Nonlinear Dynamics : міжнародна конференція

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Algorithms and JavaScript Programs in Calculation of R-Functions and Producing Their Two- and Three-Dimensional Charts
(NTU "KhPI", 2016) Uvarov, Roman A.
Usage of HTML5 canvas element suitable for two- and three-dimensional charts of the function is unveiled. Appliance of JavaScript dynamics to this element is implemented. R-functions and level curves construction details are specified. Two- and three-dimension function charts are represented.
Analysis of Traveling and Standing Waves in the DNA Model by Peyrard-Bishop-Dauxois
(NTU "KhPI", 2016) Mikhlin, Yuri V.; Goloskubova, Natalia S.
The model by Peyrard - Bishop - Dauxois (the PBD model), which describes the DNA molecule nonlinear dynamics, is considered. This model represents two chains of rigid disks connected by nonlinear springs. An interaction between opposite disks of different chains is modeled by the Morse potential. Solutions of equations of motion are obtained analytically in two approximations of the small parameter method for two limit cases. The first one is the long-wavelength limit of traveling waves, when frequencies of vibrations are small. Dispersion relations are obtained also for the long-wavelength limit by the small parameter method. The second case is a limit of high frequency standing waves in the form of out-of-phase vibration modes. Two such out-of-phase modes are obtained; it is selected one of them, which has the larger frequency. In both cases systems of nonlinear ODEs are obtained. Nonlinear terms are presented by the Tailor series expansion, where terms up to third degree by displacement are saved. The analytical solutions are compared with checking numerical simulation obtained by the Runge - Kutta method of the 4-th order. The comparison shows a good exactness of these approximate analytical solutions. Stability of the standing localized modes is analyzed by the numerical-analytical approach, which is connected with the Lyapunov definition of stability.
Analytical Approximation of Periodic Ateb-Functions via Elementary Functions
(NTU "KhPI", 2016) Andrianov, Igor V.; Olevskyi, Victor I.; Olevska, Yuliia B.
Abstract We consider the problem of analytic approximation of periodic Ateb- functions, widely used in nonlinear dynamics. Ateb-functions are the result of the following procedure. Initial ODE contains only the inertial and non-linear terms. It can be integrated, which leads to an implicit solution. To obtain explicit solutions we are led to necessity to inverse incomplete Beta functions. As a result of this inversion we obtain the special Atebfunctions. Their properties are well known, but the use of Ateb- functions is difficult in practice. In this regard, the problem arises of the Ateb functions approximation by smooth elementary functions. For this purpose in the present article the asymptotic method is used with a quantity 1 / (a + 1) as a small parameter, were a > 1 — exponent of nonlinearity. We also investigated the analytical approximation of Ate-b functions' period. Comparison of simulation results, obtained by the approximate expression, with the results of numerical solution of the corresponding Cauchy problem shows their sufficient accuracy for practical purposes, even for a = 1.
Analyzing Parallel Computation of the Functions Created with R-operations in CUDA
(NTU "KhPI", 2016) Uvarov, Roman A.
Brief overview of the recent general tasks for parallel computation on graphics processing units is represented. Adequacy of the parallel computation approach for single analytic function constructed with R-operations is carried out. Brief overview of ray tracing technique and its connection with constructive apparatus of R-functions is considered. Sample comparison calculations to show the benefits of CUDA are provided.
Applicatin of R-functions Theory to Nonlinear Vibration Problems of Laminated Shallow Shells with Cutouts
(NTU "KhPI", 2016) Kurpa, Lidiya; Timchenko, Galina; Osetrov, Andrey
In present work an effective method to research geometrically nonlinear free vibrations of elements of thinwalled constructions that can be modeled as laminated shallow shells with complex planform is applied. The proposed method is based on joint use of R-functions theory, variational methods and Bubnov-Galerkin procedure. It allows reducing an initial nonlinear system of motion equations of a shallow shell to the Cauchy problem. The mathematical formulation of the problem is performed in a framework of the refined first-order theory. The appropriate software is created within POLE-RL program system for polynomial results and using C+ + programs for splines. New problems of linear and nonlinear vibrations of laminated shallow shells with cutouts are solved. To confirm reliability of the obtained results their comparison with the ones obtained using spline-approximation and known in literature is provided. Effect o f boundary condition on cutout is studied.
Application of Solution Structure Method to Modeling Dynamic Response of Mechanical Structures
(NTU "KhPI", 2016) Tsukanov, Igor
Transient nature of the loading conditions applied to the structural components makes dynamic analysis one of the important components in the design-analysis cycle. Time-varying forces and accelerations can substantially change stress distributions and cause a premature failure of the mechanical structures. In addition, it is also important to determine dynamic response of the structural elements to the frequency of the applied loads. In this paper we describe an application of the meshfree Solution Structure Method to the structural dynamics problems. Solution Structure Method is a meshfree method which enables construction of the solutions to the engineering problems that satisfy exactly all prescribed boundary conditions. This method is capable of using spatial meshes that do not conform to the shape of a geometric model. Instead of using the grid nodes to enforce boundary conditions, it employs distance fields to the geometric boundaries and combines them with the basis functions and prescribed boundary conditions at run time. This defines unprecedented geometric flexibility of the Solution Structure Method as well as the complete automation of the solution procedure.
Application of the R-Functions Method for Nonlinear Bending of Orthotropic Shallow Shells on an Elastic Foundation
(NTU "KhPI", 2016) Kurpa, Lidiya; Lyubitska, Kateryna
Geometrically nonlinear behavior of orthotropic shallow shells subjected to the transverse load and resting on Winkler’s foundation is investigated. On base of the R-function theory and variational methods problem's solution for shells with complex plan form is proposed. The algorithm to finding upper and lower critical loads is developed. The stress-strain state of shallow shells with the complex planform is investigated including different boundary conditions, properties of material and elastic foundation.
Application of the R-Functions Theory to Problems of Nonlinear Dynamics of Laminated Composite Shallow Shells and Plates: Review
(NTU "KhPI", 2016) Kurpa, Lidiya
A review of studies performed using the R-functions theory to solve problems of nonlinear dynamics of plates and shallow shells is presented. The systematization of results and studies for the problems of free and parametric vibrations and for problems of static and dynamic stability is fulfilled. Expansion of the developed original method of discretization for nonlinear movement equations on new classes of nonlinear problems is shown. These problems include researches of vibrations of antisymmetric laminated cylindrical and spherical panels; laminated composite shallow shells with variable thicknesss of the layers; functionally graded (FG) shallow shells and others. The basic issues that arise when using RFM are described. The future prospects of using the theory of R-functions for solving problems of nonlinear dynamics of plates and shallow shells with complex form are formulated. First of all this is an algorithms development and creation of the associated software to apply multi-modes approximations; improvement of approximation tools for nonlinear problems; investigation of the cracked functionally graded shallow shells; FG panels under thermal environments; parametric vibrations, static and dynamical stability of the multilayered and FG plates and shells.
Application of the Variational-Structural Method to Investigate the Elasto-Plastic Bending of Thin Shells and Plates
(NTU "KhPI", 2016) Morachkovska, Irina; Timchenko, Galina; Lyubitska, Kateryna
The effective method basing on theory of R-functions and variational structural method is developedfor solving non-linear boundary problems. Elastic-plastic bending of thin shallow shells is considered. The problems are reduced to finding stationary points of suggested mixed variational functionals according to the initial linearization due to usage of subsequent loading and Newton-Kantorovich jointly with method of varying elastic parameters. The method is used for automatic calculations in «POLE» programming system for investigations of shell structural elements. The numerical justification of the method is given. New laws of nonlinear deformation of shallow shells and plates with complex shapes in plane are established.
Application of Vibration Correlation Technique for Open Hole Cylinders
(NTU "KhPI", 2016) Skukis, Eduards; Kalnins, Kaspars; Ozolins, Olgerts
As non-destructive method for axial buckling load determination - Vibration Correlation Technique (VCT) showed major advantages for a range of industrial application. Particular technique for validation of structural limit state in accordance to numerical model prediction for large (true) scale structures are getting the required momentum. The Vibration Correlation Technique (VCT) allows to correlate the ultimate load or instability point with rapid decrement of self-frequency response. Nevertheless this technique is still under development for thin-walled shells and plates. The current research discusses an experimental verification of extended approach, applying vibration correlation technique, for the prediction of actual buckling loads on unstiffened isotropic cylindrical shells with circular cut-outs, loaded in axial compression. Validation study include several aluminium cylinders which were manufactured and repeatedly loaded up to instability point. In order to characterize a correlation with the applied load, several initial natural frequencies and mode shapes were measured during tests by 3D laser scanner. Results demonstrate that proposed vibration correlation technique allows one to predict the experimental buckling load with high reliability, by loading up to % of ultimate load. Additional experimental tests including geometric imperfections from initial manufacturing and postbuckling mode shape are currently under development to further validation ofproposed approach.
An Approximate Analytical Solution of Vibration Problem for Imperfect FGM Shallow Shells with Time Dependent Thickness under Static Loading
(NTU "KhPI", 2016) Gristchak, V. Z.; Fatieieva, Yu. A.
This paper deals with research of nonlinear vibration of imperfect shallow shells made of functionally-graded materials (FGM) under static and dynamic loadings. The material properties are changing in the thickness direction according to the given power law distribution and the non-linear strain-displacement relationships based on the von Karman theory for moderately large normal deflections. Initial nonlinear system of differential equations transforms to singular ordinary differential equations with variable in time coefficients, which is solved by hybrid perturbation and WKB- Galerkin methods in three steps. Comparison of numerical integration of initial equation and asproximate analytical solutions are given.
Asymptotic Solution of Anisotropic Cyclic Creep Problem
(NTU "KhPI", 2016) Breslavsky, Dmitry; Mietielov, Volodymyr; Morachkovsky, Oleg; Tatarinova, Oksana; Pashchenko, Sergey
Thin-walled structural elements made from rolled metal usually demonstrate anisotropic creep behavior. Very often the model of transversally-isotropic material is suitable for its description. Due to the difficulties in direct numerical integration the case of cyclic loading demands the development of suitable method of the creep problem’s solution when the material behavior isn’t isotropic. The general problem statement as well as the constitutive equations for two-dimensional creep problem are presented. Transversally-isotropic creep material model developed by O.Morachkovsky is used. The case of substantial stress values which are greater than yield limit is studied. Addition of cyclic loading, which is essentially varying the creep response, is analyzed. Deriving of resolving system of creep equations was performed by use of the method of asymptotic expansions jointly with the method of averaging in a period of stress cycle. This system allows simulation of only the problem of static loading with constitutive equations of special type, in which the values of cyclic parts of loading are included in so-called influence functions. These equations are derived from the general form by use of asymptotic expansions of creep strain functions with further averaging in a period of oscillations. Developed method is realized as an applied C+ + software. The Finite Element Method is used for solution of boundary-value problem jointly with Finite Difference Scheme for initial one. The results of experimental investigations of creep in specimens and plates with holes made from rolled steel are discussed. The anisotropic creep curves in three directions were obtained and constants for creep flow rule were determined. Additional number of experiments was performed for the case of cyclic loading. The comparison between experimental and numerical data shows the satisfactory agreement. Due to this a number of numerical examples for cyclic creep simulation in thin plates were performed and their results are discussed.
Atomic Functions: the History of the Formation, Development and Practical Application
(NTU "KhPI", 2016) Kolodyazhny, V. M.
Atomic functions are infinitely differentiable compactly supported solutions of functional differential equations of a special type. After the first successful building of the functions performed by VL Rvachev and VA Rvachev in the 70s of the previous century, different classes of the atomic functions of one and several variables were studied, which have found application in the solution of various problems of mathematical analysis and mathematical modeling of practical problems. Generalization of atomic functions to the case of several variables associated with the expansion of their possible application to solving boundary value problems in partial derivatives had been considered, in particular, and the development of new methods for the numerical solution of such tasks. Mathematical tools based on atomic functions of several variables have the necessary properties of universality and locality, to be requested in the practice of numerical solutions of boundary value problems. The study of functional differential equations, which are used for their formation other differential operators, fo rexample, Laplace, Helmholtz, biharmonic operators et al., leads to the construction of the special form of atomic functions. The atomic functions form the classes radial basis functions that allow you to develop on their basis meshless scheme of solving boundary value problems. In comparison with the known radial basis functions atomic radial basis functions have advantages, namely, are infinitely smooth, satisfy the functional-differential equation, effectively computable, have explicit formulas for the calculation of the Fourier transform.
Breather Modes Induced by Localized RF Radiation: Analytical and Numerical Approaches
(NTU "KhPI", 2016) Belan, Victor I.; Kovalev, Alexander S.; Peretyatko, Anastasii A.
Numerical computations and collective variables approach are applied to analytical and numerical study of spatially localized excitations of one-dimensional magnetic system in external high-frequency magnetic field. It is demonstrated the hysteresis character of dependence for amplitude of local soliton-like states on external field magnitude. The system shows a variety of interesting nonlinear phenomena such as periodicity doubling and chaos.
Bursting Oscillations in Nonlinear Oscillators With Slowly Varying Excitation
(NTU "KhPI", 2016) Rakaric, Zvonko; Kovacic, Ivana
This work represents the extension of a recently published paper by the authors, where mechanical manifestations of bursting oscillations in slowly rotating systems are presented. One of these mechanical systems consists of a mass moving along a slowly rotating frame, which creates a slowly varying external excitation. Here, the aim is to link its mathematical model with Izhikevich’s methodology and classification for bursting oscillations. In this respect, the existence of a ’fold – fold’ hysteresis loop of a point-point type burster is recognised and explained in detail. Thus, in the system considered, bursting oscillations occur becouse the rate of convergence between the branches of the slow flow is relatively weak. The influence of some system’s parametrs on the characteristics of fast motion is illustrated.
Bushes of Nonlinear Normal Modes in Single-Layer Graphene
(NTU "KhPI", 2016) Chechin, G. M.; Ryabov, D. S.; Shcherbinin, S. A.
In-plane vibrations in uniformly stretched single-layer graphene (space group P6mm), which are described by the Rosenberg nonlinear normal modes (NNMs) and their bushes, are studied with the aid of group-theoretical methods developed by authors in some earlier papers. It was found that only 4 symmetry-determined NNMs (one-dimensional bushes), as well as 14 two-dimensional bushes are possible in graphene. They are exact solutions to the dynamical equations of this two-dimensional crystal. The verification of group-theoretical results with the aid of ab initio simulations based on density functional theory is discussed.
Circular Cylindrical Shell Made of Neo-Hookean-Fung Hyperelastic Material Under Static and Dynamic Pressure
(NTU "KhPI", 2016) Breslavsky, Ivan; Amabili, Marco; Legrand, Mathias
The present study is devoted to the investigation of static and dynamic behavior of the three-layered composite shell made of hyperelastic material. Such a shell can be considered as a model of human aorta. Since soft biological materials are essentially nonlinear even in the elasticity zone, not only geometrical, but also physical nonlinearity should be taken into account. The physical nonlinearity of soft biological tissues is usually modeled by certain hyperelastic law. The law chosen for this study is the combination of the Neo- Hookean law, which describes the isotropic response at small strains, and Fung exponential law, that models the stiff anisotropic response of the collagen fibers at larger strains. Each of three shell layers has its own hyperelastic constants set. These constants are determined basing on experiential data [1]. The straindeflection relations are modeled with higher-order shear deformation theory [2]. Initially, the shell is preloaded with static pressure. Since the defection in our study is large we use the expression for pressure as a follower load [3]. The static problem is solved with the help of the local models method [4]. Afterwards, the free and forced dynamical response of the preloaded shell is studied both in vacuo and with still fluid inside. The modes of interest are the first axisymmetric mode and mode with two half-waves in circumferential direction (so-called collapse mode). It is found that static pressure decreases the dynamic nonlinearity and it is quite weak. At the same time, the presence of fluid makes the softening nonlinearity stronger as in case of shells of conventional material [5].
Complicated Scenarios of Transitions to Deterministic Chaos in Non-Ideal Dynamic Systems
(NTU "KhPI", 2016) Shvets, Aleksandr Yu.; Sirenko, Vasiliy
Some of non-ideal dynamic systems are considered. It is discovered and described the complicated transition scenarios from regular to chaotic regimes and transitions between different types of chaotic regimes. Described the transition to chaos, which begins by the Feigenbaum scenario, and ends by intermittency. Also discusses the scenario of intermittency with several laminar phases and one turbulent phase. It is discovered and described transitions "hyperchaotic attractor of one type – hyperchaotic attractor of another type," which are realized according to the scenario of a generalized intermittency, but only with two rough-laminar phases.
Construction of Nonlinear Normal Modes by Shaw-Pierre via Schur Decomposition
(NTU "KhPI", 2016) Perepelkin, Nikolay V.
In the paper the simplification of construction of nonlinear normal vibration modes by Shaw-Pierre in power series form is considered. The simplification can be obtained via change of variables in the equations o f motion of dynamical system under consideration. This change of variables is constructed by means of so-called ordered Schur matrix decomposition. As the result of the transformation there is no need in solving nonlinear algebraic equations in order to evaluate coefficients of nonlinear normal mode.
Developing Structural Methods For Solving Boundary-Value Problems In Non-Smooth Boundary Domains
(NTU "KhPI", 2016) Baranov, Igor; Kravchenko, Oleg; Suvorova, Iryna
The paper presents the development of structural methods for solving boundary-value problems in complexshape domains to enhance calculation accuracy in the neighbourhood of angular points in the boundary-value problems solution domain. Structural methods allow building bases for solving mathematical physics boundary-value problems, which accurately account for the boundary conditions and geometric information on the domain form. These methods are based on using the mathematical tools of the theory of R-functions. They can dramatically extend the potentialities of variation methods when solving mathematical physics boundary-value problems in complexshape domains with different boundary conditions. The most common systems of R-operations used in practice are normalised; however, they are not smooth in point (0,0), and all smooth R-operations are not normalised. The paper presents the results of investigating the behaviour of smooth functions up to the domain boundary, which satisfy uniform Dirichlet and Neumann conditions, and the condition at which the function proper and its derivatives over the normal to a definite order are equal to zero. New approaches are offered to build basis functions that are smooth up to the non-smooth domain boundary and which meet the above-mentioned boundary conditions. The suggested new system of asymptotically normalised R-operations whose functions belong to the given smoothness class can be used to build smooth basis functions that satisfy certain boundary conditions. The approaches developed were tested for model problems, some of which were used for problems in modelling hydrodynamic fields in complex-shape domains.