Showing papers on "Parabola published in 2010"

Optimum structure to carry a uniform load between pinned supports

Abstract: Since the time of Huygens in the 17th century it has been believed that, if the weight of the structural members themselves are negligible in comparison to the applied load, the optimum structure to carry a uniformly distributed load between pinned supports will take the form of a parabolic arch rib (or, equivalently, a suspended cable). In this study, numerical layout optimization techniques are used to demonstrate that when a standard material with equal tension and compressive strength is involved, a simple parabolic arch rib is not the true optimum structure. Instead, a considerably more complex structural form, comprising a central parabolic section and networks of truss bars in the haunch regions, is found to possess a lower structural volume.

Dynamic analysis of railway transition curves

Abstract: Transition curves play an important role in ride quality, especially for high-speed railways. Accordingly, various curves have been proposed as transition forms. To evaluate the effects of the forms in transition curves, the basic requirements and forms of transition curves are first discussed, and then simulations are performed with several typical transition curves at various speeds, based on the dynamic model developed in this research. The results indicate that the dynamic effects of transition curves are determined by the curvature variation and order of the curvature derivative that is equal to zero at the curve endpoints, while the expression mode (such as algebraic or trigonometric expression) does not make much difference. From a dynamic viewpoint, the ideal transition form requires that the second derivative of its curvature is equal to zero at the curve endpoints at least, such as the 7th parabola, 9th parabola, and sinusoid, with which the vibration accelerations of the vehicle vary smoothly and the jerk can be effectively eased. Then the appropriate transition forms are recommended for ordinary and high-speed railways, respectively.

Safe-Parking Framework for Fault-Tolerant Control of Transport-Reaction Processes

Abstract: This work considers the problem of handling actuator faults in transport−reaction processes described by quasi-linear parabolic partial differential equations (PDEs) subject to input constraints. To this end, first, by exploiting the separation between the fast and slow eigenmodes of the parabolic spatial differential operator in combination with Galerkin’s method, a finite set of ordinary differential equations (ODEs) that captures the dominant dynamics of the PDE system are constructed. This finite ODE system is used to develop a Lyapunov-based model predictive controller which provides an explicit characterization of the set of initial conditions from where closed-loop stability of the parabolic PDE system is guaranteed. This control design is then subsequently used to develop a safe-parking framework for handling faults. In particular, faults which preclude the possibility of maintaining operation at the nominal equilibrium distribution, using the existing robust or reconfiguration-based fault-toleran...

The caustics of two-?and three-dimensional parabolic reflectors

Abstract: The caustic of a two-dimensional parabola illuminated off-axis is the Tschirnhausen cubic and does not contain any singular points. The caustic of a three-dimensional paraboloid of revolution similarly illuminated consists of two cusped sheets containing a hyperbolic umbilic singularity. We relate these two situations by considering a suitable sequence of illuminated elliptic paraboloids.

The tail of the maximum of Brownian motion minus a parabola

Abstract: We analyze the tail behavior of the maximum N of Brownian motion minus a parabola and give an asymptotic expansion for P(N>x) as x tends to infinity. This extends a first order result on the tail behavior, which can be deduced from Huesler and Piterbarg (1999). We also point out the relation between certain results in Groeneboom (2010) and Janson, Louchard and Martin-Lof (2010).

The solvability of the first initial-boundary problem for parabolic and degenerate parabolic equations in domains with a?conical point

Abstract: The first initial-boundary problem for second-order parabolic and degenerate parabolic equations is investigated in a domain with a conical or angular point. The means of attack is already known and uses weighted classes of smooth or integrable functions. Sufficient conditions for a unique solution to exist and for coercive estimates for the solution to be obtained are formulated in terms of the angular measure of the solid angle and the exponent of the weight. It is also shown that if these conditions fail to hold, then the parabolic problem has elliptic properties, that is, it can have a nonzero kernel or can be nonsolvable, and, in the latter case, it is not even a Fredholm problem. A parabolic equation and an equation with some degeneracy or a singularity at a conical point are considered. Bibliography: 49 titles.

Large-focal distance parabolic trough reflecting column-type light-focusing solar energy high-temperature heat collector

Abstract: The utility model discloses a large-focal distance parabolic trough column-type light-focusing solar energy heat collector which comprises a horizontal rotation component and a combined bracket, wherein the combined bracket and the horizontal rotation component are connected with each other by a hinge, the combined bracket is rigidly connected with a parabola arc-shaped strip, the inner arc surface of the parabola arc-shaped strip is attached with a correcting substrate, and the inner arc surface of the correcting substrate is closely attached with a parabolic reflecting plate. In the utility model, the whole parabolic reflecting plate and the combined bracket conduct circumferential bidirectional rotation together along the horizontal rotation component by taking the hinge as a center, therefore, the heat collector can track sunlight in east-west direction and south-north direction, wherein the sunlight is automatically tracked in the east-west direction, and manually tracked in the south-north direction. The heat collector increases the output temperature of the solar energy, and widens the application range of the solar energy.

Solar heat-collector with grillage film and parabola groove

Abstract: The invention aims to provide a frame film parabolic groove solar heat collector, comprising a parabolic groove reflective film and a heat collecting pipe, wherein the parabolic groove reflective film is supported and fixed by the a parabolic groove bracket which is movably mounted on a base to flexibly rotate on the base, the heat collecting pipe is mounted in the focus position of the parabolicgroove bracket, and the coordinate curvilinear equation of a parabolic surface bracket and the parabolic surface reflective film is F(x)=x /99. The invention is manufactured by the combined frame with simple structure, convenient and easy manufacture, high heat efficiency, and during the practical uses, based on the needs, can be manufacture as solar heat supply systems with different scales and uses to replace normal coal-burning boilers generation, air conditioners, bath, food processing, textile industries and other heat utilization.

How To Teach The Quadratic Curves Through Historical Overview

Abstract: Nowadays in school mathematics, the skill and method for solving problems are often emphasized in preference to the theoretical principles of mathematics. Students pay attention to how to make an equation mechanically before even understanding the meaning of the given problem. Furthermore they do not get to really know about the principle or theorem that were used to solve the problem, or the meaning of the answer that they have obtained. In contemporary textbooks the conic section such as circle, ellipse, parabola and hyperbola are introduced as the cross section of a cone. But they do not mention how conic section are connected with the quadratic equation or how these curves are related mutually. Students learn the quadratic equations of the conic sections introduced geometrically and are used to manipulating it algebraically through finding a focal point, vertex, and directrix of the cross section of a cone. But they are not familiar with relating these equations with the cross section of a cone. In this paper, we try to understand the quadratic curves better through the analysis of the discussion made in the process of the discovery and eventual development of the conic section and then seek for way to improve the teaching and learning methods of quadratic curves.

Paraboloid mirror light energy transmitting device

Abstract: The utility model provides a paraboloid mirror light energy transmitting device which comprises at least one first parabola concave reflecting mirror and at least one second parabola concave reflecting mirror, wherein the concaves of two parabola concave reflecting mirrors are parabola-shaped reflecting surfaces, the reflecting surfaces of the first parabola concave reflecting mirror and the second parabola concave reflecting mirror are relatively arranged, the axial lines of the reflecting surfaces of the two parabola reflecting mirrors are overlapped, the focuses of the reflecting surfaces are overlapped on a focus, the parabola-shaped reflecting surfaces of the first and the second parabola concave reflecting mirrors both adopt columnar paraboloids, and a recondensing device is further arranged on the public axial line of the first parabola concave reflecting mirror and the second parabola reflecting mirror, and is used to recondense lights gathered in the vicinity of the public axial line. The device is provided with a plurality of recondensing structures for the difficult matching of double paraboloid mirror structures, thereby solving the difficulty.

Modifiable Quasi Cubic Bézier Trigonometric Curves

Abstract: A modifiable quasi cubic curve based on functions 1,sinu,cosu,sin2u is presented in this paper.The curve is controlled by four points,and it has a lot of similar characteristics to the cubic Bezie curve,and its shape can be adjusted by a parameter,which makes the curve have more powerful expression ability.For designing free curves,the continuity condition of two-piece curves is discussed.As a result,the continuity of the curve is better than the cubic Bezier curve,two-piece curves can be C3 continuous when choosing a proper shape parameter,and the continuity condition of the curve is simpler than the cubic Bezier curve.In addition,the curve can represent elliptic curves,parabola and other conical curves without using a rational form,which is helpful for practical applications.

Initial Equilibrium State Analysis of Self-anchored Suspension Bridge with Spatial Cables

Abstract: Based on integral analysis of cable-hanger system and tower-beam system,the initial equilibrium state analysis is proposed for the self-anchored suspension bridge with spatial cablesThe main study focuses on the coupling characteristic of spatial cable-hanger system and self-equilibrium characteristic of self-anchored systemA simplified model is established in which spatial cable-hanger is simplified as parabola in vertical plane and segmental lines in horizontal plane,and simplified analytical formulas are deduced by the minimum potential energy principleSteffens-Newton algorithm is employed to solve actual cable-hanger coupling model,and FEA method is incorporated to establish numerical iterative method for entire equilibrium stateThe program SASB- IESA is writtenA numerical example is solved with the two proposed methods respectively,and their applicability is pointed out based on a comparison of the result accuracy

A smallest error algorithm for generation of parabola curve generating algorithm of parabola curve

Abstract: A smallest error algorithm for parabola curve generation is presented in this paper. A strip region can be constructed for parabola curve and in the strip region these pixels for generation of parabola is the smallest error. We apply on the strip region. the smallest error algorithm for generation of parabola if proposed and the recurrence formula is given. In the recurrence formula, pixels number is calculated. This method is better than old methods that recurrence formula is calculated in generating a pixel. The example is given.

Enhancement of directivity of source radiation using parabolic cylinder made of left-handed metamaterials

Abstract: We explore radiation of a line source within a parabolic cylinder made of left-handed metamaterials. The far field radiation patterns show the metamaterial parabolic cylinder can enhance the directivity of the line source radiation. It is found that the focal length and the feature length of the parabola affect the directivity. Further improvement of the directivity is realized by changing the shape of parabolic end by a metallic circular arc, which also reduces the total size of the parabolic cylinder. Finally, we compared the backscattering cross section (BCS) between the traditional parabolic reflector and the proposed parabolic cylinder with the same size and found the parabolic cylinder had much less BCS.

Parabolic through collector has parabolic reflective surface with crown line running through vertices of parabola and focal line, where optical radiation inlet is in direction of reflective surface of parabolic trough

Abstract: The parabolic through collector (1) has a parabolic reflective surface (2) with a crown line (4) running through the vertices of a parabola and a focal line (6). Multiple opto-electronic sensors are assigned to an optical radiation inlet, and are arranged adjacent to both the side surfaces of the sensors. The optical radiation inlet is in the direction of the reflective surface of a parabolic trough (8). An independent claim is also included for a measuring transducer for tracking a parabolic through collector.

The maximum of Brownian motion minus a parabola

Abstract: We derive a simple integral representation for the distribution of the maximum of Brownian motion minus a parabola, which can be used for computing the density and moments of the distribution, both for one-sided and two-sided Brownian motion.

Concentrator solar radiation, with multiplesindependientes parabolic mirrors.

Abstract: Concentrator solar radiation, with independent multiple parabolic mirrors. Device comprises several longitudinal mirrors (7) with parallel axes, and in turn parallel to the longitudinal axis of the receiver (1) in which focus, whose straight sections are parabolic arcs defined for a solar reference position, moving the parabola (44) the central point of the mirror (25), and having as focus the focal point (3) of the active face (2) of the receiver (1), where incident radiation reflected, the width of each mirror being bounded to narrow that It drifts way other than solar reflected in positions of the reference beams and being established separation between successive mirrors to avoid shadows from certain height of the sun.