Constructing invariant tori for two weakly coupled van der Pol oscillators

An algorithm is developed for the construction of an invariant torus of a weakly coupled autonomous oscillator. The system is put into angular standard form. The determining equations are found by averaging and are solved for the approximate amplitudes of the torus. A perturbation series is then constructed about the approximate amplitudes with unknown coefficients as periodic functions of the angular variables. A sequence of solvable partial differential equations is developed for determining the coefficients. The algorithm is applied to a system of nonlinearly coupled van der Pol equations and the first order coefficients are generated in a straightforward manner. The approximation shows both good numerical accuracy and reproducibility of the periodicities of the van der Pol system. A comparitive analysis of integrating the van der Pol system with integrating the phase equations from the angular standard form on the approximate torus shows numerical errors of the order of the perturbation parameter =0.05 for integrations of up to 10,000 steps. Applying FFT to the numerical periodicities generated by integrating the van der Pol system near the tours reveals the same predominant frequencies found in the perturbation coefficients. Finally an expected rotation number is found by integrating the phase equations on the approximate torus.Contribution of the National Institute of Standards and Technology, a Federal agency.     

A Degenerate Bifurcation Structure in the Dynamics of Coupled Oscillators with Essential Stiffness Nonlinearities

We study the degenerate bifurcations of the nonlinear normal modes(NNMs) of an unforced system consisting of a linear oscillator weaklycoupled to a nonlinear one that possesses essential stiffnessnonlinearity. By defining the small coupling parameter , we study thedynamics of this system at the limit 0. The degeneracy in the dynamics ismanifested by a 'bifurcation from infinity' where a bifurcation point isgenerated at high energies, as perturbation of a state of infiniteenergy. Another (nondegenerate) bifurcation point is generated close tothe point of exact 1:1 internal resonance between the linear andnonlinear oscillators. The degenerate bifurcation structure can bedirectly attributed to the high degeneracy of the uncoupled system inthe limit 0, whose linearized structure possesses a double zero, and aconjugate pair of purely imaginary eigenvalues. First we construct localanalytical approximations to the NNMs in the neighborhoods of thebifurcation points and at other energy ranges of the system. Then, we`connect' the local approximations by global approximants, and identifyglobal branches of NNMs where unstable and stable mode and inverse modelocalization between the linear and nonlinear oscillators take place fordecreasing energy.     

Deformationsbeständige,elastische und Relaxations-Eigenschaften der flüssigplastischen kolloiden Systeme

Zusammenfassung Es wurde gezeigt, daß es zweckmäßig ist, die strukturierten kolloiden Systeme in fest-plastische und flüssig-plastische Systeme einzuteilen, weil beide Systeme einen übereinstimmenden Eigenschaftskomplex aufweisen, sich jedoch quantitativ durch die RelaxationszeitenP >P _k im Schubspannungsbereich unterhalb der unteren FließgrenzeP < P _k unterscheiden. Es wurde darauf hingewiesen, daß für beide Systeme die Spannungsdeformationskurven sehr charakteristisch sind.Die Kurven werden unter der Bedingung der konstanten Deformationsgeschwindigkeit erhalten, wobei ihre Form von der gegenseitigen Beziehung von und abhängt und mit den Strukturelementetypen, die durch und _i charakterisiert sind, im Zusammenhang stehen.Die Methoden,die zur Messung der elastischen Deformation im breiten Bereich längs der KurveP () bei sowohl kleiner als auch größer _r entsprechend der kritischen SchubspannungP _r angewandt werden können, wurden entwickelt. Dabei wurde gezeigt, daß die Kurveine() durch das Maximum bei _m hindurchgeht.Der Einfluß von auf die kritische Deformation _r der Strukturzerstörung und auf die maximale Rückfederung _e max, die ihrerseits wiederum von der Gelkonzentration abhängen, wurde eingehend untersucht.Es wurden die Zahlenwerte der Grenzviskosität der Nachwirkung bestimmt und die Abhängigkeit der Geschwindigkeit (der Zeit) der Relaxation der elastischen Deformation von der gesamten und der elastischen Deformation ermittelt.Weiter wurde gezeigt, daß die größte elastische Deformation _e max des Systems größer als die kritische Deformation _r der Strukturzerstörung, die dem Maximum der kritischen Schubspannung der Struktur entspricht, sein kann.     

Scheme of Investigation of the Spectrum of a Family of Perturbed Operators and Its Application to Spectral Problems in Thick Junctions

We develop a scheme for the investigation of the asymptotic behavior of eigenvalues and eigenvectors of a family of self-adjoint compact operators {A: > 0} that act in different spaces and lose their compactness in the limit case 0. We prove the Hausdorff convergence of the spectrum of the operator A to the spectrum of the limit operator A₀, obtain asymptotic estimates for this convergence both to points of the discrete spectrum and to points of the essential spectrum of the operator A₀, and prove asymptotic estimates for eigenvectors of A. This scheme is applied to the investigation of the asymptotic behavior of eigenvalues and eigenfunctions of the Neumann problem in a thick singularly degenerate junction that consists of two domains connected by an -periodic system of thin rods of fixed length.     

The flow in a narrow duct with an indentation or hump on one wall

This paper deals with the flow in a narrow duct with an indentation or hump on one wall, on the assumption that , the ratio of the duct width to the length of the indentation or hump, is small. This enables the governing equations to be simplified and an analytic solution is derived on the assumption thatRe (the Reynolds number based on duct width) is of 0 (1). This simple solution breaks down whenRe is of 0 (^–1) and numerical solutions are obtained for the case whenR=Re is of 0 (1). These show, forR sufficiently large, that there are regions of reversed flow both in the indentation and on the plane wall opposite to it, and for humps, regions of reversed flow downstream.

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Strömung in einem schmalen Kanal mit einer Einbuchtung oder Wölbung an einer Wand

Zusammenfassung Diese Untersuchung befaßt sich mit der Strömung in einem schmalen Kanal mit einer Einbuchtung oder Wölbung an einer Wand, mit der Annahme, daß das Verhältnis der Kanallänge zu der Länge der Einbuchtung oder Wölbung klein ist. Dies ermöglicht, daß die beschreibenden Gleichungen vereinfacht werden können und eine analytische Lösung mit der AnnahmeRe (Re basierend auf der Kanallänge) gleich 0 (1) erhalten werden kann. Die einfache Lösung ist nicht mehr anwendbar, wennRe gleich 0 (^–1) ist. Numerische Lösungen werden für den Fall erhalten, daßR=Re gleich 0 (1) ist. Dies zeigt, daß es für hinreichend großesR Gebiete mit Gegenstrom sowohl in der Einbuchtung als auch an der flachen Wand gegenüber gibt und für den Fall einer Wölbung, daß diese stromabwärts existieren.

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Nomenclature h width of duct - h ₀ maximum height of hump/indentation - H the shape of the hump/indentation - l length of the hump/indentation - p pressure of the fluid - Q applied flow rate - Re Reynolds number based on duct width,Re=Q/ - R modified Reynolds number,R= Re - u velocity in thex-direction - v velocity in they-direction - x coordinate along the duct - y coordinate across the duct - duct width/length ratio,=h/l - modified transverse coordinate,=y/H - kinematic viscosity - ₀ skin friction ony=0 - stream function     

Propagation of a spherical wave in nonlinearly compressible and elastoplastic materials

The problem of spherical wave propagation in soil under the action of an intense uniformly decreasing load ₀(t) applied to the boundary of a cavity with radius r₀ is considered. Soil with a high stress level is modeled either by ideally nonlinearly compressible or elastoplastic material, taking account of linear irreversible unloading for the material. In contrast to [1–7], in order to describe material movement use is made of strain theory [8] with determining functions = (), _i=_i(_i), where , _i, , _i are the first and second invariants of strain and stress tensors. During material loading these functions are presented in the form of polynomials ()=(_i+₂¦¦), _ii)=(_i-_2i)_i, in which constant coefficients _i, _i=1, 2) are determined by experiment, taking account of the triaxial stressed state of soil. Solution of the problem is constructed by an analytically reversible method, with prescribed shape for the shock-wave (SW) surface in the form of a second-degree polynomial relating to time t and a numerical method of characteristics for a prescribed arbitrarily decreasing load _i(t). On the basis of the analytical equations obtained, calculations are carried out for material parameters (including loading profile) in a computer and stresses and mass velocity of plastic and elastoplastic materials are compared.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 95–100, July–August, 1986.The authors express their sincere thanks to Kh. A. Rakhmatulin for discussing the results of this work.     

Transcendentally small transversality in the rapidly forced pendulum

The rapidly forced pendulum equation with forcing sin((t/), where =<₀^p,p = 5, for ₀, sufficiently small, is considered. We prove that stable and unstable manifolds split and that the splitting distanced(t) in the ( ,t) plane satisfiesd(t) = sin(t/) sech(/2) +O( ₀ exp(–/2)) (2.3a) and the angle of transversal intersection,, in thet = 0 section satisfies 2 tan/2 = 2S _s = (/2) sech(/2) +O(( ₀ /) exp(–/2)) (2.3b) It follows that the Melnikov term correctly predicts the exponentially small splitting and angle of transversality. Our method improves a previous result of Holmes, Marsden, and Scheuerle. Our proof is elementary and self-contained, includes a stable manifold theorem, and emphasizes the phase space geometry.     

Flow model for velocity distribution in fixed porous beds under isothermal conditions

In a brief survey of the previous work the limitations of the modified Darcy equation and of the vectorial form of the Ergun equation are discussed. To include the effect of wall friction on the flows the viscous resistance term is added to the vectorial form of the Ergun equation. Using the generalized Ergun equation a one-dimensional formulation is presented for flow of fluids through packed beds taking into account the variation of porosity along the radial direction. It is found that there is a reasonable agreement between the numerical and the experimental results and it is observed that the variation of porosity with radial position has greater influence on channeling of velocity near the walls. For the assumption of constant porosity the velocity profiles exhibit similar nature as the plug flow profiles with a thin boundary layer near the wall.

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Modell der Geschwindigkeitsverteilung in einem isotherm durchströmten Festbett

Zusammenfassung In der vorliegenden Arbeit werden eingangs die Anwendbarkeitsgrenzen der modifizierten Darcy-Gleichung und der in vektorieller Form geschriebenen Ergun-Gleichung diskutiert. Um Einflüsse der Wandreibung auf eine Strömung mit in der Ergun-Gleichung berücksichtigen zu können, wird ein Reibungsterm hinzugefügt. Die so generalisierte Gleichung kann benutzt werden, um die eindimensional gerichtete Strömung durch eine Kugel schüttung zu berechnen. Eine radiale Veränderung der Schüttungsporosität ist dabei mit in die Betrachtung eingeschlossen. Das nichtlineare Grenzwertproblem wird numerisch gelöst und mit experimentellen Daten aus der Literatur verglichen. Die mit Meßwerten zufriedenstellend übereinstimmenden Rechenergebnisse zeigen, daß die radiale Porositätsverteilung in einem Festbett einen erheblichen Einfluß auf die Durchströmungsgeschwindigkeit in Wandnähe ausübt; die Berechnungen geben die Strömungsrandgängigkeit wieder. Wird die Bettporosität als unveränderlich angenommen, erhält man pfropfenströmungsähnliche Geschwindigkeitsprofile mit einer dünnen Wandgrenzschicht, in welcher die Geschwindigkeit auf den Wert null abfällt.

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Nomenclature A Tridiagonal matrix defined in Eq. (20) - a Bed radius - d_p Particle diameter - f₁ 150 (1–)²/(³d _p ² ) Darcy resistance term - f₂ 1,75(1–)/(³d_p) Parameter of resistance due to inertial effects - ¯f₁ 150(1–)²/³ - ¯f₂ 1,75(1–)/₃ - G Column vector defined in Eq. (20) - k Permeability, /f₁ - L Length of the bed - P Pressure - r Radial co-ordinate - R_p Reynolds number based on particle diameter, v₀d_p/ - , v_z Superficial velocity vector, axial component - v_1z Average superficial velocity defined in Eq. (20) - V Absolute magnitude of velocity - ¯v The average velocity - v₀ The velocity at the centre of the tube - X Column vector defined in Eq. (20) - r^* Dimensionless radial co-ordinate, r/a - p^* Dimensionless pressure, p/v ₀ ² - v _z ^* Dimensionless axial component of velocity, v_z/v₀ - ¯v^* Dimensionless average velocity defined in Eq. (20) - z^* Dimensionless axial co-ordinate, z/L Greek letters Ratio of tube radius to particle diameter, a/d_p - Porosity or void fraction - ₀ Porosity at the axis of the container - Dynamical viscosity - Kinematic viscosity - p Density - Distance from the wall of the container, defined in Eq. (16)     

Numerical solution of quasilinear singularly perturbed ordinary differential equation without turning points

In this paper we consider a quasilinear second order ordinary diferential equation with a small parameter ε, Firstly an approximate problem is constructed. Then an iterative procedure is developed. Finally we give an algorithm whose accuracy is good for arbitrary ε>0.     

Near-wall turbulence models for 3D boundary layers

Calculations of the three-dimensional boundary layer in an S shaped duct are performed with various – models. Three different near-wall models are used for the – model, of which one is using a new set of near-wall damping functions deduced from direct numerical simulations of turbulent channel flow available in the literature. The results show that it is possible to obtain damping functions giving better agreement, especially for and , with direct simulation data and experiments than with damping functions deduced from trial and error.     

Schwere symmetrische Kreisel kleiner Nutationen

Zusammenfassung Zur Integration der Eulerschen Bewegungsgleichungen schwerer symmetrischer Kreisel werden der Winkel (t) (Abb. 1) durch (t)=₀+(t) ersetzt und in sämtlichen Reihenentwicklungen von abhängiger Funktionen die Potenzen höheren als zweiten Grades vernachlässigt. Dadurch ist es möglich, die Eulerschen Winkel (t), (t) und (t) durch elementare Formeln zu beschreiben und somit sind die wesentlichsten Erscheinungen im Bewegungsablauf der schweren symmetrischen Kreisel einfach zu übersehen.     

The dynamics of resonant capture

Resonant capture describes the behavior of a weakly coupled multi-degree-of-freedom system when two or more of its uncoupled frequencies become locked in resonance. Flow on the region of phase space near the resonance (the resonance manifold) involves a region bounded by a separatrix in the uncoupled (=0) system. Capture corresponds to motions which appear to cross into the interior of the separated region for >0. We offer two approximate methods for estimating which initial conditions lead to capture: an energy method and a perturbation method based on invariant manifold theory. These methods are applied to a model problem involving the spinup of an unbalanced rotor attached to an elastic support.     

A Turbulence Model Sensitivity Study for CH4/H2 Bluff-Body Stabilized Flames

The objective of this work is to assess the performances of different turbulence models in predicting turbulent diffusion flames in conjunction with the flamelet model.The k– model, the Explicit Algebraic Stress Model (EASM) and the k– model withvaried anisotropy parameter C (LEA k– model)are first applied to the inert turbulent flow over a backward-facing step, demonstrating the quality of the turbulence models. Following this, theyare used to simulate the CH₄/H₂ bluff-body flame studied by the University of Sydney/Sandia.The numerical results are compared to experimental values of the mixture fraction, velocity field, temperature and constituent mass fractions.The comparisons show that the overall result depends on the turbulence model used, and indicate that theEASM and the LEA k– models perform better than the k– model and mimic most of the significant flow features.     

Chaos and bifurcation of phase-locking loops under periodic perturbation

This paper discusses the chaos and bifurcation for equation x+εcosxx+αsinx =εbsint. By use of the Melnikov method the conditions to have the chaotic behavior and to have subharmonic oscillations are given.     

Homogenization of the Robin problem in a thick multilevel junction

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction that is the union of a domain ₀ and a large number 2N of thin rods with variable thickness of order = (N ^–1). The thin rods are divided into two levels, depending on their length. In addition, the thin rods from each level are -periodically alternated. We investigate the asymptotic behavior of the solution as 0 under the Robin conditions on the boundaries of the thin rods. By using some special extension operators, a convergence theorem is proved.Published in Neliniini Kolyvannya, Vol. 7, No. 3, pp. 336–355, July–September, 2004.     

On stability of the Hill equation with damping

We consider the Hill equation with damping describing the parametric oscillations of a torsional pendulum excited by varying the moment of inertia of the rotating body. Using the method of a small parameter, we analytically calculate a fundamental system of solutions of this equation in the form of power series in the excitation amplitude with accuracy O(²) and verify conditions for its stability. In the first-order approximation in , we prove that the resonance domain exists only if the excitation frequency is sufficiently close to the double natural frequency of the pendulum; the corresponding equation of the stability boundary is obtained.Published in Neliniini Kolyvannya, Vol. 7, No. 2, pp. 169–179, April–June, 2004.     

Homogenization of the Process of Phase Transitions in Multidimensional Heterogeneous Periodic Media

The homogenization of the Stefan multidimensional problem is carried out in the case where the medium is a composite consisting of two different substances with an –periodic structure. The averaged problem is deduced by asymptotic methods. It is shown that its solution is the limit of solutions of –problems.     

The influence of blending polystyrenes of narrow molecular weight distribution on melt creep flow and creep recovery in elongation

Creep and creep recovery experiments in elongation were performed with melts of anionically polymerized polystyrenes (PS) and with their blends at a temperature of 150 °C. For stresses ₀ < 10 000 N/m² the samples with narrow molecular weight distribution show linear viscoelastic behavior up to the maximum Hencky strain = 3.5, achievable in a newly developed elongational rheometer for polymer melts. The compliances,D (t), of the blends are linear-viscoelastic only up to a strain limit _L . For strains beyond _L the compliance of each blend depends on the stress ₀. For a series of binary blends, prepared from the same components of narrow MWD, the linear-viscoelastic limit _L seems to be independent of the mixing ratio and stress. _L seems to be a function only of the molecular weights of the original components, the blends investigated were made from.Paper presented at the Annual Conference of the German Society of Rheology at Ulm, March 7–10, 1983.     

A solution of the E-ε model for nearly homogeneous turbulence with a mean shear

An analytical solution of the E- model for the downstream evolution of a stationary and nearly homogeneous turbulent shear flow is presented. In case that the turbulent time scale has adjusted itself to the time scale imposed by the shear, an asymptotic solution can be derived from the full solution, which shows that both E and increase downstream exponentially. By comparing this asymptotic solution with experimental data a value for the unknown constant c _l , in the -equation, is derived. Moreover, we find an expression for the downstream development of the variance of a scalar, which is also compared with experimental data. The analytical solution shows that a homogeneous shear flow with a uniform velocity gradient can only be obtained if the shear is sufficiently small. In the experiments this condition is not always satisfied. A discussion is given on how a nearly homogeneous shear flow can be obtained over a limited downstream interval by changing the initial conditions in E and , and a comparison is made with experimental data. Finally it is shown that better transverse homogeneity can be obtained by taking an exponential velocity profile instead of a linear profile.