Multiple Scales Solution Research Articles

Nonlinear forced vibration analysis of clamped functionally graded beams

Multiple time scale solutions are presented to study the nonlinear forced vibration of a beam made of symmetric functionally graded (FG) materials based on Euler–Bernoulli beam theory and von Karman geometric nonlinearity. It is assumed that material properties follow either exponential or power law distributions through the thickness direction. A Galerkin procedure is used to obtain a second-order nonlinear ordinary equation with cubic nonlinear term. The natural frequencies are obtained for the nonlinear problem. The effects of material property distribution and end supports on the nonlinear dynamic behavior of FG beams are discussed. Also, forced vibrations of the system in primary and secondary resonances have been studied, and the effects of different parameters on the frequency-response have been investigated.

Multiple Time Scales Solution of an Equation with Quadratic and Cubic Nonlinearities Having Fractional- Order Derivative

Nonlinear vibrations of quadratic and cubic system are considered. The equation of motion includes fractional order term. Multiple time scales (a perturbation method) solution of the system is developed. Effect of fractional order derivative term is discussed.

A multiple times scale solution for non-linear vibration of mass grounded system

In this paper, the vibration of a mass grounded system which includes two linear and non-linear springs in series has been considered. Since this system, depending on its parameters can oscillate symmetrically and asymmetrically, both cases have been solved using multiple times scales (MTS) method and some analytical relations have been obtained for natural frequency of oscillations. The results have been compared with previous work and good agreement has been obtained. Also forced vibrations of system in primary and secondary resonances have been studied for the first time and the effects of different parameters on the frequency-response have been investigated.

Multiple Scales Solution for a Beam with a Small Bending Stiffness

This paper considers the problem of a beam with a small bending stiffness, within the framework of a nonlinear beam model that includes both the classical cable and the linear beam as limiting cases. This problem, treated as a perturbation of the catenary solution, is solved with the multiple scales method. The resulting expressions of the beam deflection and of the internal forces, as well as those obtained with the more commonly applied matched asymptotics method, are compared with numerical results. This comparison indicates that a better accuracy can be achieved with the multiple scales approach, for a similar computational effort. These results also suggest that application of the multiple scales method to the solution of beam problems involving boundary layers extend the range of values of the small parameter, for which accurate analytical solutions can be obtained by a perturbation technique.

Oscillating flames: multiple-scale analysis

A complete multiple-scale solution is constructed for the one-dimensional problem of an oscillating flame in a tube, ignited at a closed end, with the second end open. The flame front moves into the unburnt mixture at a constant burning velocity relative to the mixture ahead, and the heat release is constant. The solution is based upon the assumption that the propagation speed multiplied by the expansion ratio is small compared with the speed of sound. This approximate solution is compared with a numerical solution for the same physical model, assuming a propagation speed of arbitrary magnitude, and the results are close enough to confirm the validity of the approximate solution. Because ignition takes place at the closed end, the effect of thermal expansion is to push the column of fluid in the tube towards the open end. Acoustics set in motion by the impulsive start of the column of fluid play a crucial role in the oscillation. The analytical solution also captures the subsequent interaction between acoustics and the reaction front, the effect of which does not appear to be as significant as that of the impulsive start, however.

Buckling and post-buckling of extensible rods revisited: A multiple-scale solution

An exact non-linear formulation of the equilibrium of elastic prismatic rods subjected to compression and planar bending is presented, electing as primary displacement variable the cross-section rotations and taking into account the axis extensibility. Such a formulation proves to be sufficiently general to encompass any boundary condition. The evaluation of critical loads for the five classical Euler buckling cases is pursued, allowing for the assessment of the axis extensibility effect. From the quantitative viewpoint, it is seen that such an influence is negligible for very slender bars, but it dramatically increases as the slenderness ratio decreases. From the qualitative viewpoint, its effect is that there are not infinite critical loads, as foreseen by the classical inextensible theory. The method of multiple (spatial) scales is used to survey the post-buckling regime for the five classical Euler buckling cases, with remarkable success, since very small deviations were observed with respect to results obtained via numerical integration of the exact equation of equilibrium, even when loads much higher than the critical ones were considered. Although known beforehand that such classical Euler buckling cases are imperfection insensitive, the effect of load offsets were also looked at, thus showing that the formulation is sufficiently general to accommodate this sort of analysis.

Multiple-Scale Analysis of Charge Transport in Semiconductor Radiation Detectors: Application to Semi-Insulating CdZnTe

The transport, trapping, and subsequent detrapping of charge in single crystals of semi-insulating cadmium zinc telluride (CdZnTe) has been analyzed using multiple-scale perturbation techniques. This method has the advantage of not only treating impulse charge generation typical in spectroscopic analysis, but also a large class of continuous generation sources more relevant to high-flux x-ray imaging applications. We first demonstrate that the multiple-scale solutions obtained for small-current transients induced by an impulse generation of charge are consistent with well-known exact solutions. Further, we use the multiple-scale solutions to derive an analytic generalization of the Hecht equation that incorporates detrapping over times much longer than the carrier transit time (i.e., delayed signal components). The method is then applied to a continuous charge generation source that approximates that of an x-ray source. The space–time solutions obtained are relevant to detector design in high-flux x-ray imaging applications. Throughout this work the multiple-scale solutions are compared with exact solutions as well as full numerical solutions of the fundamental charge conservation equations.

Nonlinear longitudinal/transversal modal interactions in highly extensible suspended cables

Recent research literature mostly deals with nonlinear resonant dynamics of low-extensible cables involving transversal modes. Herein, we aim to investigate geometrically nonlinear longitudinal/transversal modal interactions in highly extensible suspended cables, whose material properties are assumed to be linearly elastic. Depending on cable elasto-geometric properties, the spectrum of low-order planar frequencies manifests primary and secondary frequency crossover phenomena of transversal/transversal and longitudinal/transversal modes, respectively. By focusing on 1:1 internal resonances, nonlinear equations of finite-amplitude, harmonically forced and damped, cable motion are considered, fully accounting for overall inertia and displacement coupling effects. Meaningful quadratic nonlinear contributions of non-resonant, higher-order, longitudinal modes are highlighted via a multimode-based, second-order multiple scales solution. Overall coupled/uncoupled dynamic responses, bifurcations, stability and space–time-varying displacements due to longitudinal/transversal (vs. transversal/transversal) modal interactions at secondary (vs. primary) crossovers are analytically and numerically evaluated, along with the resonant longitudinal mode-induced dynamic forces.

The effects of kinematic condensation on internally resonant forced vibrations of shallow horizontal cables

This study aims at comparing non-linear modal interactions in shallow horizontal cables with kinematically non-condensed vs. condensed modeling, under simultaneous primary external and internal resonances. Planar 1:1 or 2:1 internal resonance is considered. The governing partial-differential equations of motion of non-condensed model account for spatio-temporal modification of dynamic tension, and explicitly capture non-linear coupling of longitudinal/vertical displacements. On the contrary, in the condensed model, a single integro-differential equation is obtained by eliminating the longitudinal inertia according to a quasi-static cable stretching assumption, which entails spatially uniform dynamic tension. This model is largely considered in the literature. Based on a multi-modal discretization and a second-order multiple scales solution accounting for higher-order quadratic effects of a infinite number of modes, coupled/uncoupled dynamic responses and the associated stability are evaluated by means of frequency– and force–response diagrams. Direct numerical integrations confirm the occurrence of amplitude-steady or -modulated responses. Non-linear dynamic configurations and tensions are also examined. Depending on internal resonance condition, system elasto-geometric and control parameters, the condensed model may lead to significant quantitative and/or qualitative discrepancies, against the non-condensed model, in the evaluation of resonant dynamic responses, bifurcations and maximal/minimal stresses. Results of even shallow cables reveal meaningful drawbacks of the kinematic condensation and allow us to detect cases where the more accurate non-condensed model has to be used.

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II: Internal resonance activation, reduced-order models and nonlinear normal modes

Resonant multi-modal dynamics due to planar 2:1 internal resonances in the non-linear, finite-amplitude, free vibrations of horizontal/inclined cables are parametrically investigated based on the second-order multiple scales solution in Part I [1] (in press). The already validated kinematically non-condensed cable model accounts for the effects of both non-linear dynamic extensibility and system asymmetry due to inclined sagged configurations. Actual activation of 2:1 resonances is discussed, enlightening on a remarkable qualitative difference of horizontal/inclined cables as regards non-linear orthogonality properties of normal modes. Based on the analysis of modal contribution and solution convergence of various resonant cables, hints are obtained on proper reduced-order model selections from the asymptotic solution accounting for higher-order effects of quadratic nonlinearities. The dependence of resonant dynamics on coupled vibration amplitudes, and the significant effects of cable sag, inclination and extensibility on system non-linear behavior are highlighted, along with meaningful contributions of longitudinal dynamics. The spatio-temporal variation of non-linear dynamic configurations and dynamic tensions associated with 2:1 resonant non-linear normal modes is illustrated. Overall, the analytical predictions are validated by finite difference-based numerical investigations of the original partial-differential equations of motion.

Effect of mean entropy on unsteady disturbance propagation in a slowly varying duct with mean swirling flow

The propagation of acoustic disturbances in a slowly-varying duct with mean swirling flow and non-uniform mean entropy is considered. This is representative of the region of swirling flow downstream of the rotor in aeroengine applications where variations in mean entropy can be significant. The duct is assumed to vary slowly in axial cross section and a consistent multiple-scales solution is derived. The variation in axial wavenumber and amplitude of the duct modes is determined as part of the solution. Comparisons are made between the cases of isentropic and non-isentropic flow. The cut-on/cut-off characteristics of a given mode are altered by any form of non-uniform mean entropy, with modes pushed closer to cut-off resulting in larger variations in amplitude. The greatest departure from the baseline isentropic result occurs for a positive entropy gradient. Increasing the swirl velocity at the duct inlet leads to increasing levels of mean entropy.

A uniformly valid multiple scales solution for cut-on cut-off transition of sound in flow ducts

Starting from a multiple-scales approach to model sound transmission through a slowly varying duct of arbitrary cross section, an explicit analytical solution is derived for a mode undergoing cut-on cut-off transition. The solution is a composite one, removing the singularity that appears in the WKB approximation by encompassing the inner boundary-layer solution with the behaviour far upstream and downstream. The solution should not only prove to be a useful benchmark for computational aeroacoustic models, but also enable a designer to continue to use multiple-scales theory to examine sound transmission even when cut-on cut-off transitional modes are present.

A fifth-order multiple-scale solution for Hopf bifurcations

The subject of this paper is the multiple-time-scale analysis of Hopf bifurcations up to fifth-order nonlinearities. It is shown how an asymptotic fifth-order expansion captures the change in the nature of the limit cycle from stable to unstable and viceversa. The formulation is validated by applying it to a simple mechanical system for which there exists an analytical limit-cycle solution. Applications include the pre- and post-flutter behavior of a typical section with nonlinear spring having a stable limit cycle (supercritical Hopf bifurcation) that turns into an unstable one, because of fifth-order nonlinearities.

Azimuthal Sound Mode Propagation in Axisymmetric Flow Ducts

Assuming an axisymmetric mean flow and acoustic boundary conditions, a set of three-dimensional axisymmetric equations suitable for each azimuthal sound mode of fluctuation is derived based on the three-dimensional linearized Euler equations. The independent two-dimensional model for each azimuthal sound mode is deduced by a Fourier series decomposition. A computational aeroacoustics approach is then applied to solve each two-dimensional equation system. The dispersion-relation-preserving finite difference scheme is implemented for spatial discretization, whereas the 2N storage low-dissipation and low-dispersion Runge-Kutta scheme is applied for time integration. Appropriate boundary conditions are prescribed at the various boundary regions. The numerical procedure is first validated by cases of a straight circular pipe and an annular duct subjected to a subsonic uniform mean flow, the numerical results of which show very good agreement with the analytical solutions. A further numerical example is presented for an axisymmetric inlet duct with an aeroenginelike geometry including an internal spinner. The aeroacoustic computation is based on an inviscid irrotational mean flow calculated by a second-order computational fluid dynamics solver. The acoustic solutions agree well with the existing finite element and multiple-scales solutions in the literature. Finally, the cuton, cutoff transition phenomena are investigated based on the proposed numerical approach. The transition behavior is found to be highly dependent on the mean flowfield in addition to the known geometric effect. These results demonstrate the feasibility of the proposed model and solution procedure.

Transient free‐surface flow of a viscoelastic fluid in a narrow channel

AbstractThe interplay between inertia and elasticity is examined for transient free‐surface flow inside a narrow channel. The lubrication theory is extended for the flow of viscoelastic fluids of the Oldroyd‐B type (consisting of a Newtonian solvent and a polymeric solute). While the general formulation accounts for non‐linearities stemming from inertia effects in the momentum conservation equation, and the upper‐convected terms in the constitutive equation, only the front movement contributes to non‐linear coupling for a flow inside a straight channel. In this case, it is possible to implement a spectral representation in the depthwise direction for the velocity and stress. The evolution of the flow field is obtained locally, but the front movement is captured only in the mean sense. The influence of inertia, elasticity and viscosity ratio is examined for pressure‐induced flow. The front appears to progress monotonically with time. However, the velocity and stress exhibit typically a strong overshoot upon inception, accompanied by a plug‐flow behaviour in the channel core. The flow intensity eventually diminishes with time, tending asymptotically to Poiseuille conditions. For highly elastic liquids the front movement becomes oscillatory, experiencing strong deceleration periodically. A multiple‐scale solution is obtained for fluids with no inertia and small elasticity. Comparison with the exact (numerical) solution indicates a wide range of validity for the analytical result. Copyright © 2004 John Wiley & Sons, Ltd.

Comparison of a finite element model with a multiple-scales solution for sound propagation in varying ducts with swirling flows

A multiple-scales (MS) solution is proposed to study sound propagation in slowly varying ducts with mean swirling flows. Instead of the standard linearized Euler equations, the MS method is applied to the so-called Galbrun’s equation. This equation is based on an Eulerian–Lagrangian description and corresponds to a wave equation written only in terms of the Lagrangian perturbation of the displacement. This yields simpler differential equations to solve for the MS model as well as simpler boundary conditions. In this paper, Galbrun’s equation is also solved by a mixed pressure-displacement finite element method (FEM). The proposed FEM model has already been tested in authors’ previous papers. This model is quite general and is extended here to arbitrary mean flows, including compressibility and swirling flow effects. Some MS and FEM solutions are then compared in order to validate both models.

Sequential Computation of Total Least-Squares Parameter Estimates

Conclusions This Note has developed a solution of Lambert’s problem, using a second-order approximation of the equations of motion, referenced to a point on a nearby circular orbit. The solution was obtained from a multiple-scale solution of Kepler’s problem in which the initial velocity variables were determined by an algebraic perturbation method, given the initial and final positions. The results show a significant improvement over the previous solution derived from a first-order approximation of the equations of motion. The advantages of such a method are an improved numerical accuracy as well as the coupling of in-plane and out-of-plane motion captured at second order. Acknowledgment

Sound propagation in slowly varying lined flow ducts of arbitrary cross-section

Sound transmission through ducts of constant cross-section with a uniform inviscid mean flow and a constant acoustic lining (impedance wall) is classically described by a modal expansion, where the modes are eigenfunctions of the corresponding Laplace eigenvalue problem along a duct cross-section. A natural extension for ducts with cross-section and wall impedance that are varying slowly (compared to a typical acoustic wavelength and a typical duct radius) in the axial direction is a multiple-scales solution. This has been done for the simpler problem of circular ducts with homentropic irrotational flow. In the present paper, this solution is generalized to the problem of ducts of arbitrary cross-section. It is shown that the multiple-scales problem allows an exact solution, given the cross-sectional Laplace eigensolutions. The formulation includes both hollow and annular geometries. In addition, the turning point analysis is given for a single hard-wall cut-on, cut-off transition. This appears to yield the same reflection and transmission coefficients as in the circular duct problem.

Multiple scales analysis of the Fokker–Planck equation for simple shear flow

We extend the multiple time scales formalism originally introduced for the Fokker–Planck equation by Wycoff and Balazs (Physica A 146 (1987) 175) to the case of simple shear flow. The analysis is carried out for small values of the Stokes number, St, a dimensionless measure of the inertia of a Brownian particle. The probability density is expanded in terms of Hermite functions of the fluctuation velocity, the difference between the actual particle velocity and the unperturbed flow velocity at its location, and consequently, the structure of the multiple scales hierarchy differs from the analogous equilibrium problem with a conservative forcing. The multiple scales solutions are compared to a small St expansion of the exact solutions for two canonical initial conditions. We also obtain the form of the Smoluchowski equation correct to O( St) for the effects of particle inertia. Modifications to the scheme for hydrodynamically interacting particles are discussed.

Two Asymptotic Forms of the Rotational Solution for Wave Propagation Inside Viscous Channels with Transpiring Walls

Summary In this article, a long rectangular channel is considered with two transpiring walls that are a small distance apart. The channel’s head end is hermetically closed while the aft end is either open (isobaric) or acoustically closed (choked). A mean flow enters uniformly across the permeable walls, turns, and exits from the downstream end. The slightest unsteadiness in flow velocity is inevitable and occurs at random frequencies. Small pressure disturbances are thus produced. Those waves with oscillations matching the natural frequencies of the enclosure are promoted. Inception of small pressure perturbations alters the flow character and leads to a temporal field that we wish to analyse. The mean flow is of the Berman type and can be obtained from the Navier–Stokes equations over different ranges of the cross-flow Reynolds number. The unsteady component can be formulated from the linearized momentum equation. This has been carried out in numerous studies and has routinely given rise to a singular, boundary-value, double-perturbation problem in the cross-flow direction. The current study focuses on the resulting second-order differential equation that prescribes the rotational wave motion in the transverse direction. This equation exhibits unique features that define a general class of ordinary differential equations. Due to the oscillatory behaviour of the problem, two general asymptotic formulations are derived, for an arbitrary mean-flow profile, using WKB and multiple-scales expansions. The fundamental asymptotic solutions reveal the same similarity parameter that controls the rotational wave character. The multiple-scales solution unravels the problem’s characteristic length scale following a unique, nonlinear variable transformation. The latter is derived rigorously from the problem’s solvability condition. The advantage of using a multiple-scales procedure lies in the ease of construction, accuracy, and added physical insight stemming from its leading-order term. For verification purposes, a specific mean-flow solution is used for which an exact solution can be derived. Comparisons between asymptotic and exact predictions are gratifying, showing an excellent agreement over a wide range of physical parameters.