Showing papers on "Multiple-scale analysis published in 2002"
TL;DR: In this paper, a detailed study of the forced asymmetric non-linear vibrations of circular plates with a free edge is presented, where the dynamic analogue of the von Karman equations is used to establish the governing equations.
136 citations
TL;DR: In this paper, the chaotic dynamics and global bifurcations of the suspended elastic cable under combined parametric and external excitations are investigated and the non-linear equations of motion of the elastic cable to small vibration of one support are derived.
Abstract: The chaotic dynamics and global bifurcations of the suspended elastic cable under combined parametric and external excitations are investigated. The non-linear equations of motion of the elastic cable to small vibration of one support are derived. The averaged equations are obtained by using the method of multiple scales. Based on the averaged equations, the theory of normal form and Maple program are used to obtain the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues. On the basis of the normal form, global bifurcation analysis of the parametrically and externally excited suspended elastic cable is given by a global perturbation method developed by Kovacic and Wiggins. The chaotic motion of the elastic cable is also found by numerical simulation.
69 citations
TL;DR: In this article, a perturbation method was used to determine the boundary frequencies of dynamic instability regions for various orders in a consistent and hierarchical manner, and the principal instability regions of cross-ply conical shells with simply supported boundary conditions were studied to demonstrate the performance of the linear instability approach.
61 citations
TL;DR: In this article, the effect of large amplitude on the dissipative nature as well as on the natural frequency of viscoelastic laminated plates is investigated. But the authors focus on the nonlinear and hereditary type governing equations.
45 citations
TL;DR: In this paper, the non-linear behavior of an elastic cable subjected to a harmonic excitation is investigated using Garlerkin's method and method of multiple scales, the discrete dynamical equations and a set of first order nonlinear differential equations are obtained.
42 citations
TL;DR: In this article, the effect of material damping on the non-linear free vibration of moving belts is investigated by reformulating the equations for motion of the system concerned and directly applying the method of multiple scales, which yields closed form solutions in the first order approximation to free vibration response.
30 citations
TL;DR: In this paper, a phase-plane analysis of the linearized model is used to limit residual pendulation at the goal position, and the method is extended to account for quadratic and cubic nonlinearities.
Abstract: This paper describes a method to move the load of a gantry crane to a desired position in the presence of known, but arbitrary, motion-inversion delays as well as cart acceleration constraints. The method idea is based on a phase-plane analysis of the linearized model. In order to limit residual pendulation at the goal position, the method is extended to account for quadratic and cubic nonlinearities. The method of multiple scales is used to determine an approximate solution to the nonlinear equations of motion, thus providing a more accurate measure of the frequency of the oscillations. The nonlinear approach is very successful in limiting residual oscillations to very small values (less than 1 degree of amplitude), offering a reduction, with respect to the linear case, of as much as two orders of magnitude. Finally, this method offers a rationale for the future development of a controller for suppression of load oscillations in ship-mounted cranes in the presence of arbitrary delays.
30 citations
TL;DR: In this article, it is shown that natural vibrations, localized around the inclusion, are possible in a system consisting of an infinite string on an elastic foundation-concentrated inertial inclusion which moves at a constant, subcritical velocity.
24 citations
TL;DR: In this article, the principal resonance of two-degree-of-freedom nonlinear system to narrow-band random external excitation was investigated by means of qualitative analysis and numerical results. And the effects of damping, detuning, bandwidth, and magnitudes of deterministic and random excitations were analyzed.
Abstract: The principal resonance of two-degree-of-freedom non-linear system to narrow-band random external excitation is investigated. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response are studied by means of qualitative analysis. The effects of damping, detuning, bandwidth, and magnitudes of deterministic and random excitations are analyzed. The theoretical analyses are 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 be changed from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady state solutions, saturation and jumps may exist.
17 citations
TL;DR: In this article, an alternative technique for the evaluation of non-linear normal modes is presented and applied to finite-element models, based upon the method of multiple scales, so that non linear normal modes are evaluated as asymptotic expansions starting with the linear damped vibration modes.
15 citations
TL;DR: In this paper, a perturbation-based method is proposed to accelerate the cyclic steady state (CSS) simulation of oxygen VSA. But the method is limited to a series expansion of the temperature term in the energy balance, and mass balance is ignored in this analysis.
01 Jan 2002
TL;DR: In this paper, a dynamic modeling of a cantilever beam under an axial movement of its basement is presented, which takes the coupledcubic nonlinearities of geometrical and inertial types into consideration.
Abstract: Dynamic modeling of a cantilever beam under an axial movement of its basement is present-ed.The dynamic equation of motion for the cantilever beam is established by using Kane’s equation first andthen simplified through the Rayleigh-Ritz method.Compared with the older modeling method,which lineari-zes the generalized inertia forces and the generalized active forces,the present modehng takes the coupledcubic nonlinearities of geometrical and inertial types into consideration.The method of multiple scales is usedto directly solve the nonlinear differential equations and to derive the nonlinear modulation equation for theprincipal parametric resonance.The results show that the nonlinear inertia terms produce a softening effectand play a significant role in the planar response of the second mode and the higher ones.On the otherhand,the nonlinear geometric terms produce a hardening effect and dominate the planar response of the firstmode.The validity of the present modeling is clarified through the comparisons of its coefficients with thoseexperimentally verified in previous studies.
TL;DR: In this paper, a model of a strip of cardiac tissue consisting of a one-dimensional chain of cardiac units is derived in the form of a non-linear partial difference equation.
Abstract: A model of a strip of cardiac tissue consisting of a one-dimensional chain of cardiac units is derived in the form of a non-linear partial difference equation. Perturbation analysis is performed on this equation, and it is shown that regular perturbations are inadequate due to the appearance of secular terms. A singular perturbation procedure known as the method of multiple scales is shown to provide good agreement with numerical simulation except in the neighborhood of a singularity of the slow flow. The perturbation analysis is supplemented by a local numerical simulation near this singularity. The resulting analysis is shown to predict a "spatial bifurcation" phenomenon in which parts of the chain may be oscillating in period-2 motion while other parts may be oscillating in higher periodic motion or even chaotic motion.
TL;DR: Weakly nonlinear Kelvin-Helmholtz instability for two viscous fluids streaming through porous media is investigated in this article, where the authors use the Taylor expansion through the multiple scale scheme to derive the well-known nonlinear Schrodinger equation with complex coefficients from the nonlinear characteristic equation.
Abstract: Weakly nonlinear Kelvin–Helmholtz instability for two viscous fluids streaming through porous media is investigated. The electro-gravitational stability of the horizontal plane interface is examined. A vertical or a horizontal electric field stresses the system. The linear form of equation of motion is solved in the light of the nonlinear boundary conditions. The present boundary value problem leads to construct nonlinear characteristic equation. This nonlinear characteristic equation has complex coefficients for the elevation function. The nonlinearity is kept to the third order. The method of multiple scales, in both space and time, is used. The use of the Taylor expansion through the multiple scale scheme leads to the derivation of the well-known nonlinear Schrodinger equation with complex coefficients from the nonlinear characteristic equation. This equation describes the evolution of the wave train up to cubic order, and may be regarded as the counterparts of the single nonlinear Schrodinger equation that occurs in the non-resonance case. The relation between the stratified kinematic viscosity and the porous permeability is performed in order to control the marginal state representation. This marginality is utilised in order to relax the complexity of the linear dispersion relation. Stability conditions are discussed both analytically and numerically, and stability diagrams are obtained. Regions of stability and instability are identified. It is found that the porosity of the media increases the destabilizing influence for the fluid density. In nonlinear scope, a destabilizing influence for the upper porous permeability is recorded, while a stabilizing influence is found for the lower porous permeability. Both the horizontal and vertical electric fields are still playing the same roles in linear and nonlinear examinations as in the non-porous media. Two opposite roles are presented for the variation of the stratified fluid velocity V . A stabilizing influence for V ⩽1 and a destabilizing effect for V >1 are illustrated in this examination. A dual role in the nonlinear examination is recorded for values of the wave-frequency.
TL;DR: In this article, the principal resonance of Van der Pol-Duffing oscillator to combined deterministic and random parametric excitations is investigated, and the behavior, stability and bifurcation of steady state response are studied.
Abstract: The principal resonance of Van der Pol-Duffing oscillator to combined deterministic and random parametric excitations is 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. Jumps were shown to occur under some conditions. The effects of damping, detuning, bandwidth, and magnitudes of deterministic and random excitations are analyzed. The theoretical analysis were verified by numerical results.
TL;DR: In this paper, the authors study how boundary conditions affect the multiple-scale analysis of hyperbolic conservation laws with rapid spatial fluctuations and show that the most significant difficulty occurs when one has insufficient boundary conditions to solve consistency conditions.
Abstract: We study how boundary conditions affect the multiple-scale analysis of hyperbolic conservation laws with rapid spatial fluctuations. The most significant difficulty occurs when one has insufficient boundary conditions to solve consistency conditions. We show how to overcome this missing boundary condition difficulty for both linear and nonlinear problems through the recovery of boundary information. We introduce two methods for this recovery (multiple-scale analysis with a reduced set of scales, and a combination of Laplace transforms and multiple scales) and show that they are roughly equivalent. We also show that the recovered boundary information is likely to contain secular terms if the initial conditions are nonzero. However, for the linear problem, we demonstrate how to avoid these secular terms to construct a solution that is valid for all time. For nonlinear problems, we argue that physically relevant problems do not exhibit the missing boundary condition difficulty.
TL;DR: In this paper, the von Karman equations were used to model large-amplitude vibrations of a simply supported circular flat plate subjected to harmonically varying temperature fields arising from an external heat flux.
Abstract: We consider the problem of large-amplitude vibrations of a simply supported circular flat plate subjected to harmonically varying temperature fields arising from an external heat flux (aeroheating for example). The plate is modeled using the von Karman equations. We used the method of multiple scales to determine an approximate solution for the case in which the frequency of the thermal variations is approximately twice the fundamental natural frequency of the plate; that is, the case of principal parametric resonance. The results show that such thermal loads produce large-amplitude vibrations, with associated multi-valued responses and subcritical instabilities.
Journal Article•
TL;DR: In this article, a dynamic modeling of a cantilever beam under an axial movement of its basement is presented, which takes coupled cubic nonlinearities of geometrical and inertial types into consideration.
TL;DR: In this article, the response of a two-degrees-of-freedom nonlinear system to narrow-band random parametric excitation is investigated and the effect of detunings and amplitude is analyzed.
Abstract: Response of two-degrees-of-freedom nonlinearsystem to narrow-band random parametric excitation isinvestigated. The method of multiple scales is used todetermine the equations of modulation of amplitude andphase. The effect of detunings and amplitude areanalyzed. Theoretical analyses and numerical simulationsshow that the nontrivial steady-state solution may changeform a limit cycle to a diffused limit cycle as theintensity of the random excitation increase. Under someconditions, the system may have two steady-statesolutions.
TL;DR: The method of multiple scales is used to deduce equations for three nonlinear approximations of a wave disturbance in a basin of constant depth covered with broken ice in this article.
Abstract: The method of multiple scales is used to deduce equations for three nonlinear approximations of a wave disturbance in a basin of constant depth covered with broken ice. In deducing these equations, we take into account the space and time variability of the wave profile in the expression for the velocity potential on the basin surface. These equations are used to construct uniformly suitable asymptotic expansions up to quantities of the third order of smallness for the liquid-velocity potential and elevations of the basin surface formed by a periodic running wave of finite amplitude. We analyze the dependence of the amplitude-phase characteristics of elevations of the basin surface on the thickness of ice, nonlinearity of its vertical acceleration, and the amplitude and wavelength of the fundamental harmonic.
01 Jan 2002
TL;DR: In this paper, the authors investigate slow passage through the 2:1 resonance tongue in Mathieu's equation and find that amplification or de-amplification can occur depending on the speed travelling 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 travelling through the tongue and the initial conditions. We use the method of multiple scales to obtain a slow flow approximation. The WKB method is then applied to the slow flow equations to get an analytic approximation.Copyright © 2002 by ASME
07 Jun 2002
TL;DR: In this paper, the amplitude-frequency relation for all vibration modes is analyzed in terms of terms representing the softening effect of the resonance, the static offset, and harmonics.
Abstract: Recent atomic force microscopy research has focused on dynamical methods in which AFM probes are vibrated while in contact with a specimen during scanning The nonlinear tip-sample interactions can induce nonlinear features into the dynamic response Nonlinear responses observed experimentally include the DC shift (or lift-off) and primary response softening as well as the development of subharmonics and superharmonics Here, this problem is formulated in terms of a nonlinear boundary value problem which is solved using the method of multiple scales The main result of this analysis is the amplitude-frequency relation for all vibration modes The nonlinear normal modes are comprised of terms representing the softening effect of the resonance, the static offset, and harmonics The softening effect on the primary response is shown to be a function of the particular vibration mode as expected The contact mechanics model used here is restricted to Hertzian contact, but can be generalized to more complex models Results of the primary response for various excitations are presented The amplitude-frequency behavior is dependent on the linear contact stiffness, the forcing amplitude, and contact damping It is also shown that the modes have a differing sensitivity to the nonlinearities present in the contact© (2002) COPYRIGHT SPIE--The International Society for Optical Engineering Downloading of the abstract is permitted for personal use only
TL;DR: In this article, the conditions for the emergence and stability of non-inertial over-stability for axisymmetric Taylor-Couetteow of an Oldroyd-Buid in the narrow gap limit were examined.
Abstract: SUMMARY The conditions for the emergence and stability ofnite amplitude purely elastic (non-inertial) over- stability are examined for axisymmetric Taylor-Couetteow of an Oldroyd-Buid in the narrow-gap limit. The study is a detailed account of the formulation and results published previously (Khayat, Phys. Rev. Lett. 1997; 78:4918). Theoweld is obtained as a truncated Fourier representation for velocity, pressure and stress in the axial direction, and in terms of symmetric and antisymmetric Chandrasekhar functions along the radial direction. The Galerkin projection of the various modes onto the conservation and constitutive equations leads to a closed low-dimensional nonlinear dynamical system with 20 ◦ of freedom. In contrast to our previous model that was based on the simplifying rigid-free boundary con- ditions (Khayat, Phys. Fluids A 1995; 7:2191), the present formulation incorporates the more realistic rigid-rigid boundary conditions, and is capable of capturing quantitatively theow sequence observed in the experiment of Muller et al .( J. Non-Newtonian Fluid Mech. 1993; 46:315) for a highly elastic (Boger) �uid under conditions of negligible inertia. Existing linear analysis results arerst recovered by the present formulation, which predict the exchange of stability between the circular Couetteow and oscillatory Taylor vortexow via a postcritical Hopf bifurcation as the Deborah number exceeds a critical value. The stability conditions of the limit cycle are determined using the method of multiple scales. The present nonlinear theory predicts, as experiment suggests, the growth of oscillation amplitude of the velocity and the emergence of higher harmonics in the power spectrum as the Deborah number increases. Good agreement is obtained between theory and experiment. Copyright ? 2002 John Wiley & Sons, Ltd.
01 Jan 2002
TL;DR: In this article, a new branch of application of continuous group's techniques in the investigation of nonlinear differential equations is proposed, and the principal stages of the development of perturbation theory for nonlinear DDEs are considered in short.
Abstract: The work is devoted to a new branch of application of continuous group’s techniques in the investigation of nonlinear differential equations. The principal stages of the development of perturbation theory of nonlinear differential equations are considered in short. It is shown that its characteristic features make it possible a fruitful usage of continuous group’s techniques in problems of perturbation theory.
TL;DR: In this paper, the method of multiple scales was used to introduce a small-time scale into the nonlinear diffusion equation to model the spreading of a thin liquid drop under gravity.
Abstract: The method of multiple scales is used to introduce a small-time scale into the nonlinear diffusion equation modelling the spreading of a thin liquid drop under gravity. The Lie group method is used to analyse the resulting system. An approximate group invariant solution and an approximation to the waiting-time is obtained. A mathematical description of a spreading drop with non-infinite contact angle is obtained. This application to determining an approximation to the waiting-time is novel as it combines the method of multiple scales and Lie groups.
TL;DR: In this paper, a chain of parametrically driven nonlinear oscillators with a mass impurity was studied and an equation was presented to describe the nonlinear wave of small amplitude in the chain.
Abstract: By virtue of the method of multiple scales, we study a chain of parametrically driven nonlinear oscillators with a mass impurity. An equation is presented to describe the nonlinear wave of small amplitude in the chain. In our derivation, the equation is applicable to any eigenmode of coupled pendulum. Our result shows that a nonpropagation soliton emerges as the lowest or highest eigenmode of coupled pendulum is excited, and the impurity tends to pin the nonpropagation soliton excitation.