Showing papers on "Multiple-scale analysis published in 2014"

Nonlinear dynamics of axially moving beam with coupled longitudinal–transversal vibrations

Abstract: In this study, the nonlinear vibrations of an axially moving beam are investigated by considering the coupling of the longitudinal and transversal motion. The Galerkin method is used to truncate the governing partial differential equations into a set of coupled nonlinear ordinary differential equations. By detuning the axially velocity, the exact parameters with which the system may turn to internal resonance are detected. The method of multiple scales is applied to the governing equations to study the nonlinear dynamics of the steady-state response caused by the internal–external resonance. The saturation and jump phenomena of such system have been reported by investigating the nonlinear amplitude–response curves with respect to external excitation, internal, and external detuning parameters. The longitudinal external excitation may trigger only longitudinal response when excitation amplitude is weak. However, beyond the critical excitation amplitude, the response energy will be transferred from the longitudinal motion to the transversal motion even the excitation is employed on the longitudinal direction. Such energy transfer due to saturation has the potential to be used in the vibration suppression.

Evolution of the double-jumping in pipes conveying fluid flowing at the supercritical speed

Abstract: Non-linear vibration of viscoelastic pipes conveying fluid around curved equilibrium due to the supercritical flow is investigated with the emphasis on steady-state response in external and internal resonances. The governing equation, a non-linear integro-partial-differential equation, is truncated into a perturbed gyroscopic system via the Galerkin method. The method of multiple scales is applied to establish the solvability condition in the first primary resonance and the 2:1 internal resonance. The approximate analytical expressions are derived for the frequency–amplitude curves of the steady-state responses. The stabilities of the steady-state responses are determined. The generation and the vanishing of a double-jumping phenomenon on the frequency–amplitude curves are examined. The analytical results are supported by the numerical integration results.

Analysis on nonlinear oscillations and resonant responses of a compressor blade

Abstract: This paper focuses on the nonlinear oscillations and the steady-state responses of a thin-walled compressor blade of gas turbine engines with varying rotating speed under high-temperature supersonic gas flow. The rotating compressor blade is modeled as a pre-twisted, presetting, thin-walled rotating cantilever beam. The model involves the geometric nonlinearity, the centrifugal force, the aerodynamic load and the perturbed angular speed due to periodically varying air velocity. Using Hamilton’s principle, the nonlinear partial differential governing equation of motion is derived for the pre-twisted, presetting, thin-walled rotating beam. The Galerkin’s approach is utilized to discretize the partial differential governing equation of motion to a two-degree-of-freedom nonlinear system. The method of multiple scales is applied to obtain the four-dimensional nonlinear averaged equation for the resonant case of 2:1 internal resonance and primary resonance. Numerical simulations are presented to investigate nonlinear oscillations and the steady-state responses of the rotating blade under combined parametric and forcing excitations. The results of numerical simulation, which include the phase portrait, waveform and power spectrum, illustrate that there exist both periodic and chaotic motions of the rotating blade. In addition, the frequency response curves are also presented. Based on these curves, we give a detailed discussion on the contributions of some factors, including the nonlinearity, damping and rotating speed, to the steady-state nonlinear responses of the rotating blade.

Nonlinear transverse vibration of axially accelerating strings with exact internal resonances and longitudinally varying tensions

Abstract: This work explores the steady-state periodic transverse responses with their stabilities of axially accelerating viscoelastic strings. Longitudinally varying tension due to the axial acceleration is recognized in the modeling, while the tension was approximatively assumed to be longitudinally uniform in previous investigations. Exact internal resonances are highlighted in the analysis, while the resonances have been neglected in all available works. A governing equation of transverse nonlinear vibration is derived from the generalized Hamilton principle and the Kelvin viscoelastic model on the assumption that the string deformation is not infinitesimal, but still small. The axial speed is supposed to be a small simple harmonic fluctuation about the constant mean axial speed. The method of multiple scales is applied to solve the governing equation in the parametric resonances when the axial speed fluctuation frequency approaches the first three natural frequencies of the linear generating system based on 1–3 term truncations. The amplitude, the existence conditions, and the stability are determined, and the effects of the viscosity, the mean axial speed, the axial speed fluctuation amplitude, and the axial support rigidity on the amplitude and the existence are examined via the numerical examples. It is found that the 1-term, the 2-term, and the 3-term truncations yield the qualitatively same and the quantitatively close results in the case that there exist the exact internal resonances among the first three frequencies.

Nonlinear dynamics of composite laminated cantilever rectangular plate subject to third-order piston aerodynamics

Abstract: This paper presents the analysis of the nonlinear dynamics for a composite laminated cantilever rectangular plate subjected to the supersonic gas flows and the in-plane excitations. The aerodynamic pressure is modeled by using the third-order piston theory. Based on Reddy’s third-order plate theory and the von Karman-type equation for the geometric nonlinearity, the nonlinear partial differential equations of motion for the composite laminated cantilever rectangular plate under combined aerodynamic pressure and in-plane excitation are derived by using Hamilton’s principle. The Galerkin’s approach is used to transform the nonlinear partial differential equations of motion for the composite laminated cantilever rectangular plate to a two-degree-of-freedom nonlinear system under combined external and parametric excitations. The method of multiple scales is employed to obtain the four-dimensional averaged equation of the non-automatic nonlinear system. The case of 1:2 internal resonance and primary parametric resonance is taken into account. A numerical method is utilized to study the bifurcations and chaotic dynamics of the composite laminated cantilever rectangular plate. The frequency–response curves, bifurcation diagram, phase portrait and frequency spectra are obtained to analyze the nonlinear dynamic behavior of the composite laminated cantilever rectangular plate, which includes the periodic and chaotic motions.

The nonlinear vibrations of functionally graded cylindrical shells surrounded by an elastic foundation

Abstract: This paper reports the result of an investigation on the nonlinear vibrations of functionally graded cylindrical shell surrounded by an elastic foundation, based on Hamilton’s principle, von Karman nonlinear theory, and the first-order shear deformation theory. Material properties are assumed to be temperature dependent. The surrounding elastic medium is modeled as Winkler foundation model, Pasternak foundation model, and nonlinear foundation model. Galerkin’s method is utilized to convert the governing partial differential equations to nonlinear ordinary differential equations with quadratic and cubic nonlinearities. Considering the primary resonance case, the method of multiple scales is used to study the frequency response of nonlinear vibrations and the softening/hardening behavior. Parametric effects on the nonlinear vibrations are investigated.

In-plane and out-of-plane dynamics of a curved pipe conveying pulsating fluid

Abstract: This paper investigates the in-plane and out-of-plane dynamics of a curved pipe conveying fluid. Considering the extensibility, von Karman nonlinearity, and pulsating flow, the governing equations are derived by the Newtonian method. First, according to the modified inextensible theory, only the out-of-plane vibration is investigated based on a Galerkin method for discretizing the partial differential equations. The instability regions of combination parametric resonance and principal parametric resonance are determined by using the method of multiple scales (MMS). Parametric studies are also performed. Then the differential quadrature method (DQM) is adopted to discretize the complete pipe model and the nonlinear dynamic equations are carried out numerically with a fourth-order Runge–Kutta technique. The nonlinear dynamic responses are presented to validate the out-of-plane instability analysis and to demonstrate the influence of von Karman geometric nonlinearity. Further, some numerical results obtained in this work are compared with previous experimental results, showing the validity of the theoretical model developed in this paper.

The effects of nonlinearities on the vibration of viscoelastic sandwich plates

Abstract: The nonlinear free and forced bending vibration of sandwich plates with incompressible viscoelastic core is investigated under the effects of different source of nonlinearities. For the core constrained between stiffer layers, the transverse shear strains, as well as the rotations are assumed to be moderate. The linear and quadratic displacement fields are also adopted for the in-plane and out-of-plane displacements of the core, respectively. The assumption of moderate transverse strains requires a nonlinear constitutive equation which is obtained from a single-integral nonlinear viscoelastic model using the assumed order of magnitudes for linear strains and rotations. The 5th-order method of multiple scales is directly applied to solve the equations of motion. The different-order linear partial differential equations that were obtained during the perturbation solution, are solved by the method of eigenfunction expansion and the nonhomogeneous boundary conditions are dealt with by transforming to homogeneous boundaries, or using the extended Green's formula. The effects of different system parameters on the nonlinear estimation of frequencies, damping ratios, and peak response are studied. Also, the importance of different nonlinear terms arisen from different ordering assumptions is assessed and the ranges of system parameters with higher values of error are identified.

An analytical study of time-delayed control of friction-induced vibrations in a system with a dynamic friction model

Abstract: We investigate the control of friction-induced vibrations in a system with a dynamic friction model which accounts for hysteresis in the friction characteristics. Linear time-delayed position feedback applied in a direction normal to the contacting surfaces has been employed for the purpose. Analysis shows that the uncontrolled system loses stability via. a subcritical Hopf bifurcation making it prone to large amplitude vibrations near the stability boundary. Our results show that the controller achieves the dual objective of quenching the vibrations as well as changing the nature of the bifurcation from subcritical to supercritical. Consequently, the controlled system is globally stable in the linearly stable region and yields small amplitude vibrations if the stability boundary is crossed due to changes in operating conditions or system parameters. Criticality curve separating regions on the stability surface corresponding to subcritical and supercritical bifurcations is obtained analytically using the method of multiple scales (MMS). We have also identified a set of control parameters for which the system is stable for lower and higher relative velocities but vibrates for the intermediate ones. However, the bifurcation is always supercritical for these parameters resulting in low amplitude vibrations only.

Nonlinear vibrations and stability analysis of axially moving strings having nonideal mid-support conditions

Abstract: In this study, nonlinear vibrations of an axially moving string are investigated. The main difference of this study from other studies is that there is a nonideal support between the opposite sides, which allows small displacements. Nonlinear equations of motion and boundary conditions are derived using Hamilton’s principle. Equations of motion and boundary conditions are converted to nondimensional form. Thus, the equations become independent from geometry and material properties. The method of multiple scales, a perturbation technique, is used. A harmonically varying velocity function is chosen for modeling the axial movement. String as a continuous medium is investigated in two regions. Vibrations are investigated for three different cases of the excitation frequency Ω. Stability analysis is carried out for these three cases, and stability boundaries are determined for the principle parametric resonance case. Thus, differences between ideal and nonideal boundary conditions are investigated.

Multi-pulse chaotic motions of high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate

Abstract: This paper investigates the multi-pulse global bifurcations and chaotic dynamics of the high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate by using an extended Melnikov method in the resonant case. Using the von Karman type equations, Reddy’s third-order shear deformation plate theory and Hamilton’s principle, the equations of motion are derived for the laminated composite piezoelectric rectangular plate with combined parametric excitations and transverse excitation. Applying the method of multiple scales and Galerkin’s approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:2 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used for the first time to investigate the Shilnikov type multi-pulse homoclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The necessary conditions of the existence for the Shilnikov type multi-pulse chaotic dynamics of the laminated composite piezoelectric rectangular plate are analytically obtained. Numerical simulations also illustrate that the Shilnikov type multi-pulse chaotic motions can also occur in the laminated composite piezoelectric rectangular plate. Overall, both theoretical and numerical studies demonstrate that the chaos in the Smale horseshoe sense exists for the laminated composite piezoelectric rectangular plate.

Linear dynamical analysis of fractionally damped beams and rods

Abstract: The aim of this study is to develop a general model for beams and rods with fractional derivatives. Fractional time derivatives can represent the damping term in dynamical models of continuous systems. Linear differential operators with spatial derivatives make it possible to generalize a wide range of problems. The method of multiple scales is directly applied to equations of motion. For the approximate solution, the amplitude and phase modulation equations are obtained in terms of the operators. Stability boundaries are derived from the solvability condition. It is shown that a fractional derivative influences the stability boundaries, natural frequencies, and amplitudes of vibrations. The solution procedure may be applied to many problems with linear vibrations of continuous systems.

Nonlinear-forced vibrations of piezoelectrically actuated viscoelastic cantilevers

Abstract: This paper aims to study the nonlinear-forced vibrations of a viscoelastic cantilever with a piecewise piezoelectric actuator layer on its top surface using the method of Multiple Scales. The governing equation of motion is a second-order nonlinear ordinary differential equation with quadratic and cubic nonlinearities which appear in stiffness, inertia, and damping terms. The nonlinear terms are due to the piezoelectricity, viscoelasticity, and geometry of the system. Forced vibrations of the system are investigated in the cases of primary resonance and non-resonance hard excitation including subharmonic and superharmonic resonances. Analytical expressions for frequency responses are derived, and the effects of different parameters including damping coefficient, thickness to width ratio of the beam, length and position of the piezoelectric layer, density of the beam, and the piezoelectric coefficient on the frequency-response curves are discussed for each case. It is shown that in all these cases, the response of the system follows a softening behavior due to the existence of the piezoelectric layer. The piezoelectric layer provides an effective tool for active control of vibration. In addition, the effect of the viscoelasticity of the beam on passive control of amplitude of vibration is illustrated.

Internal-external resonance of a curved pipe conveying fluid resting on a nonlinear elastic foundation

Abstract: In this study, the forced vibration of a curved pipe conveying fluid resting on a nonlinear elastic foundation is considered. The governing equations for the pipe system are formed with the consideration of viscoelastic material, nonlinearity of foundation, external excitation, and extensibility of centre line. Equations governing the in-plane vibration are solved first by the Galerkin method to obtain the static in-plane equilibrium configuration. The out-of-plane vibration is simplified into a constant coefficient gyroscopic system. Subsequently, the method of multiple scales (MMS) is developed to investigate external first and second primary resonances of the out-of-plane vibration in the presence of three-to-one internal resonance between the first two modes. Modulation equations are formed to obtain the steady state solutions. A parametric study is carried out for the first primary resonance. The effects of damping, nonlinear stiffness of the foundation, internal resonance detuning parameter, and the magnitude of the external excitation are investigated through frequency response curves and force response curves. The characteristics of the single mode response and the relationship between single and two mode steady state solutions are revealed for the second primary resonance. The stability analysis is carried out for these plots. Finally, the approximately analytical results are confirmed by the numerical integrations.

Modeling relativistic soliton interactions in overdense plasmas: A perturbed nonlinear Schrodinger equation framework

Abstract: We investigate the dynamics of localized solutions of the relativistic cold-fluid plasma model in the small but finite amplitude limit, for slightly overcritical plasma density. Adopting a multiple scale analysis, we derive a perturbed nonlinear Schr\"odinger equation that describes the evolution of the envelope of circularly polarized electromagnetic field. Retaining terms up to fifth order in the small perturbation parameter, we derive a self-consistent framework for the description of the plasma response in the presence of localized electromagnetic field. The formalism is applied to standing electromagnetic soliton interactions and the results are validated by simulations of the full cold-fluid model. To lowest order, a cubic nonlinear Schr\"odinger equation with a focusing nonlinearity is recovered. Classical quasiparticle theory is used to obtain analytical estimates for the collision time and minimum distance of approach between solitons. For larger soliton amplitudes the inclusion of the fifth-order terms is essential for a qualitatively correct description of soliton interactions. The defocusing quintic nonlinearity leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude.

Steady-state response of pipes conveying pulsating fluid near a 2:1 internal resonance in the supercritical regime

Abstract: The work investigates steady-state responses of a pipe conveying fluid with a harmonic component of flow speed superposed on a constant mean value in the supercritical regime. If the flow speed exceeds a critical value, the straight configuration of the pipe becomes unstable and bifurcates into two stable curved configurations. The transverse motion measured from each of the curved equilibrium configurations is governed by a nonlinear integro-partial-differential equation. The Galerkin method is employed to discretize the governing equation into a set of coupled nonlinear ordinary differential equations with gyroscopic terms. For the pipes in the supercritical regime, the method of multiple scales is used to determine the steady-state in the vicinity of two-to-one resonance. In the presence of the internal resonance, the subharmonic, the superharmonic and the summation, and the difference resonances exist due to the pulsating fluid. The amplitude–frequency relationships are established with the emphasis on the effects of the viscosity, the pulsating amplitude, the nonlinearity, and the mean flow speed. Some nonlinear phenomena such as the appearance of the peak or jumps pertaining to modal interaction are demonstrated. The numerical integration results are in agreement with the analytical predictions.

An enriched multiple scales method for harmonically forced nonlinear systems

Abstract: This article explores enrichment to the method of Multiple Scales, in some cases extending its applicability to periodic solutions of harmonically forced, strongly nonlinear systems. The enrichment follows from an introduced homotopy parameter in the system governing equation, which transitions it from linear to nonlinear behavior as the value varies from zero to one. This same parameter serves as a perturbation quantity in both the asymptotic expansion and the multiple time scales assumed solution form. Two prototypical nonlinear systems are explored. The first considered is a classical forced Duffing oscillator for which periodic solutions near primary resonance are analyzed, and their stability is assessed, as the strengths of the cubic term, the forcing, and a system scaling factor are increased. The second is a classical forced van der Pol oscillator for which quasiperiodic and subharmonic solutions are analyzed. For both systems, comparisons are made between solutions generated using (a) the enriched Multiple Scales approach, (b) the conventional Multiple Scales approach, and (c) numerical simulations. For the Duffing system, important qualitative and quantitative differences are noted between solutions predicted by the enriched and conventional Multiple Scales. For the van der Pol system, increased solution flexibility is noted with the enriched Multiple Scales approach, including the ability to seek subharmonic (and superharmonic) solutions not necessarily close to the linear natural frequency. In both nonlinear systems, comparisons to numerical simulations show strong agreement with results from the enriched technique, and for the Duffing case in particular, even when the system is strongly nonlinear.

Secondary resonances of a quadratic nonlinear oscillator following two-to-one resonant Hopf bifurcations

Abstract: Stable bifurcating solutions may appear in an autonomous time-delayed nonlinear oscillator having quadratic nonlinearity after the trivial equilibrium loses its stability via two-to-one resonant Hopf bifurcations. For the corresponding non-autonomous time-delayed nonlinear oscillator, the dynamic interactions between the periodic excitation and the stable bifurcating solutions can induce resonant behaviour in the forced response when the forcing frequency and the frequencies of Hopf bifurcations satisfy certain relationships. Under hard excitations, the forced response of the time-delayed nonlinear oscillator can exhibit three types of secondary resonances, which are super-harmonic resonance at half the lower Hopf bifurcation frequency, sub-harmonic resonance at two times the higher Hopf bifurcation frequency and additive resonance at the sum of two Hopf bifurcation frequencies. With the help of centre manifold theorem and the method of multiple scales, the secondary resonance response of the time-delayed nonlinear oscillator following two-to-one resonant Hopf bifurcations is studied based on a set of four averaged equations for the amplitudes and phases of the free-oscillation terms, which are obtained from the reduced four-dimensional ordinary differential equations for the flow on the centre manifold. The first-order approximate solutions and the nonlinear algebraic equations for the amplitudes and phases of the free-oscillation terms in the steady state solutions are derived for three secondary resonances. Frequency-response curves, time trajectories, phase portraits and Poincare sections are numerically obtained to show the secondary resonance response. Analytical results are found to be in good agreement with those of direct numerical integrations.

Using the extended Melnikov method to study multi-pulse chaotic motions of a rectangular thin plate

Abstract: This paper investigates the multi-pulse global heteroclinic bifurcations and chaotic dynamics for the nonlinear vibrations of a simply supported rectangular thin plate by using an extended Melnikov method in the resonant case. The rectangular thin plate is subjected to spatially and temporally varying transversal and in-plane excitations, simultaneously. The equations of motion for the rectangular thin plate are derived from the von Karman equation. Applying the method of multiple scales and Galerkin’s approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:2 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary Eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used to analyze the multi-pulse heteroclinic bifurcations and chaotic dynamics of the rectangular thin plate. The contribution of the paper is the simplification of the extended Melnikov method. The extended Melnikov function can be simplified in the resonant case and does not depend on the perturbation parameter. The necessary conditions of the existence for the Shilnikov type multi-pulse chaotic dynamics of the rectangular thin plate are analytically obtained. Numerical simulations also display that the Shilnikov type multi-pulse chaotic motions can occur in the rectangular thin plate. Overall, both theoretical and numerical studies demonstrate that the chaos for the Smale horseshoe sense exists in the rectangular thin plate.

Asymptotic Solution for the Two Body Problem with Radial Perturbing Acceleration

Abstract: In this article, an approximate analytical solution for the two body problem perturbed by a radial, low acceleration is obtained, using a regularized formulation of the orbital motion and the method of multiple scales. The results reveal that the physics of the problem evolve in two fundamental scales of the true anomaly. The first one drives the oscillations of the orbital parameters along each orbit. The second one is responsible of the long-term variations in the amplitude and mean values of these oscillations. A good agreement is found with high precision numerical solutions.