Nonlinear Dynamics–2016

Постійне посилання колекціїhttps://repository.kpi.kharkov.ua/handle/KhPI-Press/24522

Переглянути

Зараз показуємо 1 - 20 з 79

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 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.
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.
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].
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.
Improvement of Piezoelectric Energy Harvester Efficiency Through Optimal Patch Configuration
(NTU "KhPI", 2016) Gosliga, Julian S.; Ganilova, Olga A.
The aim of this paper is to explore how to improve the efficiency of a vibrating hybrid energy harvester through changing the patch configuration. The results of this work identify the patch configuration that maximises output while using the same amount of piezoelectric material. Using 6 patches was found to be the most efficient when looking at the energy output from a single cycle. Stress distributions generated using ANSYS show that this was because the patches were all located in areas of high stress. The 2 patch configuration resulted in the highest energy conversion at low frequencies (peak loss factor <50Hz) while the performance of the 6 patch configuration was characterised by high energy conversion over a wider range of frequencies.
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.
Non-Iterative Rauscher Method for 1-DOF System: a New Approach to Studying Non-Autonomous System via Equivalent Autonomous One
(NTU "KhPI", 2016) Perepelkin, Nikolay V.
In the paper a new non-iterative variant of Rauscher method is considered. In its current statement the method can be used in analysis of forced harmonic oscillations in 1-DOF nonlinear system. It is shown that three different types o f equivalent authonomous dynamical systems can be built for a given 1-DOF non-autonomous one. Two of them (1st and 2nd type) have wider set of solutions than that of the initial system. These solutions correspond to various values of amplitude and phase of external excitation. Solutions of the equivalent system of 3rd type are exclusively periodic ones. Based on the equivalent system of 3rd type such a function W(x,x') can be constructed that its level curves correspond to periodic orbits of the initial non-autonomous system. This function can be built a priori via computation of the invariant manifold of the equivalent system of 1st type. Using the same approach the Rauscher expansions cos(Qt)=C(x,x'), sin(Qt)=S(x,x') can also be constructed. It is also shown that equivalent systems can be investigated by means of harmonic balance method which allows construction o f W(x,x'), C(x,x') andS(x,x') in semi-analytical manner.
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.
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.
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.
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.
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.
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.
Simulation of Vibro-Impact Systems Using Reciprocal Mass Matrices
(NTU "KhPI", 2016) Tkachuk, Anton; Bischoff, Manfred
Computation cost of explicit time integration can be reduced substantially using the reciprocal mass matrices. General variational derivation of the method, its verification by an eigenvalue benchmark and comparison on a transient example are presented in this contribution.
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.
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.
Isogeometric Approximation Methods Using the Interlineation Operators
(NTU "KhPI", 2016) Sergienko, Ivan; Lytvyn, Oleg M.; Lytvyn, Oleg O.; Tkachenko, Alexandr; Grytsai, Olga
Interlineation of functions of two or more variables is approximation of the functions by their traces or traces of some differential operators on the fixed system of lines. The given paper presents the analysis of the building methods of interlineation operators, that preserve the differentiability class and have the same traces as the approximated function.