Asymptotic Method Of Averaging Research Articles

PROBLEMS OF EVOLUTION OF RIGID BODY MOTION SIMILAR TO LAGRANGE TOP

The problem of evolution of the rigid body rotations about a fixed point continues to attract the attention of researches. In many cases, the motion in the Lagrange case can be regarded as a generating motion of the rigid body. In this case the body is assumed to have a fixed point and to be in the gravitational field, with the center of mass of the body and the fixed point both lying on the dynamic symmetry axes of the body. A restoring torque, analogues to the moment of the gravity forces, is created by the aerodynamic forces acting on the body in the gas flow. Therefore, the motions, close to the Lagrange case, have been investigated in a number of works on the aircraft dynamics, where various perturbation torques were taken into account in addition to the restoring torque. Many works have studied the rotational motion of a heavy rigid body about a fixed point under the action of perturbation and restoring torques. The correction of the studied models is carried out by taking into account external and internal perturbation factors of various physical nature as well as relevant assumptions according to unperturbed motion. The results of reviewed works may be of interest to specialists in the field of rigid body dynamics, gyroscopy, and applications of asymptotic methods. The authors of this papers present a new approach for the investigation of perturbed motions of Lagrange top for perturbations which assumes averaging with respect to the phase of the nutation angle. Nonlinear equations of motions are simplified and solved explicitly, so that the description of motion is obtained. Asymptotic approach permits to obtain some qualitative results and to describe evolution of rigid body motion using simplified averaged equations. Thus it is possible to avoid numerical integration. The authors present a unified approach to the dynamics of angular motions of rigid bodies subject to perturbation torques of different physical nature. These papers contains both the basic foundations of the rigid body dynamics and the application of the asymptotic method of averaging. The approach based on the averaging procedure is applicable to rigid bodies closed to Lagrange gyroscope. The presented brief survey does not purport to be complete and can be expanded. However, it is clear from this survey that there is an literature on the dynamics of rigid body moving about a fixed point under the influence of perturbation torques of various physical nature. The research in this area is in connection with the problems of motion of flying vehicles, gyroscopes, and other objects of modern technology

Image-based simulation of dispersed composites and porous rocks

The article presents the results of numerical simulation of the effective elastic properties and stress concentration of the dispersed composite and porous rocks as well as the elastoplastic diagrams of tensile stress of the B4C/2024Al composite using asymptotic averaging method. 2D and 3D simulations of uniaxial tension was carried out using the image-based models of the real structure obtained by X-ray tomography. We also used model structures of the B4C/2024Al composite, consisting of an aluminum matrix with boron carbide inclusions in the form of ellipsoids with a random orientation. It turned out that in the elastic region 3D real structure and 3D ellipsoidal structure lead to the same effective moduli. On contrary, the stress concentrations are much higher in the real structure model than in the model of ellipsoidal inclusions. Also, two types of the models behave completely differently in the plastic region. Comparison of 2D and 3D calculations with experimental data shows that 2D models can be also used for determining the effective elastic moduli of B4C/2024Al composite, but the use of 3D models is preferable for porous geomaterials and dispersed composites.

Modeling Elastic Properties of Composites using Asymptotic Averaging Method with Imperfect Interface

The paper presents a modification of the asymptotic averaging method for solving the homogenization problem of elastic properties for composite materials. The elasticity of the phase interface is taken into account. The conditions of a soft imperfect interface are considered, which account for a displacement jump on the phase boundary. A review of the interface-modeling methods in composite materials is presented. The finite element method is used for the numerical implementation of the averaging method. A model of a interface finite element is proposed. Elastic properties averaging is adapted to the presence of a discontinuity in the displacement field. Application limits of the soft imperfect interface are estimated in terms of interphase layer properties. The identification of the interface parameters using experimental data is considered. Computational experiments are conducted for dispersed-reinforced and unidirectional composite with isotropic inclusion.

Numerical investigation of mesoscopic volumetric ablation of 3D braided charring composites

Owing to the complex mesoscopic structures, 3D braided charring composites for the thermal protection system (TPS) have the multi-scale and multi-physical high-temperature responses. The mesoscopic features can significantly affect the volumetric ablative behavior of the braided composites, which will be essential for the TPS design. To take into account the effect of mesoscopic structures on the volume ablation response, the thermo-physical properties of constituents within the braided composites are predicted using asymptotic averaging method. The volumetric ablation model of the braided composites considering the mesoscopic structures is proposed, which is calculated by finite volume method with a tightly coupled solving process. The effect of braid angles on the volumetric ablation responses of the braided composites are investigated. The maximum back temperature and the peak inner pressure of the braided composites decrease with the increase of the braid angle. The isothermal and isobaric surfaces of the braided composites under volumetric ablation exhibit complex characteristics and are not perpendicular to the heating direction, which may induce the complex thermal stress field within the composites. The study on mesoscopic volume ablation may be helpful to accurately predicting the ablation responses and guiding the material design of the braided charring composites.

Surface Finite Element for Imperfect Interface Modeling in Elastic Properties Homogenization

The paper presents a mathematical model of a finite element for modeling imperfect interface conditions for two contacting surfaces. The element is used in the numerical implementation of the Asymptotic Averaging Method (AAM) for the determination of effective elastic properties of composite materials under investigation. Numerical experiments are carried out to calculate the elastic properties taking into account the adhesion layer using a displacements field jump condition at the phase boundary. Results are compared with adhesion modeling using an additional bulk phase.

Dispersion in electroosmotic flow generated by oscillatory electric field interacting with oscillatory wall potentials

An analytical study is presented in this article on the dispersion of a neutral solute released in an oscillatory electroosmotic flow (EOF) through a two-dimensional microchannel. The flow is driven by the nonlinear interaction between oscillatory axial electric field and oscillatory wall potentials. These fields have the same oscillation frequency, but with disparate phases. An asymptotic method of averaging is employed to derive the analytical expressions for the steady-flow-induced and oscillatory-flow-induced components of the dispersion coefficient. Dispersion coefficients are functions of various parameters representing the effects of electric double-layer thickness (Debye length), oscillation parameter, and phases of the oscillating fields. The time–harmonic interaction between the wall potentials and electric field generates steady as well as time-oscillatory components of electroosmotic flow, each of which will contribute to a steady component of the dispersion coefficient. It is found that, for a thin electric double layer, the phases of the oscillating wall potentials will play an important role in determining the magnitude of the dispersion coefficient. When both phases are zero (i.e., full synchronization of the wall potentials with the electric field), the flow is nearly a plug flow leading to very small dispersion. When one phase is zero and the other phase is π, the flow will be sheared to the largest possible extent at the center of the channel, and such a sharp velocity gradient will lead to the maximum possible dispersion coefficient.

Vibrational Fréedericksz transition in liquid crystals

Our aim is to show the possibility of a "vibrational Fréedericksz's transition" in nematic liquid crystals (LCs) consisting of asymmetric molecules (in the absence of the reflectional symmetry for the director field). For this purpose we study the effects of the external high-frequency vibrations imposed on LCs by employing the rigorous two-timing asymptotic averaging method to the governing LC equations in their most general form. We restrict our attention only to translational mechanical vibrations of an incompressible LC medium as a whole, so our vibrations are not related to acoustical waves. We show that the averaged "vibrogenic" torque acted on the director field is mathematically equivalent to the torque caused by an external magnetic field. This equivalence leads us to the key conclusion that the high-frequency mechanical vibrations can cause the vibrational Fréedericksz's transition. Our evaluation of the relevant physical parameters shows that this phenomenon can be observed in laboratory experiments.

THE ASYMPTOTIC SOLUTION OF AN INTEGRAL EQUATION FOR MAGNETIC CURRENT IN A PROBLEM OF WAVEGUIDES COUPLING THROUGH NARROW SLOTS

Based on the asymptotic method of averaging,an approximate analytical solution of the integral equation concerning a magnetic current in slot-hole coupling apertures of electrodynamic volumes,which differ profitably from the known ones in literature, has been obtained. The formulas for the currents and characteristics scattering of transverse and longitudinal slots in common broad and narrow walls of rectangular waveguides are given. The comparison to results obtained by other methods and experimental data has been

Averaging Oscillations with Small Fractional Damping and Delayed Terms

We demonstrate the method of averaging for conservative oscillators which may be strongly nonlinear, under small perturbations including delayed and/or fractional derivative terms. The unperturbed systems studied here include a harmonic oscillator, a strongly nonlinear oscillator with a cubic nonlinearity, as well as one with a nonanalytic nonlinearity. For the latter two cases, we use an approximate realization of the asymptotic method of averaging, based on harmonic balance. The averaged dynamics closely match the full numerical solutions in all cases, verifying the validity of the averaging procedure as well as the harmonic balance approximations therein. Moreover, interesting dynamics is uncovered in the strongly nonlinear case with small delayed terms, where arbitrarily many stable and unstable limit cycles can coexist, and infinitely many simultaneous saddle-node bifurcations can occur.

Analysis of test results of composite specimens modeling rock

Formulas permitting evaluation of the representative nature of familiar tests with composite specimens are derived using the asymptotic method of averaging differential equations. It is demonstrated that to obtain effective values of the compression modulus, there is no need to increase the number of elements in the model, nor have adequate knowledge of the compression modulus of the element and the compression modulus (or stiffness) of a crack. The technique employed to apply the asymptotic method of averaging - an effective means of investigating a broad class of rocks - is briefly described.

A note on the Aris dispersion in a tube with phase exchange and reaction

An asymptotic method of averaging is applied to the extended Aris problem in which a solute is transported in a gas flow along a circular tube subject to phase exchange with and reaction in a stationary liquid film held on the wall of the tube. In addition to recovering the Aris dispersion coefficient, the second-order effects of the reaction on the advection velocity and the reaction rate itself are also derived. Dependence of these effects on the partition coefficient and the ratio of fluid amounts are examined.

SUBHARMONIC OSCILLATIONS IN HARMONICALLY EXCITED MECHANICAL SYSTEMS WITH CYCLIC SYMMETRY

The third-order subharmonic oscillations in weakly non-linear cyclic symmetric structures with multiple degrees of freedom are studied. These strongly coupled cyclic structures, in their linear approximation, are known to possess pairwise double-degenerate natural frequencies with orthogonal normal modes. The asymptotic method of averaging is used to study the nonlinear interactions between the pairs of modes with nearly identical natural frequencies when the external excitation is nearly three times the natural frequency of the modes being excited. A careful local bifurcation analysis of the averaged equations is conducted to study the effects of frequency mistuning and excitation amplitudes, as well as the modal damping in the system. Subharmonic standing and travelling wave type solutions, Hopf bifurcation from travelling wave solutions to quasiperiodic responses, period-doubling bifurcations, and Silnikov type chaos are found to exist in the averaged system.

An asymptotic model of a membrane and a string

A three-dimensional problem in the theory of elasticity of a stressed solid in a domain of small thickness (diameter) is converted into a problem in the theory of membranes (strings). The Averaged problems obtained enable one to describe membranes (strings) with inhomogeneities of a size comparable with their thickness. The problems are analysed using the asymptotic method of averaging.

Heat-Mass-Transport and Thermal Stresses in Porous Charring Materials

The aim of this paper is to develop a method of asymptotic averaging for processes occurring in porous charring materials under high temperatures. The advantage of the method is the ability to calculate not only averaged macrocharacteristics of the processes, namely internal gas generation, filtration and deforming processes, but also microcharacteristics, such as microstresses in phases of charring material, gas velocity in a pore, etc. To determine microcharacteristics, the method allows us to formulate special mathematical problems ‘on a periodic cell’. To calculate macrocharacteristics, such as pore pressure of filtrating gas, rate of charring and macrostresses, with the help of asymptotic averaging method, averaged ‘global’ equations are formulated. Here effective characteristics of porous medium (gas permeability coefficient, rate of charring, elasticity modulus, thermal expansion coefficient) are determined not empirically, as in most works on porous materials, but on the basis of solving the local problems. Solution of these problems over the periodic cell allows us to derive analytically the law of the Darcy type for a gas phase flow in porous media, to obtain an expression for intensive mass transfer between solid and gas phases, to set the form of constitutive relations for charring porous media, and also to calculate microstresses in a vicinity of a growing pore. As an example of solving a global averaged problem, the problem on one-sided high-temperature heating of a plate made of epoxy binder has been solved numerically.

Long-wave processes in a periodic medium

LONG-WAVE PROCESSES IN A PERIODIC MEDIUM V. A. Vakhnenko and V. V. Kulich UDC 532.59:517.19 The development of experimental techniques has shown that the internal structure of a medium affects wave motion and transport processes. Nonlinear effects start playing an in- creasing role with increasing wave amplitude. To the same class belong high-gradient, fast- flowing nonequilibrium processes. All these features must be taken into account in the math- ematical simulation of waves in real media. There exists a class of effects, in which the characteristic size considered L is much larger than the size of medium inhomogeneity E (L ~ ~). The drawback of assigning microprop- erties of an inhomogeneous medium creates major difficulties for direct solution of these problems. Numerical solutions are difficult due to high cost of computer time. One way of investigating an inhomogeneous medium is the spatial averaging method. The averaged description has an asymptotic nature. There exist different mathematical methods of asymptotic averaging of long-wave processes with detailed account of the structure. In the present study, for wave modeling in a barotropic periodic medium we use the asymptotic averaging method developed for a composite regular structure [I, 2]. Processes in an inhomo- geneous medium can be described by equations with fast-oscillating coefficients. For media of regular structure the fast-oscillating coefficients are periodic functions. The essence of asymptotic averaging consists of applying the multiple scale method [3] in conjuction with spatial averaging [4]. The method provides an asymptotically correct approximation to the solution. In the general case one obtains a system of integrodifferential equations. Some- times the problem can be reduced to the average characteristics of wave fields. At the same time, in solving the integrodifferential system one can find a numerical method in which one succeeds in selecting a step in the spatial coordinate substantially exceeding the period of the structure. i. >ystem of Averaged Equations. The asymptotic averaging method can be applied to de- scribe processes in a compressible inhomogeneous medium [5]. The equations of motion are conveniently written in Lagrangian coordinates. In these variables the structure of the periodic compressible medium is nonvarying, making it possible to use the averaging procedure. In the general case the coefficients and solutions of the equations, describing pro- cesses in inhomogeneous media, have discontinuous values. The equations must then be repre- sented in integral form. Under the assumption of smooth coefficients and solutions they are equivalent to the differential equation of motion. It was shown in [i, 6] that for insignif- icant differences in physical properties of medium inhomogeneity the approximate continuous solutions of the system of differential equations formally constructed by means of the asymp- totic method satisfy the integral conservation laws with a certain accuracy. This fact indi- cates the correctness of using a system of differential equations of motion in an asymptotic averaging method. The original equations of motion are the equations of a barotropic periodic medium, written down for each structural element [7]:

Torus doublings and chaotic amplitude modulations in a two degree-of-freedom resonantly forced mechanical system

This work studies the response of a weakly non-linear vibratory system with two degrees-of-freedom when the system is excited near resonance. The two linear modes are in 1:3 internal resonance. The asymptotic method of averaging and direct numerical integration are used to obtain the response of the system. Over a range of excitation frequencies and modal damping, the averaged equations in slow time are found to possess limit cycle solutions. These solutions undergo period doubling bifurcations to chaotic solutions. The averaging theory then implies the existence of amplitude modulated motions, the exact nature of modulations not being well defined. Numerical simulation of the original vibratory two degree-of-freedom system shows that the system does undergo amplitude modulated motions. For sufficiently large damping, only periodic modulations arise in the form of a 2-torus. For lower damping, the 2-torus can undergo doubling and ultimate destruction to result in a chaotic attractor. Poincare sections of steady state solutions are used to characterize the various types of amplitude modulated motions.

Effective characteristics of the deformation properties of stratified rock

1. The asymptotic method of averaging, which ensures the closeness of solutions for initial inhomogeneous and equivalent homogeneous masses having the same dimensions and the same loading is recommended for computation of the effective characteristics of the deformation properties of stratified rock.