Parallel Chords Research Articles

Smoothing of Radon projections type of data by bivariate polynomials

Given information about a function in two variables, consisting of a finite number of Radon projections, we study the problem of smoothing this data by a bivariate polynomial. It turns out that the smoothing problem is closely connected with the interpolation problem. We propose several schemes consisting of sets of parallel chords in the unit disk which ensure uniqueness of the bivariate polynomial having prescribed Radon projections along these chords. Regular schemes play an important role in both interpolation and smoothing of such kind of data. We prove that the existence and uniqueness of the best smoothing polynomial relies on a regularity property of the scheme of chords. Results of some numerical experiments are presented too.

On a conjecture about relative lengths

We need some definitions from [1]. Let C ⊂ R2 be a convex body. A chord pq of C is called an affine diameter of C, if there is no longer parallel chord in C. The ratio of |ab| to 1 2 |a′b′|, where a′b′ is an affine diameter of C parallel to ab, is called the C-length of ab, or the relative length of ab, if there is no doubt about C. We denote it by λC(ab). Denote by λn the relative length of a side of the regular n-gon. For every ab ⊂ C we have |ab| ≤ |a′b′|, where a′b′ is the affine diameter parallel to ab, hence 0 < λn = |ab| |a′b′|/2 ≤ 2. For every regular triangle (or square), since its side length equals its corresponding affine diameter, λ3 = λ4 = 2. Let C = abcde be a regular pentagon with side length 1, join the points c and e, then we know that ab is parallel to ce and λ5 = λC(ab) = |ab| |ce|/2 = 1/ cos(π5 ) = √ 5 − 1 (see Fig. 1). Let C = abcdef be a regular hexagon with side length 1, join the points c and f , then ab is parallel to c f and λ6 = |ab| |c f |/2 = 1 (see Fig. 2). A side ab of a convex n-gon P is called relatively short if λP (ab) ≤ λn , and it is called relatively long if λP (ab) ≥ λn .

Packing a planar convex body with three homothetical copies and inscribing relatively equilateral triangles

Abstract Let C be a convex body. By C-length of a segment we mean the ratio of the length of this segment to the half of the length of a longest parallel chord of C. We show that every planar convex body C permits an inscribed triangle whose sides are of equal C-length at least 3/2. Equivalently, C can be packed with three equal homothetical copies of ratio at least 3/7 which touch the boundary of C and touch pairwise themselves.

Structural optimization as a harmony of design, fabrication and economy

The main requirements of load-carrying engineering structures are safety, producibility and economy. The role and importance of fabrication constraints and costs is shown in the case of a compression tubular strut, a tubular truss of parallel chords and a welded stiffened square plate. The evaluation of optimal versions of these structures shows that, neglecting the fabrication constraints and costs, the design cannot result in up-to-date solutions.

Experimental Studies on Large-Scale Timber Trussed Systems

Trusses have been widely used for roof and bridge construction, to provide mankind with a means to meet the most indispensable of needs. Despite the extensive use of trusses, it is still believed that the significant simplifications and assumptions made to facilitate design e.g. frictionless pin-joints, are far from reality, resulting in over designed, uneconomical members. This study is a sequel to previous work, and evaluates the realistic structural behaviour of prototype timber trussed systems viz-a-viz the empirical or semi-empirical design assumptions, with the goal of seeking to optimize the very scarce timber resource in Kenya. A prototype timber fink truss was subjected to nodal point loads gradually to failure. Results obtained suggest that internal web members of the truss may be regarded as axially loaded, while continuous top and bottom chord members are mainly loaded in bending in contrast to design assumptions. Extensive deflections were recorded, and according to limit state design requirements the truss is considered to have failed in serviceability limit state of deflections, much below the recorded ultimate capacity of the truss. Generally, the results are quite similar to previous work on parallel chord bridge truss, and confirm the much held opinion that conventional analysis of timber trusses is an oversimplification that is far unrealistic, too conservative and hence against the current global trend towards sustainability and resource efficiency. Journal of Civil Engineering Research and Practice Vol.1(2) 2004: 77-90

Poloidal Magnetic Field Measurements in a Reversed Field Pinch Plasma by a Far?Infrared Polarimeter

A multichannel far‐infrared (FIR) polarimeter has been installed in RFX, a Reversed Field Pinch (RFP) plasma experiment, to measure the poloidal magnetic field profile. The polarimeter uses a CH3OH FIR laser operating at λ=118.8 μm. Faraday rotation measurements on five of six parallel diagnostic chords are used in preliminary investigations of poloidal field profiles. The experimental results are generally found in good agreement with the μ&p model predictions. The choice of the profile of μ = μ0j·B/B2 is discussed. For the reconstruction of the magnetic field profiles a numerical filamentary current equilibrium code is described, where polarimetric data are included as constraints. An alternative method based on the best‐fit of a three‐parameter μ profile to the polarimetric data is also reported. Both methods provide reliable reconstructions of the plasma magnetic field and the results indicate the existence in RFX of hollow μ profiles.

Laser Faraday rotation measurement of current density fluctuations and electromagnetic torque (invited)

Far-infrared laser polarimetry with time response up to ∼1 μs and spatial resolution ∼8 cm has been successfully implemented on the Madison Symmetric Torus reversed-field pinch. Internal magnetic field and current density fluctuations are nonperturbatively measured. This is accomplished by taking parallel polarimetry chords which measure the line-integrated magnetic field fluctuations via Faraday rotation. With suitable analysis we are able to obtain information on spatial profiles of magnetic field fluctuations and current density fluctuations. Coherent interaction between these fluctuations is also measured and observed to generate an electromagnetic fluctuation-induced torque 〈δJ×δB〉.

Multichannel visible spectroscopy diagnostic for particle transport studies in the H-1 heliac

A multichannel spectroscopy diagnostic has been developed to study cross-field particle transport in the radiation-dominated low-temperature plasmas (Te&amp;lt;100 eV) in the H-1 heliac. The optical setup covers the full plasma minor radius in the poloidal plane collecting light from ten parallel chords arranged tangentially to the flux surfaces. The light collected from the plasma is coupled into optical fibers and through interference filters into photomultipliers. Two such ten-fiber arrays are aligned parallel to one another to allow the simultaneous monitoring of two different spectral lines. The net radial electron particle flux is determined from the continuity equation by integrating over the ionization source term in the steady-state partially ionized plasma. The diagnostic measures the neutral line intensities and their ratios (in case of helium using the line ratio technique) and also measures excited neutral and ion spectral lines (in case of the argon plasma transport studies). A comparative analysis of the radial particle transport in the low- and high-confinement regimes is presented.

Messiaen’s Triadic Colouration: Modes as Interversion

About halfway through the Vingt regards sur l'enfant-esus (1944), in 'Premiere communion de la vierge', Messiaen recalls the opening bars of La nativite du Seigneur (1935).1 This introduces what might reasonably be called the Virgin's theme (Ex. 1), since both movements allude to her presence, although in Technique de mon langage musical Messiaen gives this pattern the rather impersonal designation of a cadence.2 Unadorned as it is when it appears at the outset of La nativite, its recurrence in Vingt regards, once again as Messiaen's musical image of the Virgin, is played against a scintillating chain of chords that, in a fashion typical of Messiaen, move in approximate but not strict parallel motion.3 Not only does the Virgin's theme appear in Technique (Chap. 16, 'Modes of Limited Transpositions') as the 'typical cadence' of mode 2, but this chain of chords is also listed on the same occasion as mode 2 set 'in [a] parallel succession of chords' (Ex. 2a).4 This chapter is in many ways Messiaen's fullest account of his modes, although any discussion of mode-colour relationship is curiously missing.5 At the outset he dwells on the transpositional impossibilities and tonal possibilities of his seven modes. Each of them is then presented in scalar form, although typical chords and cadence formulas are shown only for modes 2 and 3. In addition, Messiaen takes special care to arrange the modes, with the exception of 1 and 5, as parallel chord series ('parallel succession of chords' in Satterfield's translation; Ex. 2), one of which, as we have seen, 51nds its way into 'Premiere communion de la vierge'. Messiaen's remark on these parallel chord series is typically brief. None of these chord series are parallel in the strict sense, but, as he puts it, 'each voice [of the series] realizes the entire mode, starting on a different degree'. As a result of this, each of them engages on a regular basis one or more different chord types.6 Although the literature has rarely touched on these series, giving rather more attention to Messiaen's adaptation of decz-talas and birdsongs, we cannot fail to grasp their import as we listen to their insistent presence in his music. As far as his published musical works are concerned, Le banquet celeste (1928) contains the earliest use of such a parallel chord series (p. 2). This series is based on mode 2. Parallel chord series based on modes 3 and 6 soon follow in Preludes of 1929 (Technique, Exs. 302 and 352).7 By contrast, mode 4 is not used in this way until La nativite (Technique, Ex. 156), while series based on

A technique for structural modeling of front rails for sofas

The design of seat systems for sofas which utilize sinusoidal type springs in the seat foundation is complicated by the fact that the springs apply forces to the top edge of the rail in a direction perpendicular to the face of the rail. Thus the rail must be analyzed as a plate rather than as a simple beam. Finite element methods provide a means of treating rails, provided the material constants are known, but a simpler procedure is needed for practical design purposes. One such procedure, which is demonstrated in the paper, is to replace the solid wood front rail with a structurally equivalent parallel chord truss. In this model, the chords of the truss account for in- or out-of-plane loading of the rail, whereas the diagonals of the truss account for twisting of the rail. The need for material properties data is greatly simplified - only modulus of rupture, modulus of elasticity, and the modulus of rigidity for the rail as a whole are required.

The multichord far infrared polarimeter of the RFX experiment

A multichannel far infrared (FIR) polarimeter has been installed on the reversed field experiment (RFX) to measure the poloidal magnetic field profile, BΘ. The polarimeter uses a FIR laser (λ=118.8 μm) whose beam is transmitted to the machine via a 31 m long, nitrogen filled beam line. Faraday rotation measurements are reported for five of six parallel chords crossing the 1 m diameter RFX plasma on a poloidal cross section. The density profile is known from a CO2 interferometer. Low noise deuterated L-alaine doped triglycene sulphate (DLATGS) detectors are used. An accuracy of about 0.2° is obtained in the measured Faraday rotation angle, which is sufficient to estimate the characteristics of the BΘ profile. Removal of the noise induced by mechanical vibrations allowed acceptable measurements of Faraday rotation angles.

Tangent Stiffness Equations for Laterally Distributed Loaded Members

Tangent stiffness equations for a beam-column, which is subjected to either uniformly or sinusoidally distributed lateral loads, are presented. The equations have been derived by differentiating the slope-deflection equations under axial forces for a member. Thus, the tangent stiffness equations take into consideration axial forces, bowing effect, and laterally distributed loads. As a numerical example, elastic buckling behavior of parallel chord latticed beams with laterally distributed loads is investigated to compare the results obtained from the present method with those from the conventional matrix method in which the distributed loads are considered as a series of concentrated loads at additional intermediate nodes of a member. Furthermore, buckling tests were carried out to confirm the equations derived as well as to clarify the buckling behavior of space frame structures. In conclusion, it can be said that the new equations can provide a good efficient way of estimating the equilibrium paths and buckling loads. They can also lead to a significant savings in core storage and computing time required for the analysis of space frame structures.

Static and dynamic post-buckling behavior of truss structures

This paper proposes a vibration model in which the lateral inertia force is considered in order to analyze the dynamic post-buckling behavior of truss members. The post-buckling behavior of perfectly symmetrical truss-structures and symmetrical truss-structures with a small imperfection are analyzed. The structures analyzed are a plane parallel chord truss structure and a double-layer space truss structure. The load-deformation relationships obtained by static and dynamic analyses are compared. It is found that the dynamic load-displacement relationship oscillates around the static relationship after buckling. Some examples show that the buckling mode and the post-buckling load-displacement relationship differ remarkably between static and dynamic analysis.

Integration of D-shaped gradient coils in a Bruker TM circular cavity for X-band electron spin resonance imaging

A very compact design for the coil assembly of an X-band imaging apparatus is described. The design is based on D-shaped coils obtained by cutting a circular current distribution by two parallel chords. The coils make the best possible use of the space available within the particular geometry of a cylindrical Bruker TM110 cavity. Its compact design results in a maximum width of 45 mm, which is available in most electromagnets used for electron paramagnetic resonance (EPR) spectroscopy. Using nitroxide free radicals of 60 mT in linewidth and field gradients of 1.5 mT/cm, a resolution of 100 μm was obtained.

Viewpoint-invariant Weber fractions and standard contour-curvature discrimination

It is proposed that any cue for the visual discrimination of shape, in particular the discrimination of curved contours, should be such that the perceptual relationships defined by the cue are invariant under changes in observer viewpoint. Such relationships may be quantified by the Weber fraction; that is, the ratio Δc/c, where, for any particular value c of the cue, Δc is the smallest difference in c that can be detected. Eight geometric attributes of curved contours having one symmetry axis and parallel chords (a standard stimulus configuration) were examined for invariance of the Weber fraction under symmetry-preserving affine transformations of the image plane (changes in viewpoint are well approximated by affine transformations when depth is small relative to viewing distance). The attributes, each investigated in previous psychophysical studies, were equivalent-curvature, radius-of-curvature, turning-angle, arc-length-divided-by-chord-length, arc-length, maximum-deviation (sag), area, and mean-deviation. Three of the attributes, namely sag, area, and mean-deviation, satisfied the viewpoint-invariance condition; the remainder failed. These results are considered in relation to previously published empirical data on the Weber fraction for contour-curvature discrimination.

Strength and stiffness of glulam trusses with punched metal plated joints

Several Canadian wood truss manufacturers expressed a need to develop technical information to initiate the fabrication and acceptance of wood trusses made with glued-laminated timber (glulam) members and connected by punched metal plates. In collaboration with industrial participants, Forintek Canada Corp. developed a program for testing and analysis of metal plated glulam trusses. Twelve 3/12 pitched chord trusses and three parallel chord trusses, all 13.7 m long, were designed by three different plate manufacturers. All truss fabrication took place at a single truss manufacturing plant. Trusses composed of 79 mm wide and 152-229 mm deep glulam members connected by 16-gauge punched metal plates were tested to failure. The ratio of ultimate to design load averaged 3.5, and varied between 2.7 and 4.4. Both wood and plate failure modes were observed. The results suggest that the sizes of glulam members used in this study are structurally suitable for use with 16-gauge metal connector plates. Key words: wood, trusses, glued-laminated timber, truss plates, strength, stiffness.

An analytical approach for dynamic response of timber railroad bridges

In the 1970s, it was reported that there were approximately 3700 track kilometers of timber railroad bridges in the United States and Canada. For short spans, they offer an attractive alternative to other types of bridges, as they are economical, faster to construct, and easy to maintain. Current design practices do not allow an independent consideration of the effects of the dynamic loads in sizing the bridge components, because very little information is available on the subject. Dynamic tests were carried out in 1986 on timber bridge spans at two test sites using test trains consisting of a locomotive unit, two loaded hopper cars, and a caboose. This paper gives a brief description of the analytical approach employed for determining the dynamic response of timber bridge spans under railway vehicles travelling at a constant speed. The model comprises a multi-degree-of-freedom system with each vehicle having bounce, pitch, and roll movements. Two parallel chords, each having its distributed mass lumped at discrete points, were used to idealize the bridge spans. A computer program developed on this basis was used to predict the loads at the wheel–rail interfaces and the vertical displacements at the discrete points on the spans. The predicted loads at wheel–rail interfaces and the maximum vertical displacements were found to be in agreement within about 20% and 16% respectively of the measured values. The program was utilized to study the effect of speed and other factors on the dynamic response of open-deck and ballast-deck bridges. Key words: analytical approach, timber railway bridge, railway locomotive and cars, constant speed, wheel–rail interface, loads, displacements, accelerations, dynamic response.

A new integration technique for flowmeters with chordal paths

A class of volume flowmeter exists in which measurements that are essentially line integrals of the velocity field along a series of parallel chords are combined linearly to form an estimate for the mean velocity. The paper shows how a priori knowledge of the functional forms of the velocity profile may be used in a systematic manner to optimize the design of such an instrument.

Circles, chords and epicycloids

In a recent session of the Royal Institution Master Classes in Cambridge, the enthusiastic participants explored a computer graphics program, designed and written by Derek and Alan Ball. The program first draws a circle, and, given an integer N, marks N equally spaced points on it, these being cyclically labelled 1,2,…,N. The operator then inputs some rule, say n ↦ 5n + 8, and the program draws, for each n, the chord from the point marked n to the point marked 5n + 8 (modulo N). Before long, the children took N to be large and flower-like envelopes of the chords started to appear (as illustrated below). Here, we show how the equations of these envelopes can be obtained in a simple way with some general parameter k, and that for positive k they are all epicycloids, that is, curves obtained by rolling a circle around a fixed circle, the simplest one being a cardioid. A similar analysis shows that for negative k they are hypocycloids (see for example E. H. Lockwood’s Book of curves, C.U.P. 1961) except when k = 1 when parallel chords occur.

Transient natural convection in prismatic enclosures of arbitrary cross-section

Two-dimensional laminar natural convection in horizontal prismatic enclosures of arbitrary cross-section is investigated. With the Boussinesq approximation the convection equations are written in a curvilinear non-orthogonal coordinate system and are solved by the implicit finite-difference method. Transient natural convection is studied in quadrangular regions of non-conventional configurations : (a) an isosceles trapezium; (b) a portion of a circle confined between two equal and parallel chords; (c) a region with the boundaries formed by the cosine semi-wave and three segments of straight lines. The solution of the problems is obtained within the range of Grashof numbers 10 2 ⩽ Gr ⩽ 10 6 at different geometrical parameters of the enclosures. It is found that the maximum values of the stream function and Nusselt number may accomplish damping oscillations about their stationary values. The results obtained for the trapezoidal enclosures are in good agreement with the data reported by other authors.