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Advanced Vibration-Based Fault Diagnosis and Vibration Control Methods

A special issue of Sensors (ISSN 1424-8220). This special issue belongs to the section "Fault Diagnosis & Sensors".

Deadline for manuscript submissions: closed (30 June 2023) | Viewed by 17529

Special Issue Editors

School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an 710072, China
Interests: gearbox; bearing; dynamics; vibrations; finite element bearing; power loss
Special Issues, Collections and Topics in MDPI journals
Department of Engineering, Lancaster University, Lancaster LA1 4YW, UK
Interests: machine condition monitoring; smartning machine condition monitoring; smart sensing; machine learning
Special Issues, Collections and Topics in MDPI journals
School of Electronic Information Engineering, Xi'an Technological University, Xi'an 710021, China
Interests: vibraitons; control method; signal processing; fault diagnosis
School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China
Interests: advanced design methods

Special Issue Information

Dear Colleagues,

Introduction to the topic: Fault diagnosis and vibration control is the tracking of any aspect of an industrial mechanical component’s performance by reliably measured data and analytical simulations in conjunction with the heuristic experience so that the current and expected future performance of the machine for at least the most critical limit events, can be described in a proactive manner. Advanced intelligent fault diagnosis methods, together with abundant measurement and simulation information, can greatly improve the performances of various industry mechanical systems. Although a lot of works have been introduced to develop the method for determining the working status of different machines by using different detection techniques, it is still challenging and difficult to accurately determine the time-varying working status because the practical working conditions are extremely complex and stochastic. Classical statistical features obtained from the raw vibration signal fail to hold understanding and steadiness for fault detection, exclusively in multifaceted noisy situations. A proper application and integrations of big data analytics, signal processing methods, measurement system, fast computing, and machine learning can lead to the development of accurate diagnosis and prognosis methods of mechanical systems for industrial applications. Moreover, advanced vibration control methods can be helpful for preventing the failures of mechanical components.

Current challenges in the field: Despite the advancement in the field of fault diagnosis and vibration control methods, the advanced industry face following challenges while developments of accurate fault diagnosis solutions. There is limited labeled data about machine health. The developed methods work satisfactorily on the machines from which training is done. However, as soon as the working conditions such as the complex loading and operating speed conditions, the theoretical methods have low accuracy. The most key issue is that the industrial environment does not allow many investigations.

Aims of the Special Issue: In order to rapidly report and spread the latest advancements in the science of fault diagnosis, including new discoveries and valuable applied investigations, from all over the world, this Special Issue's main aim is to gather state-of-the-art research and review articles into all aspects of theoretical and applied investigations about the latest development of new sensor technologies and diagnosis methods for the detection of faults. Original research and review articles are welcome.

Potential topics include but are not limited to the following:

  • Vibration-based fault diagnosis
  • New sensor technologies
  • Vibration control methods
  • Failure mechanism of the faults in various mechanical systems
  • Vibro-acoustic diagnosis of machinery
  • Case studies and industrial applications

Prof. Dr. Jing Liu
Dr. Min Xia
Dr. Xiaohua Song
Dr. Xiaoyan Zhang
Guest Editors

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 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.

Published Papers (13 papers)

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Editorial

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6 pages, 185 KiB  
Editorial
Advanced Vibration-Based Fault Diagnosis and Vibration Control Methods
by Xiaohua Song, Jing Liu and Min Xia
Sensors 2023, 23(18), 7704; https://doi.org/10.3390/s23187704 - 06 Sep 2023
Viewed by 673
Abstract
Fault diagnosis and vibration control are the tracking of any aspect of an industry mechanical components’ performance using reliably measured data and analytical simulations in conjunction with the heuristic experience, so that the current and expected future performance of the machine for at [...] Read more.
Fault diagnosis and vibration control are the tracking of any aspect of an industry mechanical components’ performance using reliably measured data and analytical simulations in conjunction with the heuristic experience, so that the current and expected future performance of the machine for at least the most critical limit events can be described in a proactive manner [...] Full article

Research

Jump to: Editorial

14 pages, 538 KiB  
Article
Optimal Control Algorithm for Stochastic Systems with Parameter Drift
by Xiaoyan Zhang, Song Gao, Chaobo Chen and Jiaoru Huang
Sensors 2023, 23(12), 5743; https://doi.org/10.3390/s23125743 - 20 Jun 2023
Cited by 1 | Viewed by 656
Abstract
A novel optimal control problem is considered for multiple input multiple output (MIMO) stochastic systems with mixed parameter drift, external disturbance and observation noise. The proposed controller can not only track and identify the drift parameters in finite time but, furthermore, drive the [...] Read more.
A novel optimal control problem is considered for multiple input multiple output (MIMO) stochastic systems with mixed parameter drift, external disturbance and observation noise. The proposed controller can not only track and identify the drift parameters in finite time but, furthermore, drive the system to move towards the desired trajectory. However, there is a conflict between control and estimation, which makes the analytic solution unattainable in most situations. A dual control algorithm based on weight factor and innovation is, therefore, proposed. First, the innovation is added to the control goal by the appropriate weight and the Kalman filter is introduced to estimate and track the transformed drift parameters. The weight factor is used to adjust the degree of drift parameter estimation in order to achieve a balance between control and estimation. Then, the optimal control is derived by solving the modified optimization problem. In this strategy, the analytic solution of the control law can be obtained. The control law obtained in this paper is optimal because the estimation of drift parameters is integrated into the objective function rather than the suboptimal control law, which includes two parts of control and estimation in other studies. The proposed algorithm can achieve the best compromise between optimization and estatimation. Finally, the effectiveness of the algorithm is verified by numerical experiments in two different cases. Full article
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14 pages, 3670 KiB  
Article
Verification of Mechanical Properties Identification Based on Impulse Excitation Technique and Mobile Device Measurements
by Lukasz Scislo
Sensors 2023, 23(12), 5639; https://doi.org/10.3390/s23125639 - 16 Jun 2023
Cited by 5 | Viewed by 947
Abstract
The Impulse Excitation Technique (IET) is one of the most useful testing methods for evaluating or calculating some material properties. This can be useful to evaluate and confirm that the material ordered is what was delivered. In the case of unknown materials, where [...] Read more.
The Impulse Excitation Technique (IET) is one of the most useful testing methods for evaluating or calculating some material properties. This can be useful to evaluate and confirm that the material ordered is what was delivered. In the case of unknown materials, where their properties are required by simulation software, this is also a quick way to obtain mechanical properties and thus improve the simulation quality. The main drawback of the method is the requirement for a specialized sensor and acquisition system and a well-trained engineer to prepare the setup and analyze the results. The article evaluates the possibility of using a low-cost solution in the form of a mobile device microphone as a way to obtain data, which after the Fast Fourier Transform (FFT), allows to obtain frequency response graphs and use the IET method procedure to calculate the mechanical properties of the samples. The data obtained by the mobile device are compared with the data obtained by professional sensors and data acquisition systems. The results confirm that for typical homogenous materials, the mobile phone is a cheap and reliable alternative for fast, on-the-go material quality inspections and can be introduced even in small companies and on construction sites. Additionally, this kind of approach does not require specific knowledge of sensing technology, signal treatment, or data analysis and can be performed by any assigned employee, who can receive the quality check information immediately on-site. Additionally, the presented procedure allows data collection and transfer to the cloud for future references and additional information extraction. This element is fundamental for introducing sensing technologies under the Industry 4.0 concept. Full article
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30 pages, 14266 KiB  
Article
A Novel Method for Bearing Fault Diagnosis under Variable Speed Based on Envelope Spectrum Fault Characteristic Frequency Band Identification
by Di Pei, Jianhai Yue and Jing Jiao
Sensors 2023, 23(9), 4338; https://doi.org/10.3390/s23094338 - 27 Apr 2023
Cited by 3 | Viewed by 1161
Abstract
Rolling element bearing (REB) vibration signals under variable speed (VS) have non-stationary characteristics. Order tracking (OT) and time-frequency analysis (TFA) are two widely used methods for REB fault diagnosis under VS. However, the effect of OT methods is affected by resampling errors and [...] Read more.
Rolling element bearing (REB) vibration signals under variable speed (VS) have non-stationary characteristics. Order tracking (OT) and time-frequency analysis (TFA) are two widely used methods for REB fault diagnosis under VS. However, the effect of OT methods is affected by resampling errors and close-order harmonic interference, while the accuracy of TFA methods is mainly limited by time-frequency resolution and ridge extraction algorithms. To address this issue, a novel method based on envelope spectrum fault characteristic frequency band identification (FCFBI) is proposed. Firstly, the characteristics of the bearing fault vibration signal’s envelope spectrum under VS are analyzed in detail and the fault characteristic frequency band (FCFB) is introduced as a new and effective representation of faults. Then, fault templates based on FCFB are constructed as reference for fault identification. Finally, based on the calculation of the correlation coefficients between the envelope spectrum and fault templates in the extended FCFB, the bearing fault can be diagnosed automatically according to the preset correlation coefficient criterion. Two bearing VS experiments indicate that the proposed method can achieve satisfactory diagnostic accuracy. The comparison of OT and TFA methods further demonstrates the comprehensive superiority of the proposed method in the overall consideration of accuracy, diagnostic time, tachometer dependency, and automatic degree. Full article
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13 pages, 960 KiB  
Article
Nonlinear Vibration Control Experimental System Design of a Flexible Arm Using Interactive Actuations from Shape Memory Alloy
by Ximei Li, Guang Jin and Mingcong Deng
Sensors 2023, 23(3), 1133; https://doi.org/10.3390/s23031133 - 18 Jan 2023
Cited by 4 | Viewed by 1475
Abstract
The flexible arm easily vibrates due to its thin structural characteristics, which affect the operation accuracy, so reducing the vibration of the flexible arm is a significant issue. Smart materials are very widely used in the research topic of vibration suppression. Considering the [...] Read more.
The flexible arm easily vibrates due to its thin structural characteristics, which affect the operation accuracy, so reducing the vibration of the flexible arm is a significant issue. Smart materials are very widely used in the research topic of vibration suppression. Considering the hysteresis characteristic of the smart materials, based on previous simulation research, this paper proposes an experimental system design of nonlinear vibration control by using the interactive actuation from shape memory alloy (SMA) for a flexible arm. The experiment system was an interactive actuator–sensor–controller combination. The vibration suppression strategy was integrated with an operator-based vibration controller, a designed integral compensator and the designed n-times feedback loop. In detail, a nonlinear vibration controller based on operator theory was designed to guarantee the robust stability of the flexible arm. An integral compensator based on an estimation mechanism was designed to optimally reduce the displacement of the flexible arm. Obtaining the desired tracking performance of the flexible arm was a further step, by increasing the n-times feedback loop. From the three experimental cases, when the vibration controller was integrated with the designed integral compensator, the vibration displacement of the flexible arm was much reduced compared to that without the integral compensator. Increasing the number of n-times feedback loops improves the tracking performance. The desired vibration control performance can be satisfied when n tends to infinity. The conventional PD controller stabilizes the vibration displacement after the 7th vibration waveform, while the vibration displacement approaches zero after the 4th vibration waveform using the proposed vibration control method, which is proved to be faster and more effective in controlling the flexible arm’s vibration. The experimental cases verify the effectiveness of the proposed interactive actuation vibration control approach. It is observed from the experimental results that the vibration displacement of the flexible arm becomes almost zero within less time and with lower input power, compared with a traditional controller. Full article
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13 pages, 1791 KiB  
Article
Interface Design of Head-Worn Display Application on Condition Monitoring in Aviation
by Xiaoyan Zhang, Jia’ao Cheng, Hongjun Xue and Siyu Chen
Sensors 2023, 23(2), 736; https://doi.org/10.3390/s23020736 - 09 Jan 2023
Cited by 1 | Viewed by 1116
Abstract
Head-worn displays (HWDs) as timely condition monitoring are increasingly used in aviation. However, interface design characteristics that mainly affect HWD use have not been fully investigated. The aim of this study was to examine the effects of several important interface design characteristics (i.e., [...] Read more.
Head-worn displays (HWDs) as timely condition monitoring are increasingly used in aviation. However, interface design characteristics that mainly affect HWD use have not been fully investigated. The aim of this study was to examine the effects of several important interface design characteristics (i.e., the distance between calibration lines and the layouts of vertical and horizontal scale belts) on task performance and user preference between different conditions of display, i.e., HWD or head-up display (HUD). Thirty participants joined an experiment in which they performed flight tasks. In the experiment, the calibration lines’ distance was set to three different levels (7, 9 and 11 mrad), and the scale belt layouts included horizontal and vertical scale belt layouts. The scale belts were set as follows: the original vertical scale belt width was set as L, and the horizontal scale belt height as H. The three layouts of the vertical calibration scale belt used were 3/4H, H and 3H/2. Three layouts of horizontal calibration scale belts were selected as 3L/4, L and 3L/2. The results indicated that participants did better with the HWD compared to the HUD. Both layouts of vertical and horizontal scale belts yielded significant effects on the users’ task performance and preference. Users showed the best task performance while the vertical calibration scale belts were set as H and horizontal calibration scale belts were set as L, and users generally preferred interface design characteristics that could yield an optimal performance. These findings could facilitate the optimal design of usable head-worn-display technology. Full article
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13 pages, 3880 KiB  
Article
Design of Active Vibration Isolation Controller with Disturbance Observer-Based Linear Quadratic Regulator for Optical Reference Cavities
by Yuchen Qian, Yong Xie, Jianjun Jia and Liang Zhang
Sensors 2023, 23(1), 302; https://doi.org/10.3390/s23010302 - 28 Dec 2022
Cited by 4 | Viewed by 2157
Abstract
The optical reference cavity in an ultrastable laser is sensitive to vibrations; the microvibrations in a space platform affect the accuracy and stability of such lasers. In this study, an active vibration isolation controller is proposed to reduce the effect of vibrations on [...] Read more.
The optical reference cavity in an ultrastable laser is sensitive to vibrations; the microvibrations in a space platform affect the accuracy and stability of such lasers. In this study, an active vibration isolation controller is proposed to reduce the effect of vibrations on variations in the cavity length and improve the frequency stability of ultrastable lasers. Based on the decentralized control strategy, we designed a state-differential feedback controller with a linear quadratic regulator (LQR) and added a disturbance observer (DOB) to estimate the source noise. Experiments were conducted using an active vibration isolation system; the results verified the feasibility and performance of the designed controller. The accelerations along the axis (Z-, X-, Y-) directions were suppressed in the low-frequency band within 200 Hz, and the root-cumulative power spectral densities (PSDs) declined to 1.17 × 10−5, 7.16 × 10−6, and 8.76 × 10−6 g. This comprehensive vibration met the requirements of an ultrastable laser. Full article
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18 pages, 8101 KiB  
Article
Color Recurrence Plots for Bearing Fault Diagnosis
by Vilma Petrauskiene, Mayur Pal, Maosen Cao, Jie Wang and Minvydas Ragulskis
Sensors 2022, 22(22), 8870; https://doi.org/10.3390/s22228870 - 16 Nov 2022
Cited by 8 | Viewed by 1500
Abstract
This paper presents bearing fault diagnosis using the image classification of different fault patterns. Feature extraction for image classification is carried out using a novel approach of Color recurrence plots, which is presented for the first time. Color recurrence plots are created using [...] Read more.
This paper presents bearing fault diagnosis using the image classification of different fault patterns. Feature extraction for image classification is carried out using a novel approach of Color recurrence plots, which is presented for the first time. Color recurrence plots are created using non-linear embedding of the vibration signals into delay coordinate space with variable time lags. Deep learning-based image classification is then performed by building the database of the extracted features of the bearing vibration signals in the form of Color recurrence plots. A Series of computational experiments are performed to compare the accuracy of bearing fault classification using Color recurrence plots. The standard bearing vibration dataset of Case Western Reserve University is used for those purposes. The paper demonstrates the efficacy and the accuracy of a new and unique approach of scalar time series extraction into two-dimensional Color recurrence plots for bearing fault diagnosis. Full article
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15 pages, 4252 KiB  
Article
Research on the Applicability of Touchscreens in Manned/Unmanned Aerial Vehicle Cooperative Missions
by Hongjun Xue, Qingpeng Zhang and Xiaoyan Zhang
Sensors 2022, 22(21), 8435; https://doi.org/10.3390/s22218435 - 02 Nov 2022
Cited by 2 | Viewed by 1311
Abstract
The suitability of touchscreens for human–computer interaction in manned/unmanned aerial vehicle cooperative missions remains uncertain, especially in situations that are time-sensitive with variations in difficulty levels. The purpose of this study is to determine the feasibility of touchscreen applications in manned/unmanned aerial vehicle [...] Read more.
The suitability of touchscreens for human–computer interaction in manned/unmanned aerial vehicle cooperative missions remains uncertain, especially in situations that are time-sensitive with variations in difficulty levels. The purpose of this study is to determine the feasibility of touchscreen applications in manned/unmanned aerial vehicle cooperative missions and the magnitude of the effects of time pressure and task difficulty. In contrast to previous studies, a combination of performance and perceptual load measures was used to divide errors into disposition errors, undetected errors, and miscalculation errors to explore specific error mechanisms, set up typical manned/unmanned aerial vehicle cooperative human–computer interaction tasks, and set up antecedent features for potential factors. Thirty subjects participated in an experiment that required the use of touchscreens or keyboards to perform a human–computer interaction task in a simulated manned/unmanned aerial vehicle cooperative mission. Experiments were set at three task difficulties: low, medium, and high, and were matched to a set time pressure or no time pressure for two seconds for low difficulty, three seconds for medium difficulty, and four seconds for high difficulty. The results showed that the touchscreens improved the participants’ response speed at a time pressure of 2 s or less compared with the use of a general input device; however, the task error rate also increased significantly. The higher the task difficulty was, the worse the performance was and the greater the perceived workload of the participants. The application of touchscreens in dynamic environments subjected the participants to greater physical demands. The performance of participants using a keyboard was no better than that when touchscreens were used during the experiment. Moreover, touchscreens did not significantly improve participant performance. The results support the possibility of using touchscreens in manned/unmanned aerial vehicle cooperative missions. Full article
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12 pages, 16826 KiB  
Communication
Strain Response and Buckling Behavior of Composite Cylindrical Shells Subjected to External Pressure with One End Fixed and the Other under Free Boundary Conditions
by Ke-Chun Shen, Xue-Jian Liu, Yi-Hua Huang and Guang Pan
Sensors 2022, 22(18), 6781; https://doi.org/10.3390/s22186781 - 08 Sep 2022
Cited by 2 | Viewed by 1389
Abstract
This study aims to reveal the buckling behavior of filament-wound composite cylindrical shells subjected to external pressure. The boundary conditions of the cylindrical shells were one end fixed and the other free. The carbon fiber stacking sequences were [±90]2/([±20]/[±90]/[±40]/[±90]/[±60]/[±90])2/[±90]. [...] Read more.
This study aims to reveal the buckling behavior of filament-wound composite cylindrical shells subjected to external pressure. The boundary conditions of the cylindrical shells were one end fixed and the other free. The carbon fiber stacking sequences were [±90]2/([±20]/[±90]/[±40]/[±90]/[±60]/[±90])2/[±90]. Finite element software ANSYS 16.2 was used for the numerical simulation to predict the critical buckling pressure and buckling behavior of composite cylindrical shell. External hydrostatic pressure tests were conducted, where the buckling behavior and strain response were observed. Numerical simulation accurately predicted the critical buckling pressure of carbon fiber/epoxy filament composite cylindrical shells under external pressure with 3.5% deviation from the experimental results. The buckling modes simulated by the finite element method agreed well with the deformed shape observed in the experiment, which was characterized by the uniform distribution of the three hoop waves. Comparing the axial compressive strain and hoop compressive strain of the composite shell, it was found that the circumferential stiffness of the shell was weaker than the axial stiffness. In addition, a comparative study of the strains of the fixed-end and free-end metal control sleeves was carried out. The results show that the boundary conditions have a significant influence on the strain response of control sleeves. Full article
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19 pages, 8244 KiB  
Article
Comprehensive Engineering Frequency Domain Analysis and Vibration Suppression of Flexible Aircraft Based on Active Disturbance Rejection Controller
by Litao Liu and Bingwei Tian
Sensors 2022, 22(16), 6207; https://doi.org/10.3390/s22166207 - 18 Aug 2022
Cited by 3 | Viewed by 1619
Abstract
The crash of an aircraft with an almost vertical attitude in Wuzhou, Guangxi, China, on 21 March 2022, has caused a robust discussion in the civil aviation community. We propose an active disturbance rejection controller (ADRC) for suppressing aeroelastic vibrations of a flexible [...] Read more.
The crash of an aircraft with an almost vertical attitude in Wuzhou, Guangxi, China, on 21 March 2022, has caused a robust discussion in the civil aviation community. We propose an active disturbance rejection controller (ADRC) for suppressing aeroelastic vibrations of a flexible aircraft at the simulation level. The ADRC has a relatively simple structure and it has been proved in several fields to provide better control than the classical proportional-integral-derivative (PID) control theory and is easier to translate from theory to practice compared with other modern control theories. In this paper, the vibration model of the flexible aircraft was built, based on the first elastic vibration mode of the aircraft. In addition, the principle of ADRC is explained in detail, a second-order ADRC was designed to control the vibration model, and the system’s closed-loop frequency domain characteristics, tracking effect and sensitivity were comprehensively analyzed. The estimation error of the extended state observer (ESO) and the anti-disturbance effect were analyzed, while the robustness of the closed-loop system was verified using the Monte Carlo method, which was used for the first time in this field. Simulation results showed that the ADRC suppressed aircraft elastic vibration better than PID controllers and that the closed-loop system was robust in the face of dynamic parameters. Full article
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25 pages, 8571 KiB  
Article
Failure Mode Detection and Validation of a Shaft-Bearing System with Common Sensors
by Chung-Hsien Kuo, Yu-Fen Chuang and Shu-Hao Liang
Sensors 2022, 22(16), 6167; https://doi.org/10.3390/s22166167 - 17 Aug 2022
Cited by 4 | Viewed by 1196
Abstract
Failure mode detection is essential for bearing life prediction to protect the shafts on the machinery. This work demonstrates the rolling bearing vibration measurement, signals converting and analysis, feature extraction, and machine learning with neural networks to achieve failure mode detection for a [...] Read more.
Failure mode detection is essential for bearing life prediction to protect the shafts on the machinery. This work demonstrates the rolling bearing vibration measurement, signals converting and analysis, feature extraction, and machine learning with neural networks to achieve failure mode detection for a shaft bearing. Two self-designed bearing test platforms with two types of sensors conduct the bearing vibration collection in normal and abnormal states. The time-domain signals convert to the frequency domain for analysis to observe the dominant frequency between these two types of sensors. In feature extraction, principal components analysis (PCA) combines with wavelet packet decomposition (WPD) to form the two feature extraction methods: PCA-WPD and WPD-PCA for optimization. The features extracted by these two methods serve as input to the long short-term memory (LSTM) networks for classification and training to distinguish bearing states in normal, misaligned, unbalanced, and impact loads. The evaluation arguments include sensor types, vibration directions, failure modes, feature extraction methods, and neural networks. In conclusion, the developed methods with the typical lower-cost sensor can achieve 97% accuracy in bearing failure mode detection. Full article
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19 pages, 4860 KiB  
Article
Smartphone-Based and Data-Driven Superstructure State Prediction Method for Highway Bridges in Service
by Jixin Duan, Weili He, Shizhan Xu, Zhaoyuan Zhong and Liang Huang
Sensors 2022, 22(15), 5620; https://doi.org/10.3390/s22155620 - 27 Jul 2022
Cited by 2 | Viewed by 1154
Abstract
Survival analysis is a data-driven approach that is widely used in various fields of biomedical prognostic research, and it is highly reliable in the processing of time-event data. This study developed a method for evaluating the service performance of bridge superstructures using the [...] Read more.
Survival analysis is a data-driven approach that is widely used in various fields of biomedical prognostic research, and it is highly reliable in the processing of time-event data. This study developed a method for evaluating the service performance of bridge superstructures using the built-in acceleration sensor of smartphones and the prediction of survival analysis theory. It will be used to assist in the daily maintenance and repair of small and medium bridges. The effects of the upper load-bearing structure, upper general structure, bearings, deck paving, expansion joints, and frequency ratio on the deterioration of the bridge superstructure were investigated. The results show that the first-order vibration frequency of the bridge can be effectively detected by the built-in acceleration sensor of the mobile phone, but its low sensitivity and high output noise make it impossible to accurately detect the higher-order frequencies of the bridge. The upper load-bearing members, the upper general structure, the bearing, the deck pavement, and the frequency ratio are all related to the changing trend of the technical condition level of the bridge superstructure. Full article
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