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(1)AGH – UNIVERSITY OF SCIENCE AND TECHNOLOGY KRAKOW, POLAND FACULTY OF MECHANICAL ENGINEERING AND ROBOTICS DEPARTMENT OF MECHANICS AND VIBROACOUSTICS. Ph.D. Thesis A hardware implementation of neural classifiers for condition monitoring of planetary gearboxes by Dariusz Dąbrowski. Supervisor: Prof. zw. dr hab. inż. Jan Adamczyk. Kraków, 2013.

(2) AGH AKADEMIA GÓRNICZO-HUTNICZA im. Stanisława Staszica w Krakowie WYDZIAŁ INŻYNIERII MECHANICZNEJ I ROBOTYKI KATEDRA MECHANIKI I WIBROAKUSTYKI. Dariusz Dąbrowski Praca Doktorska Sprzętowa realizacja klasyfikatorów neuronowych do rozpoznawania stanu technicznego przekładni planetarnych. Promotor: Prof. zw. dr hab. inż. Jan Adamczyk. Kraków, 2013.

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(4) I would like to dedicate this work to my family, for supporting me during my PhD studies; and to Zahra, for her support and for encouraging me, despite the challenges of life..

(5) Acknowledgments I would like to express my gratitude to my supervisor, Professor Jan Adamczyk, for supervising me during my doctoral studies and for his assistance with the research presented in this thesis, and to Doctor Hactor Plasencia Mora for his valuable comments and invaluable help during my internship at the University of Guanajuato. Also, to Professor Piotr Rusek without whom the cooperation with Mexican University would not be possible. The research was partially carried out at the University of Guanajuato in Mexico, where the author was working on a study which is partially presented in Chapter 4 of this thesis. The study was partially supported by the Dean’s Scholarship for Young Scientists (grant no. 15.11.130.146), funded by the Faculty of Mechanical Engineering and Robotics AGH..

(6) Table of Contents Abstract ........................................................................................................................................i Polish summary.......................................................................................................................... ii Frequently used Symbols .......................................................................................................... iii 1. 2. 3. 4. 5. 6. Introduction ......................................................................................................................... 1 1.1. Background ..................................................................................................................1. 1.2. Theses and Objective ...................................................................................................2. 1.3. Content of the Dissertation...........................................................................................4. Literature Review ................................................................................................................ 6 2.1. Artificial Neural Networks ...........................................................................................6. 2.2. Application of Neural Classifiers in CM ...................................................................15. 2.3. Review of Gear Modeling Methods ...........................................................................17. The Object of Study........................................................................................................... 24 3.1. The Test Rig ...............................................................................................................24. 3.2. Characteristic Frequencies of a Planetary Gearbox ...................................................25. 3.3. Identification of the Characteristic Frequencies ........................................................30. 3.4. Conclusions ................................................................................................................32. Rigid-elastic Model of the Planetary Gear ........................................................................ 33 4.1. Multi-body Systems ...................................................................................................33. 4.2. Description of the Gear Multi-body Model ...............................................................37. 4.3. Tests for the Spur Gear Model ...................................................................................40. 4.4. Tests for the Planetary Gear Model ...........................................................................44. 4.5. Comparison of the Tests Results with the Experiment ..............................................48. 4.6. Conclusions ................................................................................................................50. Description of the Experiment........................................................................................... 51 5.1. The Experiment ..........................................................................................................51. 5.2. Analysis of the Experiment Results ...........................................................................53. 5.3. Conclusions ................................................................................................................56. Analysis of Diagnostic Features ........................................................................................ 57 6.1. Vibration Signal Estimates Used in CM ....................................................................57. 6.2. Diagnostic Features for Gearboxes ............................................................................59. 6.3. Analysis of the Signal Pre-processing Algorithm ......................................................60. 6.4. Presentation of Selected Diagnostic Features ............................................................62.

(7) 6.5 7. 8. 9. Conclusions ................................................................................................................67. Description of the Hardware Platform .............................................................................. 68 7.1. Programmable Logic Devices ....................................................................................68. 7.2. Architecture of FPGAs ...............................................................................................69. 7.3. Embedded Device NI Single Board RIO ...................................................................72. 7.4. Conclusions ................................................................................................................74. Implementation of the Signal Pre-processing Algorithm .................................................. 75 8.1. Digital Filters .............................................................................................................75. 8.2. Selection of Filter Parameters for Hardware Implementation ...................................79. 8.3. Implantation of the Signal Pre-processing Algorithm ...............................................85. 8.4. Conclusions ................................................................................................................87. Implementation of the Artificial Neural Network ............................................................. 88 9.1. Learning Vector Quantization Algorithm ..................................................................88. 9.2. Model of the Neural Classifier ...................................................................................90. 9.3. Implementation of the Neural Network .....................................................................93. 9.4. Conclusions ................................................................................................................99. 10 Verification of the Neural Classifier ............................................................................... 100 10.1. Training Process of the ANN ...................................................................................100. 10.2. Verification of the classifier .....................................................................................104. 10.3. Comparison Between Other Platforms.....................................................................106. 10.4. Conclusion................................................................................................................109. 11 Conclusions ..................................................................................................................... 110 11.1. Conclusions ..............................................................................................................110. 11.2. Future Developments ...............................................................................................112. Appendix A Characteristics of the electromagnetic particle break EMA-ELFA ................... 114 Appendix B Structure of the Learning Algorithm for the LVQ Neural Network .................. 115 Appendix C Implementation of the learning algorithm in MATLAB ................................... 117 Appendix D Technical drawings of the planetary gear Mercury 1-A .................................... 119 List of Figures ......................................................................................................................... 123 List of Tables .......................................................................................................................... 127 Bibliography ........................................................................................................................... 128.

(8) Abstract Gearboxes have a significant influence on the durability and reliability of a power transmission system. Continuous monitoring of their condition is crucial in view of early prediction of serious damages, which ensures safety of exploitation and economic benefits. Artificial neural networks can be used to recognize the technical state of machines. Artificial neural networks allow for a quick and effective association of the symptoms with the technical state of the machine. Extensive research shows that neural networks can be successfully used to recognize the condition of the gearboxes; they allow for detection of new failures which were not known at the time of training and can be applied for identification of a machine’s technical state in variable-speed applications. In a majority of the studies conducted so far, neural networks were implemented in the software. For dedicated engineering applications the hardware implementation of neural networks is being used increasingly more often due to their high efficiency, flexibility and ability to adapt to the working conditions. In the study a hardware implementation of an artificial neural network for condition monitoring of the planetary gearbox is presented. Firstly, the signal pre-processing algorithm was developed. Such an algorithm allows to determine the technical state vector, which consists of estimates obtained from the vibration signal indicating the condition of the object. In order to implement the algorithm, a wide analysis of the digital filter parameters was conducted. The main advantages of the presented algorithm is resistance to speed fluctuations, low consumption of hardware resources and high speed of implementation. The neural classifier was developed on the basis of the learning vector quantization algorithm. In the study some modifications of the neural network were proposed in order to improve the learning process and to reduce the hardware resources of implementation. The signal preprocessing algorithm and the neural network were implanted on the field programmable gate array and verified in the experiment. A comparison between hardware and software implementations of the developed algorithms was also presented. In the thesis a dynamic model of planetary gear allowing for simulation of the meshing forces and transmission error was presented. The model study is an important source of information about the object; it can be applied to develop and verify diagnostic methods as well as to extend the knowledge base used for the training of neural networks. To conclude, the presented system based on hardware implementation of a neural network is characterized by high performance, high reliability and low power consumption. The developed classifier can be used as an independent monitoring system, or it can be combined with data acquisition systems. As a result of the study, a library was built which allows for easy and flexible implementation of neural networks on FPGAs.. i.

(9) Polish summary Przekładnie zębate mają znaczący wpływ na trwałość i niezawodność systemów przekazu mocy. Ciągły monitoring ich stanu technicznego jest istotny ze względu na wczesne wykrywanie poważnych uszkodzeń, co zapewnia bezpieczeństwo eksploatacji oraz korzyści ekonomiczne. W celu rozpoznania stanu technicznego maszyn i urządzeń mogą być zastosowane sztuczne sieci neuronowe. Pozwalają one na szybkie i efektywne powiązanie symptomów z określonym stanem technicznym badanej maszyny. Prowadzone dotychczas szerokie badania wskazują że sieci neuronowe mogą być wykorzystane do rozpoznawania stanu technicznego przekładni zębatych pracujących w stacjonarnych jak i w zmiennych warunkach pracy, pozwalają one również na detekcję nowych uszkodzeń które nie były uwzględniane na etapie uczenia. W większości badań prowadzonych do tej pory sieci neuronowe były realizowane programowo na procesorach numerycznych. Dla dedykowanych zastosowań inżynierskich, coraz częściej stosuje się sprzętową implementacja sieci neuronowych, ze względu na dużą wydajność, elastyczność oraz możliwość dostosowania się do warunków pracy. W rozprawie doktorskiej przedstawiono sprzętową implementację sieci neuronowej zaprojektowanej do rozpoznawania stanu technicznego przekładni planetarnej. W pierwszej kolejności zbudowano algorytm wstępnego przetwarzania sygnałów diagnostycznych. Taki algorytm pozwala na wyznaczenie wektora stanu technicznego, który składa się z estymat uzyskanych z sygnału wibraokustycznego, określającego stan techniczny badanego obiektu. W celu implementacji algorytmu została przeprowadzona szeroka analiza parametrów filtrów cyfrowych. Głównymi zaletami zaprojektowanego algorytmu jest jego odporność na niewielkie zmiany prędkości obrotowej, niskie zużycie zasobów sprzętowych oraz szybkość realizacji. Klasyfikator neuronowy został oparty o sprzętową realizację sztucznej sieci neuronowej zbudowanej w oparciu o algorytm LVQ. W badaniach zaproponowano kilka modyfikacji tego algorytmu, tak aby usprawnić proces uczenia oraz zredukować zasoby sprzętowe, wymagane do implementacji. Algorytm wstępnego przetwarzania sygnałów oraz sieć neuronowa zostały zaimplementowane na układzie programowalnym FPGA i zweryfikowane podczas eksperymentu, jak również przeprowadzono porównanie realizacji algorytmów na różnych platformach sprzętowych. W pracy przedstawiono model dynamiczny przekładni planetarnej, przedstawiony model umożliwia symulację sił działających w zazębieniu oraz badanie błędu transmisji przekładni. Badania modelowe są cennym źródłem informacji na temat analizowanego obiektu, mogą być zastosowane do rozbudowy i weryfikacji metod diagnostycznych, jak również do poszerzenia bazy wiedzy stosowanej do nauki sieci neuronowych. Podsumowując, przedstawiony system bazujący na sprzętowej implementacji sieci neuronowej, charakteryzuje się dużą wydajnością, niezawodnością oraz niskim poborem mocy. Zbudowany klasyfikator może być wykorzystany jako niezależny system monitorujący stan techniczny maszyny również może współpracować z systemami pomiarowymi. W wyniku badań zbudowano bibliotekę która umożliwia łatwą i elastyczną implementację sieci neuronowych na układach programowalnych FPGA.. ii.

(10) Frequently used Symbols C – damping coefficient d – parameter representing intra-class dissipation D – parameter representing inter-class dissipation e – contact force exponent E[∙] – expectation operator fa – rotational frequency of carrier in planetary gear fm – modulation frequency in planetary gear f1 – rotational frequency for the input shaft f12 – meshing frequency for the sun and planet gear f23 - meshing frequency for the planet and ring gear F – meshing force vector F() – activation function of neuron g(x) – discrimination function G – parameter representing quantitative evaluation of data set J – inertia of gear K – contact stiffness mi – codebook vector q – vector describing location and orientation of n bodies in multi-body system w – weight vector x – input vector to neural network y – output signal from neural network  – membrane potential of neural network.. iii.

(11) Introduction. 1 1.1. Introduction Background. Gearboxes are used in a majority of mechanical power transmission systems, thus continuous monitoring of their condition is very important since serious damage that might occur during operation can be predicted. Continuous monitoring and assessment of the gearboxes’ technical state is crucial, especially in an industry where the costs of unplanned downtimes outweigh the costs of repairs; also, early prediction of failures ensures safety of exploitation. Properly registered and processed vibration signals can serve as a source of diagnostic information (Cempel, 1989). Many types of defects in mechanical objects are characterized by a specific signature in the residual signal which is generated by the object. Assessment of the technical state can be carried out on the basis of vibroacoustic signals generated by the object. In practice, usually the vibration signals and temperature of the lubricants are analyzed to assess the condition of the gearboxes. Also, it is necessary to monitor the operating parameters, such as rotational speed and loading. The main source of diagnostic information is contained in the vibration signals, and different signal processing methods are used to select information related to the condition of other elements (Jablonski & Barszcz, 2013; Raad, et al., 2008). It is necessary to build a diagnostic system in order to carry out the monitoring and diagnostic process on an object working in the industry. Such a system performs the tasks of data acquisition, signal processing, data logging and alarms in case of failures (Batko, et al., 2013). Condition-based maintenance is the most effective method of maintenance in the industry. It assumes that the potential breakdown of a machine is predicted through regular Condition Monitoring (CM). Such an approach allows to obtain large economic benefits by eliminating unplanned downtimes and repairs as well as by early planning of any component replacements. It is especially important for industries which apply machinery that must run continuously, e.g. in an opencast mine even the smallest unplanned downtime of bucket wheel excavators causes large economic losses. Methods based on maintenance at regular intervals of time or exploitation of the machines until they broke down were commonly used before the development of condition-based maintenance. Run-to-break maintenance is characterized by the longest time of exploitation between shutdowns, but failures could be catastrophic, and could even result in safety hazards. Preventive maintenance is carried out at regular time intervals which are shorter than the expected time between failures, but this method is characterized by big consumption of replacement components (Randall, 2011). Gearboxes are complex objects, and their dynamic interaction has a significant influence on the durability and reliability of the whole power transmission system. Vibrations generated by gearboxes have an influence on the environment and the functioning of the working machine, but they also contain information about the gearboxes’ condition. Applying dynamic models at the design and manufacture stage allows to find a solution that will minimize vibrations and will create the possibility of developing effective diagnostic methods. The models of the gearboxes are valuable tools for studying the influence of design, manufacturing, and exploitation features on the vibrations, which carry the most significant information about the gearboxes’ condition (Łazarz & Peruń, 2012; Dąbrowski, et al., 2000). The discussion presented in (Dąbrowski, 2008) emphasizes the need for the application of 1.

(12) Introduction nonlinear models in technical diagnosis. Modeling of dynamic phenomena occurring in gearboxes allows for the detection of faults in their early stage of degradation. The study presented in the thesis focuses on condition monitoring of Planetary Gearboxes (PG). Planetary gearboxes are distinguished by compactness of construction, high ratio, and the ability for high power transmission. PGs are used widely in the industry, especially in such sectors as automation of manufacturing, in mining, in wind turbines as well as in the power transmission systems of helicopters. 1.2. Theses and Objective. Neural networks are being used more commonly in engineering applications due to their flexibility and ability to work in real-time conditions. Recently, wide research is being conducted on the application of neural networks in condition monitoring (Czech & Lazarz, 2007; Barszcz, 2006; Bartelmus, et al., 2003; Adamczyk, et al., 1999). Neural classifiers allow for a quick and effective association of symptoms and enable to identify the technical state of the machine, where categorization of symptoms by mathematical formulas is ambiguous. It has been shown that neural networks can be successfully used for recognition of the condition of gearboxes (Czech & Lazarz, 2007; Cioch, 2005), as they allow for the detection of new failures which were not known at the time of training (Barszcz, et al., 2011), and they can be applied for identification of the technical state of a machine in variable-speed applications (Cocconcelli, et al., 20-22.06.2011; Krzyworzeka, et al., 2006). Hardware implementation of neural networks is becoming more important due to very high efficiency and adaptability to working conditions. In engineering applications, neural networks differ in the scale, topology, transfer functions and learning algorithms, that is why they require significant computational power and flexibility of implementation. Field Programmable Gate Arrays (FPGAs) have created a new possibility for carrying out Artificial Neural Networks (ANNs), and they provide robust flexibility, reprogrammable properties and parallelism of computing. Implementation of ANNs using programmable devices have been presented in many papers (Misra & Saha, 2010; Li, et al., 2006; Jamro & Wiatr, 13– 16.04.2005; Liu & Liang, 2005; Gao & Hammersfrom, 2003), the authors point out that there is still a need for new tools allowing for the implementation of neural networks in dedicated hardware architectures. The literature review, as presented in Chapter 3, shows the possibilities of applying neural networks in engineering areas, particularly in condition monitoring of rotating machinery. Neural classifiers are mostly implemented in the software, whereas the hardware implementation of neural networks, in most cases, is dedicated to specialized solutions – and the main objective of this work has been formulated in view of that fact. Objective A hardware implementation of a neural classifier for condition monitoring of a planetary gearbox. The presented study is based on the following assumptions: a neural classifier which consists of a signal pre-processing algorithm and a neural network is implemented on a 2.

(13) Introduction dedicated hardware platform. The signal pre-processing algorithm allows for determination of the input vector for the neural network and is executed point-by-point. The implemented neural network allows for assessment of the condition of the planetary gearboxes; neurons in one layer of the network are executed simultaneously. Intermediate objectives were created in order to fulfill the presented assumptions:  development and implementation of a signal pre-processing algorithm  selection of digital filter parameters for hardware implementation  selection of an architecture for the neural classifier and its implementation on field programmable gate arrays  experimental verification of the neural classifier. Crucial for the successful application of neural networks in condition monitoring is defining the technical state vector, as such a vector contains estimates indicating the condition of the object; another important aspect is to properly determine the training data set which contains the technical state vectors for the other conditions of the investigated object. The model tests can be applied to improve the training data set, especially for big and complex objects working in the industry. In a study (Czech & Lazarz, 2007) it was shown that artificial neuron networks, by using data obtained from a model and a real gearbox, offer the highest accuracy of classification. Model tests can be used as a useful source of information about gearbox dynamics. Simulation of the phenomena occurring in gearboxes allow to identify a vibration signature related to other failures of the gears, bearings and shafts. Models of gearboxes are a valuable tool to study the influence of other features on gearbox dynamics (Dąbrowski & Pakowski, 2005; Dąbrowski, et al., 2000). The models can be applied to extend the training set for artificial neural networks, especially for big objects, where investigation of other technical states is difficult and expensive. For gearbox dynamics other models can be distinguished, such as lumped parameter models, models based on mathematical descriptions of the phenomena occurring in gearboxes, and multi-body models which merge the advantages of CAD modeling and efficient numerical simulations. An additional objective was formulated based on the premises from the foregoing considerations:  development and verification of a dynamic model of a planetary gear based on the multibody dynamics method. The following assumptions for the model study were formulated: the model allows for simulation of meshing forces and transmission error for both assumed operating parameters and the manufacturing errors of gears. In the doctoral dissertation the following theses were formulated. Thesis I The signal pre-processing algorithm based on filtration and estimation of statistics from the vibration signal and artificial neural network dedicated for condition monitoring can be implemented in hardware on a single field programmable gate array chip.. 3.

(14) Introduction Thesis II Hardware implementation of artificial neural networks on field programmable gate arrays, designed for assessment of the condition of a planetary gearbox, is characterized by much higher efficiency and stability than software implementation. The formulated theses of the doctoral dissertation were proved on the basis of the study results, and supported by computer simulations and experimental investigations. 1.3. Content of the Dissertation. The presented thesis consists of 11 chapters containing documentation of the study as carried out to fulfill the objectives and to prove the thesis. In the next lines a brief description of the chapters contained in the dissertation is presented. Chapter 1 In this chapter the background, objectives and theses of the dissertation are formulated. Chapter 2 In this chapter the author presented the theory related to artificial neural networks and to modeling methods of gear dynamics. A review of the application of neural classifiers in condition monitoring, especially in the monitoring of gearboxes, is presented. Also, papers related to the hardware implementation of ANNs are reviewed. The chapter ends with a vast description of gear modeling methods, and a chronological presentation of gears models. Chapter 3 In the first part of this chapter the test rig that was used in the experimentation is presented. The second part includes an identification of the characteristic frequencies of the planetary gearbox. Chapter 4 This part of the thesis is devoted to the multi-body model of planetary gear and its verification. Firstly, the theory of the multi-body dynamics method is discussed, and afterwards the model is presented. The presented model allows for simulations of the meshing force as well as of the transmission error generated by the system. The chapter ends with conclusions which present the pros and cons of the proposed modeling method as well as suggestions for future study. Chapter 5 In this chapter the measuring system and equipment used during the experiment are presented as well as example vibration signals measured for other technical states of a gearbox. Chapter 6 In order to build a technical state vector it is necessary to define the features of the vibration signal indicating the condition of the object, so in this chapter an analysis of the diagnostic features used in the condition monitoring of gearboxes is presented. The signal pre4.

(15) Introduction processing algorithm and technical state vector for the planetary gearbox are discussed. The presented algorithm does not require significant resources of an FPGA, and it can be performed in real-time conditions. A presentation of the technical state vectors in a threedimensional space does not allow for a clear identification of the classes, that is why they were presented in a two-dimensional space by applying the Sammon mapping algorithm. Chapter 7 The developed neural classifier and signal pre-processing algorithm were implemented on the dedicated hardware platform. In this chapter the architecture and features of the field programmable gate arrays are discussed. The main advantages of FPGAs are performance, cost, time-to-market, long-term maintenance, reliability and resistance to harsh environment conditions. In the study the sbRIO-9602 device provided by National Instruments was used. Chapter 8 The proposed signal pre-processing algorithm for estimation of the input data to the neural network is based on filtration. For implementation of the digital filters on the hardware platform, a fixed-point notation of their coefficients is required. In the thesis it was proven that modification of integer word length significantly influenced the filters’ magnitude response. The filter characteristics strongly depend on the digits of precision; also, there is a certain correlation between coefficient accuracy and cut-off frequency of the filters. In the last part of this chapter the implementation of a selected filter is presented. Chapter 9 In this chapter the architecture of a neural classifier is presented. It was based on the Learning Vector Quantization (LVQ) algorithm. In the thesis some modifications of the chosen algorithm were proposed to improve the training process and hardware implementation. The neural network was tested on a PC computer. Chapter 10 In this chapter the verification of a neural classifier and a comparison of performance between other platforms is presented. The training process and its verification were carried out after implementation of the neural network. The training was based on data obtained from the experiment. The classifier was verified in two tests – a test where the technical states of the planetary gearbox considered only in the learning process were considered, and a test conducted for other conditions of the PG which were not considered in the training. The most efficient and stable implementation of the classifier was obtained on the FPGA, but for more simple algorithms, such as single neuron, software implementation on the PC was comparable. Chapter 11 To review the results presented in the thesis, the main conclusions and a discussion on the application possibilities are presented. In this chapter the possibilities of a future study are outlined.. 5.

(16) Literature Review. 2. Literature Review. 2.1. Artificial Neural Networks. To present a model of artificial neural networks it is necessary to discuss the construction and functions of a nervous system. A nervous system is built up of basic cells which are known as neurons. Neurons have the ability to transfer and change electrical signals, which can be transferred to other neurons, muscle cells or glands. A neuron can be defined as a nerve cell, and it is the basic unit of the nervous system (http://encyklopedia.pwn.pl/, 15.04.2013). A neural cell is built of a cell membrane surrounded by cytoplasm and a nucleus. The branched projections of a neuron that conduct the electrochemical stimulation received from other neural cells to the cell body are called dendrites. One dendrite is thicker and longer than the others and is covered by a myelin sheath – this is the axon. The axon allows for connection of the neurons by synaptic terminals, its length can vary from 0.1 mm to 1 m. The synaptic terminal allows for transmission of the electrochemical potential between the neurons by mediators (neurotransmitters). Figure 2.1 presents a structure built of two neurons connected by an axon.. Figure 2.1 Two neurons connected by an axon (http://stemcells.nih.gov, 15.04.2013) Neurotransmitters are a kind of chemical molecules that transfer signals between neurons. The most significant neurotransmitters are listed below (http://en.wikipedia.org/, 15.04.2013).  Dopamine – is a monoamine neurotransmitter and hormone. Dopamine plays a major role in the brain system and is responsible for reward-driven learning. 6.

(17) Literature Review  Serotonin – is a monoamine neurotransmitter. It is responsible for the regulation of mood, appetite, and sleep.  Acetylcholine – is an organic, polyatomic cation that acts as a neurotransmitter in both the peripheral nervous system and the central nervous system.  Noradrenaline – is a catecholamine with multiple roles, including that of a hormone and a neurotransmitter. One of the most important functions of norepinephrine is its role as a neurotransmitter released from the sympathetic neurons affecting the heart.  Enkephalin - is a pentapeptide involved in regulating nociception in the body. The electrochemical potential is transferred to a neuron via synaptic connections, and it depends on the mediators in a synapse. The action of a neuron is based on cumulation of the electrochemical potential from the other neurons. The incoming signals to a cell’s body change regarding the mediators of synaptic connections. If the nucleus of a neuron reaches a sufficiently large potential in a sufficiently short period of time, then excitation of the neuron occurs and transition of the electrochemical signal to other neurons via the axon connection takes place (Masters, 1996; Tadeusiewicz, 1993). Neurons exhibit other functionalities in the nervous system and have other structures; for instance, there are as many as 10000 specific types of neurons in the human brain. Three kinds of neurons can be distinguished: motor neurons, sensory neurons, and intern neurons (http://www.mind.ilstu.edu, 15.04.2013). The most extended are the Purkinje cells, as they have a volume that is a thousand times larger than the smallest neurons. Specialized neuron cells create the senses, e.g. photosensitive cells, sensorial cells, auditory cells, smell cells and taste cells. In Figure 2.2 the structures of other neurons are presented.. Figure 2.2 Different structures of neurons (www.mind.ilstu.edu, 15.04.2013) A neural network can be defined as a set of neurons with a system of synaptic connections. The connection system is extremely complex, as certain areas of the brain are connected 7.

(18) Literature Review densely while some are less, such as the brain hemispheres. In the area of the cerebral cortex the pyramidal cells are firmly connected; here also higher nervous functions take place, such as perception, memorization and awareness. The topology of the human neural network changes over time due to learning processes and aging. Numerous and effective connections appear between neurons which communicate frequently, while between neurons that communicate rarely the connections disappear (Masters, 1996; Żurada, et al., 1996). The brain is the most important organ of the nervous system of developed organisms; it has a corrugated surface, two hemispheres, and a mass of about 1.1 kg–2 kg. It is surrounded by the meninges. In the brain area the brainstem and pallium can be distinguished. The memory of biological organisms is related to neural networks, and even the most primitive organisms exhibit this characteristic. The memory of humans is the most developed and has a very high potential, but so far its nature is not fully understood. According to research (Hebb, 1949), memory in the initial phase is formed on the basis of phenomena enhancing synaptic connections between neurons which are simultaneously agitated, which is short-term memory. The second phase of memorizing involves the formation of memory based on consolidation, which consists in changes of synaptic conductance and changes in the nucleus of a neuronal cell at the genetic level (Kosiński, 2004; Tadeusiewicz, 1993). The description of the human nervous system as presented above is simplified, as from the point of view of neuroscience the nervous system is governed by complex and so far still unknown phenomena. One of the most intriguing effects of neural networks is awareness, which is the highest level of regulation of human behavior, and specifically the inner ability to directly explore the environment, acting on three levels: perceptual, conceptual-verbal and self-awareness (http://encyklopedia.pwn.pl/, 15.04.2013). The first model of a neuron was presented by McCulloch and Pitts in 1943. In this model the input signal takes a value of 0 or 1, and the output signal depends on the membrane potential and the threshold, thus it can be expressed in the form ∑ (2.1) {. ∑. where xi={0,1} is the input signal, y is the output signal, T is the threshold, wi is the weight, n is the number of neurons, and k is the time interval. The presented model is simple but has high potential. With a proper selection of weights and threshold it can be used to build logic functions, such as NOT, OR, AND, NOR or NAND. With these functions it is possible to build an arbitrarily complex logic. By introducing feedback to the presented model one can create a sequential circuit (Żurada, et al., 1996). In this model the binary states of neurons, the discreteness of time and the synchronism of neurons in a neural network were assumed. These assumptions make the presented model of a neuron an oversimplification of real biological neurons. In reality, different artificial neural systems use different models of neurons and neural networks. The most general model of a neuron is shown in Figure 2.3.. 8.

(19) Literature Review. x1 x2 xn. w1 w2. . wn. y. F. Figure 2.3 General model of a neuron This model can be expressed by the equation (2.2). (2.2) where. ∑. (2.3). In the equation above y is the output signal, F is the activation function,  represents the membrane potential of a neuron, w is the weight vector, x is the input vector, and n is the dimension of the input vector. The general model of a neuron consists of the processing element, connected with inputs by the weights, and one output element. In the model as described above, input vector x is multiplied by weight vector w, the products are summed, and the result is called the membrane potential . If the membrane potential exceeds the value of the threshold, the neuron is activated. The output signal is a function of the membrane potential, and the activation of a neuron is based on the activation function F. In Table 2.1 the other activation functions are presented. Table 2.1 Activation functions of neurons Function. Equation. Plot. y Linear. . 9.

(20) Literature Review. y {. Threshold.  y Sigma. low . high .  y (. ). Tangent. . The features of biological neurons, such as delayed reaction, period of resistance or action in a continuous time, were not considered in the presented models. The response of the model depends only on the value of current excitation. More complex models can be used to model more precisely the processes that take place in biological neural networks. These were presented in (Kosiński, 2004; Masters, 1996; Żurada, et al., 1996; Tadeusiewicz, 1993). An Artificial Neural Network (ANN) can be defined as a set of artificial neurons in which the output of each neuron is connected to the inputs of all neurons by weights as well as with neuron’s own input (Żurada, et al., 1996). In reality, neural networks are composed of neurons grouped in layers. Each output of neurons in one layer is connected to all the inputs of the neurons in the next layer. The scheme of a neural network built with three layers is presented in Figure 2.4.. Figure 2.4 Structure of an artificial neural network 10.

(21) Literature Review In order for an ANN to work properly it is necessary to select the values of the weights for each neuron. The aim of the learning process is to force a specific response of a network to the given input. Weights of neurons before the learning process are mostly chosen randomly over a specified range. During processing of the training, the weights are modified according to specific procedures. There are two basic learning methods of ANNs – supervised and unsupervised. Learning with a teacher or a supervised learning method involves comparing the network output, or known input, to the expected response of a network; then the error is calculated and the weights are modified (to minimize the error). Unsupervised learning is based on self-detection by a network of the regularity or classes on the basis of the selforganization process. In fact, in unsupervised learning it is necessary to define the purpose of learning by the teacher. The basic learning rules for artificial neural networks are listed below.  Hebbian Learning Rule – is an unsupervised learning process, the output also represents the training signal. It can be presented by the following formula (2.4) (2.5) where r is the training signal, x is the input vector, yi is the output of an ANN, wi is the weight vector and =[1-0] is the learning rate.  Perceptron Learning Rule – is a supervised learning process, the training signal is the difference between the desired and the actual response, it can be expressed by (2.6) [. ]. (2.7). where di is the desired response.  Delta Learning Rule – this rule can be applied to neurons with the continuous activation function; for supervised learning it is defined as follows [. ]. (2.8) (2.9).  Windrow-Hoff Learning Rule – this refers to supervised learning and can be used for neurons with any activation function, it is based on minimizing the mean square error between the desired response and stimulation. The Windrow-Hoff Learning Rule is defined according to the following formula. 11.

(22) Literature Review (2.10) [. ]. (2.11).  Correlation Learning Rule – for this learning rule the correction of a weight vector is proportional to the product of the input signal and desired output, given by (2.12) (2.13)  Winner-take-all Learning Rule – is used to extract statistical properties of the input signals, the weights are modified only for one neuron that has the greatest excitation. It is an unsupervised learning process and can be defined as (2.14) The most popular learning rules are presented in the thesis. Supervised learning assumes full knowledge of the response to the training set. The situation is different for unsupervised learning, e.g. in a rivalry learning process only the general purpose of the training is determined, as a neural network has to find the statistical properties of the signals by itself (Tadeusiewicz, 1993). Artificial neural networks have better performance than traditional algorithms for fuzzy data, i.e. when patterns are deeply hidden or the data exhibit an unexpected non-linearity. To train the ANNs it is necessary to divide the data set into two sets, i.e. into the learning set and the verification set. The verification set cannot be included in the training set. The basic feature of neural networks is approximation so that they can fulfill three basic tasks – classification, auto-association and prediction. Classification is the most important field of applying artificial neural networks. The classification process is based on the association of a pattern with a specific class. A pattern can be understood as a quantitative description of phenomena (Figure 2.5 presents a classification system). A pattern is given on the input of the system, after classification in an n-dimensional space, and information about the class is obtained on the output.. Figure 2.5 Classification system At the beginning stage the measured value of the phenomena has to be pre-processed by a converter to classify different types of signals. A features extractor allows for estimation of 12.

(23) Literature Review the pattern, which can be classified by a classification system (Żurada, et al., 1996). The scheme of the recognition system is presented in Figure 2.6.. Figure 2.6 Recognition system The ANNs have an auto-association feature, thus a neural network, after storing a specific pattern, is able to associate an equivalent distorted pattern with the corresponding appropriate pattern. In Figure 2.7 an example of the association is presented.. Figure 2.7 Auto-association feature of neural networks A convenient approach to the classification is its consideration in the geometric sense. The classified pattern (input vector) can be presented as a point in the space. The purpose of a classifier is to associate a pattern with the proper class (output vector) on the basis of the decision functions (Żurada, et al., 1996). Figure 2.8 presents a nonlinear decision surface for two classes.. Figure 2.8 Nonlinear decision surface for two classes 13.

(24) Literature Review The decision surfaces can be determined on the basis of discrimination functions g. Generally, these functions can be written in the form (2.15) where R is the number of classes. The equation defining the decision surface between Ri and Rj has the form .. (2.16). For a particular case for two classes (R=2) (2.17) and condition (2.15) takes the form. {. .. (2.18). Generally, classifiers use non-linear discriminant functions. There are two main approaches to training neural classifiers: parametric and nonparametric methods. The parametric methods are based on an analytical selection of weights for neurons, depending on the input signals, to create discriminant functions which separate the considered class. The non-parametric methods rely on weights, which change directly in the learning process (Osowski, 2000; Żurada, et al., 1996; Masters, 1996; Tadeusiewicz, 1993). A neural network composed of one layer of neurons with the linear activation function can be used for classification of any image that is linearly separated. To characterize the learning model of such a network it is convenient to use the space created by weights. The coordinates of this space are created by weights wi of the neurons, a hyperplane corresponds for each image x (Żurada, et al., 1996). .. (2.19). The hyperplane divides the area of weights into two parts and passes through the center of the system. It is assumed that the positive part of the hyperplane belongs to the class and the negative part does not. In the training process, if a weight belongs to the negative part of the hyperplane (2.20) then its value changes according to the following equation (2.21) 14.

(25) Literature Review One of the basic types of classifiers is a classifier based on self-organizing neural networks. Self-organizing networks were introduced by T. Kohonen (Kohonen, 2001). Self-organization is based on rivalry between the neurons of one layer of an ANN. In the process of rivalry there emerges one winner neuron whose excitation is the biggest for the presented input, next the weights of the winner neuron are modified so that its excitation is even greater, and the weights of other neurons remain unchanged. Self-organization is a natural way of learning in the brains of living organisms, as it allows for adaptation to previously unknown patterns which are grouped in the training process. Depending on the learning algorithms, several subtypes of self-organization neural networks can be distinguished, such as the Selforganizing Map (SOM) or Learning Vector Quantization (LVQ). Artificial neural networks are a model of neural networks of living organisms and they are an important tool for classification. Classifiers based on neural networks allow for association of phenomena that cannot be clearly described by mathematical formulas. 2.2. Application of Neural Classifiers in CM. Toothed gears are commonly used in various power transmission systems. Collecting information about their degradation processes, early enough, is crucial during operation. Properly registered and processed vibration signals can serve as a source of information about the condition of the machines (Randall, 2011). Faults that are not detected early enough are the reason for failures, which are dangerous to the whole power transmission system (Barszcz & Randall, 2009). Neural classifiers have wide applications in technical diagnostics, e.g. they allow for efficient and effective association of symptoms with the technical state of the object. Applying classifiers based on neural networks enables identification of the machine’s condition, where a categorization of symptoms using mathematical formulas is ambiguous. Recently, extensive research on the application of neural classifiers in condition monitoring was carried out (Barszcz, et al., 2011; Rafieea, et al., 2007; Cioch, 2005; Zhang, 2000; Adamczyk, et al., 1999). The authors indicate that ANNs are a promising alternative to conventional methods of classification. The neural networks enable transformation of a multidimensional space of symptoms into classes of the technical state of an examined object (Osowski, 2000; Cempel, 1989). A local or global transformation of the multi-dimensional space can be extremely difficult by means of mathematical formulas. The ANNs also have an auto-association feature, i.e. they allow to associate an image with a pre-remembered pattern (Tadeusiewicz, 1993). The possibility of applying neural networks in technical diagnostics was presented by J. Adamczyk et al. (Adamczyk, et al., 1999). In the study the learning vector quantization algorithm for identifying the condition of rotating machinery in the case of variable operating conditions was proposed. In this study the training set was based on selected components of the synchronous amplitude spectrum. The selection of symptoms of the technical state for rotating machinery, based on spectral analysis of vibration signals, was presented in a doctoral dissertation (Cioch, 2005). In the same thesis the application of the Sammon mapping algorithm for a presentation and selection of technical state vectors was presented. A wide study conducted by Czech at al. indicated that neural networks can be successfully used for recognition of the condition of gearboxes. In a study (Czech & Lazarz, 15.

(26) Literature Review 2007) it was shown that artificial neuron networks, by using data obtained from a model and a real gearbox, offer the highest accuracy of classification. The research was focused on the identification of tooth failure and its level of progress. The neural classifiers can be used in gearbox condition monitoring, especially in detection of the early stages of gear’s failures (Łazarz, et al., 2011; Łazarz & Czech, 2004; Czech, et al., 2000). Application of the probabilistic neural network for detecting the degree of the crack in the tooth root was presented in (Czech, 2007). A comparison of radial basis function neural networks for classification purposes in technical diagnostics was presented in (Łazarz, et al., 2–7.03.2009). Dybała et al. presented a way of applying the Counter Propagation (CP) neural network in technical diagnostics. In the paper (Dybała, 2005; Dybała & Radkowski, 2004) the feature selection method used for object condition monitoring is presented. It is based on the observation space. The authors state that “the attractiveness of using artificial neural networks in technical diagnosis consists in the possibility of application of these networks without actually having any knowledge about the mathematical model of the diagnosed object” (Dybała & Radkowski, 1998). In the study (Barszcz, 2006) the nonlinearities were taken into account for detecting defects of rotating machines. The paper presents an approach and method of classification based on NARX networks. The advantage of this method is the possibility of detecting new failures which were not known at the time of training. In another article (Barszcz, et al., 2011), the fuzzy-ART neural network is proposed as a classification system. The introduced normalization procedure improves the classification process, which was confirmed in the experiment. In the research (Bartelmus, et al., 2003) the problem of selecting input values for neural network training in machine condition monitoring was raised. In this study the factors which have an influence on the vibration generated by machines, such as design, technological, operational and change of condition during operation, were raised. In another paper (Cocconcelli, et al., 20-22.06.2011) the diagnostics of ball bearings by means of ANNs was presented. Changes of rotational speed in direct-drive motors make traditional algorithms for the diagnostics of bearings, which assume a constant rotational speed, not applicable. In the study it was proven that artificial neural networks can be applied for identifying the technical state of the ball bearing in variable-speed applications. W. Wang et al. (Wang & Kanneg, 2009) showed the possibility of applying a new integrated classifier which was tested for different gearbox conditions. The classifier was implemented for real-time conditions. In the paper (Lei, et al., 2010) a new multidimensional hybrid intelligent diagnosis method was proposed. The discussed classifiers were based on a multi-layer perceptron, radial basis function neural network, and K-nearest neighbor classification algorithm. J. Rafiee et al. (Rafieea, et al., 2007) confirmed the right of using a neural classifier for gearbox monitoring. In the study the presented multi-layer perceptron was used to identify gears and bearings faults. In this research the feature vector was improved by standard deviation of the wavelet packet coefficients. Artificial neural networks require significant computational power, therefore, they are often implemented on dedicated hardware architectures. Field programmable gate arrays provide a new approach to the implementation of ANNs. They allow for implementation of any digital function through its proper hardware configuration (Majewski, 2007). All computations on FPGAs are carried out simultaneously, therefore, they allow for better 16.

(27) Literature Review modeling of biological neural networks. The structure of FPGAs can be described as many blocks connected together via programmable interconnections. The main advantage of FPGAs is the flexibility and parallelism of computing and reconfiguration (Majewski & Zbysiński, 2007; Skahill, 2004). In the survey (Kuon, et al., 2007) the historical development of programmable logic devices and fundamental programming techniques is reviewed. In view of the features of biological neural networks, the hardware implementation of artificial neural networks is better than the software (Gao & Hammersfrom, 2003; Masters, 1996). The FPGAs provide robust flexibility and reprogrammable properties. Many scientists have made great efforts to make artificial neural networks using programmable devices (Liu & Liang, 2005). For engineering applications, neural networks change in scale, topology, transfer functions and learning algorithms. A reconfigurable approach for hardware implementation of ANNs which can fully meet real-time requirements was presented in (Li, et al., 2006). In contrast to the dominant trend of software or analog implementation (Gao & Hammersfrom, 2003), FPGAs allow for digital implementation of ANNs (Misra & Saha, 2010; Liu & Liang, 2005). E. Jamro at al. (Jamro & Wiatr, 13–16.04.2005) presented a novel Parallel-Serial Architecture for Neural Networks (PSAN) which was optimized for digital hardware implementation. The proposed architecture is especially efficient for multi-layer neural networks, and it was developed and tested for an already trained feed-forward neural network. A reconfigurable approach for hardware implementation of ANNs was proposed in (Li, et al., 2006), by comparison with other implementations it was shown that this approach displays the highest performance. 2.3. Review of Gear Modeling Methods. A gearbox has a crucial influence on the reliability of a power transmission system. A gearbox is a complex object which consists of shafts, gears, bearings and housing. The dynamic interactions that take place in a gearbox have a major effect on the vibrations and noise generated by the system. Identifying the dynamic phenomena that take place in gearboxes allows for a proper selection of the design, technological and operational features at the initial design stage. Model tests of gearboxes can be used as a source of information about gearbox dynamics, thus they allow to optimize the gearbox design due to a reduction of vibrations as well as to investigate gear failures (Łazarz & Peruń, 2012; Łazarz & Peruń, 2009). The vibrations generated by gearboxes are the main source of information about their condition. Every failure of the shafts, gears or bearings has a characteristic signature in the vibration signal. Dynamic models are especially useful for investigating the influence of other failures that occur in gearboxes. Two main variants of gear dynamic models can be distinguished: models where all phenomena occurring in a power transmission system are included, and models which take into account only phenomena inside the gearbox. In the first group, the dynamics of a motor, couples, gears and working machine is included. The second group considers physical phenomena occurring only inside the gearbox, and in these models the meshing stiffness and manufacturing of gears mostly affects the dynamics. The meshing stiffness varies in time, it depends on the number of intermeshing teeth and the deflection of a tooth by the action of normal force during meshing (across a path of action the stiffness changes as a parabola 17.

(28) Literature Review function). Depending on the backlash, the teeth may lose contact or work on opposite sides, this may induce large impact forces associated with consecutive single-sided or double-sided impacts. Wrongly designed backlash may cause teeth interferences and undercuts (Bartelmus, 1998; Müller, 1986). The model tests allow to identify dynamic phenomena occurring in a gearbox as well as to simulate vibration signals generated by gears. To model a gearbox dynamics it is possible to use lumped parameter models which are used for relatively simple gear systems (Åkerblom, 2001). In the lumped parameter models the laws governing the system are described by differential equations. A vast review of mathematical models used in gear dynamics was presented by Özguven and Houser (Nevazt Ozguvent & Houser, 1988). In the paper the authors proposed a classification of models into five groups: simple dynamic factor models, models with tooth compliance, models for gear dynamics, models for geared rotor dynamics, and models for torsional vibrations. For more complex systems, methods based on mathematical descriptions of the phenomena taking place in gearboxes or a multi-body dynamics approach are used increasingly more often. It can be stated that the first dynamic model of gears was proposed in 1868 by Walker (Fisher, 1961). It was based on an empirical dynamic factor (DF). The DF was defined as a static load divided by a dynamic load. The tooth load of gears in mesh consists of two components: a static component corresponding to the transmitted power and a dynamic component which depends on load fluctuations due to dynamic interactions. It was shown in the report (Bucktingham, 1931) that the tooth force depends mostly on the effective masses, effective errors and speed of the gears (Nevazt Ozguvent & Houser, 1988). The first springmass model was introduced by Tuplin (Tuplin, 1953): the meshing of two gears was modeled by a system with one-degree freedom, built with a movable wedge and mass. The mass models inertia of the gears, while gear errors were modeled by various shapes of a wedge. Constant meshing stiffness and gear errors were considered in the model. The presented model can be used to estimate dynamic factors at conditions below resonance. Figure 2.9 presents the gears dynamic model by Tuplin. x. ke. me. u. Figure 2.9 Gears dynamic model as presented by Tuplin In this model ke is the equivalent constant tooth meshing stiffness, me is the equivalent mass and u is the transmitted load (Nevazt Ozguvent & Houser, 1988). A model in which the meshing stiffness varied in time was presented by Strauch (Strauch, 1953). In this model the changes in meshing stiffness were due to changing from a single pair to a double pair of teeth in the engage. 18.

(29) Literature Review In models examined later the tooth stiffness was the main potential energy-storing element, as the other elements were rigid or neglected. Two examples of torsional models with tooth compliance are presented in Figure 2.10. a). b). 1 1. 1. 1 km. 2. 1. 1 e(t) km. 2. cm Figure 2.10 Model of gears with tooth compliance Where I1, I2 are the mass moment of inertia of gears, km is the tooth meshing stiffness, cm is the tooth meshing damping, and e(t) is the displacement input representing gear errors. For torsional models the system is idealized by a pair of inertias coupled by a spring, which permits relative motion. By applying torsional models it is possible to study the torsional vibrations of gears in mesh (Nevazt Ozguvent & Houser, 1988). The model developed by Bollinger and Bosh and discussed in (Dąbrowski, et al., 2000; Wilk, et al., 1999) is constructed of two masses connected by a spring-damper element. In the model the changes of meshing stiffness according to teeth number in engage and kinematic deviations were considered. Another model introduced by Kovalev allows to determine the vibrations instability area without considering dumping. A more complex model was presented by Rettig (Dąbrowski, et al., 2000). This model consists of six bodies. The gears with shafts and bearings are treated as a system with six degrees of freedom. The model assumes inertia of pinion and gear, gear and shaft masses, meshing stiffness, torsional and bending shafts stiffness and bearings stiffness. The model allows to determine the gears rotation, linear displacement of the shafts and deformation of teeth. The gears dynamic model presented by Müller (Müller, 1986) was one of the most advanced, i.e. compared to models that had been presented before. The model was based on the following assumptions: only the torsional vibrations of gears are considered, stiffness of one pair of teeth in engage can by constant or varying on the line of action; shafts, gears and gearbox housing are rigid, gearbox loading is constant; viscous damping of vibrations and elastic teeth collision, backlash and other kinds of geometrical deviations in the model were also considered. Figure 2.11 presents the ‘palisade’ model.. 19.

(30) Literature Review. u m. k x. y. v Figure 2.11 Model of gears presented by Müller In the Figure above k is the equivalent tooth stiffness, m is the equivalent mass and u is the transmitted load. Müller also introduced the dynamic model of planetary gear (Müller, 1986). In this model the meshing stiffness, stiffness and dumping of bearings, meshing phase relations and non-linearity of phenomena occurring in kinematic pairs are considered. The Lagrange method was applied to build a mathematical model. Another modeling method was presented by Radkowski at al. (Filonik, et al., 1998; Radkowski, 1996), in which the authors presented a model of gears based on the apparent interface method. This method is based on additional angular displacement of gears in relation to the actual displacement, which causes apparent overlap of the pinion and gear. The interference is compensated by elastic deformation of the teeth, which allows to calculate the meshing forces. In a study presented by Łazarz at al. (Peruń, 2010; Łazarz & Peruń, 2009; Łazarz, 2001), the gears, bearings, shafts and couplings were modeled by a spring-damper system. The developed models assumed flexibility of gear teeth, friction and damping in mesh, geometry deviations, and clearance between the teeth. The authors presented a model of the system consisting of a motor, shafts, bearings, gears and bearings. In the papers (Łazarz & Peruń, 2009; Łazarz & Peruń, 2006; Łazarz, 2002) a study on modeling methods of local and global gear faults was presented. The dynamic models allow to investigate the influence of other factors on the dynamics and vibroactivity of the gears. The presented models were verified on a test rig operating in the circulating power system (Łazarz & Peruń, 2012; Wilk, et al., 2011). The presented results can be used for analysis of the methods, which can be used to reduce the vibroactivity of the gearboxes in an early design stage. In the study it was shown that it is important to analyze the design and the technological factors in gear modeling. In research presented by Bartelmus at al. (Bartelmus & Zimroz, 2009; Bartelmus, 1998), modeling methods of high power gearboxes used in the mining industry were presented. In the study it was found that a planetary gearbox in bad condition is more susceptible to load than a gearbox in good condition. The operation parameters, such as time-varying loading and speed, are crucial for a proper modeling of the phenomena in gearboxes. In the paper (Bartelmus, et al., 2010), models of fixed axis and planetary gearboxes operating under varying load conditions were presented. In another study (Fakher, et al., 2012), a lumped parameter model was applied to investigate the influence of meshing forces, variable loads 20.

(31) Literature Review and errors on the dynamics of gearboxes. The time varying operations and teeth faults were modeled by a proper selection of the meshing stiffness. Another approach to analyzing phenomena in gearboxes is based on modeling processes in gearboxes on the basis of empirical observations. A group of models where a mathematical description of the observations is used instead of differential equations depicting physical processes are phenomenological models. A mathematical model describing the modulation mechanisms in a planetary gear was presented by Inalpolat at al. (Inalpolat & Kahraman, 2009). A vibration signal generated by the planetary gear is modeled on the basis of system parameters: the number of planets, planet position angles, and planet phasing relationships. In other research (Vicuña, 2012), the phenomenological model of vibrations generated by a planetary gear and its analysis based on the Fourier transform was presented. In this study, particularly the meshing process in a planetary gear was discussed. The vibration signals in a planetary gear system are transmitted from their source (meshing) to a sensor, which in most cases is located on a gearbox housing. In a single-stage planetary gear the signals are generated in the planet-sun (PS) and planet-ring (PR) meshing process. The signals are transmitted through a ring gear, carrier and sun gear to a sensor. In Figure 2.12 the vibration signals’ transmission paths in a planetary gear and the transmission paths proposed in the cited study are presented. a). b). Figure 2.12 Transmission paths of vibration signals in a planetary gearbox, a) meshing points, b) transmission paths (PR - planet-ring meshing, PS - planet-sun meshing, RPRi - transmission path of the vibrations generated in the planet-ring meshing process through the ring gear, RPSi - transmission path of the vibrations generated in the planet-sun meshing process through the ring gear, SCPRi - transmission path of the vibrations generated in the planet-ring meshing process through the sun gear and carrier, SCPSi - transmission path of the vibrations generated in the planet-sun meshing process through the sun gear and carrier) A planetary gearbox is a complex object – the vibration signals are generated by other elements interacting with one another and are transmitted by other paths to the signal receiving point. In the papers (Vicuña, 2012; Inalpolat & Kahraman, 2009), the classifications of planetary gears in the case of geometrical parameters and the specific structure of the vibration spectrum, depending on the meshing process, were proposed.. 21.

(32) Literature Review A relatively new approach for modeling the dynamics of gearboxes is based on the multibody dynamics method. A multi-body system is a model of a real system built with the assumption that bodies in a real system are rigid or flexible and connected by joints. The multi-body system allows for time domain integration of the solution, which captures the nonlinear effects of bearing stiffness and clearances, gear backlash, large rotations and other nonlinear phenomena (Palermo, et al., 2010). Specialized multi-body dynamics software, such as ITI-SIM, SIMPACK, LMS Virtual.Lab Motion and MSC ADAMS, allow to model threedimensional gear bodies, tooth micro-geometry, global and local tooth stiffness. In Figure 2.13 an example of the multi-body gear model is presented. a). b) Jpinion, pinion Jgear, gear. F(t). Figure 2.13 Multi-body model of gears, a) geometry, b) dynamic scheme In the Figure above, F(t) is the meshing force vector generated by a contact algorithm, J is the inertia of gear and  is the angular velocity. In the multi-body method firstly the geometry is modeled, mostly by specialized CAD software, then in multi-body software the forces, constraints and contacts are assumed. In the simulation, the integration of equations of motion, with proper parameters, is conducted. The study describing the multi-body approach for modeling the vibration of gears was presented by Dresig at al. (Dresig & Schreiber, 26– 28.10.2005). The model allows for dynamic simulations of planetary gearboxes, considering the stiffness characteristics. A new approach to modeling gear systems was presented by Ebrahimi et al. (Ebrahimi & Eberhard, 2006). It assumes that the teeth and the body of the gear wheels are rigid but are connected by elastic elements. In another paper (Palermo, et al., 2010), the methodology for calculating the bearing forces and gear noise was presented. The study, related to a multi-body dynamics model developed in MSC ADAMS software, was presented in a few publications (Sommer, et al., 2011; Han, et al., 2009; Kong, et al., 4.02.2008). Kong et al. (Kong, et al., 4.02.2008) presented dynamic simulations of gears for other technical states. Han at al. (Han, et al., 2009) presented three models: an equivalent model, a rigid-body model, and a frequency-based model, which were compared in case of simulation of the meshing forces. A model which assumes a non-linear contact algorithm between the teeth and geometric defects of the gears, such as chipped tooth and eccentric tooth, was presented by Sommer et al. (Sommer, et al., 2011). The multi-body method is 22.

(33) Literature Review useful in modeling complex systems, such as multi-stage gearboxes or power trains. A dynamic model of an off-shore wind turbine was presented by Viadero et al. (Viadero, et al., 2012). The authors presented simulation results of the non-stationary dynamic behavior of a wind turbine power train. The model was developed in ADAMS multi-body software, the tests were conducted for start-up and emergency stop. In the presented model the shafts and gears are lumped and rigid bodies. Gear meshing forces were modeled by variable stiffness spring, and tooth deflection by relative displacement of each gear center. To sum up, the first gear models were based on the empirical dynamic factor, then the lumped parameter models were developed – these can be used for gear dynamics modeling or modeling of all gearboxes working in power transmission systems. The first spring-mass models assume constant meshing stiffness and simple gear errors. Next, the time-varying meshing stiffness, depending on the number of teeth in contact and teeth deflection, was considered. The gear bodies connected by the spring-damper element allowed to investigate torsional vibrations. One of the most advanced models was introduced by Müller; it assumes, among others, the meshing stiffness, backlash, geometrical deviations, viscous damping of the vibrations, and elastic teeth collision. In recent years, lumped parameter models were developed to model power transmission systems with a gearbox and high power gearboxes working under non-stationary operations. A description of phenomena occurring in gearboxes by differential equations is especially difficult for complex systems, as sometimes it is not possible to build differential equations due to other parameters, such as friction or lubrication. The phenomenological models allow for modeling vibrations on the basis of empirical observations; they are relatively simple but require detailed knowledge about all phenomena occurring in a gearbox. Another method that is useful in modeling complex systems is based on multi-body systems. It allows to model a dynamic system on the basis of the geometry of the elements and interacting forces. The gears meshing is modeled on the basis of contacts acting between the gear bodies, and the contact force is a function of the penetration depth, thus it depends on the contact stiffness and damping parameter.. 23.

(34) The Object of Study. 3 3.1. The Object of Study The Test Rig. The experiments presented in this thesis were conducted in the Laboratory of Mechanical Diagnostics at AGH University of Science and Technology in Krakow. They were conducted on a test rig with planetary gearbox Mercury 1-A, which consists of an asynchronous motor, one-stage planetary gear and an electromagnetic particle break. The construction of the test rig is presented in Figure 3.1.. Figure 3.1 Test rig with planetary gearbox Mercury 1-A The planetary gearbox Rexnord Mercury 1-A (gear ratio 3.75, power 15 kW) consists of an input shaft supported by two roller bearings, sun gear, a carrier with three planet gears, ring gear and an output shaft supported by a pair of roller bearings. The three-phase asynchronous motor, a Siemens with a nominal rotational speed of 1424 RPM and power of 3 kW, was used. The motor and planetary gearbox are coupled by an elastic coupling; in turn, the planetary gearbox and break are coupled by a rigid coupling. To control the motor speed, the inverter with a Modbus communication protocol was used. The Modbus communication protocol allows to control all of the motor parameters, such as the rotation speed (up to 60 Hz), run-up speed and maximum motor loading. The entire system and the control panel are placed on a base equipped with vibration isolation (SENOMA SP. Z.O.O, 2012). Application of the electromagnetic particle break, EMA-ELFA type P120HV, allows for simulation of the time varying operations. The torque on the break is proportional to the current excitation and can be changed from zero to a maximum value of 120 Nm. The characteristics of the torque as the function of a current can vary by about 5%, depending on whether the current increases or falls due to electromagnetic hysteresis. The torque does not depend on the rotational speed, and it can be kept with an accuracy of 5%. The residual torque, in case of brake-off, occurs as a result of the residual magnetism circuit and bearings friction; it is less than 1% of the nominal torque. The brake reaction time is determined by the ratio of inductance of the coil to its resistance plus the magnetic delay, which occurs as a result of eddy current losses (EMA24.