Nonlinear Dynamics : міжнародна конференція

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https://repository.kpi.kharkov.ua/handle/KhPI-Press/24521

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Nonlinear Dynamics — 2010
(National Technical University "Kharkov Polytechnic Institute", 2010)
The book of Proceedings includes extended abstracts of presentations on the Third International conference on nonlinear dynamics dedicated to the 125-th anniversary of the National Technical University "Kharkov Polytechnic Institute" foundation.
Nonlinear Dynamics — 2013
(National Technical University "Kharkov Polytechnic Institute", 2013)
The book of Proceedings includes extended abstracts of presentations on the Fourth International conference on nonlinear dynamics.
Method of Determination of Natural Frequencies and Forms of Nonlinear Vibrations for Layered Cylindrical Panels
(NTU "KhPI", 2016) Marchuk, M. V.; Goriachko, T. V.; Pakosh, V. S.; Lesyk, O. F.
The technique of finding a finite number of first natural frequencies for geometrically nonlinear vibrations of layered elongated cylindrical panels at discrete consideration of components is proposed and verified. The influence of the radius of curvature on the natural frequencies of three- and five-layered panels is investigated. The dependence between the volume of filler three-layer panels and the lowest natural frequency has been established.
Three ways of treating a linear delay differential equation
(NTU "KhPI", 2016) Sah, Si Mohamed; Rand, Richard H.
This work concerns the occurrence of Hopf bifurcations in delay differential equations (DDE). Such bifurcations are associated with the occurrence of pure imaginary characteristic roots in a linearized DDE. In this work we seek the exact analytical conditions for pure imaginary roots, and we compare them with the approximate conditions obtained by using the two variable expansion perturbation method. This method characteristically gives rise to a “slow flow” which contains delayed variables. In analyzing such approximate slow flows, we compare the exact treatment of the slow flow with a further approximation based on replacing the delayed variables in the slow flow with non-delayed variables, thereby reducing the DDE slow flow to an ODE. By comparing these three approaches we are able to assess the accuracy of making the various approximations. We apply this comparison to a linear harmonic oscillator with delayed self-feedback.
Energy Exchange and Localization in Essentially Nonlinear Oscillatory Systems: Canonical Formalism
(NTU "KhPI", 2016) Gendelman, O. V.; Sapsis, T.
Over recent years, a lot of progress has been achieved in understanding of the relationship between localization and transport of energy in essentially nonlinear oscillatory systems. In this paper we are going to demonstrate that the structure of the resonance manifold can be conveniently described in terms of canonical action-angle variables. Such formalism has important theoretical advantages: all resonance manifolds may be described at the same level of complexity, appearance of additional conservation laws on these manifolds is easily proven both in autonomous and non-autonomous settings. The harmonic balance - based complexification approach, used in many previous studies on the subject, is shown to be a particular case of the canonical formalism. Moreover, application of the canonic averaging allows treatment of much broader variety of dynamical models. As an example, energy exchanges in systems of coupled trigonometrical and vibro-impact oscillators are considered.
R-Functions and WA-Systems of Functions in Modern Information Technologies
(NTU "KhPI", 2016) Kravchenko, Victor F.; Kravchenko, Oleg V.; Konovalov, Yaroslav Yu.; Pustovoit, Vladislav I.; Churikov, Dmitry V.
The review report consists of five parts. It describes the main physical applications of atomic, WA-systems and R-functions.
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.
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.
Dynamics of Multielement Agricultural Aggregates, Taking Into Account Nonholonomic Constraints and Spatial Motion
(NTU "KhPI", 2016) Andreev, Yuri; Antoshchenkov, Roman
The paper studies a discrete system of multielement agricultural aggregate composed of a tractor, hopper and seeder. Mechanical model includes these elements, which are considered as rigid bodies and perform spatial motion subject to of the wheels considering elasticity. As the geometric constrains taking into account the flat surface of the earth and the hinges connecting the units. The aggregate is controlled by angle of the steering wheel or the angle between the tractor half-frames. A feature o f the model is the account of non-holonomic constraints caused by the rolling o f the wheels. This significantly reduces the number o f degrees o f freedom and also complicates the process of forming the equations o f motion. Differential equations are automatically generated by a special system of computer algebra KiDyM based on a general dynamics equation. The gravity force, the driving force and the resistance force applied to the elements o f the aggregate defined as force interactions. The studied linear motion, maneuvers with constant and harmonic law change o f control angle.
Nonlinear dynamics of SWNTs. Energy beating and localization
(NTU "KhPI", 2016) Strozzi, Matteo; Manevitch, Leonid I.; Smirnov, Valeri V.; Pellicano, Francesco
In this paper, the nonlinear vibrations and energy exchange of single-walled carbon nanotubes (SWNTs) are investigated. The Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The SWNT deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. The circumferential flexural modes (CFMs), radial breathing modes (RBMs) and beam-like modes (BLMs) are studied. A numerical model of the SWNT dynamics is proposed. The three displacement fields are expanded in the nonlinear field by using approximate linear eigenfunctions. An energy method based on the Lagrange equations is used to reduce the nonlinear partial differential equations of motion to a set of nonlinear ordinary differential equations, which is solved using the implicit Runge-Kutta numerical method. The nonlinear energy exchange along the SWNT axis is analysed for different initial excitation amplitudes. The internal resonances between CFMs, RBMs and BLMs are investigated. The transition from energy beating to energy localization in the nonlinear field is studied.
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.
Resonance Behavior of the Forced Dissipative Spring-Pendulum System
(NTU "KhPI", 2016) Plaksiy, Kateryna Yu.; Mikhlin, Yuri V.
Dynamics of the dissipative spring-pendulum system under periodic external excitation in the vicinity of external resonance and simultaneous external and internal resonances is studied. Analysis of the system resonance behaviour is made on the base of the concept of nonlinear normal vibration modes (NNMs), which is generalized for systems with small dissipation. The multiple scales method and subsequent transformation to the reduced system with respect to the system energy, an arctangent of the amplitudes ratio and a difference of phases of required solutions are applied. Equilibrium positions of the reduced system correspond to nonlinear normal modes. So-called Transient nonlinear normal modes (TNNMs), which exist only for some certain levels of the system energy are selected. In the vicinity of values of time, corresponding to these energy levels, these TNNMs temporarily attract other system motions. Interaction of nonlinear vibration modes under resonance conditions is also analysed. Reliability of obtained analytical results is confirmed by numerical and numerical-analytical simulation.
Nonlinear Interaction of Oscillation and Rotation in Oscillator-Vibrator Systems
(NTU "KhPI", 2016) Manevich, Arkadiy I.
Synchronous stationary regimes in a model of an oscillator-vibrator system are considered. Along with the first approximation solution for averaged characteristics we obtain a refined analytical solution in the second approximation which is confirmed with excellent accuracy by numerical simulation of the original set of the motion equations. It is revealed that the non-uniformity of the rotation can cause instability of synchronous stationary regimes in the pre-resonance range which was not predicted in the previous researches.
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).
Nonlinear Dynamics of a Spinning Shaft with Non-Constant Rotating Speed
(NTU "KhPI", 2016) Georgiades, Fotios
Reported research on spinning shafts is mostly restricted to cases of constant rotational speed without examining the dynamics thatoccursduringtheir spin-up or spin-down operation. In suchcases, the motion is described by a nonlinear system of Partial Differential Equations (PDEs) coupled with an Integro-Differential Equation (IDE). The nonlinear system of PDEs with IDE, projected onto the infinite basis of the modes of the underlying linear system, results in a system of nonlinear Ordinary Differential Equations (ODEs). In this articleis appliedthe multiple scales perturbation method for dynamic analysis and the system in first order approximation takes the form of two coupled sets of pairedequations. The first pair describes torsional and rigid body rotation whilst the secondconsists of the equations describing the two lateral bending motions. Although in this system non-conservative forces are not considered in terms of damping or explicit externally applied load (torques/forces), the solution of the Is'order approximation of the first set of equations indicates that there are no periodic motions. The solution of the second set of equations of 1st order approximation coincides with the case of constant rotating speed. It isshown, that the Normal Modes in bending motions are the critical speeds of the shaft. It is shown that the frequencies in the Campbell diagram coincide with the frequencies associated with the 1st order solution of the nonlinear system. Moreover, the analytical solution of the first pair of equations is in good agreement with direct numerical simulations. This work paves the way for the development of the Nonlinear Campbell diagram that can be used to determine the dynamic behaviour of rotating structures during spin-up or spin-down operation.
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.
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.
Nonlinear Dynamics of Composite Plates and Shells
(NTU "KhPI", 2016) Kushnir, R. M.; Marchuk, M. V.
The equations of refined geometrically nonlinear theory of dynamic deformation of flexible with respect to transversal shear and compression ofplates and shells are proposed. The solutions for problems of transversal nonlinear vibrations of hinge fixedplate-strip are obtained. Numerical results are compared with the known in literature.
Nonlinear Vibration Problem of Launch Vehicle Carrying a Moving Time-Dependent Mass
(NTU "KhPI", 2016) Gristchak, D. D.
In the present study, an analytical solution for nonlinear vibration problem of launch vehicle as a flexible beam model carrying moving with constant velocity mass which is time function is proposed. The nonlinear third and fifth order terms of partial differential equation with variable in time coefficients correspond to high amplitude and mid-plane stretching beam and the right term of initial equation of the problem represents the concentrated time-dependent moving mass effect. A hybrid asymptotic approach applied for an approximate analytical solution problem at given boundary and initial conditions an given. The resulting (approximate) solution has a form of a sum where each term consists of the product of two functions according to perturbation (on parameter at nonlinear terms) and phase-integral-Galerkin technique (on singular parameter at higher derivative) methods. The results of comparison of an approximate analytical solution and direct numerical integration of initial equation have shown a good enough accuracy as for “small” as well for “large ” scalar parameters for asymptotic expansion of the desired function.
From Geometry, Kinematics and Dynamics of Billiards to the Extended Theory of Skew Collision between Two Rolling Bodies and Methodology of Vibro-Impact Dynamics
(NTU "KhPI", 2016) Hedrih (Stevanovic), Katica R.
Starting from explanation of geometry, kinematics and dynamics of game billiards, and phenomena of impact a rolling ball into different types of curved surfaces and direct and skew central collision of two rolling, same dimension, balls we open question of collision of two rolling axial symmetrically bodies with different dimensions and different forms. Use elementary approach and Petrovic's theory presented in two books “Elements of mathematical phenomenology” and “Phenomenological mappings”, extended theory of direct and skew central collision of two rolling, axially symmetric, but different dimensions and forms, bodies is formulated with all additional and new analytical expressions, theorems , to define all pre- and post- collision kinetic states. Use these new results complete methodology of vibro-impact system dynamics is formulated and applied for investigation kinetic parameters and phenomena in vibro-impact systems with successive collisions between two or a finite number of rolling bodies. Energy jumps in collisions between rolling bodies in vibroimpact system dynamics are indicated and analytically described in a number of these systems.

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