Perturbation Method Of Multiple Scales Research Articles

Time-delay feedback control of nonlinear forced vibration of graphene/piezoelectric sandwich nanoplates under a mechanical shock

In this study, the nonlinear forced vibration of graphene/piezoelectric sandwich nanosheets under mechanical impact was investigated. For the first time, linear (displacement and velocity) and nonlinear (displacement) time-delay proportional derivative (PD) controllers were applied to graphene/piezoelectric sandwich nanosheets, and the primary/secondary resonance under different time-delay parameters were analyzed. An approximate analytical solution for the nonlinear forced vibration control equation was obtained using the multiple scale perturbation method, and the stability of graphene/piezoelectric sandwich nanosheets under primary/secondary resonance conditions was discussed using the Ross Herwitz theorem. Finally, the results of this study were compared with those of other studies. The feasibility of the research methodology was demonstrated. On this basis, the research results indicate that linear displacement delay, nonlinear displacement delay, and velocity delay parameters can suppress the unsteady vibration characteristics of the main system. And comparing the linear and nonlinear displacement time-delay, it is found that the nonlinear displacement time-delay has a greater effect on the system. Specifically, the presence of nonlinear displacement time-delay changes the vibration characteristics of the curve from hard spring characteristics to soft spring characteristics. Comparison of displacement time-delay and velocity time-delay reveals that velocity time-delay gives better control over the system. In addition, the results show that the excitation amplitude, nonlocal parameters, and external voltage enhance the nonlinear characteristics of the system under the main and superharmonic resonance conditions before the time-delay control. Subharmonic resonance conditions, the system parameters make the vibration range wider. After the time-delay control, the influence of these three factors on the system vibration is significantly reduced for the three resonance conditions, especially for the non-local parameters. The results of the study can provide guidance for active control of time delay in different types of nanodevices for engineering applications.

Introduction of Local Resonators to a Nonlinear Metamaterial With Topological Features

Abstract Recent work in nonlinear topological metamaterials has revealed many useful properties such as amplitude dependent localized vibration modes and nonreciprocal wave propagation. However, thus far, there have not been any studies to include the use of local resonators in these systems. This work seeks to fill that gap through investigating a nonlinear quasi-periodic metamaterial with periodic local resonator attachments. We model a one-dimensional metamaterial lattice as a spring-mass chain with coupled local resonators. Quasi-periodic modulation in the nonlinear connecting springs is utilized to achieve topological features. For comparison, a similar system without local resonators is also modeled. Both analytical and numerical methods are used to study this system. The dispersion relation of the infinite chain of the proposed system is determined analytically through the perturbation method of multiple scales. This analytical solution is compared to the finite chain response, estimated using the method of harmonic balance and solved numerically. The resulting band structures and mode shapes are used to study the effects of quasi-periodic parameters and excitation amplitude on the system behavior both with and without the presence of local resonators. Specifically, the impact of local resonators on topological features such as edge modes is established, demonstrating the appearance of a trivial bandgap and multiple localized edge states for both main cells and local resonators. These results highlight the interplay between local resonance and nonlinearity in a topological metamaterial demonstrating for the first time the presence of an amplitude invariant bandgap alongside amplitude dependent topological bandgaps.

On a Nonlinear Locally Resonant Metamaterial With Resistance-Inductance Shunt

Abstract Numerous recent works have established the potential of various types of metamaterials for simultaneous vibration control and energy harvesting. In this paper, we investigate a weakly nonlinear metamaterial with electromechanical (EM) local resonators coupled to a resistance-inductance shunt circuit, a system with no previous examination in the literature. An analytical solution is developed for the system, using the perturbation method of multiple scales, and validated through direct numerical integration. The resulting linear and nonlinear band structures are used for parametric analysis of the system, focusing on the effect of resonator and shunt circuit parameters on band gap formation and vibration attenuation. This band structure analysis informs further study of the system through wavepacket excitation as well as spectro-spatial analysis. The voltage response of the system is studied through spatial profiles and spectrograms to observe the effects of shunt inductance, nonlinearity, and their interactions. Results describe the impact of adding a shunted inductor, including significant changes to the band structure; multiple methods of tuning band gaps and pass bands of the system; and changes to wave propagation and voltage response. The results demonstrate the flexibility of the proposed metamaterial and its potential for both vibration control and energy harvesting, specifically compared to a previously studied system with resistance-only shunt.

A study on coupled edgewise and flapwise vibration modes of wind turbine blade

The coupled flapwise and edgewise vibrations of a horizontal axis wind turbine blade (HAWT) are discussed in this paper. Kinetic and potential energies of the blade are evaluated with consideration of the effects of gravity and aerodynamic forces. Hamilton’s principle is employed to develop the nonlinear coupled modal equations of the blade. The coupling between the first order edgewise and the first two order of flapwise modes is considered. This leads to three equations with nonlinear terms. Multiple-scales perturbation method is employed to solve these equations. Furthermore, the numerical values of structural specifications of National Renewable Energy Laboratory (NREL) 5-MW reference wind turbine blade are used, for example. It is shown that first edgewise and first flapwise vibrations are dominant, while second flapwise mode of vibration is less significant in the case of transient responses. Energy transfer between resonant modes is observed in both internal and combination resonances. The amplitude–frequency curves and phase diagrams during primary and combination resonances are obtained for the steady-state responses. Combination and primary resonances are further examined by considering various factors such as geometric nonlinearity and aerodynamic force.

Coupled nonlinear Schrödinger (CNLS) equations for two interacting electrostatic wavepackets in a non-Maxwellian fluid plasma model

The nonlinear dynamics of two co-propagating electrostatic wavepackets in a one-dimensional non-magnetized plasma fluid model is considered, from first principles. The coupled waves are characterized by different (carrier) wavenumbers and amplitudes. A plasma consisting of non-thermalized (κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-distributed) electrons evolving against a cold (stationary) ion background is considered. The original model is reduced, by means of a multiple-scale perturbation method, to a pair of coupled nonlinear Schrödinger (CNLS) equations for the dynamics of the wavepacket envelopes. For arbitrary wavenumbers, the resulting CNLS equations exhibit no known symmetry and thus intrinsically differ from the Manakov system, in general. Exact analytical expressions have been derived for the dispersion, self-modulation (nonlinearity) and cross-modulation (coupling) coefficients involved in the CNLS equations, as functions of the wavenumbers (k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_1$$\\end{document}, k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_2$$\\end{document}) and of the spectral index κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document} characterizing the electron profile. An analytical investigation has thus been carried out of the modulational instability (MI) properties of this pair of wavepackets, focusing on the role of the intrinsic (variable) parameters. Modulational instability is shown to occur in most parts of the parameter space. The instability window(s) and the corresponding growth rate are calculated numerically in a number of case studies. Two-wave interaction favors MI by extending its range of occurrence and by enhancing its growth rate. Growth rate patterns obtained for different κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document} index (values) suggest that deviation from thermal (Maxwellian) equilibrium, for low κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document} values, leads to enhance MI of the interacting wave pair. To the best of our knowledge, the dynamics of two co-propagating wavepackets in a plasma described by a fluid model with κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-distributed electrons is investigated thoroughly with respect to their MI properties as a function of κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document} for the first time, in the framework of an asymmetric CNLS system whose coefficients present no obvious symmetries for arbitrary k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_1$$\\end{document} and k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_2$$\\end{document}. Although we have focused on electrostatic wavepacket propagation in non-thermal (non-Maxwellian) plasma, the results of this study are generic and may be used as basis to model energy localization in nonlinear optics, in hydrodynamics or in dispersive media with Kerr-type nonlinearities where MI is relevant.

Shear-induced wrinkling in accelerating thin elastic discs

The wrinkling instabilities produced by in-plane angular accelerations in a rotating disc are discussed here in a particular limit of relevance to very thin plates. By coupling the classical linear elasticity solution for this configuration with the Föppl–von Kármán plate buckling equation, a fourth-order boundary-value problem with variable coefficients is obtained. The singular-perturbation character of the resulting problem arises from a combination of factors encompassing both the pre-stress (due to the spinning motion) and the geometry of the annular domain. With the help of a simplified multiple-scale perturbation method in conjunction with matched asymptotics, we succeed in capturing the dependence of the critical (wrinkling) acceleration on the instantaneous speed of the disc as well as other physical parameters. We show that the asymptotic predictions compare well with the results of direct numerical simulations of the original bifurcation problem. The limitations of the formulae obtained are also considered, and some practical suggestions for improving their accuracy are suggested.

Integral Resonant Controller for Suppressing Car’s Oscillations and Eliminating its Inherent Jump Phenomenon

This work deals with modeling and controlling the oscillations of a horizontally-supported car under the effect of a nonlinear spring, a damper, and a harmonic excitation external force. The proposed controller is the integral resonant controller (IRC) which is a first order oscillator coupled the car via a linear variable differential transformer (LVDT) and a servo-controlled linear actuator(SCLA). The multiple scales perturbation method is used to obtain an approximate solution and stability analyses are carried out. Based on the stability analysis, the optimum values of the controller parameters are recommended in this work for a better control operation. Several response curves are plotted for clarifying the concept of the proposed integral resonant control algorithm.A numerical simulation of the car’s motion is achieved by numerically integrating the model equations with the 4th order Rung-Kutta algorithm. The applied controller can treat the severe jumps in the oscillation amplitude where it is suppressed at different conditions of forward and backward sweeping of the amplitude and frequency of the external excitation force.

Modulational electrostatic wave–wave interactions in plasma fluids modeled by asymmetric coupled nonlinear Schrödinger (CNLS) equations

The interaction between two co-propagating electrostatic wavepackets characterized by arbitrary carrier wavenumber is considered. A one-dimensional (1D) non-magnetized plasma model is adopted, consisting of a cold inertial ion fluid evolving against a thermalized (Maxwell–Boltzmann distributed) electron background. A multiple-scale perturbation method is employed to reduce the original model equations to a pair of coupled nonlinear Schrödinger (CNLS) equations governing the dynamics of the wavepacket amplitudes (envelopes). The CNLS equations are in general asymmetric for arbitrary carrier wavenumbers. Similar CNLS systems have been derived in the past in various physical contexts, and were found to support soliton, breather, and rogue wave solutions, among others. A detailed stability analysis reveals that modulational instability (MI) is possible in a wide range of values in the parameter space. The instability window and the corresponding growth rate are determined, considering different case studies, and their dependence on the carrier and the perturbation wavenumber is investigated from first principles. Wave–wave coupling is shown to favor MI occurrence by extending its range of occurrence and by enhancing its growth rate. Our findings generalize previously known results usually associated with symmetric NLS equations in nonlinear optics, though taking into account the difference between the different envelope wavenumbers and thus group velocities.

Nonlinear vibrations of four-degrees of freedom for piezoelectric functionally graded graphene-reinforced laminated composite cantilever rectangular plate with PPF control strategy

The nonlinear vibration suppression of the piezoelectric functionally graded graphene-reinforced laminated composite cantilever (PFG-GRLCC) rectangular plate with positive position feedback (PPF) control strategy is investigated firstly. The material properties of the graphene-reinforced structure are calculated through the Halpin–Tsai micromechanical model. Considering the transverse external excitation and the converse effect of piezoelectricity, the governing equations of motion are formulated through von Karman large deformation theory, the classical laminated plate theory, and Hamilton principle. After adding the PPF controllers, a four-degrees of freedom model for the close-loop vibration control system is achieved via Galerkin truncation technique. The average equations in the case of the primary resonance and 1:1:3:3 internal resonance can be obtained using the multiple scale perturbation (MSP) method. The amplitude–frequency response curves are studied by the numerical continuation algorithm. The detailed parametric analyses show that the PPF controller can effectively reduce the nonlinear vibration response amplitudes of the PFG-GRLCC rectangular plate. In addition, the results reveal the energy transform between the host system and the PPF controller. This work is expected to provide theoretical guidance for nonlinear large amplitude vibration reduction of graphene-reinforced structure.

Modelling and nonlinear analysis of the wave-induced vibrations of a single hybrid riser conveying two-phase flow

This paper analyses the nonlinear dynamic response of a single hybrid riser pipe to wave excitations. Detailed nonlinear equations of motion are derived for the riser pipe as it conveys two-phase flow. Hence its dynamics are described by two coupled nonlinear equations, relating to both longitudinal and transverse displacements. Wave excitations forcing the system are imposed as oscillating inline flow. In addition, considering that the riser is a structure that is vibrating in a moving fluid, the inline force is modelled with the extended Morison equation. These equations are solved by the method of multiple scale perturbation. Theoretical results show that longitudinal and transverse responses are coupled at some specified natural frequencies. Similarly, the riser pipe is also prone to primary and super-harmonic resonance excitations at some wave frequencies. This presents a complex nonlinear multi-frequency excitation problem identical to the numerical studies conducted on a typical single hybrid riser pipe based on the frequency response curves. The uncoupled primary resonance behaviour of the transverse displacements for all the void fractions studied exhibited a hardening nonlinear behaviour of non-unique responses and jump phenomenon. However, a distinctive feature of the harmonics was observed where the curves separate into two to form an isolated branch of periodic solutions. These are known as Isola. Also, the presence of internal resonance results in the transfer of energy between the planar axis resulting in undesirable axial resonance peaks which are destabilized by the presence of the nonlinear anti-resonance effect.

Size-dependent nonlinear vibration analysis of cracked graphene-platelets-reinforced-composites (GPLRC) plate under parametric excitation

The response of a size-dependent randomly distributed graphene-platelet-reinforced-composite (GPLRC) plates, with an inclined part-through surface crack, resting on viscoelastic foundation under the action of periodic parametric excitation with quadratic and cubic nonlinearities are studied in the present study. The crack is centrally located at the center of the plate while inclined at an angle β concerning the X-axis. The angled crack’s mathematical formulation is based on a line-spring model. Indeed, the assumption of the problem is considered as the crack remains a continuous straight line. Nonlinearity is considered based on Berger’s formulation and Von-Karman’s geometric relations. Using classical plate theory, Eringen’s theory, micromechanical models, and equilibrium equation, the governing equation of motion of the problem have been derived. Then the Galerkin method was applied to transform the partial differential equation into the ordinary one. Eventually, the solution procedure of the cubic–quadratic nonlinear equation was followed by the multiple-scale perturbation method. Steady-state response and nonlinear softening or hardening treatment of the problem have been presented. Significant results of the study can be emphasized as GPLs, crack’s dimension and inclination, nonlocal parameter, damping, and detuning parameter effects on nonlinear behavior and stability of the structure. Dispersing GPLs as reinforcement material in the plate layer caused improvement in structural behavior, as the amplitude response of the system with the presence of a crack is decreased, comparing to a similar system without any reinforcement and crack. In the end, current work will help in the development of nonlinear behavior, stability analysis and critical bifurcation points of size-dependent cracked GPLRC plate exposed to external parametric loading.

Dust ion-acoustic dromions in Saturn’s magnetosphere

ABSTRACTMotivated by observations of localized electrostatic wavepackets by the Cassini – and (earlier) by Voyager 1 and 2 – mission(s) in Saturn’s magnetosphere, we have investigated the existence conditions and the dynamical evolution of localized multidimensional structures in the Saturnian dusty plasma environment. To this effect, we have adopted a plasma-fluid model for dust-ion acoustic excitations, taking into account the presence of a highly energetic (suprathermal, kappa-distributed) electron population in combination with massive dust dust particulates in the background. A multiple scales perturbation method was shown to lead to a Davey--Stewartson (DS) system of evolution equations, if one assumes perpendicular carrier wave propagation across the magnetic field (direction). The system is then shown to possess two regimes mainly, known in the literature as DS-I and DS-II. In the former case, if certain conditions are fulfilled, exponentially localized solutions are obtained, known as dromions. The combined effects of various physical parameters, such as the electron spectral index, the ambient magnetic field (strength), and the dust concentration, have been examined. A numerical investigation reveals that the dromion amplitude increases with higher dust concentration, whereas it decreases for lower κe (i.e. with an increase in the suprathermal electron population component). A stronger magnetic field results in higher amplitude but narrower dromions. Our results provide a comprehensive framework for modeling modulated electrostatic wavepackets, in direct comparison with experimental data in planetary environments, in Saturn’s magnetosphere and elsewhere.

Optimal time delayed control of the combination resonances of viscoelastic graphene sheets under dual-frequency excitation

In this study, the nonlinear combination vibration of a single-layer graphene sheet (SLGS) resting on a visco-Pasternak foundation and exposed to dual-frequency excitation was investigated. The time delayed feedback proportional-derivative (PD) controller was applied to an SLGS for the first time, and three combination resonances with two different time delays were examined. The approximate analytical solutions of nonlinear vibration equations were obtained using the multiple scales perturbation method, and their stability under the condition of combination resonance was investigated. The optimal time delay for the controller of the system was analyzed and calculated. Finally, the results of this study were compared with those of other studies. The results showed that the dual-frequency excitation intensifies the nonlinear behavior of the system. Further, the influence of the feedback gain coefficients and time delays on the combination resonance of the system was examined, and the results showed that the efficiency of vibration control can be improved by adjusting the feedback gain and time delay parameters. This study aims to provide theoretical guidance for the application of viscoelastic graphene sheets.

The strain gradient-based torsional vibration analysis of micro-rotors with nonlinear flexural-torsional coupling

The powerful non-classical continuum theory of strain gradient elasticity is capable of effectively capturing small-scale effects in micro-structures. Based on this theory, a formulation is developed to investigate the coupled torsional-flexural vibrations of micro-rotors in the presence of inertia nonlinearities arising from the eccentricity and gyroscopic motion of the rotors. With the aid of a weighted-residual technique, coupled nonlinear partial differential equations of motion are truncated into a discrete model. Then, the resonance of the fundamental mode of the torsional vibration excited by the flexural vibration is investigated by utilizing the perturbation method of multiple scales. Moreover, a numerical simulation approach is conducted to confirm the validity of the analytical solution proposed by the multiple scales perturbation method. Obtained results for the resonant frequency and amplitude of the fundamental torsional mode indicate that the strain gradient theory can predict far more reliable results than the classical continuum mechanics for micro-rotors with very thin shafts.

Nonlinear dynamics of traveling beam with longitudinally varying axial tension and variable velocity under parametric and internal resonances

In this investigation, the nonlinear dynamics of an axially accelerating viscoelastic beam on a pully mounting system have been analyzed. The axial tension of the beam is modeled as a function of the traveling velocity, support stiffness parameter as well as spatial coordinate. Geometric cubic nonlinearity in the equation of motion is due to elongation in the neutral axis of the beam. The integro-partial differential equation of motion of the axially accelerating beam associated with the simply supported end conditions is solved analytically by adopting the direct perturbation method of multiple time scales. As a result, a set of complex variable modulation equations is generated, which governs the modulation of amplitude and phase. This set of modulated equations is numerically solved to explore the influence of the support stiffness parameter upon the stability and bifurcation of the beam which has not been addressed in the existing literature. Apart from this, the impact of fluctuating velocity component, viscoelastic coefficient, longitudinal stiffness parameter, internal and parametric frequency detuning parameters on the stability and bifurcation analysis is studied, revealing significant dynamic characteristics of the traveling system. The fourth-order Runge–Kutta method is applied is to find the dynamic solution of the system. The system displays stable periodic, quasi-periodic, and mixed-mode dynamic responses along with the unstable chaotic behavior for a specific set of system parameters. The results obtained through an analytical–numerical approach may help the design and operation of an axially moving beam.

Parametrically excited axially accelerating viscoelastic beam subjected to time-varying axial speed, longitudinally varying axial tension and internal resonance

The present work investigates the nonlinear characteristics of an axially moving viscoelastic beam of the Kelvin–Voigt model on a pulley mounting system. The equation of motion is developed by including the terms like external and internal damping, geometric cubic nonlinearity, Coriolis, time-dependent axial speed, and longitudinally varying axial tension. The direct perturbation method of multiple time scales (MMS) is adopted to get an approximate solution for the nonlinear integro-partial differential equation of motion which leads to a set of complex variable modulation equations. These modulated equations are solved numerically to study the trivial state stability plot considering the effect of the support stiffness parameter for the first time. In addition to this, the impact of material, and viscous damping parameters on stability boundary is also studied. Continuation algorithm is implemented to analyse the influence of the support stiffness parameter, internal and parametric detuning parameters, fluctuating velocity component, and longitudinal stiffness parameter on the stability and bifurcation of the steady state solutions. Variation in internal frequency detuning parameter and variable axial speed component reveals different kinds of equilibrium solution curves in the nonlinear travelling system. The stability and bifurcation features observed in the present nonlinear system with change in the above control parameters are interesting and also not available in the existing literature.

Nonlinear Modes in Shear Horizontal Wave Propagation–Analytical and Numerical Analysis

Nonlinear shear horizontal guided wavefield is investigated analytically in plates. The major focus is on the effect of nonlinear material parameters, excitation frequency, wave number and wave amplitude on nonlinear mode generation. The analytical solution is based on the multiple-scale perturbation method and modal decomposition approach. The approximate analytical solution is validated using the Local Interaction Simulation Approach, implemented for nonlinear shear wave propagation. The results show that the analysed shear horizontal wavefield is distorted by material nonlinearity. The resulting displacement field can be interpreted in a form of nonlinear modes.

Effective heat conductivity of a composite with hexagonal lattice of perfectly conducting circular inclusions: An analytical solution

AbstractPaper is devoted to the effective stationary heat conductivity for the fibre composite materials. We are aimed on getting on analytical expression for effective thermal conductivity coefficient. Asymptotic homogenization approach, based on the multiple scale perturbation method, is used. This allows to reduce the original boundary value problem in multiply connected domain to the sequence of boundary value problems in simply connected domains. These problems include: the local problem for the periodically repeated cell and homogenized problem with effective coefficient. It is shown that for densely packed and high contrast fibre composites, the cell problem can be solved analytically. For this aim, lubrication approach (asymptotics of thin layer) has been employed. We also generalise obtained solution to the case of medium‐sized inclusions in the framework of the Padé approximants.

Eliminate the Nonlinear Oscillations of the Modified duffing Equation by using the Nonlinear Integrated Positive Position Feedback

In this article, the nonlinear integrated positive position feedback (NIPPF) control adds to a nonlinear dynamical system modeled as the well known Duffng oscillators. This control is proposed to mitigate system nonlinear vibrations. The whole system mathematical model is analyzed by applying the multiple time scales perturbation method. The slow- ow modulation equations that govern the oscillation amplitudes of both the main system and controller are derived. The stability of the steady-state solution is presented and studied applying frequency response equations near the simultaneous primary and internal resonance cases. Before and after (NIPPF) control the nonlinear systems' steady-state amplitude are examined, the comparison is made to validate the closeness between the numerical solution and the analytical perturbative one at time-history and frequency response curves.

Quantum and Relativistic Effects on the KdV and Envelope Solitons in Ion-Plasma Waves

In this article, we investigate the linear and nonlinear behavior of ion-plasma waves in a two-component plasma containing ions and dust particles. We have studied the linear dispersion characteristics, nonlinear KdV solitons, and amplitude modulated electrostatic waves. Such ion-plasma wave modes in dust-ion plasma have been theoretically investigated by incorporating quantum diffraction and relativistic effects. Numerical analysis of the linear dispersion relation with variations of different parameters has been carried out. By employing the reductive perturbation technique the KdV equation describing the small amplitude solitary profile has been studied. Using the standard multiple scale perturbation method, a nonlinear Schrödinger (NLS) equation has been derived by including quantum and relativistic effects, which describes the nonlinear evolution of the ion-plasma waves in dust-ion plasma. The wave is found to become modulationally unstable beyond certain critical wavenumber. The critical wavenumber and the growth rate of modulational instability are shown to depend significantly on the relativistic motion of plasma particles and the quantum diffraction effect. The results presented in the article are expected to be helpful for understanding the propagation of finite amplitude ion-plasma waves in some laboratory and astrophysical plasma systems. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PACS</i> —52.59.Hq, 67.10.Jn, 52.20 −j, 52.25 −b, 52.30 −d, 52.35 Mw, 52.65 Vv