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Showing papers on "Multiple-scale analysis published in 2003"


Journal ArticleDOI
TL;DR: In this article, an approximation of the resonant non-linear normal modes of a general class of weakly nonlinear one-dimensional continuous systems with quadratic and cubic geometric nonlinearities is constructed for the cases of two-to-one, one-toone, and three-to one internal resonances.
Abstract: Approximations of the resonant non-linear normal modes of a general class of weakly non-linear one-dimensional continuous systems with quadratic and cubic geometric non-linearities are constructed for the cases of two-to-one, one-to-one, and three-to-one internal resonances. Two analytical approaches are employed: the full-basis Galerkin discretization approach and the direct treatment, both based on use of the method of multiple scales as reduction technique. The procedures yield the uniform expansions of the displacement field and the normal forms governing the slow modulations of the amplitudes and phases of the modes. The non-linear interaction coefficients appearing in the normal forms are obtained in the form of infinite series with the discretization approach or as modal projections of second-order spatial functions with the direct approach. A systematic discussion on the existence and stability of coupled/uncoupled non-linear normal modes is presented. Closed-form conditions for non-linear orthogonality of the modes, in a global and local sense, are discussed. A mechanical interpretation of these conditions in terms of virtual works is also provided.

161 citations


Journal ArticleDOI
TL;DR: In this article, the response of a microbeam-based resonant sensor to superharmonic and subharmonic electric actuations using a model that incorporates the nonlinearities associated with moderately large displacements and electric forces is investigated.
Abstract: We investigate the response of a microbeam-based resonant sensor to superharmonic and subharmonic electric actuations using a model that incorporates the nonlinearities associated with moderately large displacements and electric forces. The method of multiple scales is used, in each case, to obtain two first-order nonlinear ordinary-differential equations that describe the modulation of the amplitude and phase of the response and its stability. We present typical frequency–response and force–response curves demonstrating, in both cases, the coexistence of multivalued solutions. The solution corresponding to a superharmonic excitation consists of three branches, which meet at two saddle-node bifurcation points. The solution corresponding to a subharmonic excitation consists of two branches meeting a branch of trivial solutions at two pitchfork bifurcation points. One of these bifurcation points is supercritical and the other is subcritical. The results provide an analytical tool to predict the microsensor response to superharmonic and subharmonic excitations, specifically the locations of sudden jumps and regions of hysteretic behavior, thereby enabling designers to safely use these frequencies as measurement signals. They also allow designers to examine the impact of various design parameters on the device behavior.

130 citations


Journal ArticleDOI
TL;DR: In this paper, an approximate solution of second-order relative motion equations is presented, where the equations of motion for a Keplerian orbit in spherical coordinates are expanded in Taylor series form using reference conditions consistent with that of a circular orbit.
Abstract: An approximate solution of second-order relative motion equations is presented. The equations of motion for a Keplerian orbit in spherical coordinates are expanded in Taylor series form using reference conditions consistent with that of a circular orbit. Only terms that are linear or quadratic in state variables are kept in the expansion. The method of multiple scales is employed to obtain an approximate solution of the resulting nonlinear differential equations, which are free of false secular terms. This new solution is compared with the previously known solution of the linear case to show improvement and with numerical integration of the quadratic differential equation to understand the error incurred by the approximation. In all cases, the comparison is made by computing the difference of the approximate state (analytical or numerical) from numerical integration of the full nonlinear Keplerian equations of motion. The results of two test cases show two orders of magnitude improvement in the second-order analytical solution compared with the previous linear solution over one period of the reference orbit.

95 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigated the non-linear forced responses of shallow suspended cables and analyzed the system by discretizing the equations of motion using the Galerkin procedure and applying the method of multiple scales to the resulting system of nonlinear ordinary-differential equations to obtain approximate solutions.

90 citations


Journal ArticleDOI
TL;DR: The method of multiple scales is used to derive the fourth-order nonlinear Schrodinger equation (NSEIV) that describes the amplitude modulations of the fundamental harmonic of Stokes waves on the surface of a medium-and large-depth (compared to the wavelength) fluid layer.
Abstract: The method of multiple scales is used to derive the fourth-order nonlinear Schrodinger equation (NSEIV) that describes the amplitude modulations of the fundamental harmonic of Stokes waves on the surface of a medium-and large-depth (compared to the wavelength) fluid layer The new terms of this equation describe the third-order linear dispersion effect and the nonlinearity dispersion effects As the nonlinearity and the dispersion decrease, the equation uniformly transforms into the nonlinear Schrodinger equation for Stokes waves on the surface of a finite-depth fluid that was first derived by Hasimoto and Ono The coefficients of the derived equation are given in an explicit form as functions of kh (h is the fluid depth, and k is the wave number) As kh tends to infinity, these coefficients transform into the coefficients of the NSEIV that was first derived by Dysthe for an infinite depth

67 citations


Journal ArticleDOI
TL;DR: In this paper, a simply supported Euler-Bernoulli beam with an intermediate support is considered and non-linear terms due to immovable end conditions leading to stretching of the beam are included in the equations of motion.

52 citations


Journal ArticleDOI
Ji-Huan He1
TL;DR: In this article, a linearized perturbation method is proposed to obtain unperturbed equations by linearizing the original nonlinear equation, not by setting ϵ = 0.
Abstract: A new perturbation technique called linearized perturbation method is proposed. Contrary to the traditional perturbation techniques, the unperturbed equations is obtained by linearizing the original nonlinear equation, not by setting ϵ = 0. Therefore, the obtained results are valid not only for small parameter, but also for very large values of ϵ. The present theory is processed as simple as the straightforward expansion, while omits the secular terms completely.

41 citations


Journal ArticleDOI
TL;DR: In this paper, a study on the validity of perturbation methods, such as the method of multiple scales, the Lindstedt-Poincare method and soon, in seeking for the periodic motions of the delayed dynamic system through an example of a Duffing oscillator with delayed velocity feedback, is presented.
Abstract: The paper presents a study on the validity of perturbation methods, suchas the method of multiple scales, the Lindstedt–Poincare method and soon, in seeking for the periodic motions of the delayed dynamic systemsthrough an example of a Duffing oscillator with delayed velocityfeedback. An important observation in the paper is that the method ofmultiple scales, which has been widely used in nonlinear dynamics, worksonly for the approximate solutions of the first two orders, and givesrise to a paradox for the third-order approximate solutions of delaydifferential equations. The same problem appears when theLindstedt–Poincare method is implemented to find the third-orderapproximation of periodic solutions for delay differential equations,though it is effective in seeking for any order approximation ofperiodic solutions for nonlinear ordinary differential equations. Apossible explanation to the paradox is given by the results obtained byusing the method of harmonic balance. The paper also indicates thatthese perturbation methods, despite of some shortcomings, are stilleffective in analyzing the dynamics of a delayed dynamic system sincethe approximate solutions of the first two orders already enable one togain an insight into the primary dynamics of the system.

36 citations


Journal ArticleDOI
TL;DR: In this article, the steady-state transverse vibration of a parametrically excited axially moving string with geometric nonlinearity is investigated, and the Boltzmann superposition principle is employed to characterize the material property of the string.
Abstract: Summary. The steady-state transverse vibration of a parametrically excited axially moving string with geometric nonlinearity is investigated in this paper. The Boltzmann superposition principle is employed to characterize the material property of the string. The method of multiple scales is applied directly to the governing equation, which is a nonlinear partial-differential-integral equation. The solvability condition of eliminating the secular terms is established. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the summation resonance are obtained. Some numerical examples showing effects of the viscoelastic parameter, the amplitude of excitation, the frequency of excitation, and the transport speed are presented.

31 citations


Proceedings ArticleDOI
20 Jul 2003
TL;DR: The results provide an analytical tool to predict a microsensor response to primary, superharmonic, and subharmonic excitations, specifically the locations of sudden jumps and regions of hysteretic behavior allowing designers to examine the impact of the design parameters on the device behavior.
Abstract: We present a nonlinear model of an electrically actuated microbeam-based resonant microsensor encompassing the electrostatic force of an air gap capacitor, the restoring force of the microbeam, and the axial load applied to the microbeam. The model accounts for moderately large deflections, dynamic loads, and the coupling between the mechanical and electrical forces. It accounts for the nonlinearity in the elastic restoring forces and the electric forces. A perturbation method, the method of multiple scales, is applied to the distributed-parameter system to study the local dynamics of the sensor under primary, superharmonic, and subharmonic excitations. In each case, we obtain two first-order nonlinear ordinary-differential equations that describe the modulation of the amplitude and phase of the response and its stability, and hence the bifurcations of the response. The perturbation results are validated by comparing them to experimental results. The DC electrostatic load affects the qualitative and quantitative nature of the frequency-response curves, resulting in either a softening or a hardening behavior. The results also show that an inaccurate representation of the system nonlinearities may lead to a qualitatively and quantitatively erroneous prediction of the frequency-response curves. The results provide an analytical tool to predict a microsensor response to primary, superharmonic, and subharmonic excitations, specifically the locations of sudden jumps and regions of hysteretic behavior allowing designers to examine the impact of the design parameters on the device behavior.

30 citations


Journal ArticleDOI
TL;DR: The propagation of nonlinear shallow water waves over a random seabed is studied and the effects of spatial attenuation (localization) on harmonic generation are studied.
Abstract: The propagation of nonlinear shallow water waves over a random seabed is studied. A bathymetry which fluctuates randomly from a constant mean adds multiple scattering to resonant interactions and harmonic generation. By the method of multiple scales, nonlinear evolution equations for the harmonic amplitudes are derived. Effects of multiple scattering are shown to be represented by certain linear damping terms with complex coefficients related to the correlation function of the seabed disorder. For any finite number of harmonics, an equation governing the total wave energy is derived. By numerical solution of the amplitude equations, the effects of spatial attenuation (localization) on harmonic generation are studied.

Journal ArticleDOI
TL;DR: In this article, the Riemann-Liouville fractional derivative of a rectangular plate described by three nonlinear differential equations is considered under the conditions of the internal resonance two-to-one.
Abstract: Nonlinear 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 two-to-one. Viscous properties of the system are described by the Riemann-Liouville fractional derivative. 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 fractional derivative is represented as a fractional power of the differentiation operator. 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.

Journal ArticleDOI
TL;DR: This work considers applications to dynamic pitchfork bifurcation, pattern formation below the threshold of stability, and transient dynamics of stochastic PDEs near this deterministic bifurancations.
Abstract: For systems of partial differential equations (PDEs) with locally cubic nonlinearities, which are perturbed by additive noise, we describe the essential dynamics for small solutions. If the system is near a change of stability, then a natural separation of time-scales occurs and the amplitudes of the dominant modes are given on a long time-scale by a stochastic ordinary differential equation. We consider applications to dynamic pitchfork bifurcation, pattern formation below the threshold of stability, and transient dynamics of stochastic PDEs near this deterministic bifurcations.

Journal ArticleDOI
TL;DR: In this article, a set of nonlinear differential equations is established by using Kane's method for the planar oscillation of flexible beams undergoing a large linear motion, and the modulation equations are numerically solved to obtain the steady-state response and the stability condition of the beam.
Abstract: A set of nonlinear differential equations is established by using Kane's method for the planar oscillation of flexible beams undergoing a large linear motion. In the case of a simply supported slender beam under certain average acceleration of base, the second natural frequency of the beam may approximate the tripled first one so that the condition of 3∶1 internal resonance of the beam holds true. The method of multiple scales is used to solve directly the nonlinear differential equations and to derive a set of nonlinear modulation equations forthe principal parametric resonance of the first mode combined with 3∶1 internal resonance between the first two modes. Then, the modulation equations are numerically solved to obtain the steady-state response and the stability condition of the beam. The abundant nonlinear dynamic behaviors, such as various types of local bifurcations and chaos that do not appear for linear models, can be observed in the case studies. For a Hopf bifurcation, the 4-dimensional modulation equations are reduced onto the central manifold and the type of Hopf bifurcation is determined. As usual, a limit cycle may undergo a series of period-doubling bifurcations and become a chaotic oscillation at last.

Journal ArticleDOI
TL;DR: It is revealed that the weakly nonlinear theory can give quantitatively adequate description up to the pressure amplitude of about 3% to the equilibrium pressure.
Abstract: Nonlinear cubic theory is developed to obtain a frequency response of shock-free, forced oscillations of an air column in a closed tube with an array of Helmholtz resonators connected axially. The column is assumed to be driven by a plane piston sinusoidally at a frequency close or equal to the lowest resonance frequency with its maximum displacement fixed. By applying the method of multiple scales, the equation for temporal modulation of a complex pressure amplitude of the lowest mode is derived in a case that a typical acoustic Mach number is comparable with the one-third power of the piston Mach number, while the relative detuning of a frequency is comparable with the quadratic order of the acoustic Mach number. The steady-state solution gives the asymmetric frequency response curve with bending (skew) due to nonlinear frequency upshift in addition to the linear downshift. Validity of the theory is checked against the frequency response obtained experimentally. For high amplitude of oscillations, an effect of jet loss at the throat of the resonator is taken into account, which introduces the quadratic loss to suppress the peak amplitude. It is revealed that as far as the present check is concerned, the weakly nonlinear theory can give quantitatively adequate description up to the pressure amplitude of about 3% to the equilibrium pressure.

Proceedings ArticleDOI
01 Jan 2003
TL;DR: In this paper, a model of parametrically excited two-degree-of-freedom nonlinear system with the quadratic and cubic nonlinearities is established to explore the periodic and quasiperiodic motions as well as the bifurcations and chaotic dynamics of the system.
Abstract: In this two-part paper, we investigate nonlinear dynamics in the rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The model of parametrically excited two-degree-of-freedom nonlinear system with the quadratic and cubic nonlinearities is established to explore the periodic and quasiperiodic motions as well as the bifurcations and chaotic dynamics of the system. The method of multiple scales is used to obtain the averaged equations in the case of primary parameter resonance and 1/2 subharmonic resonance. In Part I of the companion paper, numerical approach is applied to the averaged equations to find the periodic, quasiperiodic solutions and local bifurcations. It is found that there exist 2-period, 3-period, 4-period, 5-period, multi-period and quasiperiodic solutions in the rotor-AMB system with 8-pole legs and the time-varying stiffness. The catastrophic phenomena for the amplitude of nonlinear oscillations are first observed in the rotor-AMB system with 8-pole legs and the time-varying stiffness. The procedures of motion from the transient state chaotic motion to the steady state periodic and quasiperiodic motions are also found. The results obtained here show that there exists the ability of autocontrolling transient state chaos to the steady state periodic and quasiperiodic motions in the rotor-AMB system with 8-pole legs and the time-varying stiffness.Copyright © 2003 by ASME

Journal ArticleDOI
TL;DR: In this paper, Galerkin projections are used to obtain approximate realizations of the method of multiple scales and the related method of averaging are commonly used to study slowly modulated oscillations.
Abstract: The method of multiple scales and the related method of averaging are commonly used to study slowly modulated oscillations. If the system of interest is a slightly perturbed harmonic oscillator, then these techniques can be applied easily. If the unperturbed system is strongly nonlinear (though possibly conservative), then these methods can run into difficulties due to the impossibility of carrying out required analytical operations in closed form. In this paper, we abandon the requirement of closed form analytical treatment at all stages. Instead, Galerkin projections are used to obtain approximate realizations of the method of multiple scales. This paper adapts recent work using similar ideas for approximate realizations of the method of averaging. A key contribution of the present work is in the systematic identification and removal of secular terms in the general nonlinear case, a procedure that is more difficult than for the perturbed harmonic oscillator case, and that is unnecessary for averaging. A strength of the present work is that the heuristics (Galerkin) and asymptotics (multiple scales) are kept distinct, leaving room for systematic refinement of the former without compromising the asymptotic features of the latter.

Journal ArticleDOI
TL;DR: In this article, the authors presented both linear and nonlinear stability theories for characterization of viscoelastic film flows down on the outer surface of a rotating infinite vertical cylinder and derived a generalized nonlinear kinematic model to represent the physical system.

Journal ArticleDOI
TL;DR: In this paper, the spatio-temporal evolution of patterns in the marginally unstable Ekman layer driven by an applied shear stress is considered, where both the normal and tangential components of the Earth's angular velocity are included in a tangent plane approximation of the oceanic boundary layer at latitude λ.
Abstract: We consider the spatio-temporal evolution of patterns in the marginally unstable Ekman layer driven by an applied shear stress. Both the normal and tangential components of the Earth's angular velocity are included in a tangent plane approximation of the oceanic boundary layer at latitude λ. The fluid motion in a layer of finite depth as well as one of infinite depth is considered. The linear instability in the infinite depth case is known to depend on the direction of the applied stress for λ ¬= 90°, but this dependence is weak for the stress-driven Ekman layer. By contrast, the weakly nonlinear motion exhibits for finite and infinite depths qualitatively different dynamics for different stress directions. The problem is treated by the method of multiple scales. In the case of finite depth, this leads to the Davey-Hocking--Stewartson equation, an amplitude equation of complex Ginzburg-Landau type coupled to a Poisson equation. In the case of infinite depth, it leads to the anisotropic complex Ginzburg-Landau equation for the amplitude of the roll motion. Motions in both finite and infinite depth basins are explored by numerical simulation, and are shown to lead to chaotic dynamics for the modulation envelope in most cases. The statistics and the nature of the patterns produced in this motion are discussed.

Journal ArticleDOI
W.K. Lee1, M.H. Yeo1
TL;DR: In this article, the primary resonances of a clamped circular plate with an internal resonance were investigated, in which the natural frequencies of two asymmetric modes are commensurable and the response is expressed as an expansion in terms of the linear, free oscillation modes, and its amplitude is considered to be small but finite.

Journal ArticleDOI
TL;DR: In this paper, an effective procedure using the component mode synthesis (CMS) and the method of multiple scales (MS) or the harmonic balance (HB) method for the nonlinear vibration analysis of rotor systems is proposed.
Abstract: In this paper, an effective procedure using the component mode synthesis (CMS) and the method of multiple scales (MS) or the harmonic balance (HB) method for the nonlinear vibration analysis of rotor systems is proposed. In the procedure, the system is divided into components and the differential equation for the component for each perturbation order or frequency is derived. The equation of motion for the overall system is then obtained using the CMS method. The dynamic analysis of a rotor system is carried out using the MS or the HB method. The distinguishing feature of the proposed procedure is that the nonlinear restoring force term is expressed using modal coordinates in a convenient form. The order of the modal equation of motion and the calculation time, therefore, can be reduced. In the numerical example, it is shown that the analytical methods proposed in this paper are effective for the nonlinear vibration analysis of rotor systems.

Journal ArticleDOI
TL;DR: In this article, the authors investigate slow passage through the 2:1 resonance tongue in Mathieu's equation using numerical integration, and find that amplification or de-amplification can occur depending on the speed of travel through the tongue and the initial conditions.
Abstract: We investigate slow passage through the 2:1 resonance tongue in Mathieu's equation Using numerical integration, we find that amplification or de-amplification can occur The amount of amplification (or de-amplification) depends on the speed of travel through the tongue and the initial conditions We use the method of multiple scales to obtain a slow flow approximation The Wentzel-Kramers-Brillouin (WKB) method is then applied to the slow flow equations to obtain an analytic approximation

Journal ArticleDOI
TL;DR: It is found analytically that two-mode responses can occur at each primary resonance of two-degree-of-freedom system with quadratic and cubic nonlinearities to external excitations.

Journal ArticleDOI
TL;DR: In this article, the effect of the number of nodal diameters on non-linear interactions in asymmetric vibrations of a circular plate was investigated, where the plate is assumed to have an internal resonance in which the ratio of the natural frequencies of two asymmetric modes is three to one.

Journal ArticleDOI
TL;DR: In this paper, a modulated wave train in a nonlinear mono-inductance LC circuit is studied using the method of multiple scales in the general form, and the evolution of nonlinear excitations is governed by what is called the Modified Ginzburg-Landau Equation (MGLE).
Abstract: In this paper, a modulated wave train in a nonlinear monoinductance LC circuit is studied. Using the method of multiple scales in the general form, we establish that the evolution of nonlinear excitations is governed by what we called the Modified Ginzburg–Landau Equation (MGLE). Benjamin–Feir instability for the MGLE is analyzed.

Journal ArticleDOI
TL;DR: In this paper, an analytical method is presented for evaluation of the steady state periodic behavior of non-linear systems based on the substructure synthesis formulation and a multiple scales procedure, which is applied to the analysis of nonlinear responses.

Journal ArticleDOI
TL;DR: In this paper, the principal resonance of the Duffing oscillator to a narrow-band random parametric excitation was investigated and the behavior, stability and bifurcation of steady state response were studied by means of qualitative analyses.
Abstract: The principal resonance of Duffing oscillator to narrow-band random parametric excitation was investigated. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response were studied by means of qualitative analyses. The effects of damping, detuning, bandwidth and magnitudes of deterministic and random excitations were analyzed. The theoretical analyses were verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increases, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady state solutions.

Proceedings ArticleDOI
01 Jan 2003
TL;DR: In this paper, the authors studied the forced oscillations of micromechanical resonators fabricated as clamped-clamped composite structures, where piezoelectric actuation was used to excite these structures on the input side, and piezelectric sensing was carried out on the output side.
Abstract: Forced oscillations of micromechanical 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. Each resonator structure is modeled as a beam with stepwise properties and a distributed actuation. For weak damping and weak forcing, a nonlinear analysis is conducted to obtain an approximate solution of the system. In this analysis, the method of multiple scales is used on the governing partial differential system and a solution is sought to describe oscillations about a buckled position. The different modeling assumptions made are presented and discussed.Copyright © 2003 by ASME

Journal ArticleDOI
TL;DR: In this paper, a modified version of the reconstitution method, called MMS II, was proposed, which makes series expansions unnecessary for the frequency, damping and excitation amplitude.

Proceedings ArticleDOI
01 Jan 2003
TL;DR: In this article, the amplitude-frequency relation for the various flexural modes of an atomic force microscope cantilever in contact with a vibrating sample is investigated using Hertzian contact mechanics and the amount of softening is shown to depend on the linear contact stiffness and mode number.
Abstract: The nonlinear vibration response of an atomic force microscope cantilever in contact with a vibrating sample is investigated. The tip-sample contact is modeled using Hertzian contact mechanics. The method of multiple scales is used to analyze this problem in which it is assumed that the beam remains in contact with the moving surface at all times. The primary result from this analysis is the amplitude-frequency relation for the various flexural modes. The amplitude-frequency curves exhibit softening behavior as expected. The amount of softening is shown to depend on the linear contact stiffness as well as the specific mode. The modal sensitivity to nonlinearity is the result of the nonlinearity being restricted to a single position. The mode shape greatly affects the degree to which the nonlinearity influences the frequency response. The Hertzian restriction is then loosened slightly such that variations in nonlinear contact stiffness are examined. These results depend on the linear contact stiffness and mode number as well. The nonlinear vibration response is expected to provide new insight on the nonlinear tip mechanics present in these systems.Copyright © 2003 by ASME