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21 pages, 4725 KiB  
Article
A Novel Open Circuit Fault Diagnosis for a Modular Multilevel Converter with Modal Time-Frequency Diagram and FFT-CNN-BIGRU Attention
by Ziyuan Zhai, Ning Wang, Siran Lu, Bo Zhou and Lei Guo
Machines 2025, 13(6), 533; https://doi.org/10.3390/machines13060533 - 19 Jun 2025
Viewed by 253
Abstract
Fault diagnosis is one of the most important issues for a modular multilevel converter (MMC). However, conventional solutions are deficient in two aspects. Firstly, they lack the necessary feature information. Secondly, they are incapable of performing open-circuit fault diagnosis of the modular multilevel [...] Read more.
Fault diagnosis is one of the most important issues for a modular multilevel converter (MMC). However, conventional solutions are deficient in two aspects. Firstly, they lack the necessary feature information. Secondly, they are incapable of performing open-circuit fault diagnosis of the modular multilevel converter with the requisite degree of accuracy. To solve this problem, an intelligent diagnosis method is proposed to integrate the modal time–frequency diagram and FFT-CNN-BiGRU-Attention. By selecting the phase current and bridge arm voltage as the core fault parameters, the particle swarm algorithm is used to optimize the Variational Modal Decomposition parameters, and the fault signal is decomposed and reconstructed into sensitive feature components. The reconstructed signals are further transformed into modal time–frequency diagrams via continuous wavelet transform to fully retain the time–frequency domain features. In the model construction stage, the frequency–domain features are first extracted using the fast Fourier transform (FFT), and the local patterns are captured through a combination with a convolutional neural network; subsequently, the timing correlations are analyzed using bidirectional gated loop cells, and the Attention Mechanism is introduced to strengthen the key features. Simulations show that the proposed method achieves 98.63% accuracy in locating faulty insulated gate bipolar transistors (IGBTs) in the sub-module, with second-level real-time response capability. Compared with the recently published scheme, it maintains stable performance under complex working conditions such as noise interference and data imbalances, showing stronger robustness and practical value. This study provides a new idea for the intelligent operation and maintenance of power electronic devices, which can be extended to the fault diagnosis of other power equipment in the future. Full article
(This article belongs to the Section Electromechanical Energy Conversion Systems)
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22 pages, 10212 KiB  
Article
Combining Artificial Neural Networks and Mathematical Models for Unbalance Estimation in a Rotating System under the Nonlinear Journal Bearing Approach
by Ioannis Tselios and Pantelis Nikolakopoulos
Lubricants 2024, 12(10), 344; https://doi.org/10.3390/lubricants12100344 - 6 Oct 2024
Viewed by 1237
Abstract
Rotating systems are essential components and play a critical role in many industrial sectors. Unbalance is a very common and serious fault that can cause machine downtime, unplanned maintenance, and potential damage to vital rotating machines. Accurately estimating unbalance in rotor–bearing systems is [...] Read more.
Rotating systems are essential components and play a critical role in many industrial sectors. Unbalance is a very common and serious fault that can cause machine downtime, unplanned maintenance, and potential damage to vital rotating machines. Accurately estimating unbalance in rotor–bearing systems is crucial for ensuring the reliable and efficient operation of machinery. This research paper presents a novel approach utilizing artificial neural networks (ANNs) to estimate the unbalance masses in a multidisk system based on simulation data from a nonlinear rotor–bearing system. Additionally, this study explores the effect of various operating parameters on oil film stability and vibration response through a combination of bifurcation diagrams, spectrum cascades, Poincare maps, and orbit and FFT plots. This study demonstrates the effectiveness of ANNs for unbalance estimation in a fast and accurate way and discusses the potential of ANNs in smart online condition monitoring systems. Full article
(This article belongs to the Special Issue Tribological Characteristics of Bearing System, 2nd Edition)
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18 pages, 6297 KiB  
Article
Multi-Color Phosphor-Converted Wide Spectrum LED Light Source for Simultaneous Illumination and Visible Light Communication
by Aayushi Soni, Linthish Pulikkool, Ravibabu Mulaveesala, Satish Kumar Dubey and Dalip Singh Mehta
Photonics 2024, 11(10), 914; https://doi.org/10.3390/photonics11100914 - 27 Sep 2024
Cited by 3 | Viewed by 2190
Abstract
Simultaneous illumination and communication using solid-state lighting devices like white light-emitting diode (LED) light sources is gaining popularity. The white light LED comprises a single-colored yellow phosphor excited by the blue LED chip. Therefore, color-quality determining parameters like color-rendering index (CRI), correlated color [...] Read more.
Simultaneous illumination and communication using solid-state lighting devices like white light-emitting diode (LED) light sources is gaining popularity. The white light LED comprises a single-colored yellow phosphor excited by the blue LED chip. Therefore, color-quality determining parameters like color-rendering index (CRI), correlated color temperature (CCT), and CIE 1931 chromaticity coordinates of generic white LED sources are poor. This article presents the development of multi-color phosphors excited by a blue LED to improve light quality and bandwidth. A multi-layer stacking of phosphor layers excited by a blue LED led to the quenching of photoluminescence (PL) and showed limited bandwidth. To solve this problem, a lens-free, electrically powered, broadband white light source is designed by mounting multi-color phosphor LEDs in a co-planar ring-topology. The CRI, CCT, and CIE 1931 chromaticity coordinates of the designed lamp (DL) were found to be 90, 5114 K, and (0.33, 0.33), respectively, which is a good quality lamp for indoor lighting. CRI of DL was found to be 16% better than that of white LED (WL). Assessment of visible light communications (VLC) feasibility using the DL includes time interval error (TIE) of data pattern or jitter analysis, eye diagram, signal-to-noise ratio (SNR), fast Fourier transform (FFT), and power spectral density (PSD). DL transmits binary data stream faster than WL due to a reduction in rise time and total jitter by 31% and 39%, respectively. The autocorrelation function displayed a narrow temporal pulse for DL. The DL is beneficial for providing high-quality illumination indoors while minimizing PL quenching. Additionally, it is suitable for indoor VLC applications. Full article
(This article belongs to the Special Issue Recent Advances and Future Perspectives in LED Technology)
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19 pages, 15543 KiB  
Article
Symmetry and Asymmetry of Chaotic Motion in a Crank Arm and Connecting Rod Due to the Movement of the Follower
by Louay S. Yousuf
Symmetry 2023, 15(12), 2148; https://doi.org/10.3390/sym15122148 - 3 Dec 2023
Cited by 1 | Viewed by 1452
Abstract
The symmetry and asymmetry of chaotic motion in the planar mechanism is investigated for a crank arm and connecting rod due to the motion of a flat-faced follower. The level of chaos is investigated using the conception of the Lyapunov exponent parameter and [...] Read more.
The symmetry and asymmetry of chaotic motion in the planar mechanism is investigated for a crank arm and connecting rod due to the motion of a flat-faced follower. The level of chaos is investigated using the conception of the Lyapunov exponent parameter and phase-plane diagram at different cam speeds with and without the use of coefficients of restitution. Moreover, the fast Fourier transform (FFT) of power spectrum analysis technique is used based on SNR factor values at different cam speeds and different coefficients of restitution. The wave forms and histograms of nonlinear responses are analyzed using the SolidWorks program for the crank arm, connecting rod, and flat-faced follower. There is a clearance between the flat-faced follower and its guides while the oscillation motion of the crank arm and connecting rod is described as the motion of a double pendulum. The level of chaos is increased with increases in the cam speeds and coefficients of restitution. Full article
(This article belongs to the Special Issue Nonlinear Vibrations and Chaos: Symmetry and Topics of Symmetry)
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12 pages, 3043 KiB  
Article
Vibration Characteristics and Location of Buried Gas Pipeline under the Action of Pulse Excitation
by Baoyong Yan, Jialin Tian, Xianghui Meng and Zhe Zhang
Processes 2023, 11(10), 2849; https://doi.org/10.3390/pr11102849 - 27 Sep 2023
Cited by 6 | Viewed by 1573
Abstract
In this paper, the attenuation of sound waves in underground gas pipelines and the vibration characteristics of pipelines are studied, and the feasibility and effectiveness of acoustic measurement of PE pipelines are verified. In this paper, the attenuation equation of sound waves in [...] Read more.
In this paper, the attenuation of sound waves in underground gas pipelines and the vibration characteristics of pipelines are studied, and the feasibility and effectiveness of acoustic measurement of PE pipelines are verified. In this paper, the attenuation equation of sound waves in an underground gas transmission pipeline is derived based on the propagation characteristics of gas and the vibration characteristics of the pipeline itself. In order to verify the experimental results, we conducted an experimental test on the air pipeline model and verified the feasibility and effectiveness of the acoustic measurement of the PE pipeline through the test under the action of pulse excitation. Then, we detect the background noise, design the test scheme according to the characteristics of the buried pipeline, and select the test site for field test. In the test process, we collected the test data and obtained the spectrum diagram of the test data by fast Fourier transform (FFT). By analyzing the results of the spectrogram, we find that the pulse signal can penetrate the medium composed of the pipe formation, but the amplitude of the sound will be sharply attenuated. At the same time, according to the size of the peak in the spectrum, we can determine the location of the pipe. In summary, the feasibility and effectiveness of acoustic measurement of the PE pipeline are verified through experimental tests, and attenuation equations based on acoustic wave propagation characteristics and pipeline vibration characteristics are proposed. It has important practical application value for the safety monitoring and positioning of the buried gas pipeline. Full article
(This article belongs to the Special Issue Numerical Simulation of Nonlinear Dynamical Systems)
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21 pages, 5896 KiB  
Article
A Numerical Analysis for Ball End Milling Due to Coupling Effects of a Flexible Rotor-Bearing System Using GPEM
by Chun-Jung Huang, Jer-Rong Chang, Ting-Nung Shiau and Kuan-Hung Chen
Appl. Sci. 2023, 13(12), 7252; https://doi.org/10.3390/app13127252 - 17 Jun 2023
Cited by 1 | Viewed by 1656
Abstract
In this paper, the tool-tip responses for ball end milling, due to the coupling effects of a flexible rotor-bearing system, are investigated numerically. The milling machine tool spindle is modelled as the flexible rotor-bearing system. The critical speeds, natural modes, and unbalance responses [...] Read more.
In this paper, the tool-tip responses for ball end milling, due to the coupling effects of a flexible rotor-bearing system, are investigated numerically. The milling machine tool spindle is modelled as the flexible rotor-bearing system. The critical speeds, natural modes, and unbalance responses of the system are calculated by applying the generalized polynomial expansion method. This generalized polynomial expansion method expresses the displacement as a series formed by the product of generalized coordinates and axial coordinate polynomials. According to the dynamic cutting force obtained by some scholars in the past, combined with the characteristics of the flexible rotor, the dynamic response of the tool-tip for ball end milling is numerically analyzed. The responses, including time histories, orbits, and FFT diagrams, are plotted to analyze the dynamic behaviors of the tool-tip. The coupling effects of the flexible rotor-bearing system on the system for ball end milling are first studied using the generalized polynomial expansion method. Unlike previous studies, the natural frequency varies with spindle speed and which of the different modes are included in the tool-tip response depends mainly on the spindle speed. Thanks to the gyroscopic effect, the critical speeds and responses of tool-tips can be discussed with respect to various spindle speed and tool flutes. The natural modes are accurately determined, and will excite critical speeds for certain modes, including forward and backward modes, thereby significantly affecting tool-tip response. In addition, the cutting force component associated with the tool-tip response affects the rotor-bearing system parameters, complicating the issue. Milling at higher spindle speed (2160–19,950 rpm), an important new result is found that the tool-tip oscillates with the cutting-force frequency, accompanied by a longer period vibration of the first backward mode of the rotor-bearing system. It can also be seen from the frequency spectrum analysis that, as the spindle speed increases, the peak amplitude of the first backward mode becomes larger. Milling at lower spindle speed (960, 1320 rpm), the in-plane vibration trajectory of the tool-tip gradually expands outwards clockwise around the origin until a stable loop is reached. This is because only the first backward mode of the rotor-bearing system is excited. Considering the coupling effect of the rotor-bearing system to perform the vibration analysis of the milling machine system, the parameters of the system can be designed or the spindle speed can be selected to avoid severe vibration during machining. Full article
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21 pages, 8639 KiB  
Article
Solution for Determining Modulus of Elasticity of Natural Materials Using Vibrations of Non-Uniform Circular Cross-Section Cantilevers
by Jerzy Podgórski and Bartosz Kawecki
Materials 2023, 16(10), 3868; https://doi.org/10.3390/ma16103868 - 21 May 2023
Cited by 2 | Viewed by 1612
Abstract
The article presents an original method for determining the modulus of elasticity of natural materials. A studied solution was based on vibrations of non-uniform circular cross-section cantilevers solved using Bessel functions. The derived equations, together with experimental tests, allowed for calculating the material’s [...] Read more.
The article presents an original method for determining the modulus of elasticity of natural materials. A studied solution was based on vibrations of non-uniform circular cross-section cantilevers solved using Bessel functions. The derived equations, together with experimental tests, allowed for calculating the material’s properties. Assessments were based on the measurement of the free-end oscillations in time using the Digital Image Correlation (DIC) method. They were induced manually and positioned at the end of a cantilever and monitored in time using a fast Vision Research Phantom v12.1 Camera with 1000 fps. GOM Correlate software tools were then used to find increments of deflection on a free end in every frame. It provided us with the ability to make diagrams containing a displacement–time relation. To find natural vibration frequencies, fast Fourier transform (FFT) analyses were conducted. The correctness of the proposed method was compared with a three-point bending test performed on a Zwick/Roell Z2.5 testing machine. The presented solution generates trustworthy results and can provide a method to confirm the elastic properties of natural materials obtained in various experimental tests. Full article
(This article belongs to the Special Issue Natural Fibers: Characterization, Properties and Applications)
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24 pages, 9842 KiB  
Article
Fault Diagnosis and Tolerant Control for Three-Level T-Type Inverters
by Kuei-Hsiang Chao, Long-Yi Chang and Chien-Chun Hung
Electronics 2022, 11(16), 2496; https://doi.org/10.3390/electronics11162496 - 10 Aug 2022
Cited by 4 | Viewed by 2310
Abstract
This paper proposes a fault diagnosis system for inverters based on a cerebellar model articulation controller (CMAC). First, a three-level T-type inverter was implemented and used to create a three-level T-type inverter test environment for measuring the output voltage waveforms of faulty power [...] Read more.
This paper proposes a fault diagnosis system for inverters based on a cerebellar model articulation controller (CMAC). First, a three-level T-type inverter was implemented and used to create a three-level T-type inverter test environment for measuring the output voltage waveforms of faulty power transistors on the main inverter circuit under different output frequencies. The measured waveforms were processed using a fast Fourier transform (FFT) algorithm to create frequency spectrum diagrams and extract the characteristic spectra of corresponding faulty switches. Then, the associations of the spectra were determined and applied as training data for the CMAC to detect the positions of the faulty power transistors. The test results demonstrated that the proposed induction motor fault diagnosis system is capable of fast algorithm, requires less data to train, and has excellent accuracy of identification, with an error margin of ±5%. The detection results were then processed using a fault-tolerant controller (FTC) to enhance the reliability of the proposed system. Finally, some simulations and experimental results were conducted and analyzed to validate the feasibility of the proposed FTC system. Full article
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20 pages, 9153 KiB  
Article
Dynamic Modeling and Stability Analysis for a Spur Gear System Considering Gear Backlash and Bearing Clearance
by Gang Tian, Zhihui Gao, Peng Liu and Yushu Bian
Machines 2022, 10(6), 439; https://doi.org/10.3390/machines10060439 - 1 Jun 2022
Cited by 19 | Viewed by 3300
Abstract
In practice, gear backlash and bearing clearance usually exist together in a gear system. They may induce complicated dynamic responses and degrade transmission performance. Up to now, although each of them has been researched, little attention has been paid to the coupling dynamic [...] Read more.
In practice, gear backlash and bearing clearance usually exist together in a gear system. They may induce complicated dynamic responses and degrade transmission performance. Up to now, although each of them has been researched, little attention has been paid to the coupling dynamic characteristics of gear backlash and bearing clearance. In a limited number of relevant studies, since the linear collision models they adopted are difficult to realistically depict actual collision behaviors caused by bearing clearance, these studies cannot accurately reveal the coupling dynamic characteristics of gear backlash and bearing clearance. Furthermore, system stability of the gear system considering gear backlash and bearing clearance has not been thoroughly investigated. In view of this, this paper contributes to the research on dynamic modeling and stability analysis for the spur gear system considering gear backlash and bearing clearance. A nonlinear collision model with time-varying contact stiffness/damping is suggested for describing the bearing collision behaviors. Based on the geometrical relationship of dynamic center distance, dynamic working pressure angle, and dynamic backlash, the coupling motion model of gear backlash and bearing clearance is developed. On this basis, the dynamic model of the spur gear system considering gear backlash and bearing clearance is established and verified by numerical simulations, virtual prototyping simulations and experiments. Afterwards, to thoroughly explore the complicated dynamic characteristics of the gear system considering gear backlash and bearing clearance, several important parameters, i.e., rotational speed, gear backlash and bearing clearance, are chosen as bifurcation parameters to study their influences on system stability via bifurcation diagrams, time-domain waveforms, FFT spectra, Poincaré maps, and phase diagrams. Various complicated nonlinear behaviors, such as hopping, multiple periodic motion, quasi-periodic motion, and chaotic motion, are revealed. This study can provide useful reference for the multi-clearance coupling research of the gear system in complicated working environments. Full article
(This article belongs to the Section Machines Testing and Maintenance)
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21 pages, 671 KiB  
Article
Some FFT Algorithms for Small-Length Real-Valued Sequences
by Dorota Majorkowska-Mech and Aleksandr Cariow
Appl. Sci. 2022, 12(9), 4700; https://doi.org/10.3390/app12094700 - 7 May 2022
Cited by 6 | Viewed by 2817
Abstract
This paper proposes fast algorithms for computing the discrete Fourier transform for real-valued sequences of lengths from 3 to 9. Since calculating the real-valued DFT using the complex-valued FFT is redundant regarding the number of needed operations, the developed algorithms do not operate [...] Read more.
This paper proposes fast algorithms for computing the discrete Fourier transform for real-valued sequences of lengths from 3 to 9. Since calculating the real-valued DFT using the complex-valued FFT is redundant regarding the number of needed operations, the developed algorithms do not operate on complex numbers. The algorithms are described in matrix–vector notation and their data flow diagrams are shown. Full article
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25 pages, 13274 KiB  
Article
9-DOF IMU-Based Attitude and Heading Estimation Using an Extended Kalman Filter with Bias Consideration
by Sajjad Boorghan Farahan, José J. M. Machado, Fernando Gomes de Almeida and João Manuel R. S. Tavares
Sensors 2022, 22(9), 3416; https://doi.org/10.3390/s22093416 - 29 Apr 2022
Cited by 29 | Viewed by 14108
Abstract
The attitude and heading reference system (AHRS) is an important concept in the area of navigation, image stabilization, and object detection and tracking. Many studies and works have been conducted in this regard to estimate the accurate orientation of rigid bodies. In most [...] Read more.
The attitude and heading reference system (AHRS) is an important concept in the area of navigation, image stabilization, and object detection and tracking. Many studies and works have been conducted in this regard to estimate the accurate orientation of rigid bodies. In most research in this area, low-cost MEMS sensors are employed, but since the system’s response will diverge over time due to integration drift, it is necessary to apply proper estimation algorithms. A two-step extended Kalman Filter (EKF) algorithm is used in this study to estimate the orientation of an IMU. A 9-DOF device is used for this purpose, including a 6-DOF IMU with a three-axis gyroscope and a three-axis accelerometer, and a three-axis magnetometer. In addition, to have an accurate algorithm, both IMU and magnetometer biases and disturbances are modeled and considered in the real-time filter. After applying the algorithm to the sensor’s output, an accurate orientation as well as unbiased angular velocity, linear acceleration, and magnetic field were achieved. In order to demonstrate the reduction of noise power, fast Fourier transform (FFT) diagrams are used. The effect of the initial condition on the response of the system is also investigated. Full article
(This article belongs to the Collection Advanced Techniques for Acquisition and Sensing)
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26 pages, 41236 KiB  
Article
Smart Cutting Tools Used in the Processing of Aluminum Alloys
by Dan Dobrotă, Sever-Gabriel Racz, Mihaela Oleksik, Ionela Rotaru, Mădălin Tomescu and Carmen Mihaela Simion
Sensors 2022, 22(1), 28; https://doi.org/10.3390/s22010028 - 22 Dec 2021
Cited by 6 | Viewed by 3180
Abstract
The processing of aluminum alloys in optimal conditions is a problem that has not yet been fully resolved. The research carried out so far has proposed various intelligent tools, but which cannot be used in the presence of cooling-lubricating fluids. The objective of [...] Read more.
The processing of aluminum alloys in optimal conditions is a problem that has not yet been fully resolved. The research carried out so far has proposed various intelligent tools, but which cannot be used in the presence of cooling-lubricating fluids. The objective of the research carried out in the paper was to design intelligent tools that would allow a control of the vibrations of the tool tip and to determine a better roughness of the processed surfaces. The designed intelligent tools can be used successfully in the processing of aluminum alloys, not being sensitive to coolants-lubricants. In the research, the processing by longitudinal turning of a semi-finished product with a diameter Ø = 55 mm of aluminum alloy A2024-T3510 was considered. Two constructive variants of smart tools were designed, realized, and used, and the obtained results were compared with those registered for the tools in the classic constructive variant. The analysis of vibrations that occur during the cutting process was performed using the following methods: Fast Fourier Transform (FFT); Short-Time Fourier-Transformation (STFT); the analysis of signal of vibrations. A vibration analysis was also performed by modeling using the Finite Element Method (FEM). In the last part of the research, an analysis of the roughness of the processed surfaces, was carried out and a series of diagrams were drawn regarding curved profiles; filtered profiles; Abbott–Firestone curve. Research has shown that the use of smart tools in the proposed construction variants is a solution that can be used in very good conditions for processing aluminum alloys, in the presence of cooling-lubrication fluids. Full article
(This article belongs to the Section Intelligent Sensors)
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20 pages, 57160 KiB  
Article
Improving the Performance of Steel Machining Processes through Cutting by Vibration Control
by Mihaela Oleksik, Dan Dobrotă, Mădălin Tomescu and Valentin Petrescu
Materials 2021, 14(19), 5712; https://doi.org/10.3390/ma14195712 - 30 Sep 2021
Cited by 6 | Viewed by 2018
Abstract
Machining processes through cutting are accompanied by dynamic phenomena that influence the quality of the processed surfaces. Thus, this research aimed to design, make, and use a tool with optimal functional geometry, which allowed a reduction of the dynamic phenomena that occur in [...] Read more.
Machining processes through cutting are accompanied by dynamic phenomena that influence the quality of the processed surfaces. Thus, this research aimed to design, make, and use a tool with optimal functional geometry, which allowed a reduction of the dynamic phenomena that occur in the cutting process. In order to carry out the research, the process of cutting by front turning with transversal advance was taken into account. Additionally, semi-finished products with a diameter of Ø = 150 mm made of C45 steel were chosen for processing (1.0503). The manufacturing processes were performed with the help of two tools: a cutting tool, the classic construction version, and another that was the improved construction version. In the first stage of the research, an analysis was made of the vibrations that appear in the cutting process when using the two types of tools. Vibration analysis considered the following: use of the Fast Fourier Transform (FFT) method, application of the Short-Time Fourier-Transformation (STFT) method, and observation of the acceleration of vibrations recorded during processing. After the vibration analysis, the roughness of the surfaces was measured and the parameter Ra was taken into account, but a series of diagrams were also drawn regarding the curved profiles, filtered profiles, and Abbott–Firestone curve. The research showed that use of the tool that is the improved constructive variant allows accentuated reduction of vibrations correlated with an improvement of the quality of the processed surfaces. Full article
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13 pages, 3483 KiB  
Article
Research on the Influence of Backlash on Mesh Stiffness and the Nonlinear Dynamics of Spur Gears
by Yangshou Xiong, Kang Huang, Fengwei Xu, Yong Yi, Meng Sang and Hua Zhai
Appl. Sci. 2019, 9(5), 1029; https://doi.org/10.3390/app9051029 - 12 Mar 2019
Cited by 27 | Viewed by 4716
Abstract
In light of ignoring the effect of backlash on mesh stiffness in existing gear dynamic theory, a precise profile equation was established based on the generating processing principle. An improved potential energy method was proposed to calculate the mesh stiffness. The calculation result [...] Read more.
In light of ignoring the effect of backlash on mesh stiffness in existing gear dynamic theory, a precise profile equation was established based on the generating processing principle. An improved potential energy method was proposed to calculate the mesh stiffness. The calculation result showed that when compared with the case of ignoring backlash, the mesh stiffness with backlash had an obvious decrease in a mesh cycle and the rate of decline had a trend of decreasing first and then increasing, so a stiffness coefficient was introduced to observe the effect of backlash. The Fourier series expansion was employed to fit the mesh stiffness rather than time-varying mesh stiffness, and the stiffness coefficient was fitted with the same method. The time-varying mesh stiffness was presented in terms of the piecewise function. The single degree of freedom model was employed, and the fourth order Runge–Kutta method was utilized to investigate the effect of backlash on the nonlinear dynamic characteristics with reference to the time history chart, phase diagram, Poincare map, and Fast Fourier Transformation (FFT) spectrogram. The numerical results revealed that the gear system primarily performs a non-harmonic-single-periodic motion. The partially enlarged views indicate that the system also exhibits small-amplitude and low-frequency motion. For different cases of backlash, the low-frequency motion sometimes shows excellent periodicity and stability and sometimes shows chaos. It is of practical guiding significance to know the mechanisms of some unusual noises as well as the design and manufacture of gear backlash. Full article
(This article belongs to the Special Issue Optical High-speed Information Technology)
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