Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.     

Stability boundaries for fluid-conveying pipes with flexible support under axial load

Summary The interaction of fluid velocity and axial load on the instability of a pipe is investigated. The downstream end of the pipe is free and the upstream end is pinned with a rotational spring. For an infinite spring stiffness, the system reduces to a cantilevered pipe. Flutter or divergence instability may occur, depending on the combination of velocity and load. Results are obtained by direct numerical solution of the governing equations. An approximate procedure based on a variational principle is also presented. The effects of the axial load, spring stiffness, and mass parameter are studied, and the results are compared to the standard case of a cantilevered pipe without axial load.

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Grenzen des stabilen Verhaltens eines flexibel gestützten, flüssigkeitsleitenden Rohres unter axialer Last

Übersicht Untersucht wird das Zusammenwirken der Strömung im Rohr und seiner axialen Belastung auf das Stabilitätsverhalten unter gegebenen Randbedingungen: das stromabwärtsgerichtete Rohrende ist frei von kinematischen Zwänge, das andere Ende wird mit einer Drehfeder festgehalten. Im Grenzfall einer unendlich steifen Feder entspricht dies einem fest eingespannten Rohr. In Abhängigkeit von der Kombination der Strömungsgeschwindigkeit und der Last kann entweder stabiles Flattern oder divergente Instabilität auftreten. Die Ergebnisse werden mittels direkter numerischer Verfahren aus den Systemgleichungen gewonnen. Sie werden verglichen mit einer Näherungslösung, welche auf einem Variationsprinzip basiert. Untersucht werden die Einflüsse der axialen Last, der Federsteifigkeit und der Massenverteilung, wobei die Ergebnisse mit der Lösung für ein fest eingespanntes Ausgu\rohr ohne axiale Last-wirkung verglichen werden.

     

Structure and stability of annular sheared channel flows: effects of confinement,curvature and inertial forces – waves

The structure and stability of the flows in an annular channel sheared by a rotating lid are investigated experimentally, theoretically and numerically. The channel has a square section, and a small curvature parameter: the ratio Γ of the inter-radii to the mean radius is 9.5%. The sidewalls and the bottom of the channel are integral and can rotate independently of the lid, permitting pure shear, co-rotation and counter-rotation cases. The basic flows obtained at small shear are characterized. In the absence of co-rotation, the centrifugal force linked with the curvature of the system plays an important role, whereas, when co-rotation is fast, the Coriolis force dominates. These basic flows undergo some instabilities when the shear is increased. These instabilities lead to supercritical traveling waves in the pure shear and co-rotation cases, but to weak turbulence in the counter-rotation case. The Reynolds number for the onset of instabilities, constructed with the velocity difference between the lid and bottom at mid-radius, and the height of the channel, increases from 1000 in the counter-rotation case to 1260 in the pure shear case and higher and higher values when co-rotation increases, i.e., when the Coriolis effect increases. The relevance of uni-dimensional Ginzburg-Landau models to describe the dynamics of the waves is studied. The domain of validity of these models turns out to be quite narrow.     

Optimal Forms of Shallow Arches with Respect to Vibration and Stability

Shallow, linearly elastic arches of unspecified form but with given uniform cross section and material are considered. For given span and length of the arch, two different optimization problems are formulated and solved. In the first, we determine the form of the arch which maximizes the fundamental vibration frequency. The corresponding vibration mode turns out to be either symmetric or antisymmetric. In the second, a static load with given spatial distribution is considered, and the critical value of the load magnitude for snap-through instability is maximized. This instability may occur at a limit point or a bifurcation point. Optimal forms are determined for sinusoidal loading, uniform loading, and a central concentrated load. In both types of problems, arches with simply supported or clamped ends are considered, and the maximum frequencies and critical loads obtained are compared to those for a circular arch with similar end conditions. In all the cases with simply supported ends, it is found that a circular arch is almost optimal. For clamped ends, however, it turns out that the optimal arches have zero slope at the ends and that they are much more efficient than a circular arch.     

Sensitivity Analysis and Optimal Design of Nonlinear Beams and Plates

A uniform formulation of sensitivity analysis for beams and plates is presented in terms of generalized stresses and strains. Both physical and geometric nonlinearities can be treated within this formulation. Next, optimal design problems for stress and deflection constraints are formulated and the relevant optimality conditions are derived using the concept of a linear adjoint structure. Finally, several numerical solutions of optimal design problems of beams are presented.     

Snap loads and torsional oscillations of the original Tacoma Narrows Bridge

In 1940, the original Tacoma Narrows Bridge was completed on June 10 and opened to traffic on July 1. On November 7, the deck collapsed. Before that day, significant vertical oscillations had occurred, but no torsion. The bridge as built was stable with respect to torsional motion under the winds of November 7 and previous winds with higher speeds. However, snap loads in the diagonal ties attached to the north midspan cable band helped to loosen the band, and the frictional resistance between the band and the north suspension cable passing through it was overcome. The cable began to slip through the band. For this new structural system, with longitudinal motion of the north cable, the wind speed was higher than the critical speed for torsional flutter, and torsional motion was initiated. Approximately 700 cycles of torsional oscillations occurred during the hour prior to the collapse. In the present study, the snap loads on the cable band are discussed first. Then a continuum model of the central span (deck, cables, and hangers) is formulated. The longitudinal motions of the cables are included, so that the slippage can be incorporated. Known information from the observed steady-state torsional motion is utilized with assumed forms of the vertical cable displacements, and the governing equations provide the horizontal cable displacements, the dynamic tensions in the cables, the vertical and torsional motions of the deck, and the resultant lift force and pitching moment (including damping) acting on the deck during its final hour.     

Spatio-temporal patterns in the thermoconvection of a planar nematic layer: II. Experiments

We study experimentally the evolution of thermoconvection in a laterally extended planar nematic layer, at zero or weak stabilizing magnetic field. As the applied thermal gradient is increased, a cascade of symmetry breakings occurs, towards structures of increasing spatial complexity, and ultimately towards oscillating states. The patterns are characterized optically, and simple models for the distortion of the vertical (out of plane) component of the director field are proposed. Received: 1st December 1997 / Revised: 25 May 1998 / Accepted: 2 June 1998     

Whirling of a forced cantilevered beam with static deflection. II: Superharmonic and subharmonic resonances

Secondary resonances of a slender, elastic, cantilevered beam subjected to a transverse harmonic load are investigated. The effects of nonlinear curvature, nonlinear inertia, viscous damping and static load are included. Cubic terms in the governing equations lead to subharmonic and superharmonic resonances of order three. The static displacement produced by the weight of the beam introduces quadratic terms in the governing equations, which cause subharmonic and superharmonic resonances of order two. Out-of-plane motion is possible in all of these secondary resonances when the principal moments of inertia of the beam cross section are approximately equal.