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# Fractal Dimension and Fractional Calculus in Mechanical Signal Processing

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (28 February 2024) | Viewed by 3022

## Special Issue Editors

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Guest Editor
School of Automation and Information Engineering, Xi’an University of Technology, Xi’an 710048, China
Interests: fractal dimension; underwater signal processing; sensor signal processing; denoising; feature extraction; fault diagnosis; image processing
Special Issues, Collections and Topics in MDPI journals

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China
Interests: signal detection; intelligent information processing; stochastic resonance; fault diagnosis; image enhancement; defect detection

## Special Issue Information

Dear Colleagues,

Fractal dimension and fractional calculus are commonly used mathematical tools in mechanical signal processing. Specifically, fractal dimension refers to a parameter that is used to describe the complexity of fractal structures. In mechanical fault diagnosis, the severity and type of mechanical failure can be judged by calculating the fractal dimension of the mechanical vibration signal. Simultaneously, fractional calculus is an emerging calculus. Using the theory of fractional calculus, the fractional differential equation model of mechanical failure can be established to realize the diagnosis of mechanical failure. With the continuous development of fractal dimension and fractional calculus, their application in mechanical signal processing will be more advanced and more extensive, and these mathematical tools will help to improve the accuracy and efficiency of mechanical fault diagnosis and equipment health management.

This Special Issue aims to continue to study the theory of fractal dimension and fractional calculus, as well as their application and development in mechanical signal processing. The topics for invitation submission include (but are not limited to) the following:

• Fractal dimension;
• Fractional calculus;
• Multi-scale fractal dimension;
• Fractional reciprocal;
• Fault diagnosis based on fractal theory;
• Feature extraction for nonstationary vibration signals;
• The application of fractional calculus model in mechanical signal processing;
• Ship signal feature extraction based on fractal dimension.

Dr. Yuxing Li
Prof. Dr. Shangbin Jiao
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

## Keywords

• fractal dimension
• fractional calculus
• multi-scale fractal dimension
• fractional reciprocal
• fault diagnosis based on fractal theory
• feature extraction for nonstationary vibration signals
• the application of fractional calculus model in mechanical signal processing
• ship signal feature extraction based on fractal dimension

## Published Papers (3 papers)

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# Research

13 pages, 2901 KiB
Article
Remaining Useful Life Prediction of Roller Bearings Based on Fractional Brownian Motion
by Wanqing Song, Mingdeng Zhong, Minjie Yang, Deyu Qi, Simone Spadini, Piercarlo Cattani and Francesco Villecco
Fractal Fract. 2024, 8(4), 183; https://doi.org/10.3390/fractalfract8040183 - 23 Mar 2024
Viewed by 466
Abstract
Roller bearing degradation features fractal characteristics such as self-similarity and long-range dependence (LRD). However, the existing remaining useful life (RUL) prediction models are memoryless or short-range dependent. To this end, we propose a RUL prediction model based on fractional Brownian motion (FBM). Bearing [...] Read more.
Roller bearing degradation features fractal characteristics such as self-similarity and long-range dependence (LRD). However, the existing remaining useful life (RUL) prediction models are memoryless or short-range dependent. To this end, we propose a RUL prediction model based on fractional Brownian motion (FBM). Bearing faults can happen in different places, and thus their degradation features are difficult to extract accurately. Through variational mode decomposition (VMD), the original degradation feature is decomposed into several components of different frequencies. The monotonicity, robustness and trends of the different components are calculated. The frequency component with the best metric values is selected as the training data. In this way, the performance of the prediction model is hugely improved. The unknown parameters in the degradation model are estimated by the maximum likelihood algorithm. The Monte Carlo method is applied to predict the RUL. A case study of a bearing is presented and the prediction performance is evaluated using multiple indicators. Full article
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16 pages, 10427 KiB
Article
Variable-Step Multiscale Katz Fractal Dimension: A New Nonlinear Dynamic Metric for Ship-Radiated Noise Analysis
by Yuxing Li, Yuhan Zhou and Shangbin Jiao
Fractal Fract. 2024, 8(1), 9; https://doi.org/10.3390/fractalfract8010009 - 21 Dec 2023
Cited by 8 | Viewed by 1055
Abstract
The Katz fractal dimension (KFD) is an effective nonlinear dynamic metric that characterizes the complexity of time series by calculating the distance between two consecutive points and has seen widespread applications across numerous fields. However, KFD is limited to depicting the complexity of [...] Read more.
The Katz fractal dimension (KFD) is an effective nonlinear dynamic metric that characterizes the complexity of time series by calculating the distance between two consecutive points and has seen widespread applications across numerous fields. However, KFD is limited to depicting the complexity of information from a single scale and ignores the information buried under different scales. To tackle this limitation, we proposed the variable-step multiscale KFD (VSMKFD) by introducing a variable-step multiscale process in KFD. The proposed VSMKFD overcomes the disadvantage that the traditional coarse-grained process will shorten the length of the time series by varying the step size to obtain more sub-series, thus fully reflecting the complexity of information. Three simulated experimental results show that the VSMKFD is the most sensitive to the frequency changes of a chirp signal and has the best classification effect on noise signals and chaotic signals. Moreover, the VSMKFD outperforms five other commonly used nonlinear dynamic metrics for ship-radiated noise classification from two different databases: the National Park Service and DeepShip. Full article
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19 pages, 4278 KiB
Article
Remaining Useful Life Prediction of Lithium-Ion Battery Based on Adaptive Fractional Lévy Stable Motion with Capacity Regeneration and Random Fluctuation Phenomenon
by Wanqing Song, Jianxue Chen, Zhen Wang, Aleksey Kudreyko, Deyu Qi and Enrico Zio
Fractal Fract. 2023, 7(11), 827; https://doi.org/10.3390/fractalfract7110827 - 17 Nov 2023
Viewed by 1035
Abstract
The capacity regeneration phenomenon is often overlooked in terms of prediction of the remaining useful life (RUL) of LIBs for acceptable fitting between real and predicted results. In this study, we suggest a novel method for quantitative estimation of the associated uncertainty with [...] Read more.
The capacity regeneration phenomenon is often overlooked in terms of prediction of the remaining useful life (RUL) of LIBs for acceptable fitting between real and predicted results. In this study, we suggest a novel method for quantitative estimation of the associated uncertainty with the RUL, which is based on adaptive fractional Lévy stable motion (AfLSM) and integrated with the Mellin–Stieltjes transform and Monte Carlo simulation. The proposed degradation model exhibits flexibility for capturing long-range dependence, has a non-Gaussian distribution, and accurately describes heavy-tailed properties. Additionally, the nonlinear drift coefficients of the model can be adaptively updated on the basis of the degradation trajectory. The performance of the proposed RUL prediction model was verified by using the University of Maryland CALEC dataset. Our forecasting results demonstrate the high accuracy of the method and its superiority over other state-of-the-art methods. Full article
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