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Showing papers on "Multiple-scale analysis published in 1995"


Book ChapterDOI
TL;DR: In this article, a solution method for nonlinear vibrations of cables having small initial sag-to-span ratios is discussed, where one-toone internal resonances between the in-plane and out-of-plane modes as well as primary resonances of the inplane mode are considered.
Abstract: We discuss solution methods for nonlinear vibrations of cables having small initial sag-to-span ratios. One-to-one internal resonances between the in-plane and out-of-plane modes as well as primary resonances of the in-plane mode are considered. Approximate solutions are obtained by two different approaches. In the first approach, the method of multiple scales is applied directly to the governing partial-differential equations and boundary conditions. In the second approach, the equations are first discretized, and then the method of multiple scales is applied to the resulting ordinary-differential equations. It is shown that treatment of the discretized system is inaccurate compared to direct treatment of the partial-differential system. Discrepancies between the two solutions appear even at the first level of approximation. Stability analyses of the amplitude and phase modulation equations for both methods are also performed.

90 citations


Journal ArticleDOI
TL;DR: In this paper, two approaches for determination of the nonlinear planar modes of a cantilever beam are compared and the results show that both approaches yield the same nonlinear modes because the discretization is performed using a complete set of basis functions, namely, the linear mode shapes.
Abstract: Two approaches for determination of the nonlinear planar modes of a cantilever beam are compared. In the first approach, the governing partial-differential system is discretized using the linear mode shapes and then the nonlinear mode shapes are determined from the discretized system. In the second approach, the boundary-value problem is treated directly by using the method of multiple scales. The results show that both approaches yield the same nonlinear modes because the discretization is performed using a complete set of basis functions, namely, the linear mode shapes.

71 citations


Journal ArticleDOI
TL;DR: In this paper, the free dynamics of a monocoupled nonlinear periodic system of infinite extent is analyzed by employing the notion of nonlinear normal mode, thereby reducing the response problem to the solution of an infinite set of singular nonlinear partial differential equations.
Abstract: The free dynamics of a monocoupled nonlinear periodic system of infinite extent is analyzed. The system under consideration consists of an infinite number of elastic layers with material nonlinearities, coupled by means of linear stiffnesses. It is shown that, in analogy to linear theory, the system possesses nonlinear attenuation and propagation zones (AZs and PZs) in the frequency domain. Responses in AZs correspond to standing waves with spatially attenuating or expanding envelopes, and are synchronous motions of all points of the periodic system. These motions are analytically examined by employing the notion of ‘‘nonlinear normal mode,’’ thereby reducing the response problem to the solution of an infinite set of singular nonlinear partial differential equations. An asymptotic methodology is developed to solve this set. Motions in PZs correspond to traveling waves, i.e., to nonsynchronous oscillations. These motions are analyzed by applying the method of multiple scales in space and time. Numerical computations are carried out to complement the analytical findings. The analytical and numerical methodologies developed in this work can be applied to the study of the free motions of a general class of monocoupled nonlinear periodic systems, and can be extended to investigate motions of nonlinear periodic systems with more than one coupling coordinate.

65 citations


Journal ArticleDOI
Ali H. Nayfeh1
TL;DR: In this article, a direct method based on the method of normal forms is proposed for constructing the nonlinear normal modes of continuous systems with cubic nonlinearities, and the proposed method is compared with the methods of Shaw and Pierre and King and Vakakis by applying them to three conservative systems.
Abstract: A direct method based on the method of normal forms is proposed for constructing the nonlinear normal modes of continuous systems. The proposed method is compared with the method of multiple scales and the methods of Shaw and Pierre and King and Vakakis by applying them to three conservative systems with cubic nonlinearities: (a) a hinged-hinged beam resting on a nonlinear elastic foundation, (b) a model of a relief valve (linear elastic spring attached to a nonlinear spring with a mass), and (c) a simply supported linear beam with nonlinear torsional springs at both ends. In the absence of internal resonance, the constructed nonlinear modes with all four methods are the same. The method of multiple scales seems to be the simplest and the least computationally demanding. The methods of multiple scales and normal forms are applicable to problems with and without internal resonances, whereas the present forms of the methods of Shaw and Pierre and King and Vakakis are not applicable to problems with internal...

57 citations


Journal ArticleDOI
TL;DR: The nonlinear response of a taut string to an end excitation with components both parallel and transverse to its axis is theoretically and experimentally studied in this article, where a model for the transverse vibration of the string is developed by neglecting its longitudinal inertia.
Abstract: The nonlinear response of a taut string to an end excitation with components both parallel and transverse to its axis is theoretically and experimentally studied A model for the transverse vibration of the string is developed by neglecting its longitudinal inertia The method of multiple scales is applied directly to the governing partial-differential equations and boundary conditions A continuation method is then employed to determine the constant amplitude and phase solutions and their stability Resonant responses are predicted to occur simultaneously in as many as three modes An experimental study is conducted and the results are found to be in good agreement with the theory

37 citations


Journal ArticleDOI
TL;DR: In this article, a qualitative method of analysis of suspension bridge non-linear free vibrations is proposed, which allows one to determine the types of vibrational process, to investigate the stability of each vibrational regime, to define the character of amplitude and phase dependences from the initial conditions etc.

31 citations


Journal ArticleDOI
TL;DR: In this paper, the effects of in-plane edge restraints, small initial geometric imperfections, transverse shear deformation, and transverse normal stress are considered in the structural model which satisfies the traction-free condition on the panel faces.
Abstract: The non-linear dynamic behavior of a uniformly compressed, composite panel subjected to non-linear aerodynamic loading due to a high-supersonic co-planar flow is analyzed The effects of in-plane edge restraints, small initial geometric imperfections, transverse shear deformation, and transverse normal stress are considered in the structural model which satisfies the traction-free condition on the panel faces The panel flutter equations, derived via Galerkin's Method, are solved using Arclength Continuation for the static solution and a predictor-corrector type Shooting Technique to obtain periodic solutions and their bifurcations The possibility of hard flutter is demonstrated when considering non-linear aerodynamics Furthermore, edge compression could yield multiple buckled states or coexistence of multiple periodic solutions with the stable static solution, that is, the panel could either remain buckled or flutter Edge restraints normal to the flow appear to stabilize the panel, whereas those parallel to the flow may result in a buckled-flutter-buckled transition Quasi-periodic and chaotic motions and associated Lyapunov exponents are also obtained For perfect panels, results obtained by the Shooting Technique and the Method of Multiple Scales are in agreement only within the immediate post-flutter regime Results indicate that a shear deformation theory is required for moderately thick composite panels

27 citations


Journal ArticleDOI
TL;DR: In this paper, a model of a magnetoelastic buckled beam subjected to an external axial periodic force in a periodic transversal magnetic field is investigated, and the model is described by a two-frequency parametric vibration system with self-excitation.

24 citations


Journal ArticleDOI
TL;DR: In this paper, a nonlinear analysis is presented for combination resonances in the symmetric responses of a clamped circular plate with the internal resonance, ω 3 ω 1 + 2ω 2.
Abstract: A nonlinear analysis is presented for combination resonances in the symmetric responses of a clamped circular plate with the internal resonance, ω 3 ω 1 + 2ω 2 . The combination resonances occur when the frequency of the excitation are near a combination of the natural frequencies, that is, when Ω 2ω 1 + ω 2 . By means of the internal resonance condition, the frequency of the excitation is also near another combination of the natural frequencies, that is, Ω ω 1 - ω 2 + ω 3 . The effect of two near combination resonance frequencies on the response of the plate is examined. The method of multiple scales is used to solve the nonlinear nonautonomous system of equations governing the generalized coordinates in Galerkin's procedure. For steady-state responses, we determine the equilibrium points of the autonomous system transformed from the nonautonomous system and examine their stability. It has been found that in some cases resonance responses with nonzero-amplitude modes don't exist, and the amplitudes of the responses decrease with the excitation amplitude. We integrate numerically the nonautonomous system to find the long-term behaviors of the plate and to check the validity of the analytical solution. It is found that there exist multiple stable responses resulting in jumps. In this case the long-term response of the plate depends on the initial condition. In order to visualize total responses depending on the initial conditions, we draw the deflection curves of the plate.

24 citations


Journal ArticleDOI
TL;DR: The phase shift caused by the dispersive perturbation is a remarkable feature that has never been observed in the collision process of algebraic solitons in the Benjamin-Ono equation.
Abstract: A direct perturbation theory is developed to study the effects of small perturbations on the interaction process of algebraic solitons of the Benjamin-Ono (BO) equation. Using the method of multiple scales, the modulation equations for the amplitude and the phase of each soliton are derived in the lowest approximation. As practical applications of the theory, the interaction of two solitons is investigated for the two different types of perturbations that appear in real physical systems. One is a dissipative perturbation (BO--Burgers equation) and the other is a dispersive perturbation (higher-order BO equation). In both cases, the changes of the soliton parameters due to small perturbation are calculated by numerical integrations and their characteristics are elucidated in detail. Among them, the phase shift caused by the dispersive perturbation is a remarkable feature that has never been observed in the collision process of algebraic solitons.

22 citations


Journal ArticleDOI
TL;DR: In this article, a nonlinear dynamic response of circular rings rotating with spin speed is investigated, which involves small fluctuations from a constant average value, and the existence and stability properties of these motions are first analyzed in detail.
Abstract: This work investigates nonlinear dynamic response of circular rings rotating with spin speed which involves small fluctuations from a constant average value. First, Hamilton's principle is applied and the equations of motion are expressed in terms of a single time coordinate, representing the amplitude of an in-plane bending mode. For nonresonant excitation or for slowly rotating rings, a complete analysis is presented by employing phase plane methodologies. For rapidly rotating rings, periodic spin speed variations give rise to terms leading to parametric excitation. In this case, the vibrations that occur under principal parametric resonance are analyzed by applying the method of multiple scales. The resulting modulation equations possess combinations of trivial and nontrivial constant steady state solutions. The existence and stability properties of these motions are first analyzed in detail. Also, analysis of the undamped slow-flow equations provides a global picture for the possible motions of the ring. In all cases, the analytical predictions are verified and complemented by numerical results. In addition to periodic response, these results reveal the existence of unbounded as well as transient chaotic response of the rotating ring.

Journal ArticleDOI
TL;DR: In this article, it was shown that the angular frequency ω of the incident wave is sufficiently close to both the natural frequency of mode n + 1 (ωn + 1) and twice the normal frequency of n (2ωn) thus exciting simultaneously a subharmonic mode n and a synchronous mode n+1, and the value of n is set equal to 3 in accordance with Benjamin & Ellis' (1990) observation.
Abstract: A possible mechanism for the occurrence of the phenomenon of erratic drift of bubbles in liquids subjected to acoustic waves was proposed by Benjamin & Ellis (1990) who showed that nonlinear interactions between adjacent perturbation modes expressed in terms of spherical harmonics of any order may lead to the excitation of mode 1 which is equivalent to a displacement of the bubble centroid. We show that indeed such a mechanism can give rise to a chaotic process at least under the conditions experimentally investigated by Benjamin & Ellis (1990). In fact we examine the case in which the angular frequency ω of the incident wave is sufficiently close to both the natural frequency of mode n + 1 (ωn + 1) and twice the natural frequency of mode n (2ωn) thus exciting simultaneously a subharmonic mode n and a synchronous mode n + 1. The value of n is set equal to 3 in accordance with Benjamin & Ellis' (1990) observation. A classical multiple scale analysis allows us to follow the development of these perturbations in the weakly nonlinear regime to find an autonomous system of quadratically coupled nonlinear differential equations governing the evolution of the amplitudes of the perturbations on a slow time scale. As obtained by Gu & Sethna (1987) for the Faraday resonance problem, we find both regular and chaotic solutions of the above system. Chaos is found to develop for large enough values of the amplitude of the acoustic excitation within some region in the parameter space and is reached through a period-doubling sequence displaying the typical characteristics of Feigenbaum scenario.

Journal ArticleDOI
TL;DR: In this paper, the Galerkin procedure is used to discretize the nonlinear partial differential equation and boundary conditions governing the flutter of a simply supported panel in a supersonic stream.
Abstract: The Galerkin procedure is used to discretize the nonlinear partial differential equation and boundary conditions governing the flutter of a simply supported panel in a supersonic stream. These equations have repeated natural frequencies at the onset of flutter. The method of multiple scales is used to derive five first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the excited modes. Then, the modulation equations are used to calculate the equilibrium solutions and their stability, and hence to identify the excitation parameters that suppress flutter and those that lead to complex motions. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos. The complex motions are characterized by using phase planes, power spectra, Lyapunov exponents, and dimensions.

Journal ArticleDOI
TL;DR: In this article, the free vibration of a pinned end beam undergoing large transverse deflection is examined and the equation of motion for this problem is reduced to a Duffing-type equation with a small perturbation parameter.
Abstract: The free vibration of a pinned end beam undergoing large transverse deflection is examined. The equation of motion for this problem is reduced to a Duffing-type equation with a small perturbation parameter. The method of multiple scales is used to determine a uniformly valid higher third order perturbation solution. The predictions of the non-linear frequencies are in excellent agreement with the exact solution obtained from the direct integration of elliptic integral. Also presented are the harmonic contents of the non-linear transverse displacement.

Journal ArticleDOI
TL;DR: In this paper, the response of a single-degree-of-freedom system to a nonstationary excitation was investigated by using the method of multiple scales as well as analog and digital-computer simulations.
Abstract: The response of a single-degree-of-freedom system to a nonstationary excitation is investigated by using the method of multiple scales as well as analog- and digital-computer simulations. The unexcited system has one focus and two saddle points. The system can be used to model rolling of ships in head or follower seas. The method of multiple scales is used to derive equations governing the modulation of the amplitude and phase of the response. The modulation equations are used to find the stationary solutions and their stability. The response to nonstationary excitations is found by integrating the original governing equation as well as the modulation equations. There is good agreement between the results of both approaches. For some frequency and amplitude sweeps, the nonstationary response found from integrating the original governing equation exhibits behaviors that are analogous to symmetry-breaking bifurcations, period-doubling bi furcations, chaos, and unboundedness present in the stationary case. T...

Journal ArticleDOI
TL;DR: In this article, the effect of a periodic forcing on nonlinear modulation of interfacial gravity-capillary waves propagating between two magnetic fluids of infinite depth under the influence of a constant vertical magnetic field was investigated.

Journal ArticleDOI
TL;DR: In this paper, the dynamics of two-degree-of-freedom oscillators including Rayleigh and Duffing type nonlinearities are investigated and a set of averaged equations is derived for cases of primary external resonance.

Journal ArticleDOI
TL;DR: In this article, the role of cosmic-ray modified contact discontinuities and pressure balance structures in two-fluid hydrodynamics in one Cartesian space dimension is investigated by means of analytic and numerical solution examples, as well as by weakly nonlinear asymptotics.
Abstract: The role of cosmic-ray-modified contact discontinuities and pressure balance structures in two-fluid cosmic-ray hydrodynamics in one Cartesian space dimension are investigated by means of analytic and numerical solution examples, as well as by weakly nonlinear asymptotics. The fundamental wave modes of the two-fluid cosmic-ray hydrodynamic equations in the long-wavelength limit consist of the backward and forward propagating cosmic-ray-modified sound waves, with sound speed dependent on both the cosmic-ray and thermal gas pressures; the contact discontinuity; and a pressure balance mode in which the sum ofthe cosmic ray and thermal gas pressure perturbations is zero. The pressure balance mode, like the contact discontinuity is advected with the background flow. The interaction of the pressure balance mode with the contact discontinuity is investigated by means of the method of multiple scales. The thermal gas and cosmic-ray pressure perturbations satisfy a linear diffusion equation, and entropy perturbations arising from nonisentropic initial conditions for the thermal gas are frozen into the fluid. The contact discontinuity and pressure balance eigenmodes both admit nonzero perturbations in the thermal gas, whereas the cosmic-ray-modified sound waves are isentropic. The total entropy perturbation is shared between the contact discontinuity and pressure balance eigenmodes, and examples are given in which there is a transfer of entropy between the two modes. In particular, N-wave type density disturbances are obtained which arise as a result of the entropy transfer between the two modes. A weakly nonlinear geometric optics perturbation expansion is used to study the long timescale evolution of the short-wavelength entropy wave and the thermal gas sound waves in a slowly varying, large-scale background flow. The weakly nonlinear geometric optics expansion is also used to generalize previous studies of squeezing instability for short-wavelength sound waves in the two fluid model, by including a weakly nonlinear wave steepening term that leads to shock formation, as well as the effect of long time and space dependence of the background flow. Implications of cosmic-ray-modified pressure balance structures and contact discontinuities in models of the interaction of traveling interplanetary shocks and compression and rarefraction waves with the solar wind termination shock are briefly discussed.

Journal ArticleDOI
TL;DR: In this article, the Schrodinger equation for non-linear dispersive (glass) fiber can be extended using the method of multiple scales, and a technique is developed such that perturbation terms are greatly simplified.
Abstract: Pulses propagating in a non-linear dispersive (glass) fibre can be described by the non-linear Schrodinger equation if the pulse is longer than a picosecond; for shorter pulses, this equation must be extended. In this paper we systematically derive this extended equation using the method of multiple scales. By using an inherent freedom in the method of multiple scales, a technique is developed such that perturbation terms are greatly simplified. The limits of validity of the derived equation are discussed. It is shown to be valid for pulses longer than 30 fs.

Journal ArticleDOI
TL;DR: In this article, a corrugated circular waveguide is proposed as a microwave filter and the analysis is carried out using the perturbation method of multiple scales for the case of TM modes.
Abstract: A corrugated circular waveguide is proposed as a microwave filter. The analysis is carried out using the perturbation method of multiple scales for the case of TM modes. The analysis concerns the interaction of two and four modes satisfying the resonant condition (Bragg condition) imposed by the periodicity of the waveguide wall. The coupled mode equations derived via the method of multiple scales are used to formulate the filtering problem as a two-point boundary-value problem which is solved numerically. Desirable filtering characteristics may be realised by introducing tapered as well as chirped corrugations to control the frequency response of such a wave filter. In case of two-mode interaction the side ripples can be eliminated by tapering the waveguide wall. In the four-mode interaction case, a multichannel filter may be realised by combining taper and chirp.

Journal ArticleDOI
TL;DR: In this paper, the effect of measurement error in the independent variables (also known as errors-in-variables) on parameter estimation for the Box and Cox family of transformations is investigated.

Proceedings ArticleDOI
10 Apr 1995
TL;DR: In this paper, the nonlinear response of an infinitely long cylindrical shell to a primary excitation of one of its two orthogonal flexural modes is investigated, and the resulting two sets of modulation equations are used to study the equilibrium and dynamic solutions and their stability and hence show the different bifurcations.
Abstract: The nonlinear response of an infinitely long cylindrical shell to a primary excitation of one of its two orthogonal flexural modes is investigated. The method of multiple scales is used to derive four ordinary differential equations describing the amplitudes and phases of the two orthogonal modes by (a) attacking a two-mode discretization of the governing partial differential equations and (b) directly attacking the partial differential equations. The two-mode discretization results in erroneous solutions because it does not account for the effects of the quadratic nonlinearities. The resulting two sets of modulation equations are used to study the equilibrium and dynamic solutions and their stability and hence show the different bifurcations. The response could be a single-mode solution or a two-mode solution. The equilibrium solutions of the two orthogonal third flexural modes undergo a Hopf bifitrcation. A combination ofa shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos, multiple attractors, explosive bifurcations, and crises.

Proceedings ArticleDOI
07 Nov 1995
TL;DR: In this paper, the coupled amplitude equation governing the propagation of acoustic wave in a general anisotropic solid is presented, using the method of multiple scales, which reveals the evolution of each order of harmonic accurately.
Abstract: A systematic derivation of the coupled amplitude equation governing the propagation of acoustic wave in a general anisotropic solid is presented, using the method of multiple scales. The equation is uniformly valid prior to discontinuity and reveals the evolution of each order of harmonic accurately. The higher order harmonic generation in some SAN-materials is also investigated and the results are compared with those given by successive approximation method.

Journal ArticleDOI
TL;DR: In this paper, the coherent state ansatz and the time-dependent variational principle were introduced to obtain the two partial different equations of motion from Hamiltonian, and the amplitude function satisfied a nonlinear Schrodinger equation.
Abstract: Introducing the coherent state ansatz and the time-dependent variational principle, we obtain the two partial different equations of motion from Hamiltonian. Employing the method of multiple scales, we reduce these equations into the envelope function equation and find the amplitude function satisfied a nonlinear Schrodinger equation. We get the periodic wave solution and analyse its stability in antiferromagnet KCuF3.

Journal ArticleDOI
TL;DR: In this article, the coherent state ansatz and the time-dependent variational principle were introduced and two partial differential equations of motion were obtained from the Hamiltonian, which were then reduced into the envelope function equation and the amplitude function satisfying a nonlinear Schrodinger equation.
Abstract: Introducing the coherent state ansatz and the time-dependent variational principle, two partial differential equations of motion are obtained from the Hamiltonian. Employing the method of multiple scales, these equations are reduced into the envelope function equation and the amplitude function satisfying a nonlinear Schrodinger equation is found. The periodic wave solution and analysis of its stability in the antiferromagnet Ni(C2H8N2)(2)NO2ClO4 is derived.