Right-angled Isosceles Triangle Research Articles

Exact solutions for the initial stage of dam-break flow on a plane hillside or beach

Inviscid, incompressible liquid is released from rest by a sudden dam break, accelerating under gravity over a uniformly sloping impermeable plane bed. The liquid flows downhill or up a beach. A linearised model is derived from Euler's equations for the early stage of motion, of duration $2\sqrt {H/g}$ , where H is the depth scale and $g$ is the acceleration due to gravity. Initial pressure and acceleration fields are calculated in closed form, first for an isosceles right-angled triangle on a slope of $45^{\circ }$ . Second, the triangle belongs to a class of finite-domain solutions with a curved front face. Third, an unbounded domain is treated, with a curved face resembling a steep-fronted breaking water wave flowing up a beach. The fluid goes uphill due to a nearshore pressure gradient. In all cases the free-surface-bed contact point is the most accelerated particle, exceeding the acceleration due to gravity. Physical consequences are discussed, and the pressure approximation of shallow water theory is found poor during this early stage, near the steep free surface exposed by a dam break.

Perluasan Masalah Isoperimetrik pada Bangun Ruang

In this paper, several extensions of the isoperimetric problem in solid figures are explored, focusing on oblique and right prisms with rectangular, right-angled triangular, and regular hexagonal bases. The objective of this research is to find the prism with the largest volume while keeping the surface area constant. Through manipulations of algebra and simple trigonometry, evidence is obtained that a right prism provides a larger volume than an oblique prism if their surface areas are equal. By utilizing partial derivatives of a two-variable function and the Lagrange multiplier method, conditions for the side lengths are derived to obtain the prism with the maximum volume. The results show that a cube is the solution to the isoperimetric problem, meaning it has the largest volume among prisms with rectangular bases, while for the isoperimetric solution on prisms with right-angled triangular bases, the base of the prism must be an isosceles right-angled triangle. A regular hexagonal prism has a larger volume than prisms with rectangular and right-angled triangular bases if their surface areas are the same.

Homogenized elasticity and domain of linear elasticity of 2D architectured materials

This article presents a generic method to determine analytically the equivalent homogeneous linear elastic behavior of any 2D architectured material and to determine its macroscopic linear elasticity domain. The method is applied throughout the paper in the equilateral triangular case, as an example. Additional examples of square, isosceles right-angled triangular and hexagonal architectured materials are presented as a conclusion. The proposed method combines a periodic homogenization for the equivalent homogeneous linear elastic behavior and three criteria to establish the macroscopic linear elasticity domain. The first criterion corresponds to the limit strength of the bars. The second one, which corresponds to the apparition of a periodic buckling, involving a finite number of cells, is studied on a single primitive unit cell using Bloch wave analysis. The buckling modes that appear under equibiaxial compression for the equilateral triangle architectured material are similar to experimental results reported in the literature. The third criterion is the aperiodic buckling, or macroscopic instability, which corresponds to the apparition of shear bands or a macroscopic buckling. The resultant macroscopic linear elasticity domain resulting from these three criteria is presented, for the first time, in the case of the equilateral triangle, isosceles right-angled triangle architectured material and also for two other geometries, square and hexagonal architectured material. It is shown that the theoretical properties of these architectured materials are, in an Ashby chart, in the domain of light but relatively strong materials, a domain in which man-made materials are currently poorly represented.

Ratchet Effect in Triangle-Shaped Vertical Nb/TiO/Nb Josephson Junctions

We fabricate vertical Nb/TiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sub> /Nb Josephson junctions with the shape of right-angled isosceles triangles of different sizes. The TiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sub> layer is primarily metallic, indicating that our junctions belong to the superconductor-normal metal-superconductor (S-N-S) junction. When applying bipolar dc currents across the junctions in parallel magnetic fields, there are many obvious differences between the positive and negative parts of the current-voltage characteristics. We attributed this asymmetry behavior (or ratchet behavior) mainly to the asymmetric pinnings for Josephson vortex penetration at the asymmetric edges (the right-angled corner and the hypotenuse) of the triangled junction, in addition to the effect of current-induced magnetic self-fields. For comparison with superconducting thin-film ratchets, the rectified dc voltages of our junctions with applied ac currents are investigated at different temperatures, magnetic fields, and driving frequencies. The waveforms of the time-resolved voltage with sinusoid currents are also measured. The ratchet effect in the junctions decreases monotonically with increasing temperature, increases to the peak first, and then decreases with the increasing magnetic field. It remains almost unchanged in the driving frequency range of 0.1 to 100 kHz. Compared with Abrikosov vortex ratchets, the ratchet effect in the junctions is much more sensitive to temperatures far from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> , demonstrating the larger temperature-regulating window. This article shows the distinct influence of the asymmetric junction boundaries on the characteristics of the junctions under certain conditions, which can be useful for further understanding and inspiring new applications of Josephson junctions.

A Dual Orthogonal Fed Monopole Antenna for Circular Polarization Diversity

A monopole antenna with circular polarization diversity and high isolation is proposed in this paper. The proposed antenna consists of two bent monopole antennas and a partial ground plane. Two monopole antennas are implemented on opposite planes with respect to the ground plane. The ground plane is a right-angled isosceles triangle, and the antenna is located at one end of the ground plane. Circular polarization (CP) characteristics are obtained using this structural asymmetry. In the proposed structure, the resonant frequency at S11 is controlled by the antenna length, and the null frequency at S21 (isolation) is controlled using the bending angle of the monopole and the distance between the two antennas. From the measured results, the CP operating bandwidth of the proposed antenna is 13.8% (1.76–2.02 GHz). The measured isolation is –50 dB at the resonant frequency and –20 dB within the operating bandwidth.

Distribution of shear fragments classified by morphology

In studies on metal shell under internal explosive loading, unlike brittle metals, ductile metals usually exhibit shear fracture. Existing literature mostly focuses on slip fracture caused by unidirectional shear fracture, in fact, the shell fractures in the two directions of maximum shear stress, making morphology of the fragments diversified, considering fragment morphology in the distribution may achieve a better description. In previous studies, Poisson mixtures and Weibull mixtures model are only used for fragment distribution statistics as a functional form, and parameters (weight coefficient and distribution modulus) in the function lack physical process description. In this paper, an experimental study on the fragmentation of 50SiMnVB steel shell under TNT internal explosive loading was carried out. Through the observation and analysis of recovered fragments, the shear fragments with cross-sections of parallelogram, right-angled trapezoid, and isosceles right-angled triangle are divided into three types: A, B, and C. According to “A-type fragments created by the parallel through-the-thickness fractures, B-type and C-type fragments created by interaction of intersecting fractures,” the morphological parameters of fragments in the distribution model were derived, including fragment size (width, length, thickness, mass, etc.), shape parameters, and weight coefficient. Weibull mixture distribution model is used to describe the distribution of the total fragments, whereas Weibull distribution is used to describe the distribution of each type of fragment. Through the comparison of theoretical calculations and experimental data, the applicability of the theoretical model in the field of “shear fracture of ductile steel shell under internal explosive loading” is verified.

Error analysis and correction of multi-sensor cluster methods for acoustic emission source localization

The existing multi-sensor cluster acoustic emission (AE) source localization method has good positioning performance, but the parallel assumption in this method could cause positioning error. This paper focused on the analysis of the positioning error of multi-sensor cluster methods with sensor arrangements of isosceles right-angled triangle and triangular pyramid. Meanwhile, two and three-dimensional amendment algorithms for the two sensor arrangements were proposed. Pencil lead break experiments and numerical examples were used to verify the rationality of error source analysis and the accuracy of the amendment algorithms. Results show that the multi-sensor cluster methods can only accurately locate the AE sources in special positions, such as the AE sources satisfying θi = π/4 in the multi-sensor cluster method with a sensor arrangement of isosceles right-angled triangle and the AE sources satisfying cosθij=3/3 in the multi-sensor cluster method with a sensor arrangement of triangular pyramid. The results of pencil lead break experiment show that the two-dimensional amendment algorithm can accurately locate the AE sources in two-dimensional isotropic structure. For the two-dimensional anisotropic structure, the positioning result of the two-dimensional amendment algorithm is 27.8% higher than that of the multi-sensor cluster method with a sensor arrangement of isosceles right-angled triangle. The results of numerical examples show that the positioning errors of the multi-sensor cluster method with a sensor arrangement of triangular pyramid and the three-dimensional amendment algorithm are 24.6 mm and 0, respectively. Due to the correction of the positioning error caused by the parallel assumption, the latter has better positioning performance. Therefore, the amendment algorithms of the multi-sensor cluster methods have certain engineering application value in AE monitoring of two and three-dimensional structures.

РАСЧЕТЫ К ОДНОЙ ИЛЛЮСТРАЦИИ ТЕОРЕМЫ ПИФАГОРА

On the sides of a right-angled isosceles triangle, squares external to this triangle are built. The task is to construct a straight line on a plane that divides a figure consisting of these three squares into two equal-sized (that is, equal area) figures. Some parameters determined by this line are calculated, in particular, the areas of parts of the original squares in half-planes. The corresponding results are illustrated by various drawings, tables and graphs.

Courant-Sharp Eigenvalues for the Equilateral Torus, and for the Equilateral Triangle

We address the question of determining the eigenvalues $\lambda\_n$ (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with $n$ nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results for the right-angled isosceles triangle and for the hemiequilateral triangle.

Influence of the Joint Geometry on the Stress Intensity Factor of the Fillet Welded Cruciform Joint Subjected to Tension and Bending

In this paper is analyzed the cruciform welded joint subjected to axial tension and bending in order to determine the influence of the joint geometry on the fracture mechanics parameters. The stress intensity factors are calculated analytically, with help of the programming package Mathematica®, based on the linear elastic fracture mechanics concept. Existence of the weld and loading was taken into account through the corresponding correction factors. Results obtained in this work show that increase of the weld size for the isosceles right-angled triangle leads to decrease of the stress intensity factor, while the stress intensity factor is increasing with weld's width w increase. Ratio of thicknesses B/T of the two plates that are welded to each other does not impose significant influence on the stress intensity factor. The working life and the fatigue crack growth curve of the two welds were analyzed as well.

CONICAL BEAM WINDMILL-SHAPED ARRAY ANTENNA DESIGN BASED ON QUARTER-MODE SUBSTRATE INTEGRATED WAVEGUIDE

This paper presents a compact conical beam antenna based on quarter-mode substrate integrated waveguide (QMSIW). The antenna array is consisted of six isosceles right-angled triangle metallic patch printed on the upper side of the substrate in a windmill-shape and diagonal periodic metallic via holes drilled along the hypotenuse of the triangle patch. The planer structure exhibits conical beam radiation. The typical radiation is obtained at 5.2GHz with conical beam radiation pattern. The measured gain is about 5.36dBi and the radiation e-ciency is around 50%.

Terahertz Wave Emission from Intrinsic Josephson Junctions in Bi2Sr2CaCu2O8+δ

We have studied the conditions for the THz radiation in many systems with different shapes and dimensions such as rectangles, squares, cylinders and (equilateral and isosceles including right-angled isosceles) triangles and rectangles with non-uniform width (asymmetric rectangles) fabricated from single crystals of Bi2Sr2CaCu2O8+δ, which are known as a typical intrinsic Josephson junction system. As a result two conditions have been deduced to be fulfilled: one is the ac-Josephson effect always working at each intrinsic Josephson junction. Another one is the standing wave formation of the THz waves excited inside the mesa due to the cavity resonance for the radiation. The former condition is always necessary for the radiation but the latter may have only the secondary role for the radiation because the Q-value of the resonance cavity in some cases seems to be quite low and of order of unity. Therefore, the radiation frequency can be controlled by the voltage effectively and widely varied by the ac-Josephson effect only. In addition, we show a new phenomenon observed at high current region and a comprehensive physical picture for the extraordinary THz radiation phenomenon is given.

Terahertz Wave Emission from Intrinsic Josephson Junctions in Bi2Sr2CaCu2O8+δ

We have studied the conditions for the THz radiation in many systems with different shapes and dimensions such as rectangles, squares, cylinders and (equilateral and isosceles including right-angled isosceles) triangles and rectangles with non-uniform width (asymmetric rectangles) fabricated from single crystals of Bi2Sr2CaCu2O8+δ, which are known as a typical intrinsic Josephson junction system. As a result two conditions have been deduced to be fulfilled: one is the ac-Josephson effect always working at each intrinsic Josephson junction. Another one is the standing wave formation of the THz waves excited inside the mesa due to the cavity resonance for the radiation. The former condition is always necessary for the radiation but the latter may have only the secondary role for the radiation because the Q-value of the resonance cavity in some cases seems to be quite low and of order of unity. Therefore, the radiation frequency can be controlled by the voltage effectively and widely varied by the ac-Josephson effect only. In addition, we show a new phenomenon observed at high current region and a comprehensive physical picture for the extraordinary THz radiation phenomenon is given.

Packing unequal circles using formulation space search

In this paper we present a heuristic algorithm for the problem of packing unequal circles in a fixed size container such as the unit circle, the unit square or a rectangle. We view the problem as being one of scaling the radii of the unequal circles so that they can all be packed into the container. Our algorithm is composed of an optimisation phase and an improvement phase. The optimisation phase is based on the formulation space search method whilst the improvement phase creates a perturbation of the current solution by swapping two circles. The instances considered in this work can be categorised into two: instances with large variations in radii and instances with small variations in radii. We consider six different containers: circle, square, rectangle, right-angled isosceles triangle, semicircle and circular quadrant. Computational results show improvements over previous work in the literature.

Nodal domains of a non-separable problem—the right-angled isosceles triangle

We study the nodal set of eigenfunctions of the Laplace operator on the right-angled isosceles triangle. A local analysis of the nodal pattern provides an algorithm for computing the number νn of nodal domains for any eigenfunction. In addition, an exact recursive formula for the number of nodal domains is found to reproduce all existing data. Eventually, we use the recursion formula to analyse a large sequence of nodal counts statistically. Our analysis shows that the distribution of nodal counts for this triangular shape has a much richer structure than the known cases of regular separable shapes or completely irregular shapes. Furthermore, we demonstrate that the nodal count sequence contains information about the periodic orbits of the corresponding classical ray dynamics.

A heuristic for the circle packing problem with a variety of containers

In this paper we present a heuristic algorithm based on the formulation space search method to solve the circle packing problem. The circle packing problem is the problem of finding the maximum radius of a specified number of identical circles that can be fitted, without overlaps, into a two-dimensional container of fixed size. In this paper we consider a variety of containers: the unit circle, unit square, rectangle, isosceles right-angled triangle and semicircle. The problem is formulated as a nonlinear optimization problem involving both Cartesian and polar coordinate systems. Formulation space search consists of switching between different formulations of the same problem, each formulation potentially having different properties in terms of nonlinear optimization. As a component of our heuristic we solve a nonlinear optimization problem using the solver SNOPT. Our heuristic improves on previous results based on formulation space search presented in the literature. For a number of the containers we improve on the best result previously known. Our heuristic is also a computationally effective approach (when balancing quality of result obtained against computation time required) when compared with other work presented in the literature.

Heat transfer comparison between a vertical rectangular cavity and an isosceles right-angled triangular cavity of equal cross-sectional area

This paper addresses the heat transfer performance of natural convection flows in three different, (but related) cavities in the form of: a square, isosceles right-angled triangle, and vertical rectangle with aspect ratio 2:1. The isosceles right-angled triangular cavity is derived from a square cavity when cut in half diagonally, whereas the vertical rectangular cavity is derived from a square cavity when cut in half vertically. In the three cavities, the left vertical wall is the common wall heated. The buoyant air flow is characterized by height-based Rayleigh numbers ranging from a conduction-dominant to up to 106 for the laminar natural convection regime. Employing the finite volume method, the velocity and temperature fields as well as the mean convective coefficients evaluated at the common heated vertical wall are numerically determined for the isosceles right-angled triangular cavity. For this cavity, flow streamlines and temperature contours are presented in graphical form and some numerical results are validated against published experimental measurements. A one-to-one comparison for the heat transfer performance of the three interconnected cavities is reported in tabulated form.

Delta lithography method to increase CD uniformity and throughput of SLM-based maskless lithography

To increase the CD uniformity and throughput of SLM-based maskless lithography, the delta lithography method (DLM) has been developed. The basic idea of the DLM is to create a delta configuration by using a rectangular point array. The DLM enables irradiation in a honeycomb structure with equilateral triangles or a square structure with right-angled isosceles triangles. To verify the DLM, lithography simulations and a lithography experiment to fabricate a photomask were performed. The pattern appeared to be well defined with extremely low LER. The potential of the DLM to increase the CD uniformity and throughput is confirmed.

Explicit efficient coupling impedance formulas for the right-angled isosceles triangle and application to a circularly polarised truncated corners square patch antenna

Formulas for the four possible microstrip coupling impedances between two perimeter ports on a right-angled isosceles triangular microstrip segment are derived by integration of the associated Green's function. The formulas are economised by introducing closed forms of infinite series to give new efficient computational formulas, which are verified experimentally. The variation of input impedance with offset, for a circularly polarised truncated corners square patch antenna, is determined using the segmentation method. Predicted and simulated results shows very good agreement. The area of the perturbation segments increases as a function of offset and this increase reduces the effect of manufacturing errors. A simple matching network for the offset feed is designed for an antenna operating at 2.45 GHz. Predicted, simulated and practical results show good agreement.

Two-Axis Anemometer with Acoustic Reflector Using Single Pair of Loudspeaker and Microphone

In this paper, we describe a two-axis anemometer with an acoustic reflector using a single pair of a loudspeaker (SP) and a microphone (MIC). The two-axis anemometer proposed in this study consists of a SP, a MIC and an acoustic reflector, which form a right-angled isosceles triangle. This anemometer can measure spatial mean wind velocity and direction by calculating each wind component along the coordinate axis defined in each propagation path; this anemometer is targeted for use in air-conditioning systems of high efficiency. The sound probe consists of a SP, a MIC, an acoustic reflector and each propagation path. By using the acoustic reflector, an additional sound path becomes available. In this case, the time of flight (TOF) can be measured using a single SP–MIC pair because of the difference in TOF between direct and reflected waves. Some experiments were carried out in a wind tunnel to measure wind velocity and direction at various wind velocities. The sound probe is placed on a turntable in a wind tunnel. Although the actual wind direction is constant, the apparent direction relative to the sound probe can be changed by rotating a turntable. The measured wind velocities and directions in the experiments showed only small changes with the wind direction change. The feasibility and accuracy of the proposed two-axis anemometer for wind velocity and direction measurements were confirmed.