Search Results (68)

The Ritz method is applied to an in-plane vibration analysis to obtain accurate frequencies of isotropic annular plates. The method is formulated in a manner that allows all combinations of free boundary conditions, two types of supported (constraining only either radial or circumferential displacement) boundary conditions, and clamped boundary conditions. Admissible functions for the two displacement components are chosen as products of trigonometric functions in the circumferential coordinate and special algebraic polynomials in the radial coordinate, enabling all possible boundary-condition combinations to be satisfied. In the numerical study, after the solution’s accuracy is verified through convergence and comparison tests, extensive and accurate frequency parameters are presented to cover all combinations of the four in-plane boundary conditions along the outer and inner edges of the annular plates. Full article

Mathematics is a discipline that forms the foundation of many fields and should be learned gradually, starting from early childhood. However, some subjects can be difficult to learn due to their abstract nature, the need for attention and planning, and math anxiety. Therefore, in this study, a system that contributes to mathematics teaching using computer vision approaches has been developed. In the proposed system, users can write operations directly in their own handwriting on the system interface, learn their results, or test the accuracy of their answers. They can also test themselves with random questions generated by the system. In addition, visual graph generation has been added to the system, ensuring that education is supported with visuals and made enjoyable. Besides the character recognition test, which is applied on public datasets, the system was also tested with images obtained from 22 different users, and successful results were observed. The study utilizes CNN networks for handwritten character detection and self-created image processing algorithms to organize the obtained characters into equations. The system can work with equations that include single and multiple unknowns, trigonometric functions, derivatives, integrals, etc. Operations can be performed, and successful results can be achieved even for users who write in italicized handwriting. Furthermore, equations written within each closed figure on the same page are evaluated locally. This allows multiple problems to be solved on the same page, providing a user-friendly approach. The system can be an assistant for improving performance in mathematics education. Full article

This study proposes a low-cost indoor positioning system based on a single Light Detection and Ranging (LiDAR) sensor and several fixed reflective reference points. Distances are obtained by trigonometric measurement, and positions are computed by trilateration. In static tests the average error was 7.4 mm. When the target moves at walking speed, small survey errors of the reference points cause the average error to increase to 21.8 mm. Finally, the proposed Reference Point Update Method (RPUM) that continuously corrects reference point coordinates using a moving average of recent residuals reduces the dynamic error from 208.71 mm to 20.34 mm, which is about 90% improvement. The method used in this paper requires no additional hardware and runs in real time. Full article

In this paper, we investigate the physical mechanics of vibration wave detection in distributed acoustic sensing (DAS) systems with the aim of enhancing the interpretation of the quantitative wavefield. We investigate the nonlinear relationship of DAS gauge length and pulse width on the seismic wave response, and the result is explained by the trigonometric relationship of backscattered Rayleigh wave phases. We further demonstrate the influence of spiral winding on DAS performance and also build phase response models for P-waves and S-waves in helically wound cables. These models suggest that the winding angle controls the measurement interval spacing and the angle of wave incidence. Additionally, integration of structural reinforcement improves the amplitude response characteristics and SNR. The experimentally inspired results show, using simulations and field tests, that the same vibration sources can give helically wound cables with larger winding angles the largest phase amplitudes, which would substantially exceed that of straight cables. SNR increased significantly (approximately 10% to 30%). The efficacy of the method was also checked using experiments for different vibration amplitudes and frequencies. Such results provide evidence for the design and installation of fiber-optic cables for use in practical engineering applications involving safety monitoring. Full article

Accurate and efficient evaluation of the intersection area between two axis-aligned ellipses is essential in applications where the coordinate system or underlying geometry naturally imposes alignment. However, most existing numerical integration techniques are designed for arbitrarily oriented ellipses, and their generality typically requires adaptive refinement or solving higher-degree algebraic intersection formulations, leading to greater computational cost than necessary in the axis-aligned case. This study introduces two analytically derived, fixed-cost Gauss–Legendre quadrature formulations for computing the intersection area in the axis-aligned configuration. The first is a sine-mapped Gauss–Legendre quadrature, which applies a trigonometric transformation to improve conditioning near endpoint singularities while retaining constant-time evaluation. The second is an enhanced two-panel affine-normalized formulation, which splits the intersection domain into two sub-intervals to increase local accuracy while maintaining a fixed computational cost. Both methods are benchmarked against adaptive Simpson integration, polygonal discretization, and Monte Carlo sampling over 10,000 randomly generated ellipse pairs. The two-panel formulation achieves a mean relative error of 0.003% with runtimes more than twenty times faster than the adaptive reference and remains consistently more efficient than the polygonal and Monte Carlo approaches while exhibiting comparable or superior numerical behavior across all tested regimes. Full article

In this study, we consider a (2+1)-dimensional integrable Boussinesq equation, where the Hirota method of positive logarithmic transformation is used to convert it into a bilinear form. We proceeded by employing different test functions, through which we obtained breather solutions, two-wave solutions, lump-periodic solutions, and new interaction solutions. The resulting soliton dynamics for the governing model are also derived using the enhanced modified extended tanh function method, where varieties of solutions, such as trigonometric, hyperbolic, and rational forms, were obtained. The derived solutions may hold significant potential for explaining real-world physical phenomena in fields like mathematical physics, plasma physics, and nonlinear optics. The accuracy and reliability of the solutions were tested by substituting them back into the original equation using Python, highlighting the method’s robustness, precision, and reliability. By choosing appropriate physical parameters, we showcased the rich diversity and dynamic behavior of the obtained soliton structures. In other words, the graphical representations in 3D, contour, and 2D were provided for some of the obtained results. The modulation instability analysis and gain spectrum of the model are also provided. The importance of the obtained results in the area of (2+1)-dimensional integrable equation application was also highlighted. Full article

This paper presents a novel analytical approach for the efficient design of a particular class of 2D FIR filters, having a frequency response with an elliptically shaped support in the frequency plane. The filter design is based on a Gaussian shaped prototype filter, which is frequently used in signal and image processing. In order to express the Gaussian prototype frequency response as a trigonometric polynomial, we developed it into a Fourier series up to a specified order, given by the imposed approximation precision. We determined analytically a 1D to 2D frequency transformation, which was applied to the factored frequency response of the prototype, yielding directly the factored frequency response of a directional, elliptically shaped 2D filter, with specified selectivity and an orientation angle. The designed filters have accurate shapes and negligible distortions. We also designed a 2D uniform filter bank of elliptical filters, which was then applied in decomposing a test image into sub-band images, thus proving its usefulness as an analysis filter bank. Then, the original image was accurately reconstructed from its sub-band images. Very selective directional elliptical filters can be used in efficiently extracting straight lines with specified orientations from images, as shown in simulation examples. A computationally efficient implementation at the system level was also discussed, based on a polyphase and block filtering approach. The proposed implementation is illustrated for a smaller size of the filter kernel and input image and is shown to have reduced computational complexity due to its parallel structure, being much more arithmetically efficient compared not only to the direct filtering approach but also with the most recent similar implementations. Full article

Accurate precipitation forecasting remains a critical challenge due to the nonlinear and multifactorial nature of rainfall dynamics. This is particularly important in arid regions like Tamanghasset, where precipitation is the primary driver of agricultural viability and water resource management. This study evaluates the performance of several time series models for monthly rainfall prediction, including the autoregressive integrated moving average (ARIMA), Exponential Smoothing State Space Model (ETS), Seasonal and Trend decomposition using Loess with ETS (STL-ETS), Trigonometric Box–Cox transform with ARMA errors, Trend and Seasonal components (TBATS), and neural network autoregressive (NNAR) models. Historical monthly precipitation data from 1953 to 2020 were used to train and test the models, with lagged observations serving as input features. Among the approaches considered, the NNAR model exhibited superior performance, as indicated by uncorrelated residuals and enhanced forecast accuracy. This suggests that NNAR effectively captures the nonlinear temporal patterns inherent in the precipitation series. Based on the best-performing model, rainfall was projected for the year 2021, providing actionable insights for regional hydrological and agricultural planning. The results highlight the relevance of neural network-based time series models for climate forecasting in data-scarce, climate-sensitive regions. Full article

Addressing the urgent need for lightweight and reusable energy-absorbing materials in aviation impact resistance, this study introduces an innovative multi-directional braided metamaterial design enabled by 4D printing technology. This approach overcomes the dual challenges of intricate manufacturing processes and the limited functionality inherent to traditional textile preforms. Six distinct braided structural units (types 1–6) were devised based on periodic trigonometric functions (Y = A sin(12πX)), and integrated with shape memory polylactic acid (SMP-PLA), thereby achieving a synergistic combination of topological architecture and adaptive response characteristics. Compression tests reveal that reducing strip density to 50–25% (as in types 1–3) markedly enhances energy absorption performance, achieving a maximum specific energy absorption of 3.3 J/g. Three-point bending tests further demonstrate that the yarn amplitude parameter A is inversely correlated with load-bearing capacity; for instance, the type 1 structure (A = 3) withstands a maximum load stress of 8 MPa, representing a 100% increase compared to the type 2 structure (A = 4.5). A multi-branch viscoelastic constitutive model elucidates the temperature-dependent stress relaxation behavior during the glass–rubber phase transition and clarifies the relaxation time conversion mechanism governed by the Williams–Landel–Ferry (WLF) and Arrhenius equations. Experimental results further confirm the shape memory effect, with the type 3 structure fully recovering its original shape within 3 s under thermal stimulation at 80 °C, thus addressing the non-reusability issue of conventional energy-absorbing structures. This work establishes a new paradigm for the design of impact-resistant aviation components, particularly in the context of anti-collision structures and reusable energy absorption systems for eVTOL aircraft. Future research should further investigate the regulation of multi-stimulus response behaviors and microstructural optimization to advance the engineering application of smart textile metamaterials in aviation protection systems. Full article

This study proposes an asymmetric pitch control technique for electric oil pumps with symmetric gear-type pumps in order to reduce noise and vibration. For vane pump noise reduction, mechanical asymmetric pitch arrangements of each vane are widely used. However, the mechanical asymmetric pitch arrangement approach is not applicable in gear-type pumps due to structural limitations. The proposed asymmetric pitch control method provides similar effects to the mechanical asymmetric pitch arrangement by employing instantaneous motor torque controls for an electric oil pump with a gear-type pump. The magnitude of motor torque for each pump tooth is determined with an asymmetric pitch formula, which has been widely used for mechanical vane pumps in previous studies and patents. A formula for the shape of instantaneous motor torque is proposed for the analysis of pressure fluctuations of pumps, which is a combination of trigonometric and exponential functions. The calibration factors for the magnitude and shape can be adjusted according to the characteristics of a given pump. The experimental results for a 400 W electric pump show that the proposed method reduced and dispersed the noise peak by approximately 4 dB(A) in comparison with the normal control, and affected hydraulic performance with a less than 1% decrease in flow rate in not only pump-level but also gearbox-level test environments. Full article

This paper presents a second examination of trigonometric step sizes and their impact on Warm Restart Stochastic Gradient Descent (SGD), an essential optimization technique in deep learning. Building on prior work with cosine-based step sizes, this study introduces three novel trigonometric step sizes aimed at enhancing warm restart methods. These step sizes are formulated to address the challenges posed by non-smooth and non-convex objective functions, ensuring that the algorithm can converge effectively toward the global minimum. Through rigorous theoretical analysis, we demonstrate that the proposed approach achieves an $O\left({\displaystyle \frac{1}{\sqrt{T}}}\right)$ convergence rate for smooth non-convex functions and extend the analysis to non-smooth and non-convex scenarios. Experimental evaluations on FashionMNIST, CIFAR10, and CIFAR100 datasets reveal significant improvements in test accuracy, including a notable $2.14\%$ increase on CIFAR100 compared to existing warm restart strategies. These results underscore the effectiveness of trigonometric step sizes in enhancing optimization performance for deep learning models. Full article

Trigonometric functions are widely used to express relationships between the sides and angles of triangles, being fundamental in a wide variety of fields in science and engineering. Previous research indicates that the Gudermann function connects hyperbolic and trigonometric functions without requiring complex numbers, while the Mercator projection maps the Earth’s spherical surface onto a cylindrical plane. This article presents four key contributions derived from hyperbolic functions, with the main proof applying Newton’s first law in the static case of cables. First, a new method relates right triangle formulas to the sides of a right triangle, facilitating vector decomposition along the X and Y axes. Second, a right triangle function with a hyperbolic angle is proposed, relating the three sides of a right triangle and the hyperbolic angle, offering an alternative to the Pythagorean theorem. Third, the law of hyperbolic cosines and the law of hyperbolic tangents is applied to trigonometric problems. Fourth, the hyperbolic Mollweide’s formula is used to solve oblique triangles. These results demonstrate the potential of hyperbolic transformations in engineering and mathematical contexts, for both education and research. Future investigations should include experimental and analytical tests to further extend the applications to all branches based on mathematics. Full article

In this paper, a new sensorless approach is proposed to address the speed and position estimation of the Synchronous Reluctance Machine (SynRM). The design of the sensorless control algorithm is developed on the basis of the modified SynRM mathematical model employing a simple sliding mode observer (SMO) and a modified EMF observer that are connected in series. All variables of the modified SynRM model are expressed in the arbitrary rotating frame, which is the so-called estimated γδ reference frame. The derived modified rotor flux terms contain angle error information in the form of trigonometric functions. Initially, the modified rotor flux is expressed as a function of saliency and the stator current i_d, including the angular deviation between the dq and γδ reference frames, which are rotating at synchronous and estimated speeds, respectively. A suitably designed SMO is utilized to estimate the modified stator flux components in the γδ reference frame. Once the SMO operates in sliding mode, the derived equivalent control inputs of the flux/current observer are used to obtain the required angular position and speed information of rotor by means of the modified EMF and Speed/Position observer. Only measures of stator voltages and currents are required for the speed and position estimation. In addition, Lyapunov Candidate Functions (LCFs) have been applied to determine the sliding mode existence conditions and the gains of the modified EMF observer. The SynRM observer–controller system is tested and evaluated in a wide speed range, even at very low speeds, in the presence of torque load disturbances. Simulation results demonstrate the overall efficacy and robustness of the proposed sensorless approach. Moreover, simulation tests verify the fast convergence and high performance of the modified EMF/speed/angle observer. Full article

We discuss determining a finite sample linear time-invariant (FS-LTI) system’s impulse response function, $h\left[n\right],$ in the frequency domain when the input testing function is a uniform window function with a width of L and the output is limited to a finite number of effective samples, M. Assuming that the samples beyond M are all zeros, the corresponding infinite sample LTI (IS-LTI) system is a marginally stable system. The ratio of the discrete Fourier transforms (DFT) of the output to input of such an FS-LTI system, $H_0\left[k\right]$, cannot be directly used to find $h\left[n\right]$ via inverse DFT (IDFT). Nevertheless, $H_0\left[k\right]$ contains sufficient information to determine the system’s impulse response function (IRF). In the frequency-domain approach, we zero-pad the output array to a length of N. We present methods to recover $h\left[n\right]$ from $H_0\left[k\right]$ for two scenarios: (1) $N\ge \max (L,M+1)$ and N is a coprime of L, and (2) $N\ge L+M+1$. The marginal stable system discussed here is an artifact due to the zero-value assumption on unavailable samples. The IRF obtained applies to any LTI system up to the number of effective data samples, M. In demonstrating the equivalence of $H_0\left[k\right]$ and $h\left[n\right]$, we derive two interesting DFT pairs. These DFT pairs can be used to find trigonometric sums that are otherwise difficult to prove. The frequency-domain approach makes mitigating the effects of interferences and random noise easier. In an example application in radar remote sensing, we show that the frequency-domain processing method can be used to obtain finer details than the range resolution provided by the radar system’s transmitter. Full article

The Material Generation Optimization (MGO) algorithm is an innovative approach inspired by material chemistry which emulates the processes of chemical compound formation and stabilization to thoroughly explore and refine the parameter space. By simulating the bonding processes—such as the formation of ionic and covalent bonds—MGO generates new solution candidates and evaluates their stability, guiding the algorithm toward convergence on optimal parameter values. To improve its search efficiency, this paper introduces an Enhanced Material Generation Optimization (IMGO) algorithm, which integrates a Quadratic Interpolated Learner Process (QILP). Unlike conventional random selection, QILP strategically selects three distinct chemical compounds, resulting in increased diversity, a more thorough exploration of the solution space, and improved resistance to local optima. The adaptable and non-linear adjustments of QILP’s quadratic function allow the algorithm to traverse complex landscapes more effectively. This innovative IMGO, along with the original MGO, is developed to support applications across three phases, showcasing its versatility and enhanced optimization capabilities. Initially, both the original and improved MGO algorithms are evaluated using several mathematical benchmarks from the CEC 2017 test suite and benchmarks to measure their optimization capabilities. Following this, both algorithms are applied to the following three well-known engineering optimization problems: the welded beam design, rolling element bearing design, and pressure vessel design. The simulation results are then compared to various established bio-inspired algorithms, including Artificial Ecosystem Optimization (AEO), Fitness–Distance-Balance AEO (FAEO), Chef-Based Optimization Algorithm (CBOA), Beluga Whale Optimization Algorithm (BWOA), Arithmetic-Trigonometric Optimization Algorithm (ATOA), and Atomic Orbital Searching Algorithm (AOSA). Moreover, MGO and IMGO are tested on a real Egyptian power distribution system to optimize the placement of PV and the capacitor units with the aim of minimizing energy losses. Lastly, the PV parameters estimation problem is successfully solved via IMGO, considering the commercial RTC France cell. Comparative studies demonstrate that the IMGO algorithm not only achieves significant energy loss reduction but also contributes to environmental sustainability by reducing emissions, showcasing its overall effectiveness in practical energy optimization applications. The IMGO algorithm improved the optimization outcomes of 23 benchmark models with an average accuracy enhancement of 65.22% and a consistency of 69.57% compared to the MGO method. Also, the application of IMGO in PV parameter estimation achieved a reduction in computational errors of 27.8% while maintaining superior optimization stability compared to alternative methods. Full article