Note on Chaos in Three Degree of Freedom Dynamical System with Double Pendulum

The nonlinear response of a three degree of freedom vibratory system with double pendulum in the neighbourhood internal and external resonances is investigated. The equations of motion have bean solved numerically. In this type system one mode of vibration may excite or damp another one, and for except different kinds of periodic vibration there may also appear chaotic vibration. To prove the character of this vibration and to realise the analysis of transitions from periodic regular motion to quasi-periodic and chaotic, the following have been constructed: bifurcation diagrams and time histories, phase plane portraits, power spectral densities, Poincaré maps and exponents of Lyapunov. These bifurcation diagrams show many sudden qualitative changes, that is, many bifurcations in the chaotic attractor as well as in the periodic orbits.     

Bifurcation Analyses in the Parametrically Excited Vibrations of Cylindrical Panels

We consider parametrically excited vibrations of shallow cylindrical panels. The governing system of two coupled nonlinear partial differential equations is discretized by using the Bubnov–Galerkin method. The computations are simplified significantly by the application of computer algebra, and as a result low dimensional models of shell vibrations are readily obtained. After applying numerical continuation techniques and ideas from dynamical systems theory, complete bifurcation diagrams are constructed. Our principal aim is to investigate the interaction between different modes of shell vibrations under parametric excitation. Results for system models with four of the lowest modes are reported. We essentially investigate periodic solutions, their stability and bifurcations within the range of excitation frequency that corresponds to the parametric resonances at the lowest mode of vibration.     

Numerical study of the parametric evolution of bifurcations in the three-dimensional ring problem of N bodies

We study the dynamics of a massless particle in an annular configuration of N bodies, N − 1 of which have equal masses m and are located in equal distances on a fictitious circle and one has mass βm and is located at the center of the circle. Our interest is focused on the bifurcation points from planar to three-dimensional families of symmetric periodic orbits in the above problem. We study numerically the evolution of these bifurcation points with respect to the variation of the mass parameter β. In particular we investigate the continuous evolution of bifurcation points for values of β from 2 up to 1000. The two distinct cases of the system’s behavior at β = 2 and 1000 are examined comparatively and various conclusions are drawn regarding the overall dynamical evolution of the three-dimensional system as the relative mass of the central body grows.     

New analytical approach to energy pumping problem in strongly nonhomogeneous 2dof systems

We present new analytical approach to the problem of energy pumping in strongly non- homogeneous nonlinear two-degree-of-freedom (2DOF) systems with single anchor spring under condition of initial impact. Energy pumping is a passive, almost irreversible transfer of mechanical energy from the main substructure of the system to the light auxiliary attachment. The mechanism of energy pumping in the system under consideration is a resonance capture. The approach is based on application of Laplace transformation to the principal asymptotic approximation of the equations of motion in complex form and using the power expansion of the solution in terms of time. Obtained temporal dependence of the system energetic characteristics gives a tool for estimation of energy pumping efficiency. In particular, we show that the system without an anchor spring in attachment is more efficient than the system with such a spring. Numerical simulations confirm the analytical results.     

The Dynamics of Stretchable Rods in the Inertial Case

Nonlinear amplitude equations for the near-threshold behavior of twisted extensible elastic rods under tension with inertial and dissipative dynamics are derived. In the inertial case localized solutions to the amplitude equations are derived and a linear stability criterion for the pulse solutions is obtained using the Hamiltonian formulation of the problem.     

The application of the Lattice Boltzmann method to the one-dimensional modeling of pulse waves in elastic vessels

The one-dimensional nonlinear equations for the blood flow motion in distensible vessels are considered using the kinetic approach. It is shown that the Lattice Boltzmann (LB) model for non-ideal gas is asymptotically equivalent to the blood flow equations for compliant vessels at the limit of low Knudsen numbers. The equations of state for non-ideal gas are transformed to the pressure-luminal area response. This property allows to model arbitrary pressure-luminal area relations. Several test problems are considered: the propagation of a sole nonlinear wave in an elastic vessel, the propagation of a pulse wave in a vessel with varying mechanical properties (artery stiffening) and in an artery bifurcation, in the last problem Resistor–Capacitor–Resistor (RCR) boundary conditions are considered. The comparison with the previous results shows a good precision.     

On the use of some numerical damping methods of spurious oscillations in the case of elastic wave propagation

The use of finite element and finite difference methods of spatial and temporal discretization for solving structural dynamics problems gives rise to purely numerical errors. Among the many numerical methods used to damp out the spurious oscillations occurring in the high frequency domain, it is proposed here to analyse and compare the well-known Bulk Viscosity method, which modifies the stresses calculations and a method recently presented by Tchamwa and Wielgosz, which is based on a modification of an explicit time integration algorithm. The efficiency of both methods is evaluated in a 2-D axisymmetric compressive test.     

单自由度冲击扰动下模拟磁头／磁盘系统接触回复特性的实验研究

采用自制的试验装置研究了单自由度冲击扰动下磁头 /磁盘系统的接触回复特性 ,分析了不同载荷下球形模拟磁头跳跃振动回复系数和接触时间与输入速度之间的关系 .结果表明 :当载荷较小时 ,回复系数随着输入速度的减小而增加 ,而当载荷较大时 ,回复系数随着输入速度的减小而减小 .在不同载荷条件下 ,接触时间均随输入速度的减小而缩短 ,但小载荷下接触时间缩短的幅度较大 .球形模拟磁头的回复特性和接触时间与载荷、输入速度及润滑膜特性有关 ,增大载荷和选择合适的润滑剂均可抑制跳跃振动     

Gas bubble oscillations in a fluid in the presence of 2:1 resonance between the radial and deformation oscillation frequencies

Small nonlinear oscillations of an ellipsoidal bubble in a fluid in the presence of 2:1 frequency resonance between the radial and ellipsoidal modes are considered. The equations of motion are reduced to Hamiltonian form. The quadratic and cubic terms are taken into account in the expansion of the Hamiltonian. The Hamilton function is transformed to the normal form using the invariant normalization method in the first approximation. This makes it possible to construct an analogy between the system considered and the well-known problem of a pendulous spring. The radial and ellipsoidal bubble oscillation modes correspond to the vertical and horizontal coordinates of a material point, respectively. In the absence of resonance the solution of the nonlinear equations differs from the solution of the linear equations by only a small (quadratic in the amplitude) change in the oscillation frequency. In the resonance case the radial and ellipsoidal oscillation modes periodically change places and the energy of one mode is converted into that of the other. The interest in the system in resonance is associated with precisely this fact. The question of the dissipation effect in real media is considered. The decay rate depends significantly on the physical properties of the material and, in certain special cases, can be small enough for the energy transfer effect to manifest itself.     

A semi-implicit collocation method: Application to thermal convection in 2D compressible fluids

A semi-implicit pseudo-spectral collocation method using a third-order Runge-Kutta numerical scheme for the full Navier-Stokes equations is described. The Courant-Friedrichs-Lewy condition is overcome by the implicit handling of a diffusive term, as suggested by Harned and Kerner. All such terms are solved with an iterative scheme in the Fourier space. Simulation of thermal convection in 2D compressible fluids is made by expanding variables on a Fourier-Chebyshev basis. We give some examples of sub- and supersonic steady solutions in the case where the heat flux at the upper boundary is governed by a black body.     

Review of the 1st Spectral Line Shapes in Plasmas code comparison workshop

A review is given of the first workshop dedicated to the detailed comparison of various approaches to the calculation of spectral line shapes in plasmas. A standardized set of case problems was specified in advance, together with the prescribed atomic data and assumptions to be used. In this brief review, motivations for the case problems chosen are outlined, followed by a discussion of selected results. Plans for the next workshop are discussed in the conclusion.     

Time-dependent natural convection in a square cavity: Application of a new finite volume method

A new finite volume (FV) approach with adaptive upwind convection is used to predict the two-dimensional unsteady flow in a square cavity. The fluid is air and natural convection is induced by differentially heated vertical walls. The formulation is made in terms of the vorticity and the integral velocity (induction) law. Biquadratic interpolation formulae are used to approximate the temperature and vorticity fields over the finite volumes, to which the conservation laws are applied in integral form. Image vorticity is used to enforce the zero-penetration condition at the cavity walls. Unsteady predictions are carried sufficiently forward in time to reach a steady state. Results are presented for a Prandtl number (Pr) of 0-71 and Rayleigh numbers equal to 10³, 10⁴ and 10⁵. Both 11 × 11 and 21 × 21 meshes are used. The steady state predictions are compared with published results obtained using a finite difference (FD) scheme for the same values of Pr and Ra and the same meshes, as well as a numerical bench-mark solution. For the most part the FV predictions are closer to the bench-mark solution than are the FD predictions.     

Finite scale theory: The role of the observer in classical fluid flow

In this paper I review a coarse-grained fluid theory named the finite scale theory and describe the development of its numerical analog, implicit large eddy simulation (ILES). The derivation, interpretation and properties of the finite scale equations are discussed and connections to other physical theory and numerical methods are elucidated.     

A note on numerical simulation of vortical structures in shock diffraction

In numerical simulation of the Euler equations, the slipstream or shear layer that appears behind a diffracted shock wave may develop small discrete vortices using fine computational meshes. Similar phenomena were also observed in the simulation of a Mach reflection that is accompanied by a shear layer. However, these small vortices have never been observed in any shock-tube experiment, although the wave pattern and the shape of the main vortex agree very well with visualization results. Numerical solutions obtained with coarse grids may agree better with experimental photos than those with very fine grids because of the pollution of the small vortices. This note tries to investigate the effect of viscosity on the small vortices by comparing the solutions of the laminar Navier-Stokes equations and the turbulence model. It is found that the small vortices are still observed in the solution of the laminar Navier-Stokes equations, although they can be suppressed by using the turbulence model. Numerical and experimental factors that are responsible for the deviation of the laminar solutions from experimental results are discussed. The secondary vortex in shock diffraction is successfully simulated by solving the Navier-Stokes equations.Received: 28 March 2003, Accepted: 6 May 2003, Published online: 11 June 2003     

Rarefied Pure Gas Transport in Non-isothermal Porous Media: Validation and Tests of the Model

Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This paper presents a numerical method based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described, e.g. by a digitized 3D image. The homogenization procedure—neglecting the effect of gas density gradients on heat transfer through the solid—leads to closure problems in for the obtention of effective properties; they are then simplified using a Galerkin method based on a 21-element basis set. The kinetic equations are then discretized in space with a finite- volume scheme. The method is validated against experimental data in the case of a closed test tube. It shows to be coherent with past approaches of thermal transpiration. Then, it is applied to several 3D images of increasing complexity. Another validation is brought by comparison with other distinct numerical approaches for the evaluation of the Darcian permeability tensor and of the Knudsen diffusion tensor. Results show that thermal transpiration has to be described by an effective transport tensor which is distinct from the other tensors.     

A continuum theory for wave propagation in laminated composites/ Case 2: Propagation parallel to the laminates

A continuum theory with microstructure for wave propagation in laminated composites, proposed in a previous work concerning normal propagation, is extended herein to the case of propagation parallel to the laminates.Model construction is based upon an asymptotic scheme in which dominant signal wavelengths are assumed large compared to typical composite microdimensions. A hierarchy of models is defined by the order of truncation of the obtained asymptotic sequence. To estimate accuracy, the phase velocity spectrum is investigated. As in the previous case, retention of all terms in the asymptotic sequence is found to yield the exact pectrum of Rytov. Spectral collation of the lowest order dispersive theory with exact first mode data gives excellent agreement.Based upon asymptotic expansions, a simplified first-order theory is also developed. Transient pulse data obtained from the latter exhibits good correlation with experimental results. In addition, representative calculations of microcomponent velocity distributions and interface shear stress are carried out using the simplified theory.

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Zusammenfassung Eine Kontinuumtheorie mit Mikrostruktur für die Wellenfortschreitung in lamellierten zusammengesetzten Materialien wird erweitert für den Fall einer Wellenfortschreitung parallel zu den Lamellen. Eine solche Kontinuumtheorie war in eine früheren Arbeit für Ausbreitung senkrecht zu den Lamellen vorgeschlagen worden.Der Bau des Models ist begründet auf einen asymptotischen Schema, in welchem vorherrschende Signalwellenlängen als gross verglichen mit der typischen Mikrodimension der Lamellen angenommen wird. Eine Hierarchie von Modellen wurde nach der Ordnung der Abschneidung der erhaltenen asymptotischen Reihe definiert. Um Genaugkeit abzuschätzen, wird das Phasengeschwindigkeitsspektrum untersucht. Genau wie im vorhergehenden Falle, fanden wir, dass das Beibehalten alle Terme in der asymptotischen Reihe das exakte Spektrum von Rytov liefert. Die spektrale Zuordnung der niedersten Ordnung Dispersionstheorie zu den exakten ersten Modusdaten zeigt eine ausgezeichnete Übereinstimmung.Ebenfalls wurde eine vereinfachte erste Ordnung Theorie entwickelt, welche auf einer asymtotischen Erweiterung beruht. Die Durchgangspulsdaten, welche von den letzteren erhalten wurden, zeigen eine gute Korrelation zu den experimentellen Ergebnissen. Weiterhin werden mit Hilfe der vereinfachten Theorie einige repräsentative Berechnungen der Geschwindigkeitsverteilungen der Mikrokomponenten und der Schubspannung in der Zwischenschicht durchgeführt.

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Research was sponsored by the National Science Foundation under Grant NSF GK-34017 and the Air Force Office of Scientific Research under AFOSR Grant AF-AFOSR 70-1957.

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Associate Professor, University of California, San Diego.

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Graduate Student, University of California, San Diego.     

The Determinants of Stock Price Volatility: An Industry Study

The currently ongoing IT-revolution is a great challenge for economists. The industry displays ever arising new technologies, unstable market shares, long-term swings, and short-term volatility of stock prices. Yet, to study those phenomena empirically one is constrained by a lack of data. The U.S. auto industry, for which long-term time series are available, has shared a similar experience since its early development. This paper studies how long-term swings and short-term stock price volatility in the U.S. auto industry is related to innovative efforts and switching of market shares of firms. The early period of the life-cycle of the industry was characterized by high product innovation, high market share instability, volatile stock prices, and the later period by fewer firms, process innovation, more stable market shares and less stock price volatility. In this paper we focus on the transition period leading from the first to the second period and study the relation of innovative effort, market share fluctuations and stock price dynamics. After presenting stylized facts on the life-cycle of the industry we introduce a dynamic model that is able to replicate some of the stylized facts. The dynamic model admits heterogeneous firms and encompasses both evolutionary as well as optimizing approaches.     

A one-dimensional moving grid solution for the coupled non-linear equations governing multiphase flow in porous media. 1: Model development

A straightforward moving grid finite element method is developed to solve the one-dimensional coupled system of non-linear partial differential equations (PDEs) governing two- and three-phase flow in porous media. The method combines features from a number of self-adaptive grid techniques. These techniques are the equidistribution, the moving grid finite element and the local grid refinement/coarsening methods. Two equidistribution criteria, based on solution gradient and curvature, are employed and nodal distributions are computed iterativcly. Using the developed approach, an intermingle-free nodal distribution is guaranteed. The method involves examination of a single representative gradient to facilitate the application of moving grid algorithms to solve a non-linear coupled set of PDEs and includes a feature to limit mass balance error during nodal redistribution. The finite element part of the developed algorithm is verified against an existing finite difference model. A numerical simulation example involving a single-front two-phase flow problem is presented to illustrate model performance. Additional simulation examples are given in Part 2 of this paper. These examples include single and double moving fronts in two- and three-phase flow systems incorporating source/sink terms. Simulation sensitivity to the moving grid parameters is also explored in Part 2.     

A numerical study of bifurcation in laminar flow in curved ducts

The bifurcation phenomenon whereby multiple-vortex secondary flow occurs in place of the normal two-vortex flow in laminar flow in curved ducts has previously been studied numerically by several researchers. However, the various results have been conflicting on many points. The present paper describes a set of numerical experiments conducted to study the effect of numerical accuracy on the solution. The results show that the transition from two- to four-vortex structure depends strongly on the differencing scheme and to a lesser extent on the grid size. The study also shows that as the Reynolds number of the flow increases, a two-vortex structure is re-established via a path which involves strongly asymmetric secondary flow patterns. These results are in agreement, at least qualitatively, with recent experimental theoretical and numerical results.     

Helical and Localised Buckling in Twisted Rods: A Unified Analysis of the Symmetric Case

We review the geometric rod theory for the case of a naturallystraight, linearly elastic, inextensible, circular rod suffering bendingand torsion but no shear. Our primary focus is on the post-bucklingbehaviour of such rods when subjected to end moment and tension.Although this is a classic problem with an extensive literature, datingback to Kirchhoff, the usual approach tends to neglect the physicalinterpretation of solutions (i.e., rod configurations) to the modelsproposed. Here, we explicitly compute geometrical properties of buckledrods. In a unified approach, making use of Kirchhoff's dynamic analogy,both the classical helical and the more recently investigated localisedbuckling are considered. Special attention is given to a consistenttreatment of concepts of link, twist and writhe.