Parabolic Arcs Research Articles

Foot clearance strategy for step-over-step stair climbing in transfemoral amputees

Stair ascent is a particularly challenging task for transfemoral amputees. The aim of this clinical note was to describe the kinematic features of foot clearance in transfemoral amputee who can ascend stairs using a step-over-step strategy. The marker trajectories of the first metatarsophalangeal joint (Mt1) and clearance height were measured in two transfemoral amputees who could (TF1) and could not (TF2) climb stairs using a step-over-step strategy. The Mt1 marker trajectories of the TF1 moved backward in the early swing phase, and the trajectory followed an off-centered parabolic arc to achieve a similar clearance height as able-bodied subjects. TF2 could not climb the stairs without tripping in each step. An effective compensatory strategy to avoid tripping during stair climbing may be to use the hip joint for a backward extension and rapid flexion of the prosthetic leg during the early swing phase. The foot clearance strategy in transfemoral amputees who can climb stairs using a step-over-step strategy will help us better understand adaptive prosthetic control and thus develop more effective gait rehabilitation programs.

On a rigid inclusion pressed between two elastic half spaces

A solution to the problem of a rigid cylindrical inclusion pressed between two elastic half spaces is obtained using the distributed dislocation technique. The solution is compared with previously published analytical and numerical results for a rigid cylindrical inclusion bounded by two parabolic arcs with rounded corners. A simplified solution to the problem based on the classical contact theory and well-known results for crack problems is also suggested and validated. The simplified solution agrees well with analytical results in the case when the length of the opening around inclusion is much larger than the length of the contact zone.

The effect of arc length on the least-volume fin under sensible and latent heat loads

The analysis of fins using arc length to determine the minimum volume for a given heat transfer rate is demonstrated by a unified formulation for fully and partially wet surfaces of longitudinal and pin fins. The shape of the optimum profile, a circular arc, was identical for both fin geometries under the same surface conditions. The optimum shape for partially wet surfaces had two different circular arcs for dry and wet surfaces. Unlike the previous study, the base thickness of circular profiles was less than that of parabolic profiles, regardless of the surface conditions. The minimum fin volume for the circular profile was always less than the corresponding parabolic profile. The difference in volume between these profiles reached a maximum for fully wet surfaces, and was also dependent upon the psychrometric properties and heat rate constraint. The optimum wet circular arc resulted in a maximum 156% savings in fin material, compared with the parabolic arc for pin fins; this percentage was up to 135% for longitudinal fins. Nevertheless, the savings for longitudinal and pin fins were only 25.9% and 22.0%, respectively, for dry surfaces; these results differed from published values.

Power-law models of totally anisotropic scattering

The interstellar scattering responsible for pulsar parabolic arcs, and for intra-day variability of compact radio quasars, is highly anisotropic in some cases. We numerically simulate these observed phenomena using totally anisotropic, power-law models for the electron density fluctuations which cause the scattering. By comparing our results to the scattered image of PSR B0834+06 and, independently, to dual-frequency light curves of the quasar PKS1257-326, we constrain the nature of the scattering media on these lines of sight. We find that models with spectral indices slightly below \beta=3, including the one-dimensional Kolmogorov model, are broadly consistent with both data sets. We confirm that a single physical model suffices for both sources, with the scattering medium simply being more distant in the case of B0834+06. This reinforces the idea that intra-day variability and parabolic arcs have a common cause in a type of interstellar structure which, though obscure, is commonplace. However, the implied gas pressure fluctuations are large compared to typical interstellar pressures, and the magnetic stresses are much larger still. Thus while these scattering media may be commonplace, their underlying dynamics appear quite extraordinary.

Generalized Gaussian quadrature rules over regions with parabolic edges

This paper presents a generalized Gaussian quadrature method for numerical integration over regions with parabolic edges. Any region represented by R 1={(x, y)| a≤x≤b, f(x)≤y≤g(x)} or R 2={(x, y)| a≤y≤b, f(y)≤x≤g(y)}, where f(x), g(x), f(y) and g(y) are quadratic functions, is a region bounded by two parabolic arcs or a triangular or a rectangular region with two parabolic edges. Using transformation of variables, a general formula for integration over the above-mentioned regions is provided. A numerical method is also illustrated to show how to apply this formula for other regions with more number of linear and parabolic sides. The method can be used to integrate a wide class of functions including smooth functions and functions with end-point singularities, over any two-dimensional region, bounded by linear and parabolic edges. Finally, the computational efficiency of the derived formulae is demonstrated through several numerical examples.

The Thermal Distortion of a Funnel Mold

This article investigates the thermal distortion of a funnel mold for continuous casting of thin slabs and explores the implications on taper and solidification of the steel shell. The three-dimensional mold temperatures are calculated using shell-mold heat flux and cooling water profiles that were calibrated with plant measurements. The thermal stresses and distorted shape of the mold are calculated with a detailed finite-element model of a symmetric fourth of the entire mold and waterbox assembly, and they are validated with plant thermocouple data and measurements of the wear of the narrow-face copper mold plates. The narrow-face mold distorts into the typical parabolic arc, and the wide face distorts into a “W” shape owing to the large variation in bolt stiffnesses. The thermal expansion of the wide face works against the applied narrow-face taper and funnel effects, so the effect of thermal distortion must be considered to accurately predict the ideal mold taper.

Influence of Fluid Flow Distribution in Micro-Channel Arrays to Phase Transition Processes

Microstructured heat exchangers are well suited for such phase transition processes as evaporation of liquids due to their heat transfer capabilities, being two to three orders of magnitude higher than those of conventional heat transfer devices. Controlling liquid evaporation inside micro-channels to provide full evaporation in a stable way is not trivial. In most cases, such instabilities as slug flow, bubbly flow, or vapor clogging occur, based on cross-talk possibilities between the individual micro-channels of a channel array, normally caused by open void inlet structures. Therefore, fluid inlet distribution is inhomogeneous, which results, in the best case, in a parabolic shape of a stable evaporation frontline. The parabolic shape occurs due to the residence time distribution of the fluid, generated by shorter path length in the array center and longer ones in the outer areas of the micro-channel array. Computational fluid dynamics simulation approves this result. Such a frontline can be kept stable when the process parameters are well controlled. Small deviations of the inlet parameters may lead to strong disturbances of the evaporation process, destabilizing it. When changing the inlet fluid distribution system to provide the most equal flow distribution possible, the span of the parabolic shape of the evaporation frontline can be reduced drastically. Finally, a stable evaporation frontline perpendicular to flow direction can be obtained. This status is no longer very sensitive to process deviations. This article presents an optimized micro-channel device for the optical investigation of phase transition phenomena. The device allows the exchange of integrated micro-channel arrays to investigate different designs for their suitability. It is separated into three independent sections, which can be heated or cooled individually. Therefore, very strict and rapid temperature jumps can be obtained within relatively short distances. The micro-channel array foils used for the experiments have been manufactured by mechanical micro-machining. Thus, the cross-sections of the micro-channels are always rectangular. Hydraulic diameter and length of the micro-channels, as well as the shape of the inlet and outlet voids, can be varied. Using a simple triangular or rectangular open inlet void, a stable evaporation line was generated, showing a parabolic shape. Depending on the mass flow and the size and shape of the inlet void, the span of the parabolic arc was influenceable.

The Investigation of Arch Model Acting in Mass-Flow Hoppers

This paper presents the experimental results of mass-flow hopper arch geometry investigation, which was conducted using a variable geometry plane-flow bin. The cohesive arches formed under different critical outlet openings and hopper half-angles were measured using a 360° two-dimensional laser line scan system. This system was employed to obtain the complete surface profile of each arch across the width of the outlet by moving the rotating laser along the total length of the outlet. The test results were analyzed using Matlab, adopting stationary wavelet transformation de-noising to decrease the signal noise generated during the testing process. The geometric data for each single line scan was smoothed and combined to present a three-dimensional arch surface profile shown to be in good agreement with the observed experimental arch profiles. The angle η at the intersection of the arch with the hopper walls was then calculated by running a Matlab program and a new angle η' is introduced to the arch shape study. The detailed results are discussed in the paper. Arch geometry models, such as the parabolic arc and circular arc arch models developed, respectively, by Walker [ and Enstad [ are reviewed and their relevance is discussed based on the experimental results presented in this paper.

Refractive convergent plasma lenses explain extreme scattering events and pulsar scintillation

ABSTRACT We propose convergent plasma lenses, possibly from current sheets, as a generic solution to strong interstellar scattering. These lenses resolve the overpressure problem by geometric alignment as noted by Goldreich & Shridhar. Our new model further quantitatively explains properties of extreme scattering events and pulsar parabolic arcs. It makes quantitative predictions testable by VLBI on scattering events. It differs conceptually from previous models by occurring through rare, localized underdense sheets. Such sheets are thermally and kinematically stable, and could be consequences of reconnection. The apparent diffractive effects are a result of coherent interference of refractive images. We propose that these lenses can be used for precision distance determination to pulsars, enabling accurate gravity source localization.

Pointed drawings of planar graphs

We study the problem how to draw a planar graph crossing-free such that every vertex is incident to an angle greater than π. In general a plane straight-line drawing cannot guarantee this property. We present algorithms which construct such drawings with either tangent-continuous biarcs or quadratic Bézier curves (parabolic arcs), even if the positions of the vertices are predefined by a given plane straight-line drawing of the graph. Moreover, the graph can be drawn with circular arcs if the vertices can be placed arbitrarily. The topic is related to non-crossing drawings of multigraphs and vertex labeling.

Differentiability of fractal curves

A self-similar set that spans can have no tangent hyperplane at any single point. There are lots of smooth self-affine curves, however. We consider plane self-affine curves without double points and with two pieces. There is an open subset of parameter space for which the curve is differentiable at all points except for a countable set. For a parameter set of codimension one, the curve is continuously differentiable. However, there are no twice differentiable self-affine curves in the plane, except for line segments and parabolic arcs.

Auriculotherapy and Acupuncture in Space Sickness

I would like to tell readers about my recent experience of auriculotherapy. After taking off from Merignac, the plane flew in parabolic arcs (figure 1), executing 31 of these in the space of 3 h at an altitude of 10 km. A parabolic flight includes three phases: Figure 1 The different phases of a parabolic flight (downloaded from http://www.eurospacecenter.be). Finally, the plane follows a normal trajectory for 2 min, a period of time that is necessary for piloting reasons and to allow the passengers to recover. After this, the plane begins the next parabolic arc (figure 1). Three fighter pilots take turns piloting, as it is physically too difficult for a single pilot to carry out the entire parabolic flight on his own. Space sickness is extremely violent, and I had never experienced anything like it before. It is in no way comparable to sea or motion sickness. One enters a ‘state of shock’ with a drop in arterial pressure, motionlessness, a pale grey face and sudden uncontrollable vomiting. Those not affected ‘fly’ to the rescue of …

The regular shape of stratovolcanoes: A DEM-based morphometrical approach

We studied the shape of the most regular-shaped stratovolcanoes of the world to mathematically define the form of the ideal stratovolcano. Based on the Shuttle Radar Topographic Mission data we selected 19 of the most circular and symmetrical volcanoes, which incidentally all belong to subduction-related arcs surrounding the Pacific. The selection of volcanoes benefitted from the introduction of a new definition of circularity which is more robust than previous definitions, being independent of the erosional dissection of the cone. Our study on the shape of stratovolcanoes was based on the analysis of the radial elevation profiles of each volcano. The lower half section of the volcanoes is always well fitted by a logarithmic curve, while the upper half section is not, and falls into two groups: it is fitted either by a line (“C-type”, conical upper part) or by a parabolic arc (“P-type”, parabolic/concave upper part). A quantitative discrimination between these groups is obtained by fitting their upper slope with a linear function: C-type volcanoes show small, whereas P-type volcanoes show significant negative angular coefficient. The proposed threshold between the two groups is − 50 × 10 − 4 °/m. Chemical composition of eruptive products indicates higher SiO 2 and/or higher H 2O content for C-type volcanoes, which could imply a higher incidence of mildly explosive (e.g. strombolian) eruptions. We propose that this higher explosivity is responsible for forming the constant uppermost slopes by the deposition of ballistic tephra and its subsequent stabilisation at a constant angle. By contrast, P-type volcanoes are characterized by a smaller SiO 2 and H 2O content, which can be responsible for a higher incidence of effusive events and/or a lower incidence of upper flank-forming (i.e. mild) explosive eruptions. Therefore, the concave upper flanks of these volcanoes may be shaped typically by lava flows. Based on this hypothesis, we propose that the morphometric analysis of the elevation profile of stratovolcanoes can provide insights into their dominant eruptive style.

Catching fly balls in VR: A test of the OAC, LOT and trajectory prediction strategies

Professional baseball players somehow manage to intercept and catch fly balls successfully. Three perceptual strategies have been proposed to explain the outfielder's ability and to predict the fielder's interception path. (1) TP: The trajectory prediction strategy (Saxberg, 1987) proposes that the fielder perceives the ball's initial conditions, computes its trajectory, and runs directly to the predicted landing point. (2) OAC: The optical acceleration cancellation strategy (Chapman, 1968) proposes that the fielder continuously controls their radial motion (toward the ball) by running to null the vertical optical acceleration of the ball. Transverse motion (lateral) is controlled independently by holding the bearing direction of the ball constant. (3) LOT: The linear optical trajectory strategy proposes that the fielder runs so as to keep the ball's rising optical trajectory straight rather than curved; this maintains a constant ratio between the tangents of the ball's elevation and azimuth angles (tana/tanb = c). These strategies are normally highly correlated, but they can be tested in VR by manipulating the ball's trajectory. Brown University players were tested in an immersive virtual environment (12 x 12 m) while wearing a head-mounted-display (60 x 40 deg, 50–70 ms latency) with trackers on the head and glove. The ball's trajectory was manipulated by increasing or decreasing the gravitational constant (g) at the peak of flight. Trials in which gravity was changed were matched with normal trials in which the ball traveled the same distance on a parabolic arc. Participants made rapid and continuous adjustments to compensate for the change in gravity, inconsistent with TP. Moreover, their radial and transverse velocities varied independently, inconsistent with LOT. Behavior was most consistent with the combination of OAC and constant bearing strategies. We describe a dynamical model that implements this control strategy.

Determining Direction for Optimization of Movable Wing Tip Strake

S TRAKES are lifting surfaces featuring low aspect ratios and highly swept sharp leading edges conventionally located at the fuselage–wing junction. The benefits of using strakes on various airplane configurations have been known since the early 1970s. By imposing controlled flow separation and creating powerful leadingedge vortices that then subject both the strake and the main wing to high rotational velocities, strakes generate significant amounts of additional lift. Movable tip strake is an extension of the strake concept first proposed by the author [1]. Wing-fuselage strakes have been the subject of numerous studies; see, for example, [2–12]. The reader is referred to [1] for an extensive review of strake research. Movable wing tip strake is a novel design idea. Other researchers have studiedfixedwing tip strakes in the past, notablyMa [13], Traub et al. [14], and Staufenbiel and Vitting [15]. The author proposed making them movable in flight. Then the advantageous effect of the strakes on the aircraft performance is achieved without having to set the aircraft to a high , accompanied by a high CD. Instead, the optimal angle of operation of the strakes is attained by simply setting independently the strakes relative to thewing. This setting angle then represents an additional variable for controlling the airplane configuration. First, a simple semidelta movable tip strake (MTS) shown in Fig. 1 was tested in combination with a rectangular baseline wing [1]. The results showed that this configuration outperformed by a factor of 2.24 a modified wing obtained by extending the span of the baseline wing and maintaining the same airfoil and having the same aspect ratio ( 2 S ) as the wing with movable tip-mounted strakes. By deflecting the strakes up or down, the L=D benefit shifted to a lower, or higher, respectively, of the main wing, as it would be expected, because the flow conditions on the strake depend on both the of the wing and ds. In subsequent studies, the author studied various planform shapes of MTSs with the objective of maximizing their positive effect as measured by theL=D of thewing-strake configuration for a givenCL [16–21]. Ninevarious strake planforms,MTS1–9,were tested. These included strakes with straight leading edges swept at angles from 60 to 80 deg, a strake for which the leading edgewas a parabolic arc, and a series of strakes featuring two straight-line portions of the leading edge swept at different angles, thus resulting in a leading-edge break point. It has been found that the best performance is achievedwith the MTS4, which had a cropped double-delta strake planform featuring an inboard portion swept at 80 deg and an outboard portion swept at 45 deg with the transition, or the leading-edge break point, located at 57.5% of the strake root chord. This strake is shown in Fig. 2. When used at moderate-to-high of the wing and set to the neutral setting with respect to the wing chord plane, that is, at ds 0 deg, this strake improved the L=D by approximately 26%. This beneficial effect of the strake persisted at both positive and negative settings of the strake (ds > 0 deg, the strake leading edge up, and ds < 0 deg, the strake leading edge down) and the optimal point corresponding to the L=D max shifted to a lower or higher of the main wing, respectively, as it would be expected. This confirmed the supposition that ds would represent a new and useful variable for wing configuration control. To continue to optimize the movable strake design, it appeared of interest to examine the effect of the leading-edge break point position, xBP, while maintaining the inboard and outboard sweep angles of MTS4. Apparently, by moving the break point forward (and, consequently, slightly inboard), the area of the inboard, highly swept portion of the strake decreases and that of the outboard, moderately swept portion increases. Moving the break point backward (and somewhat outboard) accomplishes the opposite effect. The relative sizes of the inboard and outboard segments of the strake presumably have an effect on the formation and disposition of the generated leading-edge vortices and the resulting wing-strake aerodynamics. This study has been commenced with the goal to investigate those effects.

Applying inversion to construct planar, rational spirals that satisfy two-point G 2 Hermite data

A method of two-point G 2 Hermite interpolation with spirals is proposed. To construct a sought for curve, the inversion is applied to an arc of some other spiral. To illustrate the method, inversions of parabola are considered in detail. The resulting curve is 4th degree rational. The method allows the matching of a wide range of boundary conditions, including those which require an inflection. Although not all G 2 Hermite data can be matched with a spiral generated from a parabolic arc, introducing one intermediate G 2 data solves the problem. Expanding the method by involving other spirals arcs is also discussed.

100 μas RESOLUTION VLBI IMAGING OF ANISOTROPIC INTERSTELLAR SCATTERING TOWARD PULSAR B0834+06

We have invented a novel technique to measure the radio image of a pulsar scattered by the interstellar plasma with 0.1 mas resolution. We extend the "secondary spectrum" analysis of parabolic arcs by Stinebring et al. (2001) to very long baseline interferometry and, when the scattering is anisotropic, we are able to map the scattered brightness astrometrically with much higher resolution than the diffractive limit of the interferometer. We employ this technique to measure an extremely anisotropic scattered image of the pulsar B0834+06 at 327 MHz. We find that the scattering occurs in a compact region about 420 pc from the Earth. This image has two components, both essentially linear and nearly parallel. The primary feature, which is about 16 AU long and less than 0.5 AU in width, is highly inhomogeneous on spatial scales as small as 0.05 AU. The second feature is much fainter and is displaced from the axis of the primary feature by about 9 AU. We find that the velocity of the scattering plasma is 16+-10 km/s approximately parallel to the axis of the linear feature. The origin of the observed anisotropy is unclear and we discuss two very different models. It could be, as has been assumed in earlier work, that the turbulence on spatial scales of ~1000 km is homogeneous but anisotropic. However it may be that the turbulence on these scales is homogeneous and isotropic but the anisotropy is produced by highly elongated (filamentary) inhomogeneities of scale 0.05-16 AU.

A general expression for geosynthetic strain due to deflection

ABSTRACT: Giroud [Geosynthetics International, 2, No. 3, 635–641, 1995] presented expressions for the strain within a geosynthetic layer assuming it to take the shape of a circular or parabolic arc. A more realistic situation would be one in which the deflected shape consists of both circular and parabolic arcs. This technical note presents an expression for the geosynthetic strain for a combination of parabolic and circular shapes of the deflected geosynthetic, which is commonly observed in many field applications of geosynthetics. The special simplified cases of this general equation, expected in field situations, are discussed in detail. The variation of geosynthetic strain with relative deflection, defined as a ratio of rut depth (maximum deflection, r) to the initial length (L0) of the geosynthetic, is presented to show the application limits of special simplified cases of the general equation. It is observed that the geosynthetic strain significantly depends on the shape of the deflected geosynthetic, especially at large rut depths. It is therefore recommended that the deflected geosynthetic shape considered must be exactly the same as the one observed or expected in the field application under consideration, especially at large strain levels. It is also shown that the approximate expression presented by Giroud holds for all possible shapes considered in this note, for small strains.

The use of parabolic arcs in matching curved boundaries by point transformations for some higher order triangular elements

This paper is concerned with curved boundary triangular elements having one curved side and two straight sides. The curved elements considered here are the 6-node (quadratic), 10-node (cubic), 15-node (quartic) and 21-node (quintic) triangular elements. On using the isoparametric coordinate transformation, these curved triangles in the global ( x , y ) coordinate system are mapped into a standard triangle: { ( ξ , η ) / 0 ⩽ ξ , η ⩽ 1 , ξ + η ⩽ 1 } in the local coordinate system ( ξ , η ) . Under this transformation curved boundary of these triangular elements is implicitly replaced by quadratic, cubic, quartic and quintic arcs. The equations of these arcs involve parameters, which are the coordinates of points on the curved side. This paper deduces relations for choosing the parameters in quartic and quintic arcs in such a way that each arc is always a parabola which passes through four points of the original curve, thus ensuring a good approximation. The point transformations which are thus determined with the above choice of parameters on the curved boundary and also in turn the other parameters in the interior of curved triangles will serve as a powerful subparametric coordinate transformation for higher order curved triangular elements with one curved side and two straight sides.

The dynamics of parabolic flight: Flight characteristics and passenger percepts

Flying a parabolic trajectory in an aircraft is one of the few ways to create freefall on Earth, which is important for astronaut training and scientific research. Here we review the physics underlying parabolic flight, explain the resulting flight dynamics, and describe several counterintuitive findings, which we corroborate using experimental data. Typically, the aircraft flies parabolic arcs that produce approximately 25 s of freefall (0 g) followed by 40 s of enhanced force (1.8 g), repeated 30–60 times. Although passengers perceive gravity to be zero, in actuality acceleration, and not gravity, has changed, and thus we caution against the terms “microgravity” and “zero gravity.” Despite the aircraft trajectory including large (45°) pitch-up and pitch-down attitudes, the occupants experience a net force perpendicular to the floor of the aircraft. This is because the aircraft generates appropriate lift and thrust to produce the desired vertical and longitudinal accelerations, respectively, although we measured moderate (0.2 g) aft-ward accelerations during certain parts of these trajectories. Aircraft pitch rotation (average 3°/s) is barely detectable by the vestibular system, but could influence some physics experiments. Investigators should consider such details in the planning, analysis, and interpretation of parabolic-flight experiments.