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Showing papers on "Multiple-scale analysis published in 2008"


Journal ArticleDOI
TL;DR: In this paper, the nonlinear dynamics of a hinged-hinged pipe conveying pulsatile fluid subjected to combination and principal parametric resonance in the presence of internal resonance is investigated.

132 citations


Journal ArticleDOI
TL;DR: In this paper, an axially moving visco-elastic Rayleigh beam with cubic nonlinearity is considered, and the governing partial-differential equation of motion for large amplitude vibration is derived through geometrical, constitutive, and dynamical relations.

101 citations


Journal ArticleDOI
TL;DR: In this paper, a general nonlinear-comprehensive modeling framework for piezoelectrically actuated microcantilevers is presented and validated experimentally.
Abstract: Nanomechanical cantilever sensors (NMCSs) have recently emerged as an effective means for label-free chemical and biological species detection. They operate through the adsorption of species on the functionalized surface of mechanical cantilevers. Through this functionalization, molecular recognition is directly transduced into a micromechanical response. In order to effectively utilize these sensors in practice and correctly relate the micromechanical response to the associated adsorbed species, the chief technical issues related to modeling must be resolved. Along these lines, this paper presents a general nonlinear-comprehensive modeling framework for piezoelectrically actuated microcantilevers and validates it experimentally. The proposed model considers both longitudinal and flexural vibrations and their coupling in addition to the ever-present geometric and material nonlinearities. Utilizing Euler-Bernoulli beam theory and employing the inextensibility conditions, the coupled longitudinal-flexural equations of motion are reduced to one nonlinear partial differential equation describing the flexural vibrations of the sensor. Using a Galerkian expansion, the resulting equation is discretized into a set of nonlinear ordinary differential equations. The method of multiple scales is then implemented to analytically construct the nonlinear response of the sensor near the first modal frequency (primary resonance of the first vibration mode). These solutions are compared to experimental results demonstrating that the sensor exhibits a softening-type nonlinear response. Such behavior can be attributed to the presence of quadratic material nonlinearities in the piezoelectric layer. This observation is critical, as it suggests that unlike macrocantilevers where the geometric hardening nonlinearities dominate the response behavior, material nonlinearities dominate the response of microcantilevers yielding a softening-type response. This behavior should be accounted for when designing and employing such sensors for practical applications.

99 citations


Journal ArticleDOI
TL;DR: In this paper, the transverse vibrations of simply supported axially moving Euler-Bernoulli beams are investigated using the method of multiple scales, a perturbation technique.

96 citations


Journal ArticleDOI
TL;DR: In this article, Chen, Goldenfeld, and Oono used the renormalization group (RG) method for singular perturbation problems and showed that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincare-Birkhoff normal forms for these systems up to and including terms of O ( ϵ 2 ), where ϵ is the perturbations parameter.

76 citations


Journal ArticleDOI
TL;DR: In this article, a transversal nonlinear vibration of an axially moving viscoelastic string supported by a partial visco-elastic guide is analyzed for both non-resonance and principal parametric resonance.

75 citations


Journal ArticleDOI
TL;DR: In this article, the solvability condition for a linear gyroscopic continuous system under small nonlinear time-dependent disturbances is investigated for the method of multiple scales applied to gyroscopy continua.

54 citations


Journal ArticleDOI
TL;DR: In this paper, the axial speed is characterized as a simple harmonic variation about the constant mean speed, and the instability conditions are presented for axially accelerating viscoelastic beams constrained by simple supports with rotational springs in parametric resonance.
Abstract: Stability is investigated for an axially accelerating viscoelastic beam. The material time derivative is used in the viscoelastic constitutive relation, not simply the partial time derivative. The method of multiple scales is applied directly to the governing equation without discretization. When the axial speed is characterized as a simple harmonic variation about the constant mean speed, the instability conditions are presented for axially accelerating viscoelastic beams constrained by simple supports with rotational springs in parametric resonance. The finite difference schemes are developed to solve numerically the equation of axially accelerating viscoelastic beams with fixed supports for the instability regions in the principal parametric resonance. The numerical calculations confirm the analytical results. Numerical examples show the effects of the constraint stiffness, the mean axial speed, and the viscoelasticity.

51 citations


Journal ArticleDOI
TL;DR: In this article, the non-linear response of a simple rigid disk-rotor supported by active magnetic bearings (AMB) without gyroscopic effects is studied under multi-excitation forces.

46 citations


Journal ArticleDOI
TL;DR: In this paper, the global bifurcations and chaotic dynamics of a string-beam coupled system subjected to parametric and external excitations are investigated in detail, and the theory of normal form is utilized to find the explicit formulas of normal-form associated with one double zero and a pair of pure imaginary eigenvalues.
Abstract: The global bifurcations and chaotic dynamics of a string-beam coupled system subjected to parametric and external excitations are investigated in detail in this paper. The governing equations are firstly obtained to describe the nonlinear transverse vibrations of the string-beam coupled system. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degrees-of-freedom. Using the method of multiple scales, parametrically and externally excited system is transformed to the averaged equation. The case of 1:2 internal resonance between the modes of the beam and string, principal parametric resonance for the beam and primary resonance for the string is considered. Based on the averaged equation, the theory of normal form is utilized to find the explicit formulas of normal form associated with one double zero and a pair of pure imaginary eigenvalues. The global perturbation method is employed to analyze the global bifurcations and chaotic dynamics of the string-beam coupled system. The analysis of the global bifurcations indicates that there exist the homoclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation of the string-beam coupled system. These results obtained here mean that the chaotic motions can occur in the string-beam coupled system. Numerical simulations also verify the analytical predications.

42 citations


Journal ArticleDOI
TL;DR: In this article, a rigorous derivation of non-linear equations governing the dynamics of an axially loaded beam is given with a clear focus to develop robust low-dimensional models, where a structure is subjected to a uniformly distributed axial and a thrust force.
Abstract: A rigorous derivation of non-linear equations governing the dynamics of an axially loaded beam is given with a clear focus to develop robust low-dimensional models. Two important loading scenarios were considered, where a structure is subjected to a uniformly distributed axial and a thrust force. These loads are to mimic the main forces acting on an offshore riser, for which an analytical methodology has been developed and applied. In particular, non-linear normal modes (NNMs) and non-linear multi-modes (NMMs) have been constructed by using the method of multiple scales. This is to effectively analyse the transversal vibration responses by monitoring the modal responses and mode interactions. The developed analytical models have been crosschecked against the results from FEM simulation. The FEM model having 26 elements and 77 degrees-of-freedom gave similar results as the low-dimensional (one degree-of-freedom) non-linear oscillator, which was developed by constructing a so-called invariant manifold. The comparisons of the dynamical responses were made in terms of time histories, phase portraits and mode shapes.

Journal ArticleDOI
TL;DR: In this article, 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 paper, we investigate transient and steady nonlinear dynamics in rotor-active magnetic bearings (AMBs) 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. 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-AMBs system with 8-pole legs and the time-varying stiffness. The catastrophic phenomena for the amplitude of transient nonlinear oscillations are first observed in the rotor-AMBs 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 auto-controlling transient state chaos to the steady state periodic and quasiperiodic motions in the rotor-AMBs system with 8-pole legs and the time-varying stiffness.

Journal ArticleDOI
TL;DR: In this paper, the nonlinear oscillations and dynamic behavior of a rigid disk-rotor supported by active magnetic bearings (AMB), without gyroscopic effects, were investigated.
Abstract: This paper is concerned with the nonlinear oscillations and dynamic behavior of a rigid disk-rotor supported by active magnetic bearings (AMB), without gyroscopic effects. The nonlinear equations of motion are derived considering a periodically time-varying stiffness. The method of multiple scales is applied to obtain four first-order differential equations that describe the modulation of the amplitudes and the phases of the vibrations in the horizontal and vertical directions. The stability and the steady-state response of the system at a combination resonance for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical nonlinear behaviors, including multiple-valued solutions, jump phenomenon, hardening, and softening nonlinearity. A numerical simulation using a fourth-order Runge-Kutta algorithm is carried out, where different effects of the system parameters on the nonlinear response of the rotor are reported and compared to the results from the multiple scale analysis. Results are compared to available published work.

Journal ArticleDOI
TL;DR: In this article, a geometrically exact formulation for unsharable and inextensible elastic beams subject to support motions is obtained, and the third-order perturbation of the equation of motion is then determined in a form amenable to an asymptotic treatment.
Abstract: In this paper, the nonlinear characteristics of the parametric resonance of simply supported elastic beams are investigated. Considering a geometrically exact formulation for unsharable and inextensible elastic beams subject to support motions, the integral-partial-differential equation of motion is obtained. The third-order perturbation of the equation of motion is then determined in a form amenable to an asymptotic treatment. The method of multiple scales is used to obtain the equations that describe the modulation of the amplitude and phase of parametric-resonance motions. The stability and bifurcations of the system are investigated considering, in particular, the frequency-response function. Furthermore, experimental results are shown to confirm the theoretically predicted stability and bifurcations.

Journal ArticleDOI
TL;DR: In this article, nonlinear transverse vibrations of an Euler-Bernoulli beam with multiple supports are considered, where the beam is supported with immovable ends and the immovable end conditions cause stretching of neutral axis and introduce cubic nonlinear terms to the equations of motion.
Abstract: In this study, nonlinear transverse vibrations of an Euler-Bernoulli beam with multiple supports are considered. The beam is supported with immovable ends. The immovable end conditions cause stretching of neutral axis and introduce cubic nonlinear terms to the equations of motion. Forcing and damping effects are included in the problem. The general arbitrary number of support case is considered at first, and then 3-, 4-, and 5-support cases are investigated. The method of multiple scales is directly applied to the partial differential equations. Natural frequencies and mode shapes for the linear problem are found. The correction terms are obtained from the last order of expansion. Nonlinear frequencies are calculated and then amplitude and phase modulation figures are presented for different forcing and damping cases. The 3:1 internal resonances are investigated. External excitation frequency is applied to the first mode and responses are calculated for the first or second mode. Frequency-response and force-response curves are drawn.

Journal ArticleDOI
TL;DR: In this paper, a nonlinear parametric vibration of axially accelerating viscoelastic beams is investigated via an approximate analytical method with numerical confirmations, based on nonlinear models of a finite-small-stretching slender beam moving at a speed with a periodic fluctuation, a solvability condition is established via the method of multiple scales for subharmonic resonance.
Abstract: Nonlinear parametric vibration of axially accelerating viscoelastic beams is investigated via an approximate analytical method with numerical confirmations. Based on nonlinear models of a finite-small-stretching slender beam moving at a speed with a periodic fluctuation, a solvability condition is established via the method of multiple scales for subharmonic resonance. Therefore, the amplitudes of steady-state periodic responses and their existence conditions are derived. The amplitudes of stable steady-state responses increase with the amplitude of the axial speed fluctuation, and decrease with the viscosity coefficient and the nonlinear coefficient. The minimum of the detuning parameter which causes the existence of a stable steady-state periodic response decreases with the amplitude of the axial speed fluctuation, and increases with the viscosity coefficient. Numerical solutions are sought via the finite difference scheme for a nonlinear partial-differential equation and a nonlinear integro-partial-differential equation. The calculation results qualitatively confirm the effects of the related parameters predicted by the approximate analysis on the amplitude and the existence condition of the stable steady-state periodic responses. Quantitative comparisons demonstrate that the approximate analysis results have rather high precision.

Journal ArticleDOI
01 Sep 2008
TL;DR: In this paper, the primary resonances of a simply supported rotating shaft with stretching non-linearity are studied, and the effects of eccentricity and the damping coefficient are investigated on the steady-state response of the rotating shaft.
Abstract: In this paper, primary resonances of a simply supported rotating shaft with stretching non-linearity are studied. Rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The equations of motion are derived with the aid of Hamilton's principle and then transformed to the complex form. To analyse the primary resonances, the method of multiple scales is directly applied to the partial differential equation of motion. The frequency—response curves are plotted for the first two modes. It is shown that these resonance curves are of the hardening type. The effects of eccentricity and the damping coefficient are investigated on the steady-state response of the rotating shaft.

Journal ArticleDOI
TL;DR: In this article, the coupled nonlinear differential equations of the non-linear dynamical two-degree-of-freedom vibrating system including quadratic and cubic nonlinearities are studied.

Journal ArticleDOI
TL;DR: In this article, the non-linear behavior of a single-link flexible visco-elastic Cartesian manipulator is studied by using D'Alembert's principle and generalized Galarkin method and the response obtained using method of multiple scales are compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement.
Abstract: In this present work, the non-linear behavior of a single-link flexible visco-elastic Cartesian manipulator is studied. The temporal equation of motion with complex coefficients of the system is obtained by using D’Alembert's principle and generalized Galarkin method. The temporal equation of motion contains non-linear geometric and inertia terms with forced and non-linear parametric excitations. It may also be found that linear and non-linear damping terms originated from the geometry of the large deformation of the system exist in this equation of motion. Method of multiple scales is used to determine the approximate solution of the complex temporal equation of motion and to study the stability and bifurcation of the system. The response obtained using method of multiple scales are compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. The response curves obtained using viscoelastic beams are compared with those obtained from a linear Kelvin–Voigt model and also with an equivalent elastic beam. The effect of the material loss factor, amplitude of base excitation, and mass ratio on the steady state responses for both simple and subharmonic resonance conditions are investigated.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the stability of a controlled van der Pol-Duffing oscillator with nonlinear feedback control in the vicinity of non-resonant Hopf bifurcations.

Journal ArticleDOI
TL;DR: In this article, the authors investigated large-amplitude, in-plane beam vibration using numerical simulations and a perturbation analysis applied to the dynamic elastica model, where the self-weight of the beam is included in the equations.
Abstract: Large-amplitude, in-plane beam vibration is investigated using numerical simulations and a perturbation analysis applied to the dynamic elastica model. The governing non-linear boundary value problem is described in terms of the arclength, and the beam is treated as inextensible. The self-weight of the beam is included in the equations. First, a finite difference numerical method is introduced. The system is discretized along the arclength, and second-order-accurate finite difference formulas are used to generate time series of large-amplitude motion of an upright cantilever. Secondly, a perturbation method (the method of multiple scales) is applied to obtain approximate solutions. An analytical backbone curve is generated, and the results are compared with those in the literature for various boundary conditions where the self-weight of the beam is neglected. The method is also used to characterize large-amplitude first-mode vibration of a cantilever with non-zero self-weight. The perturbation and finite difference results are compared for these cases and are seen to agree for a large range of vibration amplitudes. Finally, large-amplitude motion of a postbuckled, clamped–clamped beam is simulated for varying degrees of buckling and self-weight using the finite difference method, and backbone curves are obtained.

Journal ArticleDOI
Magdy A. Sirwah1
TL;DR: In this article, the authors considered the formation of resonant waves of two modes on the interface between fluids that arisen as a result of second-harmonic resonance and derived a more general system of non-linear partial differential equations with complex coefficients governing the amplitudes of the resonant wave.
Abstract: This paper considers, in the frame work of the model of two superposed layers of viscous-potential incompressible magnetic fluids, the problem on formation of resonant waves of two modes on the interface between fluids that arisen as a result of second-harmonic resonance. The fluids moving with uniform velocities parallel to their interface are stressed by a tangential magnetic field. The analysis includes the linear, as well as the non-linear effects where the analytical solutions are constructed using the method of multiple scales, in both space and time, and hence the solvability conditions correspond to the uniform (convergent) solutions are obtained. The solvability conditions are then exploited to derive a more general system of non-linear partial differential equations with complex coefficients governing the amplitudes of the resonant waves. These equations are examined for solutions corresponding to sinusoidal wavetrains consequently different kinds of instabilities are demonstrated. The stability criterion in each case is derived and discussed both analytically and graphically.

Journal ArticleDOI
01 Mar 2008
TL;DR: In this paper, a two-degree-of-freedom non-linear system with quadratic and cubic nonlinearities and parametric excitation in the horizontal and vertical directions is investigated.
Abstract: The method of multiple scales is applied to investigate the non-linear oscillations and dynamic behaviour of a rotor-active magnetic bearings (AMBs) system, with time-varying stiffness. The rotor-AMB model is a two-degree-of-freedom non-linear system with quadratic and cubic non-linearities and parametric excitation in the horizontal and vertical directions. The case of principal parametric resonance is considered and examined. The steady-state response and the stability of the system at the principal parametric resonance case for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical non-linear behaviours including multiple-valued solutions, jump phenomenon, hardening and softening non-linearity. Different effects of the system parameters on the non-linear response of the rotor are also reported. Results are compared with available published work.

Journal ArticleDOI
TL;DR: In this paper, the parameter stability and global bifurcations of a strong nonlinear system with parametric excitation and external excitations are investigated in detail using the method of multiple scales.

Journal ArticleDOI
01 Oct 2008
TL;DR: In this paper, the influence of uncertainty in choosing the fractional parameters on the character of nonlinear damped vibrations of suspension bridges is investigated, and the method of multiple scales is used as a method of solution.
Abstract: Nonlinear free damped vibrations of a suspension bridge with a bisymmetric stiffening girder are considered under the conditions of the internal resonance one-to-one, i.e., when natural frequencies of two dominating modes - a certain mode of vertical vibrations and a certain mode of torsional vibrations - are approximately equal to each other. Damping features of the system are defined by fractional derivatives with fractional parameters (the orders of the fractional derivatives) changing from zero to one. It is assumed that the amplitudes of vibrations are small but finite values, and the method of multiple scales is used as a method of solution. The influence of uncertainty in choosing the fractional parameters on the character of nonlinear damped vibrations of suspension bridges is investigated.

Proceedings ArticleDOI
01 Jan 2008
TL;DR: In this paper, the primary resonance of single mode forced, undamped, bending vibration of nonuniform sharp cantilevers of rectangular cross-section, constant width, and convex parabolic thickness variation was reported.
Abstract: This paper reports the primary resonance of single mode forced, undamped, bending vibration of nonuniform sharp cantilevers of rectangular cross-section, constant width, and convex parabolic thickness variation The case of nonlinear curvature, moderately large amplitudes, is considered The method of multiple scales is applied directly to the nonlinear partial-differential equation of motion and boundary conditions The frequency-response is analytically determined, and numerical results show a softening effect of the geometrical nonlinearitiesCopyright © 2008 by ASME

Journal ArticleDOI
TL;DR: In this article, the primary resonance of a Duffing oscillator with two distinct time delays in the linear feedback control under narrow-band random excitation was analyzed and the analytical results were in well agreement with the numerical results.
Abstract: The paper presents analytical and numerical results of the primary resonance of a Duffing oscillator with two distinct time delays in the linear feedback control under narrow-band random excitation. Using the method of multiple scales, the first-order and the second-order steady-state moments of the primary resonance are derived. For the case of two distinct time delays, the appropriate choices of the combinations of the feedback gains and the difference between two time delays are discussed from the viewpoint of vibration control and stability. The analytical results are in well agreement with the numerical results.

Journal ArticleDOI
TL;DR: In this paper, the principal resonance of a cantilever with a contact end was investigated using the Derjaguin-Muller-Toporov theory and the Lyapunov-linearized stability theory.
Abstract: In this paper, the principal resonance is investigated for a cantilever with a contact end. The cantilever is modeled as an Euler–Bernoulli beam, and the contact is modeled by the Derjaguin–Muller–Toporov theory. The problem is formulated as a linear nonautonomous partial-differential equation with a nonlinear autonomous boundary condition. The method of multiple scales is applied to determine the steady-state response. The equation of response curves is derived from the solvability condition of eliminating secular terms. The stability of steady-state responses is analyzed by using the Lyapunov-linearized stability theory. Numerical examples are presented to highlight the effects of the excitation amplitude, the damping coefficient, and the coefficients related to the contact.

Journal ArticleDOI
TL;DR: In this paper, the effect of electro-magnetic and mechanical parameters for the stabilities of solutions and the bifurcation is further analyzed, and critical conditions of stability are also obtained.
Abstract: Based on the Maxwell equations, the nonlinear magneto-elastic vibration equations of a thin plate and the electrodynamic equations and expressions of electromagnetic forces are derived. In addition, the magneto-elastic combination resonances and stabilities of the thin beam-plate subjected to mechanical loadings in a constant transverse magnetic filed are studied. Using the Galerkin method, the corresponding nonlinear vibration differential equations are derived. The amplitude frequency response equation of the system in steady motion is obtained with the multiple scales method. The excitation condition of combination resonances is analyzed. Based on the Lyapunov stability theory, stabilities of steady solutions are analyzed, and critical conditions of stability are also obtained. By numerical calculation, curves of resonance-amplitudes changes with detuning parameters, excitation amplitudes and magnetic intensity in the first and the second order modality are obtained. Time history response plots, phase charts, the Poincare mapping charts and spectrum plots of vibrations are obtained. The effect of electro-magnetic and mechanical parameters for the stabilities of solutions and the bifurcation are further analyzed. Some complex dynamic performances such as period-doubling motion and quasi-period motion are discussed.

Journal ArticleDOI
TL;DR: An analytical approach to determine the steady state response of a damped and undamped harmonically excited oscillator with no linear term and with cubic non-linearity is presented in this article.