Showing papers on "Multiple-scale analysis published in 2009"
TL;DR: In this article, the free vibrations of an in-extensional simply supported rotating shaft with nonlinear curvature and inertia are considered, and the results of perturbation method are validated with numerical simulations.
89 citations
TL;DR: In this article, the authors considered a three-part system, where one is subjected to an elastic foundation, and two are free to vibrate, and the equations of orders one and epsilon were derived using the method of multiple scales.
80 citations
TL;DR: The creation, stability, and interaction of ILMs are investigated, and it is shown that they can form bound states, and that under certain conditions one ILM can split into two.
Abstract: We study intrinsic localized modes (ILMs), or solitons, in arrays of parametrically driven nonlinear resonators with application to microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). The analysis is performed using an amplitude equation in the form of a nonlinear Schrodinger equation with a term corresponding to nonlinear damping (also known as a forced complex Ginzburg-Landau equation), which is derived directly from the underlying equations of motion of the coupled resonators, using the method of multiple scales. We investigate the creation, stability, and interaction of ILMs, show that they can form bound states, and that under certain conditions one ILM can split into two. Our findings are confirmed by simulations of the underlying equations of motion of the resonators, suggesting possible experimental tests of the theory.
65 citations
TL;DR: In this article, the bifurcations and chaotic dynamics of a simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate are studied for the first time.
Abstract: The bifurcations and chaotic dynamics of a simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate are studied for the first time, which are simultaneously forced by the transverse, in-plane excitations and the excitation loaded by piezoelectric layers. Based on the Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the composite laminated piezoelectric rectangular plate are derived by using the Hamilton’s principle. The Galerkin’s approach is used to discretize partial differential governing equations to a two-degreeof-freedom nonlinear system under combined the parametric and external excitations. The method of multiple scales is employed to obtain the four-dimensional averaged equation. Numerical method is utilized to find the periodic and chaotic responses of the composite laminated piezoelectric rectangular plate. The numerical results indicate the existence of the periodic and chaotic responses in the averaged equation. The influence of the transverse, in-plane and piezoelectric excitations on the bifurcations and chaotic behaviors of the composite laminated piezoelectric rectangular plate is investigated numerically.
62 citations
TL;DR: In this article, an inviscid Burgers-Hilbert equation was proposed as a model equation for non-zero linearized wave with constant frequency and the authors derived a cubically nonlinear, quasi-linear, non-local asymptotic equation for weakly nonlinear solutions.
Abstract: Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers-Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi-linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two-dimensional inviscid, incompressible fluid flows. Thus, the Burgers-Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers-Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near-identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers-Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small-amplitude smooth solutions of the Burgers-Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.
62 citations
TL;DR: In this paper, the nonlinear free transverse vibrations of a nano-beam on simple supports are investigated based on nonlocal elasticity theory and the governing equation is proposed by considering geometric nonlinearity due to finite stretching of the beam.
Abstract: The nonlinear free transverse vibrations of a nano-beam on simple supports are investigated based on nonlocal elasticity theory. The governing equation is proposed by considering geometric nonlinearity due to finite stretching of the beam. The method of multiple scales is applied to the governing equation to evaluate the nonlinear natural frequencies. Numerical examples are presented to demonstrate the analytical results and highlight the contributions of the nonlinear term and nonlocal effect.
47 citations
TL;DR: In this article, a new perturbation algorithm combining the Method of Multiple Scales and Lindstedt-Poincare techniques is proposed for the first time, which combines the advantages of both methods.
Abstract: A new perturbation algorithm combining the Method of Multiple Scales and Lindstedt-Poincare techniques is proposed for the first time. The algorithm combines the advantages of both methods. Convergence to real solutions with large perturbation parameters can be achieved for both constant amplitude and variable amplitude cases. Three problems are solved: Linear damped vibration equation, classical duffing equation and damped cubic nonlinear equation. Results of Multiple Scales, new method and numerical solutions are contrasted. The proposed new method produces better results for strong nonlinearities.
41 citations
TL;DR: In this paper, a parametric resonance of axially moving beams with time-dependent speed is analyzed based on the Timoshenko model and the Hamilton principle is employed to obtain the governing equation, which is a nonlinear partial differential equation due to the geometric nonlinearity caused by the finite stretch of the beam.
Abstract: In this paper, parametric resonance of axially moving beams with time-dependent speed is analyzed, based on the Timoshenko model. The Hamilton principle is employed to obtain the governing equation, which is a nonlinear partial-differential equation due to the geometric nonlinearity caused by the finite stretch of the beam. The method of multiple scales is applied to predict the steady-state response. The expression of the amplitude of the steady-state response is derived from the solvability condition of eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by using the Lyapunov linearized stability theory. Some numerical examples are presented to demonstrate the effects of speed pulsation and the nonlinearity in the first two principal parametric resonances.
41 citations
TL;DR: In this paper, nonlinear vibrations under weak and strong external excitations of axially moving beams are analyzed based on the Timoshenko model, accounting for the geometric nonlinearity caused by finite stretching of the beams.
39 citations
TL;DR: In this paper, nonlinear vibrations of a general model of continuous system are considered. The model consists of arbitrary linear and cubic operators, and the equation of motion is solved by the method of multiple scales (a perturbation method).
35 citations
TL;DR: In this article, the authors survey the recent advances in the dynamics and control of time-delay systems, with emphasis on the singular perturbation methods, such as the method of multiple scales, the method averaging, and two newly developed methods, energy analysis and the pseudo-oscillator analysis.
Abstract: This review article surveys the recent advances in the dynamics and control of time-delay systems, with emphasis on the singular perturbation methods, such as the method of multiple scales, the method of averaging, and two newly developed methods, the energy analysis and the pseudo-oscillator analysis. Some examples are given to demonstrate the advantages of the methods. The comparisons with other methods show that these methods lead to easier computations and higher accurate prediction on the local dynamics of time-delay systems near a Hopf bifurcation.
TL;DR: In this article, the formation and propagation of optical solitons in an asymmetric double quantum-well structure was investigated using a standard method of multiple scales and a nonlinear Schr\"odinger (NLS) equation with some high-order correction terms that describe effects of linear and differential absorption, nonlinear dispersion, delay response of nonlinear refractive index, and third-order dispersion of a probe field.
Abstract: We investigate the formation and propagation of optical solitons in an asymmetric double quantum-well structure. Using a standard method of multiple scales we derive a nonlinear Schr\"odinger (NLS) equation with some high-order correction terms that describe effects of linear and differential absorption, nonlinear dispersion, delay response of nonlinear refractive index, and third-order dispersion of a probe field. We show that in order to make slowly varying envelope approximation be valid an excitation scheme of interband transition should be adopted. We also show that for realistic quantum-well parameters the probe field with time length of picosecond or shorter must be used to make dispersion and nonlinear lengths of the system be smaller than absorption length, only by which a shape-preserving propagation of optical solitons is available. In addition, we clarify validity domains for the perturbed NLS equation as well as the high-order NLS equation and provide various optical soliton solutions in different regimes both analytically and numerically. We demonstrate that the solitons obtained have ultraslow propagating velocity and can be generated under very low input light intensity.
TL;DR: In this article, a weakly nonlinear model with square and cubic terms in both structural stiffness and regenerative terms was proposed to represent self-excited vibrations in machining, and an approximate solution was derived by using the method of multiple scales.
Abstract: Chatter is an instability condition in machining processes characterized by nonlinear behavior, such as the presence of limit cycles, jump phenomenon, subcritical Hopf and period doubling bifurcations. Although the use of nonlinear techniques has provided a better understanding of chatter, neither a unifying model nor an exact solution has yet been developed due to the intricacy of the problem. This work proposes a weakly nonlinear model with square and cubic terms in both structural stiffness and regenerative terms, to represent self-excited vibrations in machining. An approximate solution is derived by using the method of multiple scales. In addition, a qualitative analysis of the effect of the nonlinear parameters on the stability of the system is performed. The structural cubic term gives a better representation of the nonlinear behavior, whereas the square term represents a distant attractor in the stability chart. Instability due to subcritical Hopf bifurcations is established in terms of the eigenvalues of the model in normal form. An important contribution of this analysis is the representation of hysteresis in terms of new lobes within the conventional stability limits, useful in restoring stability. This analysis leads to a further understanding of the nonlinear behavior of regenerative chatter.
TL;DR: In this paper, the nonlinear response of a shallow suspended cable with multiple internal resonances to the primary resonance excitation is investigated, and the effects of the excitation amplitude on the frequency-response curves of the cable are also analyzed.
TL;DR: In this article, a perturbation-incremental scheme (PIS) is proposed to investigate the periodic solution derived from Hopf bifurcation due to time delay in a system of first-order delayed differential equations.
Abstract: A new method, called perturbation-incremental scheme (PIS), is presented to investigate the periodic solution derived from Hopf bifurcation due to time delay in a system of first-order delayed differential equations. The method is summarized as three steps, namely linear analysis at critical value, perturbation and increment for continuation. The PIS can bypass and avoid the tedious calculation of the center manifold reduction (CMR) and normal form. Meanwhile, the PIS not only inherits the advantages of the method of multiple scales (MMS) but also overcomes the disadvantages of the incremental harmonic balance (IHB) method. Three delayed systems are used as illustrative examples to demonstrate the validity of the present method. The periodic solution derived from the delay-induced Hopf bifurcation is obtained in a closed form by the PIS procedure. The validity of the results is shown by their consistency with the numerical simulation. Furthermore, an approximate solution can be calculated in any required accuracy.
TL;DR: In this paper, a new perturbation algorithm combining the method of multiple scales (MS) and Lindstedt-Poincare techniques is applied to an equation with quadratic and cubic nonlinearities.
Abstract: The new perturbation algorithm combining the method of multiple scales (MS) and Lindstedt–Poincare techniques is applied to an equation with quadratic and cubic nonlinearities. Approximate analytical solutions are found using the classical MS method and the new method. Both solutions are contrasted with the direct numerical solutions of the original equation. For the case of strong nonlinearities, solutions of the new method are in good agreement with the numerical results, whereas the amplitude and frequency estimations of classical MS yield high errors. For strongly nonlinear systems, exact periods match well with the new technique while there are large discrepancies between the exact and classical MS periods. Copyright © 2009 John Wiley & Sons, Ltd.
TL;DR: In this paper, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled and the beam was resting on an elastic foundation and simply supported at both ends, and the effects of the concentrated mass on nonlinear vibrations were investigated.
Abstract: In this study, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled. The beam was resting on an elastic foundation and simply supported at both ends. Effects of the concentrated mass on nonlinear vibrations were investigated. Sinusoidal and parabolic type functions were used as curvature functions. Equations of motion have cubic nonlinearities because of elongations during vibrations. Damping and harmonic excitation terms were added to the equations of motion. Method of multiple scales, a perturbation technique, was used for solving integro-differential equation analytically. Natural frequencies were calculated exactly for different mass ratios, mass locations, curvature functions, and linear elastic foundation coefficients. Amplitude-phase modulation equations were found by considering primary resonance case. Effects of nonlinear terms on natural frequencies were calculated. Frequency–amplitude and frequency–response graphs were plotted. Finally effects of concentrated mass and chosen curvature function on nonlinear vibrations were investigated.
TL;DR: In this article, the authors investigated the non-resonant response and two types of primary resonances of the forced van der Pol-Duffing oscillator at non-reonant bifurcations of codimension two by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold.
Abstract: Non-resonant bifurcations of codimension two may appear in the controlled van der Pol–Duffing oscillator when two critical time delays corresponding to a double Hopf bifurcation have the same value. With the aid of centre manifold theorem and the method of multiple scales, the non-resonant response and two types of primary resonances of the forced van der Pol–Duffing oscillator at non-resonant bifurcations of codimension two are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. It is shown that the non-resonant response of the forced oscillator may exhibit quasi-periodic motions on a two- or three-dimensional (2D or 3D) torus. The primary resonant responses admit single and mixed solutions and may exhibit periodic motions or quasi-periodic motions on a 2D torus. Illustrative examples are presented to interpret the dynamics of the controlled system in terms of two dummy unfolding parameters and exemplify the periodic and quasi-periodic motions. The analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation.
TL;DR: In this article, the effect of the application of an alternating magnetic field on the large transverse vibration of a cantilever beam with tip mass is investigated using D'Alembert's principle, which is reduced to its nondimensional temporal form by using the generalized Galerkin method.
Abstract: In this work the effect of the application of an alternating magnetic field on the large transverse vibration of a cantilever beam with tip mass is investigated. The governing equation of motion is derived using D'Alembert's principle, which is reduced to its nondimensional temporal form by using the generalized Galerkin method. The temporal equation of motion of the system contains nonlinearities of geometric and inertial types along with parametric excitation and nonlinear damping terms. Method of multiple scales is used to determine the instability region and frequency response curves of the system. The influences of the damping, tip mass, amplitude of magnetic field strength, permeability, and conductivity of the beam material on the frequency response curves are investigated. These perturbation results are found to be in good agreement with those obtained by numerically solving the temporal equation of motion and experimental results. This work will find extensive applications for controlling vibration inflexible structures using a magnetic field.
TL;DR: In this paper, a current-conducting thin plate in a constant transverse magnetic field is considered, and nonlinear magneto-elastic vibration equations of the thin plate with two opposite sides simply supported are derived by the Galerkin method.
TL;DR: In this article, the dynamics of the workpiece of a lathe is simulated in the presented paper, and the governing equations of the rotating Rayleigh beam are derived by Hamilton's principle.
TL;DR: In this article, an active control method is applied to the system based on cubic velocity feedback and the results of perturbation solution have been verified through numerical simulations, where different effects of the system parameters have been reported.
Abstract: This paper is concerned with the nonlinear dynamics and vibration control of an electromechanical seismograph system with time-varying stiffness. The instrument consists of an electrical part coupled to mechanical one and is used to record the vibration during earthquakes. An active control method is applied to the system based on cubic velocity feedback. The electromechanical system is subjected to parametric and external excitations and modeled by a coupled nonlinear ordinary differential equations. The method of multiple scales is used to obtain approximate solutions and investigate the response of the system. The results of perturbation solution have been verified through numerical simulations, where different effects of the system parameters have been reported.
TL;DR: In this paper, a three-to-one internal resonance case is investigated for the vibrations of multiple stepped beams with cubic nonlinearities and a general approximate solution is found using the method of multiple scales (a perturbation technique).
Abstract: In this study, the vibrations of multiple stepped beams with cubic nonlinearities are considered. A three-to-one internal resonance case is investigated for the system. A general approximate solution to the problem is found using the method of multiple scales (a perturbation technique). The modulation equations of the amplitudes and the phases are derived for two modes. These equations are utilized to determine steady state solutions and their stabilities. It is assumed that the external forcing frequency is close to the lower frequency. For the numeric part of the study, the three-to-one ratio in natural frequencies is investigated. These values are observed to be between the first and second natural frequencies in the cases of the clamped-clamped and clamped-pinned supports, and between the second and third natural frequencies in the case of the pinned-pinned support. Finally, a numeric algorithm is used to solve the three-to-one internal resonance. The first mode is externally excited for the clamped-clamped and clamped-pinned supports, and the second mode is externally excited for the pinned-pinned support. Then, the amplitudes of the first and second modes are investigated when the first mode is externally excited. The amplitudes of the second and third modes are investigated when the second mode is externally excited. The force-response, damping-response, and frequency-response curves are plotted for the internal resonance modes of vibrations. The stability analysis is carried out for these plots.
01 Jan 2009
TL;DR: In this article, an analytical model for vibrations in a cracked rectangular plate is proposed, which is one of the results from a program of research on vibration based damage detection in aircraft panel structures.
Abstract: This study proposes an analytical model for vibrations in a cracked rectangular plate as one of the results from a programme of research on vibration based damage detection in aircraft panel structures. This particular work considers an isotropic plate, typically made of aluminium, and containing a crack in the form of a continuous line with its centre located at the centre of the plate, and parallel to one edge of the plate. The plate is subjected to a point load on its surface for three different possible boundary conditions, and one examined in detail. Galerkin's method is applied to reformulate the governing equation of the cracked plate into time dependent modal coordinates. Nonlinearity is introduced by appropriate formulations introduced by applying Berger's method. An approximate solution technique, the method of multiple scales, is applied to solve the nonlinear equation of the cracked plate. Results are presented in terms of natural frequency versus crack length and plate thickness, and the nonlinear amplitude response of the plate is calculated for one set of boundary conditions and three different load locations, over a practical range of external excitation frequencies.
01 Jan 2009
TL;DR: In this paper, a Galerkin-based reduced-order model is used to discretize the distributed-parameter model of the considered shallow arch, and the forced vibration response of the arch to a combined DC and AC harmonic load is determined when excited near its fundamental natural frequency.
Abstract: In this paper, we investigate theoretically and experimentally the static and dynamic behaviors of electrostatically actuated clamped-clamped micromachined arches when excited by a DC load superimposed to an AC harmonic load. A Galerkin based reduced-order model is used to discretize the distributed-parameter model of the considered shallow arch. The natural frequencies of the arch are calculated for various values of DC voltages and initial rises of the arch. The forced vibration response of the arch to a combined DC and AC harmonic load is determined when excited near its fundamental natural frequency. For small DC and AC loads, a perturbation technique (the method of multiple scales) is also used. For large DC and AC, the reduced-order model equations are integrated numerically with time to get the arch dynamic response. The results show various nonlinear scenarios of transitions to snap-through and dynamic pull-in. The effect of rise is shown to have significant effect on the dynamical behavior of the MEMS arch. Experimental work is conducted to test polysilicon curved microbeam when excited by DC and AC loads. Experimental results on primary resonance and dynamic pull-in are shown and compared with the theoretical results.© 2009 ASME
TL;DR: In this article, the evolution of resonant oscillations in closed containers shaped like a cone or a bulb is investigated theoretically, and a perturbation scheme that combines the method of multiple scales with a Duffing-type expansion yields a set of coupled nonlinear ordinary differential equations for the slow variation of the amplitude and phase of the fast resonants.
TL;DR: In this paper, the problem of suppressing the vibrations of a hinged-hinged flexible beam that is subjected to primary and principal parametric excitations is tackled, and different control laws are proposed, and saturation phenomenon is investigated to suppress the vibrations.
Abstract: The problem of suppressing the vibrations of a hinged–hinged flexible beam that is subjected to primary and principal parametric excitations is tackled. Different control laws are proposed, and saturation phenomenon is investigated to suppress the vibrations of the system. The dynamics of the beam are modeled with a second-order nonlinear ordinary-differential equation. The method of multiple scales is used to derive two-first ordinary differential equations that govern the time variation of the amplitude and phase of the response. These equations are used to determine the steady-state responses and their stability. The results of perturbation solution have been verified through numerical simulations, where different effects of the system parameters on the steady-state amplitude and on saturation phenomena at resonance have been reported.
TL;DR: In this article, the active feedback control of piezoelectric sandwich beams at large vibration amplitudes is studied and the feedback parameters' effects on the frequency, phase and time responses of sandwich beams under various types of boundary conditions are investigated.
Abstract: Mathematical modelling of the active feedback control of piezoelectric sandwich beams at large vibration amplitudes is developed. The proportional and derivative potential feedback controls via piezo-sensor and actuator layers are used. The dynamics of the sandwich beam are modelled by a nonlinear partial differential equation with feedback gain coefficients dependent. Based on Galerking's method, a second order nonlinear differential equation with strong nonlinearities is obtained. The method of multiple scales is used and the amplitude frequency and phase dependent relationships are derived. The feedback parameters' effects on the frequency, phase and time responses of sandwich beams under various types of boundary conditions are investigated. With respect to the system and on the feedback parameters, an instable zone in the upper nonlinear frequency branch can be obtained. Critical amplitude, frequency and phase values of the resulting instable zone are analytically given. The stability of the obtained...
TL;DR: In this paper, a thin liquid layer of a non-Newtonian film falling down an inclined plane that is subjected to non-uniform heating has been considered and a nonlinear evolution equation is derived by applying the long-wave theory, and the equation governs the evolution of a power-law film flowing down a nonuniformly heated inclined plane.
Abstract: A thin liquid layer of a non-Newtonian film falling down an inclined plane that is subjected to nonuniform heating has been considered. The temperature of the inclined plane is assumed to be linearly distributed and the case when the temperature gradient is positive or negative is investigated. The film flow is influenced by gravity, mean surface tension, and thermocapillary forces acting along the free surface. The coupling of thermocapillary instability and surface-wave instabilities is studied for two-dimensional disturbances. A nonlinear evolution equation is derived by applying the long-wave theory, and the equation governs the evolution of a power-law film flowing down a nonuniformly heated inclined plane. The linear stability analysis shows that the film flow system is stable when the plate temperature decreases in the downstream direction while it is less stable for increasing temperature along the plate. Weakly nonlinear stability analysis using the method of multiple scales has been investigated and this leads to a secular equation of the Ginzburg-Landau type. The analysis shows that both supercritical stability and subcritical instability are possible for the film flow system. The results indicate the existence of finite-amplitude waves, and the threshold amplitude and nonlinear speed of these waves are influenced by thermocapillarity. The nonlinear evolution equation for the film thickness is solved numerically in a periodic domain in the supercritical stable region, and the results show that the shape of the wave is influenced by the choice of wave number, non-Newtonian rheology, and nonuniform heating.
TL;DR: In this article, a quantum above-barrier reflection of a Bose-Einstein condensate by a one-dimensional rectangular potential barrier, or by a potential well, for nonlinear Schroedinger equation (Gross-Pitaevskii equation) with a small nonlinearity is considered.
Abstract: We consider a quantum above-barrier reflection of a Bose-Einstein condensate by a one-dimensional rectangular potential barrier, or by a potential well, for nonlinear Schroedinger equation (Gross-Pitaevskii equation) with a small nonlinearity. The most interesting case is realized in resonances when the reflection coefficient is equal to zero for the linear Schroedinger equation. Then the reflection is determined only by small nonlinear term in the Gross-Pitaevskii equation. A simple analytic expression has been obtained for the reflection coefficient produced only by the nonlinearity. An analytical condition is found when common action of potential barrier and nonlinearity produces a zero reflection coefficient. The reflection coefficient is derived analytically in the vicinity of resonances which are shifted by nonlinearity.