Showing papers on "Multiple-scale analysis published in 2006"
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TL;DR: In this paper, the dynamic behavior of flexible rotor systems subjected to base excitation (support movements) is investigated theoretically and experimentally, focusing on behavior in bending near the critical speeds of rotation.
Abstract: The dynamic behavior of flexible rotor systems subjected to base excitation (support movements) is investigated theoretically and experimentally. The study focuses on behavior in bending near the critical speeds of rotation. A mathematical model is developed to calculate the kinetic energy and the strain energy. The equations of motion are derived using Lagrange equations and the Rayleigh-Ritz method is used to study the basic phenomena on simple systems. Also, the method of multiple scales is applied to study stability when the system mounting is subjected to a sinusoidal rotation. An experimental setup is used to validate the presented results.
98 citations
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TL;DR: In this paper, the effects of viscoelasticity on free vibration and stability of axially moving beam constrained by simple supports with torsion springs were investigated. But the authors focused on the stability of the axial speed of the beam.
Abstract: Vibration and stability are investigated for an axially moving beam constrained by simple supports with torsion springs. A scheme is proposed to derive natural frequencies and modal functions from given boundary conditions of an elastic beam moving at a constant speed. For a beam constituted by the Kelvin model, effects of viscoelasticity on the free vibration are analyzed via the method of multiple scales and demonstrated via numerical simulations. When the axial speed is characterized as a simple harmonic variation about the constant mean speed, the instability conditions are presented for axially accelerating viscoelastic beams in parametric resonance. Numerical examples show the effects of the constraint stiffness, the mean axial speed, and the viscoelasticity.
85 citations
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TL;DR: In this article, a refined integro-partial differential model is developed for a clamped?clamped composite beam structure and used for studying the nonlinear transverse vibrations of these resonators.
Abstract: Free and forced oscillations of piezoelectric, microelectromechanical resonators fabricated as clamped?clamped composite structures are studied in this effort. Piezoelectric actuation is used to excite these structures on the input side and piezoelectric sensing is carried out on the output side. A refined integro-partial differential model is developed for a clamped?clamped composite beam structure and used for studying the nonlinear transverse vibrations of these resonators. This model accounts for the longitudinal extension due to transverse vibrations, distributed actuation and axially varying properties across the length of the structure. Free oscillations about a post-buckled position are studied, and for weak damping and weak forcing, the method of multiple scales is used to obtain an approximate solution for the response to a harmonic forcing. Analytical predictions are also compared with experimental observations. The model development and the analysis can serve as a basis for analysing the responses of other composite microresonators.
84 citations
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TL;DR: In this paper, the attenuation caused by weak damping of harmonic waves through a discrete, periodic structure with frequency nominally within the propagation zone is studied, where the period of the structure consists of a linear stiffness and a weak linear/nonlinear damping.
59 citations
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TL;DR: In this article, the authors applied multiple scales to attack the nonlinear partial differential equation and the boundary conditions, which leads to the modulation equations for the primary resonance of either the first or third symmetric mode.
56 citations
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TL;DR: In this article, the effects of viscosity on free damped vibrations of a rectangular plate described by three nonlinear differential equations are considered when the plate is being under the conditions of the internal resonance one-to-one, and the internal additive or difference combinational resonances.
Abstract: Non-linear free damped vibrations of a rectangular plate described by three non-linear differential equations are considered when the plate is being under the conditions of the internal resonance one-to-one, and the internal additive or difference combinational resonances. Viscous properties of the system are described by the Riemann-Liouville fractional derivative of the order smaller than unit. The functions of the in-plane and out-of-plane displacements are determined in terms of eigenfunctions of linear vibrations with the further utilization of the method of multiple scales, in so doing the amplitude functions are expanded into power series in terms of the small parameter and depend on different time scales, but the fractional derivative is represented as a fractional power of the differentiation operator. It is assumed that the order of the damping coefficient depends on the character of the vibratory process and takes on the magnitude of the amplitudes’ order. The time-dependence of the amplitudes in the form of incomplete integrals of the first kind is obtained. Using the constructed solutions, the influence of viscosity on the energy exchange mechanism is analyzed which is intrinsic to free vibrations of different structures being under the conditions of the internal resonance. It is shown that each mode is characterized by its damping coefficient which is connected with the natural frequency of this mode by the exponential relationship with a negative fractional exponent. It is shown that viscosity may have a twofold effect on the system: a destabilizing influence producing unsteady energy exchange, and a stabilizing influence resulting in damping of the energy exchange mechanism.
56 citations
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TL;DR: In this paper, the Galerkin method is used to discretize the governing nonlinear integral-partial-differential equation and the method of multiple scales is applied to obtain the modulation equations in the case of primary resonance.
34 citations
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TL;DR: In this article, the amplitude and phase modulation equations of a shallow curved beam are analyzed for the case of parabolic curvature and sinusoidal curvature, respectively, and two-to-one internal resonances between any two modes of vibration are studied.
Abstract: Vibrations of shallow curved beams are investigated. The rise function of the beam is assumed to be small. Sinusoidal and parabolic curvature functions are examined. The immovable end conditions result in mid-plane stretching of the beam which leads to nonlinearities. The beam is resting on an elastic foundation. The method of multiple scales, a perturbation technique, is used in search of approximate solutions of the problem. Two-to-one internal resonances between any two modes of vibration are studied. Amplitude and phase modulation equations are obtained. Steady state solutions and stability are discussed, and a bifurcation analysis of the amplitude and phase modulation equations are given. Conditions for internal resonance to occur are discussed, and it is found that internal resonance is possible for the case of parabolic curvature but not for that of sinusoidal curvature.
32 citations
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TL;DR: In this paper, a general methodology is developed for obtaining multi-degree-of-freedom discretized models for structures in planar motion, where the motion of each substructure is represented by a finite number of substructure admissible functions.
Abstract: Nonlinear normal modes for elastic structures have been studied extensively in the literature. Most studies have been limited to small nonlinear motions and to structures with geometric nonlinearities. This work investigates the nonlinear normal modes in elastic structures that contain essential inertial nonlinearities. For such structures, based on the works of Crespo da Silva and Meirovitch, a general methodology is developed for obtaining multi-degree-of-freedom discretized models for structures in planar motion. The motion of each substructure is represented by a finite number of substructure admissible functions in a way that the geometric compatibility conditions are automatically assured. The multi degree-of-freedom reduced-order models capture the essential dynamics of the system and also retain explicit dependence on important physical parameters such that parametric studies can be conducted. The specific structure considered is a 3-beam elastic structure with a tip mass. Internal resonance conditions between different linear modes of the structure are identified. For the case of 1:2 internal resonance between two global modes of the structure, a two-mode nonlinear model is then developed and nonlinear normal modes for the structure are studied by the method of multiple time scales as well as by a numerical shooting technique. Bifurcations in the nonlinear normal modes are shown to arise as a function of the internal mistuning that represents variations in the tip mass in the structure. The results of the two techniques are also compared.
32 citations
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TL;DR: In this paper, the dynamical behavior of a simple rigid disk-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects, and the steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency response function method.
Abstract: The dynamical behavior of a parametrically excited simple rigid disk-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects. The principal parametric resonance case is considered and studied. The motion of the rotor is modeled by a coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought applying the method of multiple scales. A reduced system of four first-order ordinary differential equations are determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency response function method. The numerical results show that the system behavior includes multiple solutions, jump phenomenon, and sensitive dependence on initial conditions. It is also shown that the system parameters have different effects on the nonlinear response of the rotor. Results are compared to previously published work.
32 citations
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TL;DR: In this article, the effects of shear deformation and rotary inertia on the large amplitude vibration of a doubly clamped microbeam are investigated, and the results show that nonlinearity acts in the direction of increasing the natural frequency of the doubly-clamped microbeams.
Abstract: In this paper, the large amplitude free vibration of a doubly clamped microbeam is considered. The effects of shear deformation and rotary inertia on the large amplitude vibration of the microbeam are investigated. To this end, first Hamilton’s principle is used in deriving the partial differential equation of the microbeam response under the mentioned conditions. Then, implementing the Galerkin’s method the partial differential equation is converted to an ordinary nonlinear differential equation. Finally, the method of multiple scales is used to determine a second-order perturbation solution for the obtained ODE. The results show that nonlinearity acts in the direction of increasing the natural frequency of the doubly clamped microbeam. Shear deformation and rotary inertia have significant effects on the large amplitude vibration of thick and short microbeams.
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TL;DR: In this paper, a non-linear partial-differential equation governing the transverse motion of axially moving viscoelastic beams excited by the vibration of the supporting foundation is derived from the dynamical, constitutive equations and geometrical relations.
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TL;DR: In this article, the stability of transverse parametric vibration of axially accelerating viscoelastic beams is investigated and the stability conditions are obtained for combination and principal parametric resonance.
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TL;DR: In this paper, the Galerkin's method is employed to simplify the governing equations to a set of ordinary differential equations with two degrees of freedom for the nonlinear transverse vibrations of a string-beam coupled system.
Abstract: In this paper, research on nonlinear dynamic behavior of a string-beam coupled system subjected to parametric and external excitations is presented The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system The Galerkin's method is employed to simplify the governing equations to a set of ordinary differential equations with two degrees-of-freedom The case of 1:2 internal resonance between the modes of the beam and string, principal parametric resonance for the beam, and primary resonance for the string is considered The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system Based on the averaged equation obtained here, the techniques of phase portrait, waveform, and Poincare map are applied to analyze the periodic and chaotic motions It is found from numerical simulations that there are obvious jumping phenomena in the resonant response–frequency curves It is indicated from the phase portrait and Poincare map that period-4, period-2, and periodic solutions and chaotic motions occur in the transverse nonlinear vibrations of the string-beam coupled system under certain conditions
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TL;DR: In this article, the nonlinear equations of motion of planar bending vibration of an inextensible viscoelastic carbon nanotube (CNT)-reinforced cantilevered beam are derived.
Abstract: The nonlinear equations of motion of planar bending vibration of an inextensible viscoelastic carbon nanotube (CNT)-reinforced cantilevered beam are derived. The viscoelastic model in this analysis is taken to be the Kelvin–Voigt model. The Hamilton principle is employed to derive the nonlinear equations of motion of the cantilever beam vibrations. The nonlinear part of the equations of motion consists of cubic nonlinearity in inertia, damping, and stiffness terms. In order to study the response of the system, the method of multiple scales is applied to the nonlinear equations of motion. The solution of the equations of motion is derived for the case of primary resonance, considering that the beam is vibrating due to a direct excitation. Using the properties of a CNT-reinforced composite beam prototype, the results for the vibrations of the system are theoretically and experimentally obtained and compared.
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TL;DR: In this paper, the authors explored the advantages and characteristics of nonlinear butyl rubber (type IIR) isolators in vibratory shear by comparison with linear isolators, and developed phenomenological models of the effective storage modulus and effective loss factor of a rubber isolator material as a function of excitation amplitude.
Abstract: The purpose of this study is to explore the advantages and characteristics of nonlinear butyl rubber (type IIR) isolators in vibratory shear by comparison with linear isolators. It is known that the mechanical properties of viscoelastic materials exhibit significant frequency and temperature dependence, and in some cases, nonlinear dynamic behavior as well. Nonlinear characteristics in shear deformation are reflected in mechanical properties such as stiffness and damping. Furthermore, even when the excitation amplitude is small the response amplitude may often be large enough that nonlinearities cannot be ignored. The treatment involves developing phenomenological models of the effective storage modulus and effective loss factor of a rubber isolator material as a function of excitation amplitude. The transmissibility of a nonlinear viscoelastic isolator is compared with that of a linear isolator using an equivalent linear damping coefficient. Forced resonance vibration and impedance tests are used to characterize nonlinear parameters and to measure the normalized transmissibility. It is found that as the excitation amplitude of the nonlinear viscoelastic isolator increases, the response amplitude decreases and the transmissibility is improved over that of the linear isolator for excitation frequency that exceeds a particular value governed by the temperature and excitation amplitude. The method of multiple scales and numerical simulations are used to predict the response characteristics of the isolator based on the phenomenological modeling under different values of system parameters.
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TL;DR: In this paper, the authors applied the method of multiple scales to obtain periodic solutions of a two-pulley belt system with clearance-type nonlinearity, and evaluated the validity of the perturbation method for such strong non-linearity.
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TL;DR: In this article, the global bifurcations in modal interactions of an imperfect circular plate with one-to-one internal resonance were investigated, and the equations governing nonlinear oscillations of the plate were reduced to a system of non-autonomous ordinary differential equations via Galerkin's procedure.
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TL;DR: In this paper, a control law based on quantic velocity feedback is proposed to suppress the vibrations of the first mode of a cantilever beam when subjected to primary and principal parametric excitations.
Abstract: A non-linear control law is proposed to suppress the vibrations of the first mode of a cantilever beam when subjected to primary and principal parametric excitations. The dynamics of the beam are modeled with a second-order non-linear ordinary-differential equation. The model accounts for viscous damping air drag, and inertia and geometric non-linearities. A control law based on quantic velocity feedback is proposed. The method of multiple scales method is used to derive two-first ordinary differential equations that govern the evolution of the amplitude and phase of the response. These equations are used to determine the steady state responses and their stability. Amplitude and phase modulation equations as well as external force–response and frequency–response curves are obtained. Numerical simulations confirm this scenario and detect chaos and unbounded motions in the instability regions of the periodic solutions.
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TL;DR: In this paper, a simply supported damped Euler-Bernoulli beam with immovable end conditions is considered and the concept of non-ideal boundary conditions is applied to the beam problem.
Abstract: A simply supported damped Euler-Bernoulli beam with immovable end conditions are considered. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the boundaries are assumed to allow small deflections and moments. Approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique.
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TL;DR: In this paper, the principal resonance of a 2dof nonlinear oscillator due to bounded random excitations is investigated and the modulation of response amplitude and phase are derived by the method of multiple scales Steadystate moments for the response amplitude of the system are determined through the linearized Ito differential equation.
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TL;DR: In this paper, an analytical study of the simultaneous principal parametric resonances of two coupled Duffing equations with time delay state feedback is proposed and the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control.
Abstract: This paper presents an analytical study of the simultaneous principal parametric resonances of two coupled Duffing equations with time delay state feedback. The concept of an equivalent damping related to the delay feedback is proposed and the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. The method of multiple scales is used to determine a set of ordinary differential equations governing the modulation of the amplitudes and phases of the two modes. The first order approximation of the resonances are derived and the effect of time delay on the resonances is investigated. The fixed points correspond to a periodic motion for the starting system and we show the frequency–response curves. We analyse the effect of time delay and the other different parameters on these oscillations. The stability of the fixed points is examined by using the variational method. Numerical solutions are carried out and graphical representations of the results are presented and discussed. Increasing in the time delay τ given decreasing and increasing in the regions of definition and stability respectively and the first mode has decreased magnitudes. The multivalued solutions disappear when decreasing the coefficients of cubic nonlinearities of the second mode α3 and the detuning parameter σ2 respectively. Both modes shift to the left for increasing linear feedback gain v1 and the coefficient of parametric excitation f1 respectively.
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TL;DR: In this paper, the authors apply fundamental aspects of fracture mechanics to define an elliptical crack, and the local stress field and loading conditions, arbitrarily located at some point in the plate, and then derive an analytical expression for this that can be incorporated into the PDE for an edge loaded plate with various possible boundary conditions.
Abstract: Recent NATO funded research on methods for detection and interpretation methodologies for damage detection in aircraft panel structures has motivated work on low-order nonlinear analytical modelling of vibrations in cracked isotropic plates, typically in the form of aluminium aircraft panels. The work applies fundamental aspects of fracture mechanics to define an elliptical crack, and the local stress field and loading conditions, arbitrarily located at some point in the plate, and then derives an analytical expression for this that can be incorporated into the PDE for an edge loaded plate with various possible boundary conditions. The plate PDE is converted into a nonlinear Duffing-type ODE in the time domain by means of a Galerkin procedure and then an arbitrarily small perturbation parameter is introduced into the equation in order to apply an appropriate solution method, in this case the method of multiple scales. This is used to solve the equation for the vibration in the cracked plate for the chosen boundary conditions, which, in turn, leads to an approximate analytical solution. The solution is discussed in terms of the perturbation approximations that have been applied and highlights the phenomenology inherent within the problem via the specific structures of the analytical solution.
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TL;DR: In this article, the primary and subharmonic resonance of order one-third of a cantilever beam under state feedback control with a time delay is investigated. But the effect of time delay on the resonance is investigated only from the viewpoint of vibration control.
Abstract: The primary and subharmonic resonance of order one-third of a cantilever beam under state feedback control with a time delay are investigated. Using the method of multiple scales, we obtain two slow flow equations for the amplitude and phase. The first-order approximate solution is derived and the effect of time delay on the resonance is investigated. The concept of an equivalent damping, related to the delay feedback, is proposed and an appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. The fixed points corresponding to the periodic motion of the starting system are determined, and the frequency-response and external excitation-response curves are shown. Bifurcation analysis is conducted in order to examine the stability of the system.
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TL;DR: In this paper, the effect of dry friction on the response of a system that implements a vibration absorber is discussed, which is basically a plant with a PMDC motor excited by a harmonic forcing term and coupled with a quadratic nonlinear controller.
Abstract: Application of saturation to provide active nonlinear vibration control was introduced not long ago. Saturation occurs when two natural frequencies of a system with quadratic nonlinearities are in a ratio of around 2:1 and the system is excited at a frequency near its higher natural frequency. Under these conditions, there is a small upper limit for the high-frequency response and the rest of the input energy is channeled to the low-frequency mode. In this way, the vibration of one of the degrees of freedom of a coupled 2 degrees of freedom system is attenuated. In the present paper, the effect of dry friction on the response of a system that implements this vibration absorber is discussed. The system is basically a plant with a permanent magnet DC (PMDC) motor excited by a harmonic forcing term and coupled with a quadratic nonlinear controller. The absorber is built in electric circuitry and takes advantage of the saturation phenomenon. The method of multiple scales is used to find approximate solutions. Various response regimes of the closed-loop system as well as the stability of these regimes are studied and the stability boundaries are obtained. Especial attention is paid on the effect of dry friction on the stability boundaries. It is shown that while dry friction tends to shrink the stable region in some parts, it enlarges other parts of the stable region. To verify the theoretical results, they have been compared with numerical solution and good agreement between the two is observed.
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TL;DR: In this article, the authors derived the modulational instability criterion from the discrete multiple scales approach, which is a criterion for the disintegration of the initial modulated waves into a train of pulses.
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TL;DR: In this article, the principal resonance of the Duffing oscillator to combined deterministic and random external excitation was investigated and the theoretical analysis were verified by numerical results using the double peak probability density function.
Abstract: The principal resonance of Duffing oscillator to combined deterministic and random external excitation was investigated. The random excitation was taken to be white noise or harmonic with separable random amplitude and phase. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The one peak probability density function of each of the two stable stationary solutions was calculated by the linearization method. These two one-peak-density functions were combined using the probability of realization of the two stable stationary solutions to obtain the double peak probability density function. The theoretical analysis are verified by numerical results.
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TL;DR: In this paper, multiple-scale analysis is employed for the analysis of plane wave refraction at a nonlinear slab, and it is demonstrated that the perturbation method will lead to a nonuniformly valid approximation to the solution of the nonlinear wave equation.
Abstract: Multiple-scale analysis is employed for the analysis of plane wave refraction at a nonlinear slab. It will be demonstrated that the perturbation method will lead to a nonuniformly valid approximation to the solution of the nonlinear wave equation. To construct a uniformly valid approximation, we will exploit multiple- scale analysis. Using this method, we will derive the zeroth- order approximation to the solution of the nonlinear wave equation analytically. This approximate solution clearly shows the effects of self-phase modulation (SPM) and cross-phase modulation (XPM) on plane wave refraction at the nonlinear slab. In fact, the obtained zeroth-order approximation is very accurate and there is not any need for derivation of higher-order approximations. As will be shown, the proposed method can be generalized to the rigorous study of nonlinear wave propagation in one-dimensional photonic band-gap structures.
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TL;DR: In this article, the authors analyzed the dynamics of a single bubble in a time-varying pressure field, for parameter ranges representative of those experimental conditions, using the method of multiple scales (MMS).
Abstract: In recent experimental work, Chatterjee and Arakeri have demonstrated that an imposed acoustic field of sufficiently high strength and frequency can suppress or control cavitation. In this paper, we analytically study the equation governing the dynamics of a single bubble in a time-varying pressure field, for parameter ranges representative of those experimental conditions. The governing equation is strongly nonlinear and intractable in general; however, for the parameter ranges of interest, we are able to nondimensionalize and scale the governing equation into a form that, though still strongly nonlinear, is amenable to analysis using the method of multiple scales (MMS) based on an arbitrarily chosen “small” parameter ∊. Removal of secular terms, a key step in the MMS, raises an interesting issue which we discuss. Second order MMS gives the slow average evolution of the bubble radius. Numerical solutions of the original equation match the MMS approximation well on time scales of \(\cal O\)(1/∊). The MMS approximation also provides insight into the roles played by relevant physical parameters in the system. Our results provide theoretical support for the abovementioned experimental results.
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TL;DR: In this article, the nonlinear localized excitations and gap multi-instability phenomena in the array of optical fibers are investigated analytically by means of the method of multiple scales combined with the averaging method.
Abstract: The nonlinear localized excitations and gap multi-instability phenomena in the array of optical fibers are investigated analytically by means of the method of multiple scales combined with the averaging method. The perturbative analysis in nonlinear discrete systems leads to a system of four coupled nonlinear oscillated equations without second members. The method of multiple scales is used to show the different possibilities of instability. The averaging method is considered to show how the angular frequency received the perturbation with the evolution of time.