Fractal and Fractional最新文献

{"title":"The Early Diagnosis of Rolling Bearings’ Faults Using Fractional Fourier Transform Information Fusion and a Lightweight Neural Network","authors":"Fengyun Xie, Gang Li, Chengjie Song, Minghua Song","doi":"10.3390/fractalfract7120875","DOIUrl":"https://doi.org/10.3390/fractalfract7120875","url":null,"abstract":"In response to challenges associated with feature extraction and diagnostic models’ complexity in the early diagnosis of bearings’ faults, this paper presents an innovative approach for the early fault diagnosis of rolling bearings. This method combined concepts from frequency domain signal analysis with lightweight neural networks. To begin, vibration signals from rolling bearings were collected using vibration sensors, and the mean square value was utilized as an indicator for accurate early fault signal extraction. Subsequently, employing the fractional Fourier transform, the time domain signal was converted into a frequency domain signal, which provided more detailed frequency feature information. The fusion process combined amplitude frequency and phase frequency information, and was visualized as a Gram angle field map. The lightweight neural network Xception was selected as the primary fault diagnosis tool. Xception, a convolutional neural network (CNN) variant, was chosen for its lightweight design, which maintains excellent performance while significantly reducing model parameters. The experimental results demonstrated that the Xception model excelled in rolling bearing fault diagnosis, particularly when utilizing fused information datasets. This outcome underscores the advantages of combining information fusion and the Xception model to enhance the accuracy of early rolling bearing fault diagnosis, and offers a viable solution for health monitoring and fault diagnosis in industrial settings.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"58 10","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138982518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Load Distribution Analysis and Contact Stiffness Prediction of the Dual-Drive Ball Screw Pair Considering Guide Rail Geometric Error and Slide Position","authors":"Zhifeng Liu, Weiliang Zuo, Baobao Qi, C. Chen, Jinyan Guo, Dong Li, Shan Gao","doi":"10.3390/fractalfract7120873","DOIUrl":"https://doi.org/10.3390/fractalfract7120873","url":null,"abstract":"The dual-drive ball screw pair serves as a crucial element within the fixed gantry machine tool with cross-rail movement. When in service, the dual-drive ball screw pair experiences variations in axial load, impacting the contact load distribution of the ball screw pair. A calculation model for determining the axial load offset of the dual-drive ball screw pair is proposed to investigate the variation in axial load. The impact of the geometric error associated with the guide rail and the position of the slide are considered. This paper presents the contact load distribution model for the dual-drive ball screw pair. This study investigates the contact load and contact angle distribution of the dual-drive ball screw pair during the machine tool in service. Additionally, based on fractal theory, the stiffness models of individual micro-convex body and contact surfaces have been established. This study provides a comprehensive analysis of the contact stiffness of the ball screw pair, considering the influence of guide rail geometric error and slide position. In addition, the three-dimensional surface morphology of ball screw pair is obtained by experiments. This paper investigates the contact stiffness distribution of dual-drive ball screw pair during service.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"515 1","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138982902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Multi-Task Learning Framework of Stable Q-Compensated Reverse Time Migration Based on Fractional Viscoacoustic Wave Equation","authors":"Zongan Xue, Yanyan Ma, Shengjian Wang, Huayu Hu, Qingqing Li","doi":"10.3390/fractalfract7120874","DOIUrl":"https://doi.org/10.3390/fractalfract7120874","url":null,"abstract":"Q-compensated reverse time migration (Q-RTM) is a crucial technique in seismic imaging. However, stability is a prominent concern due to the exponential increase in high-frequency ambient noise during seismic wavefield propagation. The two primary strategies for mitigating instability in Q-RTM are regularization and low-pass filtering. Q-RTM instability can be addressed through regularization. However, determining the appropriate regularization parameters is often an experimental process, leading to challenges in accurately recovering the wavefield. Another approach to control instability is low-pass filtering. Nevertheless, selecting the cutoff frequency for different Q values is a complex task. In situations with low signal-to-noise ratios (SNRs) in seismic data, using low-pass filtering can make Q-RTM highly unstable. The need for a small cutoff frequency for stability can result in a significant loss of high-frequency signals. In this study, we propose a multi-task learning (MTL) framework that leverages data-driven concepts to address the issue of amplitude attenuation in seismic records, particularly when dealing with instability during the Q-RTM (reverse time migration with Q-attenuation) process. Our innovative framework is executed using a convolutional neural network. This network has the capability to both predict and compensate for the missing high-frequency components caused by Q-effects while simultaneously reconstructing the low-frequency information present in seismograms. This approach helps mitigate overwhelming instability phenomena and enhances the overall generalization capacity of the model. Numerical examples demonstrate that our Q-RTM results closely align with the reference images, indicating the effectiveness of our proposed MTL frequency-extension method. This method effectively compensates for the attenuation of high-frequency signals and mitigates the instability issues associated with the traditional Q-RTM process.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"143 ","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138982884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Synchronization of Julia Sets in Three-Dimensional Discrete Financial Models","authors":"Zhongyuan Zhao, Yongping Zhang, Dadong Tian","doi":"10.3390/fractalfract7120872","DOIUrl":"https://doi.org/10.3390/fractalfract7120872","url":null,"abstract":"When aiming to achieve consistency in fractal characteristics between different models, it is crucial to consider the synchronization of Julia sets. This paper studies the synchronization of Julia sets in three-dimensional discrete financial models. First, three-dimensional discrete financial models with different model parameters are proposed and their Julia sets are presented. According to the model forms, two kinds of synchronous couplers that can achieve synchronization of Julia sets between different models are designed by changing the synchronization parameters. The proposed synchronization method is theoretically derived and the efficiency of different synchronous couplers are compared. Finally, the effectiveness is verified by Julia sets graphics. This method has reference value for theoretical research into financial models in the field of fractals.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"156 ","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139010868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Otsu Image Segmentation Algorithm Based on Hybrid Fractional-Order Butterfly Optimization","authors":"Yu Ma, Ziqian Ding, Jing Zhang, Zhiqiang Ma","doi":"10.3390/fractalfract7120871","DOIUrl":"https://doi.org/10.3390/fractalfract7120871","url":null,"abstract":"To solve the drawbacks of the Otsu image segmentation algorithm based on traditional butterfly optimization, such as slow convergence speed and poor segmentation accuracy, this paper proposes hybrid fractional-order butterfly optimization with the Otsu image segmentation algorithm. G-L-type fractional-order differentiation is combined with the algorithm’s global search to improve the position-updating method, which enhances the algorithm’s convergence speed and prevents it from falling into local optima. The sine-cosine algorithm is introduced in the local search step, and Caputo-type fractional-order differentiation is used to avoid the disadvantages of the sine-cosine algorithm and to improve the optimization accuracy of the algorithm. By dynamically converting the probability, the ratio of global search to local search is changed to attain high-efficiency and high-accuracy optimization. Based on the 2-D grayscale gradient distribution histogram, the trace of discrete matrices between classes is chosen as the fitness function, the best segmentation threshold is searched for, image segmentation is processed, and three categories of images are chosen to proceed with the segmentation test. The experimental results show that, compared with traditional butterfly optimization, the convergence rate of hybrid fractional-order butterfly optimization with the Otsu image segmentation algorithm is improved by about 73.38%; meanwhile, it has better segmentation accuracy than traditional butterfly optimization.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"42 27","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138588363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Comparative Analysis of Laplace Residual Power Series and a New Iteration Method for Fitzhugh-Nagumo Equation in the Caputo Operator Framework","authors":"A. Alshehry","doi":"10.3390/fractalfract7120867","DOIUrl":"https://doi.org/10.3390/fractalfract7120867","url":null,"abstract":"In this paper, I utilize the Laplace residual power series method (LRPSM) along with a novel iteration technique to investigate the Fitzhugh-Nagumo equation within the framework of the Caputo operator. The Fitzhugh-Nagumo equation is a fundamental model for describing excitable systems, playing a crucial role in understanding various physiological and biological phenomena. The Caputo operator extends the conventional derivative to handle non-local and non-integer-order differential equations, making it a potent tool for modeling complex processes. Our study involves transforming the Fitzhugh-Nagumo equation into its Laplace domain representation, applying the LRPSM to derive a series solution. We then introduce a novel iteration technique to enhance the solution’s convergence properties, enabling more accurate and efficient computations. This approach offers a systematic methodology for solving the Fitzhugh-Nagumo equation with the Caputo operator, providing deeper insights into excitable system dynamics. Numerical examples and comparisons with existing methods demonstrate the accuracy and efficiency of the LRPSM with the new iteration technique, showcasing its potential for solving diverse differential equations involving the Caputo operator and advancing mathematical modeling in various scientific and engineering domains.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"64 6","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138595490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Study on the Influence of Sample Size on the Mechanical and Integrity Characteristics of Coal Measure Sandstone under High Strain Rate","authors":"Enbing Yi, Zhenhua Li","doi":"10.3390/fractalfract7120869","DOIUrl":"https://doi.org/10.3390/fractalfract7120869","url":null,"abstract":"In order to explore the size effect of the mechanical and damage characteristics of coal measure sand stones under dynamic load, uniaxial impact compression tests were carried out on coal-bearing sand stones with a diameter of 50 mm and a length–diameter ratio of L/R = 0.5, 0.8, 1, 1.2, 1.5, 1.8, and 2 by using the Hopkinson pressure bar test system. The size effect law of the mechanical properties and energy dissipation of coal-bearing sandstone under a high strain rate were investigated. Then, the mercury injection test was carried out on the fragments at different positions, and the electron microscope scanning test was carried out on the fragments near the end of the transmission rod. Based on the area damage definition method and normalization treatment, the integrity model of coal measure sandstone, considering the influence of the length–diameter ratio, was established. The results showed that the peak strength and dynamic elastic modulus of coal measure sandstone increased first and then decreased with the increase in length–diameter ratio under impact compression load, and they reached the maximum when the length–diameter ratio was 1.2. The dynamic peak strain increased gradually with the increase in length–diameter ratio. The energy of coal-bearing sandstone showed strong size effect, that is, the total absorbed energy, elastic energy, and dissipated energy increased with the increase in length–diameter ratio, and the size effect of total absorbed energy was the most obvious. Under the same impact pressure, the porosity of coal-bearing sand stones with seven kinds of length–diameter ratios near the incident end was roughly the same. But when the length–diameter ratio was greater than 0.5, the porosity decreased gradually with the increase in the distance from the incident end. And the larger the length–diameter ratio, the more obvious the decreasing trend. When the length–diameter ratio was smaller, the size of the holes and cracks and the cluster density were larger. The integrity model of coal measure sandstone, considering the influence of the length–diameter ratio, showed that the larger the length–diameter ratio, the better the relative integrity of coal-bearing sandstone.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"15 10","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138594288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Models for Simulation of Fractal-like Particle Clusters with Prescribed Fractal Dimension","authors":"O. Tomchuk","doi":"10.3390/fractalfract7120866","DOIUrl":"https://doi.org/10.3390/fractalfract7120866","url":null,"abstract":"This review article delves into the growing recognition of fractal structures in mesoscale phenomena. The article highlights the significance of realistic fractal-like aggregate models and efficient modeling codes for comparing data from diverse experimental findings and computational techniques. Specifically, the article discusses the current state of fractal aggregate modeling, with a focus on particle clusters that possess adjustable fractal dimensions (Df). The study emphasizes the suitability of different models for various Df–intervals, taking into account factors such as particle size, fractal prefactor, the polydispersity of structural units, and interaction potential. Through an analysis of existing models, this review aims to identify key similarities and differences and offer insights into future developments in colloidal science and related fields.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"3 2","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138601178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fractal Features in Terrain Restoration of Jiuzhai Valley, a World Natural Heritage Site in China","authors":"Zan Zou, Yue Du, Huixing Song","doi":"10.3390/fractalfract7120863","DOIUrl":"https://doi.org/10.3390/fractalfract7120863","url":null,"abstract":"Jiuzhai Valley, a World Natural Heritage Site, was significantly damaged by an earthquake in 2017. However, case studies on the restoration of World Natural Heritage sites are lacking. This study aimed to use the box-counting method to analyze fractal characteristics of the terrain in Shuzheng Valley. Research data were used to conduct artificial intervention restoration of the earthquake-damaged terrain. Our results showed that (i) the travertine terrain shows self-similarity at different scales. The fractal dimension was related to terrain complexity: the more complex the terrain, the higher the fractal-dimension value; (ii) a combined form of fractal generator elements at the same scale was related to terrain complexity—differences in the spatial combination of the fractal generator elements can be compared based on fractal dimension; and (iii) the newly restored dam terrain also showed fractal characteristics whose spatial combination form was similar to that of the surrounding terrain. The complexity of the terrain’s fractal element combination may be related to the influence of surrounding environmental factors and the different ecological functional requirements. This study provides basic data for the near natural restoration of the Sparkling Lake travertine terrain after an earthquake and proposes new concepts and strategies for restoring World Natural Heritage Site terrains.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"119 43","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138599557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimized Multiloop Fractional-Order Controller for Regulating Frequency in Diverse-Sourced Vehicle-to-Grid Power Systems","authors":"Amira Hassan, Mohamed M. Aly, Mohammed A Alharbi, Ali Selim, B. Alamri, M. Aly, A. Elmelegi, Mohamed Khamies, Emad A. Mohamed","doi":"10.3390/fractalfract7120864","DOIUrl":"https://doi.org/10.3390/fractalfract7120864","url":null,"abstract":"A reduced power system’s inertia represents a big issue for high penetration levels of renewable generation sources. Recently, load frequency controllers (LFCs) and their design have become crucial factors for stability and supply reliability. Thence, a new optimized multiloop fractional LFC scheme is provided in this paper. The proposed multiloop LFC scheme presents a two-degree-of-freedom (2DOF) structure using the tilt–integral–derivatives with filter (TIDN) in the first stage and the tilt–derivative with filter (TDN) in the second stage. The employment of two different loops achieves better disturbance rejection capability using the proposed 2DOF TIDN-TDN controller. The proposed 2DOF TIDN-TDN method is optimally designed using the recent powerful marine predator optimizer algorithm (MPA). The proposed design method eliminates the need for precise modeling of power systems, complex control design theories, and complex disturbance observers and filter circuits. A multisourced two-area interlinked power grid is employed as a case study in this paper by incorporating renewable generation with multifunctionality electric vehicle (EV) control and contribution within the vehicle-to-grid (V2G) concept. The proposed 2DOF TIDN-TDN LFC is compared with feature-related LFCs from the literature, such as TID, FOTID, and TID-FOPIDN controllers. Better mitigated frequency and tie-line power fluctuations, faster response, lower overshot/undershot values, and shorter settling time are the proven features of the proposed 2DOF TIDN-TDN LFC method.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"142 17","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138598738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}