Double Angle Research Articles

Group decision-making algorithm with sine trigonometric r,s,t-spherical fuzzy aggregation operators and their application

r, s, t-spherical fuzzy (r, s, t-SPF) sets provide a robust framework for managing uncertainties in decision-making, surpassing other fuzzy sets in their ability to accommodate diverse uncertainties through the incorporation of flexible parameters r, s, and t. Considering these characteristics, this article explores sine trigonometric laws to enhance the applicability and theoretical foundation for r, s, t-SPF setting. Following these laws, several aggregation operators (AOs) are designed for aggregation of the r, s, t-SPF data. Meanwhile, the desired characteristics and relationships of these operators are studied under sine trigonometric functions. Furthermore, we build a group decision-making algorithm for addressing multiple attribute group decision-making (MAGDM) problems using the developed AOs. To exemplify the applicability of the proposed algorithm, we address a practical example regarding laptop selection. Finally, parameter analysis and a comprehensive comparison with existing operators are conducted to uncover the superiority and validity of the presented AOs.

Utilizing sine trigonometric q-spherical fuzzy rough aggregation operators for group decision-making and their role in digital transformation

q-spherical fuzzy rough set (q-SFRS) is also one of the fundamental concepts for addressing more uncertainties in decision problems than the existing structures of fuzzy sets, and thus its implementation was more substantial. The well-known sine trigonometric function maintains the periodicity and symmetry of the origin in nature and thus satisfies the expectations of the experts over the multi-parameters. Taking this feature and the significance of the q-SFRSs into consideration, the main objective of the article is to describe some reliable sine trigonometric laws for SFSs. Associated with these laws, we develop new average and geometric aggregation operators to aggregate the q-spherical fuzzy rough numbers. Then, we presented a group decision-making strategy to address the multi-attribute group decision-making problem using the developed aggregation operators. To verify the value of the defined operators, a MAGDM strategy is provided along with applications for selecting a Cloud Service Provider and a Digital Transformation Vendor for digital transformation. Moreover, a comparative study is also performed to present the effectiveness of the developed approach.

Bowtie design for thermal expansion mimicry and mitigation

A geometric bowtie design for the purpose of mimicking (matching) and/or mitigating (reducing) the thermal expansion effects between two laterally opposing structures is investigated. A bowtie design is that of two opposing overlapping triangles with farthest sides parallel. This design embodies thermal centering concepts which utilize the relationships of the trigonometric identity known as the law of sines for a triangle from which beneficial thermal properties can be engineered. For validation. The bowtie design is embodied using two separated Zerodur plane mirrors, symmetrically opposed, and each seated in Maxwell-style kinematic couplings of spheres and vee-groves. The vee-grooves form a wheel-like pattern on an aluminum baseplate. The intersection of the vee-grooves is the common overlapping thermal center of both mirror mounts. The refractive-index corrected interferometric measurement of the displacement between these two mirrors shows that even during non-isothermal non-equilibrium transitions and states, the nominal ∼90 μm thermally induced (13 °C) expansion of an aluminum baseplate can be mitigated, being reduced more than 98 % to ∼1.4 μm by mimicking the thermal expansion properties of the mirrors’ structural material. Engineering thermally-invariant kinematic bowtie designs using materials having higher thermal expansion coefficients is also described.

Modified Bracketing Iterative Method for Solving Nonlinear Equations

Non-linear equations, depicted as curves in numerous everyday situations, have long piqued the curiosity of researchers and engineers due to their frequent occurrence in practical problems. Despite attempts to tackle these equations both analytically and numerically, analytical methods often fall short when the equation's degree exceeds five, prompting the adoption of numerical approaches to yield approximate solutions. Consequently, this study places emphasis on segmenting intervals into smaller sub-intervals, with a particular focus on employing the Regula-Falsi method to integrate these segmented intervals, thereby enhancing its convergence rate. Furthermore, by utilizing the Regula-Falsi formula for interval segmentation, the number of iterations and computational time required are minimized. Additionally, the effectiveness of the proposed method is verified through numerical experiments involving various equation types, including algebraic, trigonometric, exponential, logarithmic, and transcendental equations, comparing the outcomes with established methods. The findings demonstrate that the proposed algorithm not only efficiently segments intervals but also enhances accuracy and reduces errors when these segmented intervals are utilized in conventional bracketing methods.

Analisis Kesalahan Siswa dalam Menyelesaikan Soal Identitas Trigonometri di Kelas XI MIA SMAK St. Dominikus Tambolaka

This study aims to determine student errors and the causes of student errors in solving trigonometric identity problems in class XI MIA SMAK St. Dominic Tambolaka. The type of research used is qualitative descriptive. The subjects of the study were 18 class XI MIA students at SMAK St. Dominikus Tambolaka. Data collection techniques use written tests and interviews. The results showed the percentage of errors made by students in solving math problems related to trigonometric identity. The causes of student errors include rushing to read math problems, not having the ability to focus attention on learning, not mastering trigonometric identity material, not understanding the formula that should be used, not being careful in carrying out the calculation process, carelessly doing problems because of previous mistakes, and rushing in working on the solving process.

Study on soliton behaviors of the complex nonlinear Fokas–Lenells equation utilizing different methods

In this paper, we study the Fokas–Lenells equation, which is a derivation form of the nonlinear Schrödinger equation and can be used to describe nonlinearity in the propagation of optical pulses. To seek solutions for this nonlinearity, the Logistic method has been proposed in order to drive acceptable and understandable results regarding physical meaning. This method is defined based on the summation of an ordinary differential equation. Moreover, for further study, the Bernoulli [Formula: see text]-expansion method is used by considering hypothetical solutions as a function of the Bernoulli equation. Subsequently, solutions of the Fokas–Lenells equation are successfully acquired, displaying efficient results in hyperbolic, trigonometric, and exponential equations. The importance of these results appears in the definition of the wave function under the consideration of the appropriate coefficients. Hence, computer simulation is used for better understanding of the generated results. The dynamic behaviors of these solutions are demonstrated in 3D graphs, contour maps, and line plots, under applying various parameters, solutions of the waves show different soliton behaviors, including bright-dark, bright, and breather solitons. The results indicate that the exerted methods are novel, reliable, and effective approaches which can be employed in a wide range of nonlinear differential equations. These methods and their begotten results are far from the complexities of mathematical structures and therefore go beyond previous efforts in the literature.

AUTOMATIC ASSEMBLY OF INTERLOCKING BUILDING CUBES BY USING VISUAL SERVOING APPROACH

Most of the robotic arms are set up for operation by the teach-and-repeat (T&R) technique, also known as coordinate-based control, using pre-recorded coordination values to generate robotic arm trajectories. This technique is inefficient because it requires teaching a robotic arm every single coordinate point to complete an operation. It is a manual, tedious and time-consuming task. Moreover, this task needs to be repeated if there are slight changes in the operation. This paper studied the visual servoing approach to replace the T&R technique in assembling different interlocking building cubes (IBC) combinations. The proposed method uses an image-based visual servo (IBVS) with a dynamic look-and-move system, and the eye-to-hand system, which requires computing the incremental joint angles of the robotic arm, a novel kinematic formula with a combination of Trigonometry formulas has been proposed in this paper. The experimental results show the feasibility of using a visual servoing approach to complete the assembly tasks without relying on the T&R technique and achieved an average success rate of 90% and an average of 40.2 seconds for task duration. Thus, this eliminates the tedious and time-consuming tasks of teaching the coordinate points to the robotic arms. Further improvement on the servo motor, camera and computer processor can be done to increase the accuracy, computation time and task duration in future.

Using Mathematical Thinking in Solving Trigonometric Problems within Mason’s Cognitive Framework

The paper focuses on the synergy between mathematical thinking and solving trigonometric problems within Mason’s cognitive framework. It presents and analyzes concepts related to mathematical thinking, especially Mason’s definition of the thinking process into three stages: input, impact, and evaluation. Applying this framework to solving trigonometric equations, the paper illustrates how mathematical thinking can help approach problems in an organized and rigorous manner, while opening opportunities for creativity and exploration. Discussing mathematical thinking, the paper defines it as a creative process involving prediction, induction, interpretation, description, abstraction, and reasoning. Aligned with Mason’s definition of mathematical thinking, the paper describes how this thinking aids students in understanding complex structures and solving problems quickly and flexibly.

Compression strength of reinforced steel angles

A common reinforcing detail for strengthening open web steel joists is the addition of a rod or plate to an existing angle to form a built-up member. Design standards typically include a limit on the spacing of connections between the components of built-up compression members. The limits are based primarily on more common built-up members such as double angles. Reinforced angles are different than double angles because, among other things, the components are in direct contact limiting potential buckling modes. Prior physical testing is limited; thus, it is unclear if the spacing limits are truly applicable to reinforced angles. The limits can govern the weld spacing for reinforced angles in joists, potentially resulting in more welding than necessary. This experimental and numerical study investigates the compression strength of reinforced angles and how the strength varies with weld spacing. Fifteen reinforced angle specimens were fabricated and tested with variation in reinforcement type, weld spacing, and member effective length. Numerical analyses were performed to expand the range of investigation. Weld spacing had essentially no effect on the experimental strength of the test specimens when the weld spacing was less than or equal to 305 mm (12 in.) on center. Strengths computed per current design provisions were found to be overly conservative. Recommendations to modify the provisions were made. These results lead to a better understanding of the behavior of reinforced angles and enable more efficient reinforcement of open web steel joists.

Fracture load in double keyhole notch PLA-Cu2O nanocomposites manufactured via compression molding and 3D printing: An experimental and numerical study

Polylactic acid (PLA) polymer has garnered significant attention due to its biocompatibility. The incorporation of copper oxide (Cu2O) nanoparticles into this polymer is expected to enhance its antibacterial, electrical, and thermal properties. This modification can potentially improve the performance of PLA in the fields of prosthetics manufacturing or printed circuit fabrication. However, the current research is rather focused on the mechanical properties of the PLA-Cu2O nanocomposites. This research is thus aimed to analyze PLA-Cu2O (97-3 wt%) nanocomposites with a double keyhole notch configuration both experimentally and numerically. Scanning electron microscopy (SEM), X-ray diffraction (XRD), energy dispersive X-ray spectroscopy (EDX), X-ray mapping of elemental distribution(X-map), and thermogravimetric analysis (TGA) were employed to explore the morphology, crystallinity, homogeneity, purity, and thermal stability of the nanocomposite. The specimens were fabricated through two different processes: the classical method of compression molding and the innovative method of 3D printing. The results revealed the superior mechanical performance of the 3D-printed nanocomposite at a 0° raster angle, while the mechanical properties gradually decreased for raster angles of 45° and 90°. The experimental test also indicated a decline in the maximum fracture load of specimens with a double keyhole notch and constant notch inclination angle by raising the notch radius. This behavior was also observed by increasing the notch inclination angle at constant notch radius. The numerical results were similar to the experimental findings. Moreover, the nanocomposite manufactured through the classical method exhibited higher critical fracture load compared to their 3D-printed counterparts with the same geometry.

A Three-Phase Multilevel Inverter Synthesized with 31 Levels and Optimal Gating Angles Based on the GA and GWO to Supply a Three-Phase Induction Motor

A three-phase multilevel inverter (MLI), synthesized with 31 levels in regard to its output voltage, is used to provide the AC supply to a three-phase, squirrel cage induction motor. The gating angles required for the 30 power switches on the MLI are optimized using both the genetic algorithm (GA) and the grey wolf optimizer (GWO), in which the optimal angles are determined through solving the trigonometric equations taken from Fourier analysis to target the minimum total harmonic distortion (THD) at the MLI output. A simulation model and an experimental prototype are developed for performance analysis and validation. The results demonstrate that the MLI is effectively able to produce 31 levels of three-phase AC output voltage, with the THD not exceeding 5% when loaded with a resistive load and a three-phase induction motor. The voltage and current are measured and recorded for different loads and operating conditions, including the amount of energy consumed by the load. The results of the frequency analysis demonstrate that most of the triple harmonics, which can harm the efficiency of the inverter, are cancelled.

Using the Girard-Newton-Waring Formula to Evaluate Trigonometric Sums: A Blast from the Past

Summary This article shows how two classical results about polynomials with trigonometric roots may be used together to create interesting new formulas for trigonometric sums. Readers are encouraged to find further new trigonometric formulas by applying the ideas in this article to other polynomials having trigonometric roots.

Experiments on fire-protected and hot-dip galvanised steel bolted connections

Connections are of high significance in the stability and integrity of steel structures. Connections play a key role in structures exposed to fire because they are supposed to support the entire elements against a variety of loads. In steel beams, a connection is generally implemented to join (at least) two perpendicular members. A thorough understanding of the heat transfer in structural members is a prerequisite to a reliable design. In the case of R30 requirements, a large number of members can be left unprotected if they are hot-dip galvanised. If hot-dip galvanising of a structure is not sufficient to provide required fire resistance, one part of the structure could be hot-dip galvanised while another part to be protected with a reactive fire protection system. When connecting a protected beam to an unprotected hot-dip galvanised beam, information regarding the heat transfer between these beams is required.Additional heating in a connection due to hot dip galvanisation needs to be taken into account when verifying the load-bearing capacity of the component protected by reactive fire protection according to EN 1993-1-2. This study investigates the effect of heat development and transfer in full-scale steel I-beams from a hot-dip galvanised secondary beam to an intumescent-coated main beam through various bolted connections such as fin plates, double angles, and stiffeners with lugs. Fire protection was applied to the main beams, while the secondary beams were hot-dip galvanised. A thorough experimental, numerical and theoretical study was implemented as presented in this paper, the results of which are deemed to provide significant input into improving safety, hence mitigating the risk of steel beams.

Manifold inversion on prediction dihedral angle from vicinal coupling constant with 3-sphere approach

Dihedral angles are predicted from vicinal coupling constant 3JHH[Hz] with 3-sphere approach, sphere or torus trigonometric equations of circle 1, 2 and circle inversion 3-5 for all cis-, trans-ee, trans-aa stereochemistry. The existence of circle inversion was demonstrated with conformational analysis on five and six membered rings. The sign and the stereochemistry result from vicinal coupling constant under trigonometric equations confirmed by algebraic equations, Hopf and Lie algebra theory. 3-Sphere, a hypersphere in 4D enable for all stereochemistry calculation dihedral angles under magnetic wave (NMR data -3JHH[Hz]), and all isomers from only one vicinal coupling constant.

Retraction Note: Trigonometric polynomial solutions of Bernouilli trigonometric polynomial differential equations

Retraction Note: Trigonometric polynomial solutions of Bernouilli trigonometric polynomial differential equations

Optimal modeling of line paths based on multibeam bathymetric system

Multibeam ocean bathymetry is one of the major issues in exploring the ocean. For problems such as multibeam line ocean bathymetry, this paper starts from the related theory, analyzes the inherent geometric features of the problem, comprehensively establishes a multibeam line model with generalized significance within a certain error range, and according to the method of least squares, gives a model of the line with a fixed opening angle of the multibeam transducer in a certain range of the sea area. Firstly, from the geometrical point of view, the interpreted diagram and the schematic diagram given in the title are combined and simplified into a problem of solving a triangle. The trigonometric equations are listed through the sine theorem and solved algebraically. A mathematical model is developed about the coverage width and adjacent strips. Then, from the question, we know that we need to solve the modeling differences caused by the different angles of the measuring lines in different directions, i.e., to elevate the two-dimensional problem of solving triangles to a three-dimensional problem of three-dimensional geometry. By finding the angle between the coverage width and the horizontal plane , we know the relationship between it and the angle between the direction of the survey line and the slope angle , so as to establish the model of the coverage width of multibeam bathymetry. Summarized by the verification of the previous conclusions, this paper obtains: when controlling the width of two adjacent survey lines exactly meets the overlap rate of 10% with the previous survey line, and the survey line direction is perpendicular to the seafloor slope in the projection, the measurement length is the shortest at this time. Using the above summary as a constraint, the shortest path is calculated to be 66 nautical miles using the trigonometric method.

Study of Algorithms to Calculate Real Root of Transcendental Equations

This study deals with a comparative analysis of algorithms for solving transcendental equations. It includes algorithms for RF-EXP, RF-Halley, Arc-sine, Tanh and RF-ArcTanh methods to check the accuracy, number of iterations, and errors for the solution of the trigonometric, logarithmic, and exponential equations. Several numerical examples are presented to illustrate the algorithms’ efficacy which are programmed in MATLAB. The findings indicate that Arc-sine and RF-ArcTanh, RF-Halley, and Tanh are in favour of exponential equations, trigonometric equations, and logarithmic equations respectively.

Optimum design method for structural configuration and fiber arrangement for fiber-reinforced composites

Fiber-reinforced composite materials, exemplified by CFRP, offer the possibility of achieving lightweight, high-stiffness, and high-strength structures by continuously and evenly distributing fibers. While topology and orientation optimization methods have been developed for anisotropic materials in the past, there remains a gap in design methods that consider manufacturability, especially for continuous fiber materials. In this study, we propose a design method that takes into account manufacturability, focusing on the aspects of continuity and uniformity in fiber-reinforced composite optimum design. Specifically, we introduce a two-stage optimization approach. In the first stage, we conduct concurrent optimization of topology and fiber orientation. We utilize a level-set function to represent topological configuration, while for orientation, we introduce a “double angle vector”, which enables us to consider fiber properties such as angular periodicity. These design variables are updated by solving partial differential equations based on reaction–diffusion equations. In the second stage, leveraging the optimal orientations obtained in the first stage, we optimize the path-line generation for the manufacture of continuous fiber materials. We introduce a scalar function representing path lines and formulate an optimization problem to ensure that the path lines are both evenly spaced and continuous. The update of design variables in this state is also achieved via solving the partial differential equation. Through the development of this two-stage optimization method, we aim to create an optimal structure with manufacturable continuous fiber materials, incorporating both the topology and fiber orientation that satisfy the requirements of continuity and uniformity.

Evaluation of tracheal diameter and angles in fetuses with double aortic arch using prenatal ultrasound: implications for postnatal management.

This study aims to analyze the value of prenatal ultrasound in the screening, diagnosis, and treatment of double aortic arch (DAA) malformations. A retrospective analysis was conducted on 31 fetal cases with double aortic arch anomalies over a 12-year period from June 1, 2011 to June 1, 2023. The assessment included combined measurements of fetal tracheal internal diameter Z-score and DAA pinch angle, along with ultrasonographic findings, associated anomalies, genetic abnormalities, postnatal CTA images, and long-term postnatal outcomes. Of the 31 fetal double aortic arch cases, 15 were right aortic arch dominant, 2 were left aortic arch dominant, and 14 had a balanced double arch. Genetic testing was performed on 19 cases, revealing abnormalities in 2 cases, including one Turner syndrome, and one carrier of ichthyosis gene with heterozygous deletion. Out of the total cases, 29 were delivered, and 2 cases were terminated. Prenatal diagnosis accurately identified 29 cases (29/31, 93.5%), which was confirmed by postnatal pathological anatomy, echocardiography, surgery or CTA. Fetal tracheal internal diameter Z-scores were significantly smaller in the symptomatic group than in the asymptomatic group (-1.27 ± 0.49 vs -0.68 ± 0.60, P = 0.018). The area under the curve was 0.776 (95% confidence interval, 0.593-0.960) using a tracheal internal diameter z-score cutoff of -0.73 with a sensitivity of 90% and specificity of 64.7%. The double arch pinch angle was significantly smaller in the symptomatic group than in the asymptomatic group [52.50° (38.25° to 59.00°) vs 60.00° (53.50° to 70.50°), P = 0.035]. The area under the curve was 0.744 (95% confidence interval, 0.554-0.935), and the sensitivity for determining the presence or absence of symptoms was 90% when the cutoff value was 62.5°, with a specificity of 47.1%. Fifteen cases opted for surgery with favorable surgical outcome. Prenatal echocardiography demonstrates good diagnostic efficacy for fetal double aortic arch. It is also essential to detect the presence of other underlying intra- and extracardiac malformations and genetic abnormalities. There is a significant difference in prenatal tracheal internal diameter Z-scores and double arch pinch angle between asymptomatic and symptomatic DAA infants. Symptomatic infants require early surgery, while asymptomatic infants should be monitored.

Minimization of Common Mode Voltage and Capacitor Voltage Unbalance of 3-Phase 4-Wire 3-Level NPC inverter using 4D-SVM

This paper elucidates the implementation of Four-dimensional Space Vector Modulation (4D-SVM) for reduction of Common Mode Voltage (CMV) and Capacitor Voltage Unbalance (CVU) of 3-phase 4-wire 3-level Neutral Point Clamped inverter (3L-NPCI). CMV and CVU are major drawbacks of neutral point clamped inverter circuit; It leads to affect the performance of inverter in terms of bearing failure, unequal voltage stress across various switching devices, the magnitude of load current increases, increases in total harmonic distortion (THD), tripping of ground fault relays and malfunction of various sensitive electronics and control unit. The 4D-SVM algorithm is developed to reduce the CMV and CVU; with help of the nearest state vector (NSV) control method CMV range is reduced and by proper utilization of large and small vector capacitor voltage unbalance is minimized. 4D-SVM for 3-phase 4-wire 3-level includes 81 switching state vectors with a non-redundant switching concept. This algorithm is implemented without using any trigonometric equations, lookup tables, and angle determinations. This proposed system provides a comprehensive analysis of 4D-SVM for reduction of CMV range and CVU for the 3L-NPCI system. The minimization of CMV and CVU is achieved using MatLab-Simulink and FPGA processor for simulation and experimental results respectively.