm is given. The formulated theorem, proof, and five remarks exhibit advantages of our proposed novel approach

Mathematical modeling of functionally graded porous geometrically nonlinear micro/nano cylindrical panels

Relevance. The study investigates the problem of stress-strain state and stability of porous functional-gradient size-dependent cylindrical panels. The composition and properties of alloys can differ and significantly affect the performance characteristics of products. Therefore, the research of material properties is relevant and contributes to the creation of new types of products demanded by the oil and gas industry. Aim. Development of a new model and creation of accurate methods for analyzing the stress-strain state of porous functional-gradient size-dependent micro/nano cylindrical panels taking into account geometrical nonlinearity. Methods. The method of variational iterations – the extended Kantorovich method is used to analyze the stress-strain state of cylindrical panels. The validity of the results is ensured by comparing the solutions obtained by the method of variational iterations in the first and second approximations with the solutions obtained by the authors, by the Bubnov–Galerkin method in higher approximations, by the finite difference method of the second order of accuracy, for which their convergence is investigated depending on a number of partitions of the integration area in the finite difference method and the number of series terms in the expansion of the basic functions in the Bubnov–Galerkin method. The results obtained by these methods are compared with the solutions obtained by other authors. It should be noted that the solutions obtained by the method of variational iterations for bending of functionally graded cylindrical panels under the action of transverse uniformly distributed load can be considered accurate. Results and conclusions. The authors have constructed the model of porous functional-gradient size-dependent cylindrical panels. Its use will allow studying the properties of alloys for producing drill pipes. The influence of material porosity type, porosity index, functional-gradient index, boundary conditions, size-dependent parameter, curvature parameters on the stress-strain state of cylindrical panels was analyzed using the developed method of variational iterations

Stability Improvement of Flexible Shallow Shells Using Neutron Radiation

Microelectromechanical systems (MEMS) are increasingly playing a significant role in the aviation industry and space exploration. Moreover, there is a need to study the neutron radiation effect on the MEMS structural members and the MEMS devices reliability in general. Experiments with MEMS structural members showed changes in their operation after exposure to neutron radiation. In this study, the neutron irradiation effect on the flexible MEMS resonators’ stability in the form of shallow rectangular shells is investigated. The theory of flexible rectangular shallow shells under the influence of both neutron irradiation and temperature field is developed. It consists of three components. First, the theory of flexible rectangular shallow shells under neutron radiation in temperature field was considered based on the Kirchhoff hypothesis and energetic Hamilton principle. Second, the theory of plasticity relaxation and cyclic loading were taken into account. Third, the Birger method of variable parameters was employed. The derived mathematical model was solved using both the finite difference method and the Bubnov–Galerkin method of higher approximations. It was established based on a few numeric examples that the irradiation direction of the MEMS structural members significantly affects the magnitude and shape of the plastic deformations’ distribution, as well as the forces magnitude in the shell middle surface, although qualitatively with the same deflection the diagrams of the main investigated functions were similar

A New Approach to Identifying an Arbitrary Number of Inclusions, Their Geometry and Location in the Structure Using Topological Optimization

In the present paper, a new approach to identifying an arbitrary number of inclusions, their geometry and their location in 2D and 3D structures using topological optimization was proposed. The new approach was based on the lack of initial information about the geometry of the inclusions and their location in the structure. The numerical solutions were obtained by the finite element method in combination with the method of moving asymptotes. The convergence of the finite element method at the coincidence of functions and their derivatives was analyzed. Results with an error of no more than 0.5%, i.e., almost exact solutions, were obtained. Identification at impact on the plate temperature and heat flux by solving the inverse problem of heat conduction was produced. Topological optimization for identifying an arbitrary number of inclusions, their geometry and their location in 2D problems was investigated

On the economical solution method for a system of linear algebraic equations

The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in ℝ3 is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error of O(hx12+hx22). The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported

MATHEMATICAL MODELING OF POROUS GEOMETRICALLY NONLINEAR METAL NANO-PLATES TAKING INTO ACCOUNT MOISTURE

Link for citation: Krysko A.V., Kalutsky L.A., Zakharova A.A., Krysko V.A. Mathematical modeling of porous geometrically nonlinear metal nano-plates taking into account moisture. Bulletin of the Tomsk Polytechnic University. Geo Аssets Engineering, 2023, vol. 334, no. 9, рр. 36-48. In Rus. The relevance.  The study of stress-strain behaviour and bearing capacity of porous metallic nanoplates, especially under extreme conditions and taking into account large deformations, is of great importance. These structures have a wide range of practical applications, for example, to clean solid fractions in wells during their setting up and to ensure fluid flow during operation. In addition, porous metal filters in the form of plates can serve as effective filters for removing solids from production wellbores, especially in the bottomhole zone. The versatility of these materials extends to various industries including aerospace, medical and instrumentation, indicating their potential to solve critical problems and advance technologies in various fields. The main aim of the research is to develop a new model of porous nanoplates, taking into account moisture, which would describe the real work of the studied objects in the oil and gas industry and other industries; to construct the efficient and fast methods for studying porous metallic nanoplates. Methods: variational iterations method, an extended Kantorovich method, which has high accuracy of solution of nonlinear problems and fast performance. The correctness of application of this method is conditioned by the proof of its convergence theorems belonging to the authors. In addition, the obtained solutions are compared with the solutions obtained by the Bubnov–Galerkin method in higher approximations and by the finite difference method of the second order of accuracy, as well as with the solutions obtained by other authors. Results. A model of porous flexible nanoplates is constructed taking into account moisture. Nano effects are described by the modified moment theory of elasticity. The method of variational iterations is further developed for the study of the stress-strain state of porous metallic nanoplates at large deflections. The paper analyzes the types of material porosity, size-dependent nano parameter, moisture distribution, porosity index and boundary conditions on the bearing capacity of porous metal plates. The type of porous material with the highest bearing capacity is identified

Mathematical models of higher orders: shells in temperature fields

This book offers a valuable methodological approach to the state-of-the-art of the classical plate/shell mathematical models, exemplifying the vast range of mathematical models of nonlinear dynamics and statics of continuous mechanical structural members. The main objective highlights the need for further study of the classical problem of shell dynamics consisting of mathematical modeling, derivation of nonlinear PDEs, and of finding their solutions based on the development of new and effective numerical techniques. The book is designed for a broad readership of graduate students in mechanical and civil engineering, applied mathematics, and physics, as well as to researchers and professionals interested in a rigorous and comprehensive study of modeling non-linear phenomena governed by PDEs

Quantifying Chaos by Various Computational Methods. Part 2: Vibrations of the Bernoulli–Euler Beam Subjected to Periodic and Colored Noise

In this part of the paper, the theory of nonlinear dynamics of flexible Euler–Bernoulli beams (the kinematic model of the first-order approximation) under transverse harmonic load and colored noise has been proposed. It has been shown that the introduced concept of phase transition allows for further generalization of the problem. The concept has been extended to a so-called noise-induced transition, which is a novel transition type exhibited by nonequilibrium systems embedded in a stochastic fluctuated medium, the properties of which depend on time and are influenced by external noise. Colored noise excitation of a structural system treated as a system with an infinite number of degrees of freedom has been studied

Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems

The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the classical problems governed by difference and differential equations (Hénon map, hyperchaotic Hénon map, logistic map, Rössler attractor, Lorenz attractor) and with the use of both Fourier spectra and Gauss wavelets. It has been shown that a modification of the neural network method makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyperchaos, and others. The aim of the comparison was to evaluate the considered algorithms, study their convergence, and also identify the most suitable algorithms for specific system types and objectives. Moreover, an algorithm of calculation of the spectrum of Lyapunov exponents based on a trained neural network has been proposed. It has been proven that the developed method yields good results for different types of systems and does not require a priori knowledge of the system equations