Nonlinear Dynamics–2016

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https://repository.kpi.kharkov.ua/handle/KhPI-Press/24522

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Perturbed Rotations of a Rigid Body Close to the Lagrange Case under the Action of Unsteady Perturbation Torques
(NTU "KhPI", 2016) Akulenko, L. D.; Kozachenko, T. A.; Leshchenko, D. D.; Zinkevich, Ya. S.
Perturbed rotations of a rigid body close to the Lagrange case under the action of perturbation torques slowly varying in time are investigated. Conditions are presented for the possibility of averaging the equations of motion with respect to the nutation angle and the averaged system of equations of motion is obtained. In the case of the rotational motion of the body in the linear-dissipative medium the numerical integration of the averaged system of equations is conducted.
Taking Account of Nonlinear Properties of Subsystems in Problems of Dynamic Interaction of Structures with Loads, Bases and Flows
(NTU "KhPI", 2016) Kulyabko, Vladimir; Chaban, Vyacheslav; Makarov, Andrey; Yaroshenko, Denis
This paper describes the additional features of discrete models of various DOF systems to solve nonlinear dynamical problems of complex-compound buildings and structures including elements of significant flexibility (bridges, pylons and supports of power transmission lines, pipe line crossings, guyed masts etc.). Qualitative and quantitative differences between linear solutions (which are popular among FEM designers) and nonlinear solutions (depending on geometrical, physical and constructive nonlinearities) are discussed. It is analyzed the time-history models of different combinations of mention structures with the adjacent subsystems, damping devices (well-known and the most-recently-used), static and dynamic loads and effects (including moving loads). There is also presented experimental and theoretical approach of damages determination for rod elements in the spatial structure by the dynamic diagnosis method (e.g. for bridges crane with the big span).
Some Stationary Deformation Problems for Compound Shells of Revolution
(NTU "KhPI", 2016) Grigorenko, Yaroslav; Bespalova, Elena; Yaremchenko, Natalia
A common approach to solving stationary deformation problems for compound systems composed of shells of revolution with different geometry and structure is developed. The approach is based on the use of shell models with different level of rigor and of the general numerical-analytical technique for solving corresponding problems. The examples of studying the subcritical stress-strain state, vibrations, and dynamical instability of complex form systems are presented, features of their deformation are noted.
Numerical Analysis of Stress-Strain State of Orthotropic Plates in the Form of Arbitrary Convex Quadrangle
(NTU "KhPI", 2016) Grigorenko, A. Ya.; Pankratiev, S. A.; Yaremchenko, S. N.
A numerical and analytical approach to solving problems of the stress-strain state of quadrangular orthotropic plates of complex shape has been proposed. Two-dimensional boundary value problem was solved using spline collocation and discrete orthogonalization methods after applying the appropriate domain transform. The influence of geometric shape of plate in different cases of boundary conditions on the displacement and stress fields is considered according to the refined theory. The results were compared with available data from other authors.
Nonlinear Analysis of Gas Flow in Compressors Stage Based on CFD-Method
(NTU "KhPI", 2016) Karpik, A.; Vorobiev, Yu.
The numerical simulation of a three-dimensional viscous flow in cascade of the axial compressor of low pressure of the gas-turbine engine is presented. The results of a flow in the first stage o f the compressor in nonstationary three-dimensional statement are obtained in the solver F. Velocity and pressure fields are received as a result.
Nonlinear Sonic Vacua
(NTU "KhPI", 2016) Vakakis, Alexander
We will present recent results on a special class of dynamical systems designated as nonlinear sonic vacua. These systems are non-linearizable, and have zero speed of sound (in the sense of classical acoustics). Accordingly, their dynamics and acoustics are highly degenerate and tunable with energy, enabling new and highly complex nonlinear phenomena. Two examples of sonic vacua will be discussed. The first is uncompressed ordered granular media, which, depending on their local state, behave either as strongly nonlinear and non-smooth dynamical systems (in the absence of strong local compression), or as almost linear coupled oscillators (under strong local compression, e.g., in the primary fronts of propagating solitary pulses) [1,2]. The second example concerns a spring-mass lattice in the plane. In the small energy limit this seemingly simple system is 'transformed' by geometric nonlinearity to a nonlinear sonic vacuum with surprising properties, such as strong nonlocality (despite o f only next-neighbor interactions in the lattice!), orthogonal nonlinear normal modes, and accelerating propagating fronts [3,4]. Interesting applications of nonlinear sonic vacua will be discussed, including intense energy cascading from low-to-high frequencies and long-to-short wavelengths, resembling “mechanical turbulence ”. We will discuss the implications of these findings on the design of dynamical systems for predictive passive energy management.
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.
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.
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.
Parameter Analysis of Vibroimpact Machines Dynamics With Variable Mass and Stiffness
(NTU "KhPI", 2016) Tkachuk, M. M.; Kostenko, Iurii; Grabovsky, Andriy; Tkachuk, M. A.
Vibroimpact machines operate under high repeated loadings. This poses certain requirements with regards to the strength and durability of their components that have to be met by the design. In order to predict the magnitude and time distribution of the acting forces the dynamics of the vibroimpact systems need to be studied. A two-body mechanical system is considered as a model of a shake-out machine designed for Azovmash company and used for extraction of steel casts from the mold. The first body is the suspended frame with the shake-out grid driven by an unbalanced vibrator, while the second one is the cast. Its sand mold is damaged at every impact with the shake-out grid, which results in the gradual loss of mass. The previously developed rigid body dynamics model is extended in order to account for the variable mass factor. Two approaches for the time evolution of the cast mass has been taken. The first one suggests that mass is a predefined linear or piecewise linear function of time. Alternatively the mass is treated as an unknown variable and was determined in the course of solving the equations of motion. A constitutive law for the mass reduction based on the energy dissipated at each impact is proposed. It has been shown that this model results in more adequate description of the shake-out process compared to the fixed-law mass evolution. In addition to the variable mass the influence of stiffness characteristics has been investigated. Nonlinear double springs with variable stiffness and length difference suspending the shake-out platform are considered. The survey on the combined effect of mass and stiffness parameters on the dynamics of the modelled shake-out machine allowed to determine the loads sustained by its structural elements and to make the rational design with the required strength. In particular, it has been shown how to detune the machine from resonance frequencies, in particular from the discovered dangerous subharmonic regimes due to variable mass and stiffness.
Liquid Nonlinear Oscillations in the U-Tube System
(NTU "KhPI", 2016) Kazachkov, A. R.; Lykah, V. A.; Minakova, K. A.; Syrkin, E. S.; Tkachenko, O. Y.
Dynamics of oscillation processes in a siphon U-tube is studied for the system of connected vessels filled with homogeneous liquid. The equations and phase paths describing the motion of viscous and non-viscous liquids are given, oscillation frequencies are considered. Oscillations are nonlinear in general case, but they turn into linear by setting specific parameter values of the system. Phase portraits are obtained and their dependences on parameters of the system are analyzedfor both linear and non-linear cases.
Investigation of the Parametric Vibrations of Laminated Plates by RFM
(NTU "KhPI", 2016) Kurpa, Lidiya; Mazur, Olga; Tkachenko, Victoria
The R-functions theory is applied to study free vibration and dynamic instability of the symmetrically laminated plates subjected to combined static and periodic axial forces. It is assumed that subcritical state of the plate may be inhomogeneous. Theoretical formulation is made on the classical plate theory (CTP). The developed approach is based on combined application of Ritz’s method, Galerkin procedure, R-functions theory and Bolotin’s method. The buckling, instability zones and response curves for laminated plate with different external cutouts are presented and discussed. Effects of plate geometrical parameters, parking of layers, mechanical characteristics of the material on buckling, natural frequencies and parametric resonance are also studied.
Modelling of Creep and Oscillations in Material Described by Armstrong-Frederick Equations
(NTU "KhPI", 2016) Breslavsky, Dmitry; Altenbach, Holm; Naumenko, Konstantin; Tatarinova, Oksana
Different structural elements at high temperatures and cyclic loading demonstrate essential creep behavior. Due to variety of materials which are used in modern industrial applications the different forms of creep response have to be analyzed. The one of them presents in materials are characterized by creep processes with essential recovery, which is expressed by strain decreasing after the unloading. Such material behavior is described by well-known Armstrong-Frederick model. The case of cyclic loading leading to forced oscillations at high temperature is studied. The Armstrong- Frederick creep model contains two equations: first for creep strain rate function as well as the second for socalled backstress evolution. The problem is solved by two time scales methods with subsequent averaging in a period of oscillations. The solution was performed for the hyperbolic creep strain rate function which satisfactory describes the high-temperature behavior of advanced steel with primary creep conditions. The stress function is presented by expansion in Fourier series. Asymptotic solution of creep equations was obtained and by use of the procedure of averaging in the period the new model describing ‘slow’ creep motion has been derived. The analytical forms of influence functions for both equations of the model expressing the role of cyclical loading were obtained. Numerical examples which demonstrate the cyclic creep behavior in advanced steel X20CrMoV12-l are presented and discussed.
Stability, Bifurcation and Transitions of the Nonlinear Molecular Chain In Electric Field
(NTU "KhPI", 2016) Lykah, Victor; Syrkin, Eugene
The rotational dynamics of molecular adsorbed chain in longitudinal electric field is studied theoretically. The nonlinear dynamic equations are obtained with accounting of quadrupolar interactions between molecules. A new dimensionless parameter (relation of electric and intermolecular interactions) is introduced in the system of the dynamical equations. It is shown that topology of the energy relief on the angle space is transformed in dependence on the electric intensity. The rotational reordering in electric field is complex phenomenon that have several stages. One of the intermediate states is an indifferent equilibrium net (orientation melting). The stable state is found in strong electric fields.
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.
Fuzzy Evaluations For Kinematic Characteristics of Nonlinear Second Harmonics of Shear Waves in Transversely Isotropic Elastic Medium
(NTU "KhPI", 2016) Storozhev, Sergey V.
The analytic representation for the amplitude characteristics of the nonlinear second harmonic of horizontally polarized bulk shear waves is obtained in transversely isotropic elastic medium. With the use of heuristic principle of generalization in the fuzzy sets theory a fuzzy evaluation of the amplitude levels of nonlinear anharmonic perturbations is constructed on the assumption that the approximate experimental values for the elastic modules of the second and third order for the medium describes the normal trapezoidal fuzzy intervals.
The R-Functions Method in the Creep and Creep Damage Problems of Piecewise-Homogeneous Bodies of Revolution With Meridional Section of Any Shape
(NTU "KhPI", 2016) Sklepus, S. N.
We consider the axisymmetric problem of creep and creep damage for piecewise-homogeneous bodies of revolution with meridional section of any shape. We develop a method for the solution of the nonlinear initial boundary-value problem based on the combined application of the R -functions method and the Runge- Kutta- Merson method. The structures of the solution for the main types of boundary conditions are constructed. We present an example of calculation of creep, creep damage and long-term strength for a two-layer cylinder.
Structural Modeling of Elastoplastic Deformation Processes of the Bodies of Non-classical Shape
(NTU "KhPI", 2016) Sizova, N. D.
An approach based on the theory of small elastoplastic deformations is proposed to study the aims of the stress-strain state determining of finned cylindrical and conical bodies of finite sizes. We consider small elastoplastic deformations described by the nonlinear equations system, for linearization of which the variable elastic parameters method is applied. Approximate solution of the linearized elasticity problem at each k-th iteration is made with the use of the R-functions theory in the form of a single analytical expression. Determination of the stress-strain state, the plasticity areas and analysis of the results obtained has been performed with the POLE software package.
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.
Kinematic Characteristics of Nonlinear Second Harmonic for Guided Torsion Elastic Waves in Transversely Isotropic Cylinder
(NTU "KhPI", 2016) Moiseyenko, Igor A.; Storozhev, Valeriy I.; Sidash, Oksana Yu.
The results of theoretical numerical-analytical investigation of some kinematic characteristics of the nonlinear second harmonics for monochromatic axisymmetric normal elastic torsional waves along the axial direction in transversely isotropic cylinder of circular cross section with a rigid fixed of lateral surface. Frequency-parametric analysis of amplitude and shape of wave motion into the second harmonic for normal waves of the investigated type was conducted for the cylinder made from a gadolinium using the proposed method. The regularities inherent in wave processes of this type are described.

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