Equilateral Research Articles

OPERATION FEATURES OF AN ULTRASONIC FLOW TRANSDUCER WITH A COMPLEX TRAJECTORY OF THE MEASURING PATH

To measure the flowrate of fuel and energy resources, different methods are used, which determine the range of measuring instruments. Instruments based on ultrasonic measurement methods are widely used in measurement practice due to the provision of high accuracy, a wide range of measured flowrates, the absence of additional pressure losses and the design simplicity. Such instruments are easily integrated into automated readout systems for collecting and transmitting information. The asymmetry of the measured medium flow is a significant problem in the use of single-channel ultrasonic meters. This issue is successfully solved by using multi-channel flow transducers.
 The disadvantages of multi-channel ultrasonic flow transducers include a significant complication of the hydraulic channel design, the need to use more complex mathematical algorithms for processing the obtained output signal. This requires the use of high performance electronic elements, first of all microprocessors.
 The use of a transducer with multiple reflection of one measuring beam from the wall of the measuring path acts as an alternative to multi-channel measurement.
 The aim of the work is to create a mathematical model of an ultrasonic flow transducer with a complex measuring path trajectory. For research, a scheme with sounding along three chords was chosen, which implements the time-pulse measurement method, it provides for determining the flow rate by the difference in the time of passage of the measuring path along the flow and against it. In this case, the projections of the chords on the cross section of the measuring section create an equilateral triangle. An analytical expression for the conversion response is obtained for the case of several beam reflections in different planes.
 The authors reproduced the hydraulic part of the investigated flow transducer using simulation modeling implemented on the basis of the finite element method. Transducer’s operation simulation in the range of measured flow rates in the conditions of an ideally formed profile of the gas flow velocity in the transducer inlet cross-section has been carried out. The results obtained confirmed the absence of influence of the measuring circuit elements (beam reflectors) on the flow profile and the measuring beam trajectory.

Ordered Sn distribution adjacent to the precipitate-matrix interface in a Mg–9.8wt.%Sn alloy

The major interface between β′-Mg 3 Sn precipitate plate and the α-Mg matrix in a Mg-9.8wt.% alloy has been investigated using aberration-corrected scanning transmission electron microscopy and first-principles calculations. It is found that Sn atoms orderly distribute in the single layer of the α-Mg matrix immediately adjacent to the broad surface of β′ at the early stage of ageing. These Sn atoms substitute Mg atoms located at the centers of equilateral triangles constituted by three Mg columns in the outmost layer of β′. First-principles calculations suggest that the ordered Sn distribution is energetically favored and it not only decreases the interfacial energy of the β′-matrix interface but also hinders the occurrence of 1/3<01 1 ¯ 0> α shear that thickens the β′ plate.

Ethnomathematics and creativity study in the construction of batik based on fractal geometry aided by GeoGebra

This study aims to describe geometric objects that used by students on constructing fractal batik using Geogebra, procedure that used to construct fractal batik design, and students creativity on the process of constructing fractal batik. The qualitative descriptive research was applies including data collection, data separation, data analysis and conclusions. The research data were obtained from 97 students of tadris mathematics IAIN Kediri. The research results showed that fractal batik was constructed from a single geometric shape and combination of 2, 3, and 4 single geometric shapes through steps (a) made basic patterns using geometric shapes in Geogebra, (b) Made New Tools to perform repetitions (iteration), (c) Determined the type of transformation that used to repeat the basic patterns, and (d) Constructed geometric fractal batik. Based on the creativity indicators fluency, 3 types of geometric fractals can be obtained, namely (a) Fractals with Repetition and Enlargement (FPB), (b) Fractals with Repetition and Change Position (FPS), and (c) Fractals with Mix Repetition (FPC). Based on the flexibility indicator, there are 32 basic geometric shapes that develop basic patterns by applying 15 types of transformation consist of Single Transformation, Double Transformation, Triple Transformation and Quadruple Transformation. Meanwhile, on the originality indicator, P3 is the basic shape that has been mostly developed into geometric fractal batik which is a combination of equilateral triangles and squares.

A web of confocal parabolas in a grid of hexagons

If one erects regular hexagons upon the sides of a triangle T, several surprising properties emerge, including: (i) the triangles which flank said hexagons have an isodynamic point common with T, (ii) the construction can be extended iteratively, forming an infinite grid of regular hexagons and flank triangles, (iii) a web of confocal parabolas with only three distinct foci interweaves the vertices of hexagons in the grid. Finally, (iv) said foci are the vertices of an equilateral triangle.

Investigation on heat transfer and flow-induced vibration of three cylinders in equilateral triangle arrangement

Flow-induced vibration (FIV) and heat transfer of three circular cylinders were numerically investigated in the reduced velocity range of 3 ≤ U⁎ ≤ 15. To obtain the vibration and heat transfer characters, the influence of FIV and Richardson number (0 ≤ Ri ≤ 1) on the heat transfer performance are studied. It's found that the rise of Ri can lead to the suppression of FIV and narrow the reduced velocity (U⁎) range of lock-in. The amplitude and lift coefficient of two downstream cylinders are higher than that of 1st cylinder (upstream one) for most of U⁎. The FIV can significantly enhance the heat transfer of 2nd and 3rd cylinders (downstream cylinders), the highest increments of averaged Nusselt number are 9.74% (2nd cylinder) and 9.78% (3rd cylinder), which can be obtained when Ri = 0.5 and U⁎ = 5. Meanwhile, the distribution of local Nusselt number shows that the heat transfer mainly occurs at the upstream side of each oscillator. The heat transfer characteristics on the surface of each cylinder are closely influenced by the thickness of thermal boundary, which determines the gradient of temperature around cylinders.

An IFS for the Stretched Level-n Sierpinski Gasket

Broadly, fractals are sets that exhibit a repeating pattern at multiple scales. One important fractal set is the Sierpinski Gasket (SG) which is made up of nested equilateral triangles. A variation of the classic Sierpinski gasket is to create n-levels with the equilateral triangles. Another variation is to stretch the points of intersection for the triangles in SG into line segments of length 0 &lt; α &lt; 1/3. When one combines these variations, one arrives at the stretched level-n Sierpinski gaskets (SSGn) which are the focus of this work. We give an introduction to iterated function systems (IFS) and determine an IFS which generates SSG3. We then describe how one can acquire the IFS for SSGn in general, and conclude with a theorem which determines the Hausdorff dimension for SSGn.

Energy absorption characteristics and multi-objective optimization of a novel reentrant hierarchical honeycomb bumper system

This paper aims to introduce the hierarchy design into sandwich beams and bumper systems to improve their mechanical performance. Two kinds of reentrant hierarchical sandwich beams are proposed and their energy absorption characteristics have been systematically investigated. Based on its excellent energy absorption ability, the reentrant hierarchical honeycombs with equilateral triangle substructures (RHT) core are introduced to the design of the bumper system, and the hierarchical bumper system exhibits better mechanical properties than the traditional bumper and the aluminum foam-filled bumper under different loading conditions. Then, the multi-objective optimization model of the hierarchical bumper system is established by combining the optimal Latin hypercube design method (OLHD) and the response surface model (RSM). The non-dominated sorting genetic algorithm-II (NSGA-II) algorithm is adopted to optimize the design parameters. The optimization results show that the objectives are optimized efficiently, and it performs remarkable performances in bumper system design.

Equilibrium points in restricted problems on S2 and H2

In this paper, we consider the restricted four-body problem on S2 and the restricted three-body problem on H2. In the first case, the primary particles are considered to be rotating around the vertical axis. In the latter case, the primaries move at a hyperbolic relative equilibrium. If the primary particles locate at height z0 = 0 and they form an isosceles triangle on S2, then the number of symmetric equilibrium points of the restricted four-body problem depends on the angular velocity, the configuration, and the masses of the primaries. For this case, the number of equilibrium points is 0, 2, 4, 6, or 8. If the primaries are at height z0 ∈ (0, 1), then the primaries form an equilateral triangle and the number of symmetric equilibrium points is 8, 11, 14, 17, or 20. The number depends on the position of the primaries, and it is higher than the previous case due to the higher number of symmetry axes of the triangle formed by the primaries. On the other hand, for the restricted three-body problem on H2, the number of equilibrium points is 3, considering that the primaries move at a hyperbolic relative equilibrium. We analyzed the general (non-symmetric) case, showing the locations of the equilibrium points for the given positions of the primaries. We also studied the stability of these equilibrium points.

On the Robin spectrum for the equilateral triangle* *Dedicated to Michael Berry for his 80th birthday.

The equilateral triangle is one of the few planar domains where the Dirichlet and Neumann eigenvalue problems were explicitly determined, by Lamé in 1833, despite not admitting separation of variables. In this paper, we study the Robin spectrum of the equilateral triangle, which was determined by McCartin in 2004 in terms of a system of transcendental coupled secular equations. We give uniform upper bounds for the Robin–Neumann gaps, showing that they are bounded by their limiting mean value, which is hence an almost sure bound. The spectrum admits a systematic double multiplicity, and after removing it we study the gaps in the resulting desymmetrized spectrum. We show a spectral gap property, that there are arbitrarily large gaps, and also arbitrarily small ones, moreover that the nearest neighbour spacing distribution of the desymmetrized spectrum is a delta function at the origin. We show that for sufficiently small Robin parameter, the desymmetrized spectrum is simple.

Analysis and Experiment on the Impact of Various Hook Angle Factors on Spindle Picking Performance

As the core component of the cotton picker, the structural characteristics of the spindle directly affect not only its picking performance but also different components of the cotton picker. Therefore, in order to reveal the influence of different angle parameters of hook teeth on the performance of the spindle and the optimal hook tooth angle parameters, this study analyzes the working principle of the picking mechanism of the cotton picker and establishes the force balance equation for the process of picking cotton from the spindle. By analyzing the structural characteristics of the spindle, the best section shape of the spindle head is the equilateral triangle section. The spindle with different angle parameters was made by a metal 3D printer, and the testbed for the dynamic picking performance of the spindle was constructed. The single-factor test was carried out with tooth groove angle before the tooth angle and tooth inclination angle as experimental factors, and the rejection rate of seed cotton, picking time, and picking force as evaluation indexes. The results showed that the tooth groove angle and the tooth inclination angle were negatively correlated with the rejection rate of seed cotton (P &lt; 0.01), and the tooth inclination angle was positively correlated with the picking time (p &lt; 0.01). Further analysis shows that in order to reduce the ejection rate of seed cotton and reduce the picking time, the selection range of the tooth inclination angle should be 55~65°, and the large the tooth groove angle should be selected. Finally, the optimum hook angle parameters are determined as follows: tooth groove angle is 70° before the tooth angle is 89°, and the tooth inclination angle is 65°. The results are of practical significance to optimize the structure of picking hook tooth to improve the picking performance of the spindle and high efficiency of the cotton picker.

Ethnomathematic Value in Traditional Building in Kampung Budaya Bogor Jawa Barat

Mathematic value in traditional building have been overdued typically for traditional building in sundaness tribe. The goal of this research to discover ethnomathematics value in some building of sunda in kampung budaya in Bogor West Java. This study uses a qualitative method. The sampling technique was carried out purposively, namely the head of kampung Budaya and the key person of kampung budaya sunda sindang Barang Bogor Jawa Barat. Data were collected by in-depth interviews. The sample was selected through a purposive technique. The results were carefully examined through triangulation and source triangulation techniques. The results show that in leuit as the place for saving paddies contain numerous ethnomathematical including in the leuit building, especially on the roof of leuit which is in front there are mathematical elements such as a triangle facing each other but in the middle between the two there is a square and there is an equilateral triangle. While in traditional Sundanese houses there is woven on the walls where the wicker contains mathematical values ​​ranging from straight lines, parallel lines, perpendicular lines, and horizontal lines. Besides that, on the door, there are four right triangles where one pair is opposite, but the other pair is positioned upside down to the other.

Theoretical design and analysis of multichannel plasmonic switch based on triangle resonator combined with silver bar

A multichannel plasmonic switch consisted of four metal–insulator–metal (MIM) waveguides and an equilateral triangle resonator (ETR) with a silver bar is proposed. Calculated by coupled mode theory (CMT) and standing wave theory, structural transmittance characteristics are numerically studied by finite difference time domain (FDTD) method. And the results show that the transmission spectrum of the ETR with a silver bar, coupled with a MIM waveguide, can be adjusted by increasing the height and width of the bar. As for the switch, varying with structural parameters of the bar, the transmission spectra in three ports can be tuned by rotating the bar into different angles. And this multichannel plasmonic switch owes a high on/off contrast ratio (R = 36 dB). The proposed switch has potential application value in the area of telecommunication and all-optical signal procession.

Dynamic behavior analysis of tethered satellite system based on Floquet theory

In this paper, an n-star general dynamic model of tethered satellite system with closed-loop configuration is provided. An analytical method for periodic solution stability of the general dynamic model is proposed based on Floquet theory, which proved that the periodic solution stability of the system depends on the maximum modulus for the eigenvalue of a matrix related to the Jacobian matrix. The periodic solution stability of a 3-star system with equilateral triangle as the initial configuration is analyzed as an example based upon the analytical method, and the results are verified by numerical simulation. The critical spin angular velocity caused by the tether mass and the parameter variation of the 3-star system is analyzed. The results show that the analytical method of periodic solution stability can solve the critical stable spin angular velocity of the tethered satellite system accurately, and the 3-star system can guarantee stable spin in the case of the spin angular velocity is higher than the critical spin angular velocity.

The effect of electron dose on positive polymethyl methacrylate resist for nanolithography of gold bowtie nanoantennas

Plasmonic structures, such as bowtie nanoantennas, may be used in Surface Enhanced Raman Spectroscopy (SERS). Nanoantennas can be employed to amplify the biomolecular and chemical reactions, which is useful for biomedical applications. The electric field created by nanoantennas are optimized when the resonant wavelength of the probed laser light closely matches the resonant wavelength of the plasmonic structure. In this work, we fabricated several bowtie nanoantennas with varying geometric spacing for use with a 532 nm wavelength laser line in Raman Spectroscopy. The fabrication utilized nanolithography by electron beam lithography on a Raith Voyager, development, deposition, and metal lift-off. This study explored a specific bowtie nanoantenna geometry of 270 nm equilateral sides triangle pairs with 3 varying gap sizes, 50 nm, 20 nm, and 10 nm, and the effect of varying electron beam doses on the final structure of the nanoantenna. The results presented here, will show that the working dose factor range is 6.5–10.3 (650–10,300 μC/cm2) for 120 nm thick polymethyl methacrylate (PMMA), and with a 44.78% increase in dose, the footprint area increases between 5.9% and 10.7%.

Simulation Design of Low-Profile Equilateral Triangle Microstrip Patch Antenna Operating at 28 GHz

This study is established for designing an Equilateral Triangular Microstrip Patch Antenna (ETMPA) operating at 28GHz suitable for 5G application technologies. The design procedure is performed by considering inset fed techniques and implementing both Computer Simulation Technology (CST) and High Frequency Simulator Structure (HFSS) in order to assess their performance through the computation of antenna parameters. In order to arrive at the aforementioned objectives, different substrate materials with different heights and various conductor patch material types are utilized and their effect on the antenna radiation performance is investigated. The simulation results reveal that a minimum low-profile (ETMPA) of the order of (4.59 mm3) with reliable radiation performances is generally achieved by Taconic-TLX-9 dielectric substrates of height (h=0.2mm) and with copper patch material. Based on the specifications provided above, the results show that the S11, VSWR, efficiency, directivity, gain and bandwidth with both CST and HFSS techniques are, respectively, (-26.24, -14.60), (1.102, 1.53), (81%, 87.6%), (6.31, 6.52) dB, (5.39, 5.94) dB and (0.708 GHz, 0.503 GHz). Besides, a good agreement between the presented optimum antenna parameters is also observed as compared to the ones previously obtained experimentally or theoretically by other researchers for different patch shapes operating at 28 GHz. In addition, the computed results of the antenna parameters display generally that the CST provides better results than HFSS as compared to the ones experimentally obtained by previous researchers. Finally, the proposed antenna design has the advantage of size reduction in comparison to most of the other works and it is one of the 5G technology requirements.

The Siebeck–Marden–Northshield Theorem and the Real Roots of the Symbolic Cubic Equation

The isolation intervals of the real roots of the symbolic monic cubic polynomial x^3 + a x^2 + b x + c are determined, in terms of the coefficients of the polynomial, by solving the Siebeck–Marden–Northshield triangle—the equilateral triangle that projects onto the three real roots of the cubic polynomial and whose inscribed circle projects onto an interval with endpoints equal to stationary points of the polynomial.

Simulation study on the flow behaviors of weak electrolyte solutions in nano-sized and micron-sized pores considering electric double layer interactions

ABSTRACT The interactions between the ions in the weak electrolyte solutions and the charged solid wall result in the formation of an electric double layer (EDL) in nano-sized and micron-sized pores. However, in nano-sized pores, EDLs may be overlapping. Herein, a widely applicable Poisson-Nernst-Planck equation combined with the Navier–Stokes equation was adopted for the flow of weak electrolyte solutions under pressure-driven conditions. Meanwhile, the finite element method was utilized to solve the governing equations and investigate the influence of EDL interactions on flow behaviors in different pore radii (25 nm~5 μm). The results showed that as the pore radius increased, the influence of EDL interactions on fluid flow behaviors first increased and then decreased. The influence of EDL interactions on fluid flow behaviors was the most noticeable when the pore radius was close to the EDL thickness. Furthermore, the influence of EDL interactions appeared to be negligible in the pore radius greater than 5 μm. When EDL interactions were ignored, the volume flow in the pores of the circular, square, and equilateral triangle was increased by about 13%. This study provides new insights into the flow behaviors of weak electrolyte solutions in nano-sized and micron-sized pores.

Tracing the Oval—Characterisation and Construction of Original Danish Eighteenth-Century Strainers from Portraits by Jens Juel

Few examples of original strainers for paintings from the eighteenth century or earlier still exist as they have commonly been replaced due to their often-weak construction and inability to expand. Several original strainers, however, are still present in paintings by Danish portrait painter Jens Juel (1745–1802). These preserved strainers provide rare evidence of shape, construction, availability, format, and the production of strainers in the late eighteenth century. By visual analysis of twenty-two paintings, of which seven strainers are preserved, this study characterises the original strainers and their tool marks. Rare markings found on the surface hypothesised to be related to their construction is evaluated in relation to theory of polycentric ovals and layouts on the construction of oval shapes in treatises from the sixteenth to eighteenth centuries. The strainers are very similar and of simple construction with tool marks that correspond with the development in the late eighteenth century. The oval strainers are a construction with given symmetry axes, with two 60° equilateral triangles that the centres of the arcs form when aligned with the major axes. Such a layout would have been published by Christiaan Huygens (1629–1695) and Amédeé François Frézier (1682–1773) by the time of construction by someone in the wood working crafts in Copenhagen, who supplied strainers to Juel’s studio.

A linear-time crystal-growth algorithm for discretization of continuum approximation

Continuous approximation is regarded as a scalable and insightful method for acquiring near-optimum solutions to various location problems. A continuous approximation solution is nonetheless only a continuous density function and a further discretization procedure is needed to obtain a discrete location solution for engineering practice. Inspired by the process of “crystal growth”, this paper proposes a constructive heuristic algorithm as an alternative to the classic meta-heuristic disk algorithm (Ouyang and Daganzo, 2006) in discretizing a continuous solution from a continuum approximation location model. The main idea of this algorithm is to rasterize the space into a set of small cells (either regular triangles or squares) and repeatedly grow a core cell into a full-service area according to a certain visiting sequence. Thus, it has only linear time complexity proportional to the number of cells in the space. Numerical examples are conducted to test the performance of the proposed algorithm. The results indicate that this algorithm can solve discrete facility locations more efficiently and exhibits robust performance compared with the disk algorithm.

Performance evaluation of a conjugate gradient method considering step length computation techniques in geometry fitting of coordinate measuring machine data

This paper focuses on performance evaluation of Hestenes–Stiefel conjugate gradient (CG) method along with two well-known line search conditions, which are weak Wolfe conditions (WWC) and strong Wolfe conditions (SWC), and a new step length computation algorithm in geometry fitting of five primitive geometries (i.e., circle, square, equilateral triangle, ellipse and rectangle). A coordinate measuring machine (CMM) is used to acquire 2D profiles of these geometries. Total number of function evaluations, when the CG method-step length computation technique combination being used successfully completes the geometry fitting, are considered to be performance criteria of the combination. It has been seen that the step length computation techniques for the line search crucially affect the success and performance of the Hestenes–Stiefel CG method in the geometry fitting. Therefore, they require a special care for optimal computational efficiency in addition to the line search direction.