SYSTEM AND METHOD FOR ESTIMATING REMAINING USEFUL LIFE OF A

BEARING

The present invention relates to remaining useful life esti- mation, and more particularly relates to a system and method for estimating remaining useful life of bearings.

Bearings are used in several machinery in an industry for the purpose of reducing friction between two rotating parts. These bearings also restrict relative motion between the ro- tating parts to a desired motion. However, bearings may fail unexpectedly due to factors such as poor lubrication and con- tamination within a structure of the bearing. For example, lubrication within the bearing may fail during an operational phase of the bearing due to excessive temperatures. Contami- nation may occur due to entry of foreign particles, moisture etc. into the structure of the bearing. The above factors may lead to failure modes such as corrosion, spalling, pitting, electrical erosion, plastic deformation and so on, in the bearing. As a result, an expected fatigue life of the bearing is reduced and eventually resulting in failures. Consequent- ly, bearing failures may cause unexpected downtime of the ma- chinery, resulting in production and financial loss. Accurate predictions of bearing failures can help prevent unexpected downtimes. However, estimating remaining useful life of a bearing is challenging due to the difficulties in estimating effects of contamination and lubrication. Further, the re- maining useful life also varies based on application and de- sign of the bearing.

In light of the above, there exists a need for a system and method for estimating remaining useful life of a bearing based on a scenario associated with the bearing. The term ' scenario ' as used herein may refer to various factors in- cluding such as applications , design and other operating con- ditions such as ambient temperature , lubrication type etc .

Disclosed herein is a method for estimating remaining useful life of a bearing, in accordance with an embodiment of the present invention . The method comprises receiving operational data associated with the bearing from at least one source in real-time . In an embodiment , the operational data comprises at least one of temperature signature , vibration signature , current signature and angular velocity associated with the bearing .

The method further comprises determining one or more failure modes associated with the bearing based on one or more fea- tures associated with the operational data . In an embodiment , the step of determining the one or more failure modes associ- ated with the bearing comprises analyzing the operational da- ta using a plurality of signal processing pipelines . Each of the signal processing pipelines generate a response indica- tive of the one or more features associated with the opera- tional data . Further, the one or more features from the re- sponse are classified into one or more failure modes using a classification model .

Advantageously, use of a plurality of signal proces sing pipe- lines enables determination of multiple features as sociated with the operational data .

The method further comprises identifying a best-suited virtu- al model corresponding to the bearing from a plurality of virtual models , based on the one or more features . Each of the virtual models indicates a behavior of the bearing for a predefined set of conditions . In an embodiment , the step of identifying the best-suited virtual model corresponding to the bearing comprises updating the plurality of virtual mod- els based on the one or more features and corresponding fail- ure modes . Further, simulation instances are based on each of the updated virtual models . The simulation instances are fur- ther executed in the simulation environment to generate simu- lation results . Further, the best-suited virtual model is identified based on the simulation results . In a further em- bodiment , the step of identifying the best-suited virtual model comprises computing a reliability coefficient associat- ed with each of the virtual models . Further, the virtual mod- el corresponding to the highest reliability coefficient is selected as the best-suited virtual model .

Advantageously, the present invention dynamically selects a best-suited virtual model for the bearing from a plurality of virtual models , based on real-time operational data . In other words , the bearing model is continuously adapted based on re- al-time behavior of the bearing . Consequently, manual efforts required in configuring a bearing model is reduced .

The method further comprises computing a remaining useful life of the bearing based on the best-suited virtual model . In an embodiment , the step of computing the remaining useful life of the bearing comprises determining one or more key performance indicators associated with the bearing using the best-suited virtual model . Further , the remaining useful life is computed based on the one or more key performance indica- tors determined . In a further embodiment , the method compris- es outputting the remaining useful life in real-time on an output device .

Advantageously, the present invention computes remaining use- ful life of the bearing in real-time based on the best-suited virtual model . Therefore , if additional failure modes occur a new best-suited virtual model is selected and updated to es- timate the real-time key performance indicators .

Disclosed herein is also an apparatus for estimating remain- ing useful life of a bearing, in accordance with an embodi- ment of the present invention . The apparatus comprises one or more processing units , and a memory unit communicatively cou- pled to the one or more processing units . The memory unit comprises a bearing monitoring module stored in the form of machine-readable instructions executable by the one or more processing units . The bearing monitoring module is configured to perform method steps described above .

Disclosed herein is also a system for estimating remaining useful life of a bearing, in accordance with an embodiment of the present invention . The system comprises one or more sources capable of providing operational data associated with a bearing and an apparatus as described above , communicative- ly coupled to the one or more sources . The term ' sources ' as used herein, refer to electronic devices configured to obtain and transmit the operational data to the apparatus . Non- limiting examples of sources include sensing units , control- lers and edge devices .

The object of the present invention is also achieved by a computer-readable medium, on which program code sections of a computer program are saved, the program code sections being loadable into and/or executable by a processor which per- forms the method as described above when the program code sections are executed .

The realization of the invention by a computer program prod- uct and/or a non-transitory computer-readable storage medium has the advantage that computer systems can be easily adopted by installing computer program in order to work as proposed by the present invention.

The computer program product can be, for example, a computer program or comprise another element apart from the computer program. This other element can be hardware, for example a memory device, on which the computer program is stored, a hardware key for using the computer program and the like, and/or software, for example a documentation or a software key for using the computer program.

The above-mentioned attributes, features, and advantages of the present invention and the manner of achieving them, will become more apparent and understandable (clear) with the fol- lowing description of embodiments of the invention in con- junction with the corresponding drawings. The illustrated em- bodiments are intended to illustrate, but not limit the in- vention.

The present invention is further described hereinafter with reference to illustrated embodiments shown in the accompany- ing drawings, in which:

FIG 1 illustrates a block diagram of a system for esti- mating remaining useful life of a bearing, in ac- cordance with an embodiment of the present inven- tion;

FIGS 2A illustrate structure of a ball bearing;

FIG 2B illustrates a defect in the ball bearing;

FIG 3 shows a flowchart of a method for estimating re- maining useful life of a bearing, in accordance with an embodiment of the present invention; FIG 4 shows a diagrammatic representation of processing of a vibrational signature using a plurality of signal processing pipelines , in accordance with an embodiment of the present invention;

FIG 5 illustrates a block diagram of a signal processing pipeline in accordance with an exemplary embodiment of the present invention;

FIG 6 illustrates a workflow representing selection of a best-suited virtual model , in accordance with an embodiment of the present invention; and

FIG 7 shows a representation of a convolutional neural network, in accordance with an embodiment of the present invention .

Hereinafter, embodiments for carrying out the present inven- tion are described in detail . The various embodiments are de- scribed with reference to the drawings , wherein like refer- ence numerals are used to refer to like elements throughout . In the following description, for purpose of explanation, nu- merous specific details are set forth in order to provide a thorough understanding of one or more embodiments . It may be evident that such embodiments may be practiced without these specific details .

FIG 1 illustrates a block diagram of a system 100 for esti- mating remaining useful life of a bearing 105 , in accordance with an embodiment of the present invention . In the present disclosure , the term 'bearing ' is used to refer to an assem- bly comprising an outer race , cage , rolling elements , inner race and shield that are mounted on a rotating shaft . Non- limiting examples of bearings include deep groove ball bear- ings , cylindrical roller bearings , a tapered roller bearing, thrust bearing, angular contact ball bearing, needle ball bearing and the like. In the present embodiment, the system is configured to estimate remaining useful life of ball bear- ings. In the present embodiment, the bearing 105 comprises an inner race, an outer race and a plurality of rolling elements disposed in a gap between the inner race and the outer race. The bearing 105 further comprises a cage positioned between the inner race and the outer race for maintaining a symmetric radial gap between the rolling elements. An example of a bearing is described later in the present disclosure with reference to FIG 2A.

The system 100 comprises an apparatus 110 communicatively coupled to one or more edge devices 115. The one or more edge devices 115 are connected to the apparatus 110 via a network 120, for example, local area network (LAN), wide area network (WAN), Wi-Fi, etc. Each of the edge devices 115 is configured to receive operational data from at least one sensing unit 125 associated with the bearing 105.

The sensing unit 125 generates an output signal indicative of an operational parameter associated with the bearing 105 in real-time. In an embodiment, the sensing unit 125 may include a vibration sensor configured to generate a vibration signa- ture corresponding to vibrations in the bearing 105. The vi- bration signature may include a time-domain acceleration sig- nal. In another embodiment, the sensing unit 125 may include a temperature sensor configured to generate a temperature signature associated with the bearing 105. The temperature signature may include a time-domain temperature signal. In another embodiment, the sensing unit 125 may include a cur- rent sensor such as a current transformer, configured to ob- tain a current signature associated with the bearing 105. The current signature includes a time-series current signal. The sensing unit 125 is further communicatively coupled to the apparatus. In yet another embodiment, the sensing unit 125 may include a speed sensor configured to obtain an angular velocity associated with the bearing 105. In an embodiment, the operational data are obtained through data acquisition interfaces on the edge device 115. The edge device 115 pro- vides the operational data in real-time to the apparatus 110.

In addition, the edge device 115 is also configured to pro- vide one or more bearing parameters associated with the bear- ing 105 to the apparatus 110. The one or more bearing parame- ters include, but are not limited to, a standard bearing num- ber, a ball size, bearing static load, density of material, angular velocity, internal clearance, bearing diameter, num- ber of rolling elements, radius of rolling element, diameter of inner race, diameter of outer race, fatigue load limit, and type of lubricant used in the bearing 105. It must be un- derstood that the standard bearing number is indicative of certain specifications such as a ball size, bearing static load, density of material, internal clearance, bearing diame- ter, number of rolling elements, radius of rolling element, diameter of inner race, diameter of outer race, fatigue load limit, bearing width and the like, as provided by a manufac- turer of the bearing 105. Therefore, in an embodiment, the standard bearing number associated with the bearing 105 may be provided in place of the above bearing parameters.

The one or more bearing parameters may be stored in a memory of the edge device 115 or may be input to the edge device 115 by an operator. For example, the edge device 115 may be com- municatively coupled to a client device 130. Non-limiting ex- amples of client devices include, personal computers, work- stations, personal digital assistants, human machine inter- faces. The client device 130 may enable the user to input values for the one or more bearing parameters through a web- based interface. Upon receiving the one or more bearing pa- rameters from the user, the edge device 115 transmits a re- quest for analyzing a defect in the bearing 105 to the appa- ratus 110. The defects occur due to initiation of a failure mode in the bearing 105. Non-limiting examples of failure modes include spalling, pitting, plastic deformation, abra- sion, electrical erosion and corrosion typically on an outer race of the bearing 105. The defects may occur due to pres- ence of contaminants or due to properties of a lubricant used within the bearing 105. The request comprises the one or more bearing parameters along with the operational data.

In the present embodiment, the apparatus 110 is deployed in a cloud computing environment. As used herein, "cloud computing environment" refers to a processing environment comprising configurable computing physical and logical resources, for example, networks, servers, storage, applications, services, etc., and data distributed over the network 120, for example, the internet. The cloud computing environment provides on- demand network access to a shared pool of the configurable computing physical and logical resources. The apparatus 110 may include a module that estimates remaining useful life of a given bearing based on the corresponding operational data and the one or more bearing parameters. Additionally, the ap- paratus 110 may include a network interface for communicating with the one or more edge devices 115 via the network 120.

The apparatus 110 comprises a processing unit 135, a memory- unit 140, a storage unit 145, a communication unit 150, the network interface 155 and a standard interface or bus 160, as shown in FIG IB. The apparatus 110 can be a computer, a work- station, a virtual machine running on host hardware, a micro- controller, or an integrated circuit. As an alternative, the apparatus 110 can be a real or a virtual group of computers (the technical term for a real group of computers is "clus- ter", the technical term for a virtual group of computers is "cloud").

The term 'processing unit' 135, as used herein, means any type of computational circuit, such as, but not limited to, a microprocessor, microcontroller, complex instruction set com- puting microprocessor, reduced instruction set computing mi- croprocessor, very long instruction word microprocessor, ex- plicitly parallel instruction computing microprocessor, graphics processor, digital signal processor, or any other type of processing circuit. The processing unit 135 may also include embedded controllers, such as generic or programmable logic devices or arrays, application specific integrated cir- cuits, single-chip computers, and the like. In general, a processing unit 135 may comprise hardware elements and soft- ware elements. The processing unit 135 can be configured for multithreading, i.e., the processing unit 135 may host dif- ferent calculation processes at the same time, executing the either in parallel or switching between active and passive calculation processes.

The memory unit 140 may be volatile memory and non-volatile memory. The memory unit 140 may be coupled for communication with the processing unit 135. The processing unit 135 may ex- ecute instructions and/or code stored in the memory unit 140. A variety of computer-readable storage media may be stored in and accessed from the memory unit 140. The memory unit 140 may include any suitable elements for storing data and ma- chine-readable instructions, such as read only memory, random access memory, erasable programmable read only memory, elec- trically erasable programmable read only memory, a hard drive, a removable media drive for handling compact disks, digital video disks, diskettes, magnetic tape cartridges, memory cards, and the like.

The memory unit 140 comprises a bearing monitoring module 165 in the form of machine-readable instructions on any of the above-mentioned storage media and may be in communication to and executed by the processing unit 135. The bearing monitor- ing module 165 comprises an analysis module 170, a model identification module 175, a life estimation module 180 and a notification module 185. The analysis module 170 determines one or more failure modes associated with the bearing based on one or more features associated with the operational data. The model identification module 175 is configured for identi- fying a best-suited virtual model corresponding to the bear- ing from a plurality of virtual models based on the one or more features. Each of the virtual models indicates a behav- ior of the bearing for a predefined set of conditions. The life estimation module 180 is configured for computing a re- maining useful life of the bearing based on the best-suited virtual model. The notification module 185 is configured for outputting the computed remaining useful life of the bearing 105 on an output device. In the present embodiment, the out- put device may be the client device 130.

The storage unit 145 comprises a non-volatile memory which stores default bearing parameters associated with standard bearing numbers. The storage unit 145 includes a database 195 that comprises default values of bearing parameters and one or more look-up tables comprising predetermined values for factors that vary with operating conditions of the bearing 105. The bus 160 acts as interconnect between the processing unit 135, the memory unit 140, the storage unit, and the net- work interface 155. The communication unit 150 enables the apparatus 110 to receive requests from the one or more edge devices 115. The communication module may support different standard communication protocols such as Transport Control Protocol/Internet Protocol (TCP/IP), Profinet, Profibus, and Internet Protocol Version (IPv).

Those of ordinary skilled in the art will appreciate that the hardware depicted in FIG 1A and IB may vary for different im- plementations. For example, other peripheral devices such as an optical disk drive and the like, Local Area Network (LAN)/ Wide Area Network (WAN)/ Wireless (e.g., Wi-Fi) adapter, graphics adapter, disk controller, input/output (I/O) adapt- er, network connectivity devices also may be used in addition or in place of the hardware depicted. The depicted example is provided for the purpose of explanation only and is not meant to imply architectural limitations with respect to the pre- sent disclosure.

FIG 2A illustrates structure of a ball bearing 200. The ball bearing 200 comprises an outer race 210 of diameter D, a plu- rality of balls 215 each of radius r _ball , a cage 220 and an inner race 225 of diameter d. The plurality of balls 215 are disposed in a gap between the outer race 210 and the inner race 225. The cage 220 maintains a symmetric radial spacing between the balls 215. a is the angle of contact between the ball and the outer race. Here, d _m is the pitch diameter of the bearing 200.

FIG 2B illustrates a defect of defect size ddefect on a wall of the outer race 210 in contact with the balls 210. In the pre- sent example, the defect is an indentation in the outer race 210 due to pitting. The defect size may be defined as a dis- tance travelled by a ball of the ball bearing 200 between en- tering and leaving the defect. The defect subtends an angle ^θ center st the center of a ball 215A among the plurality of balls 215 as shown. Similarly, defects may be also form in the inner race of the bearing. The defects may be due to misalignment, eccentrici- ty, loose foot, contamination, lubrication etc. Each of the faults is associated with one or more characteristic frequen- cies in a vibration signature associated with the bearing. The characteristic frequencies may be predetermined based on correlations derived based on experimental data generated corresponding to each of the defects.

In an embodiment, defects are artificially introduced into a component of the bearing or a similar bearing in order to em- ulate effects of the fault. Further, vibration signatures as- sociated with the defected component is measured using vibra- tion sensors. The vibration signatures are analyzed in fre- quency domain in order to establish correlations between the fault and failing frequencies in the vibration signature. For example, misalignments in the bearing may be indicated by peak amplitudes at 3x frequencies in a Fast Fourier Transform of the vibration signature. Similarly, correlations between fault in a component of the bearing and corresponding failing frequencies are predetermined through experimentation. For example, defects in the outer race is characterized by pres- ence of Ball Pass Frequency of Outer race (BPFO) generated when a ball comes in contact with the defect on the outer race, where: where B is diameter of the ball, P is the pitch diameter, N is the number of balls in the bearing, S is the angular ve- locity of the bearing in Hz and β is the contact angle asso- ciated with the defect. Similarly, defects in the inner race is characterized by Ball Pass Frequency of Inner race (BPFI ) generated when a ball comes in contact with the defect in the inner race , where :

Likewise , defects in the cage is characterized by train or cage frequency generated when a ball comes in contact with the defect in the cage , where :

Train or Cage Frequency

Further, defects in a ball of the bearing is characterized by ball spin frequency generated when a defected ball touches the outer race or the inner race , where :

Ball Spin Frequency

As shown, each of the failing frequencies mentioned above are dependent on bearing dimensions and the angular velocity of the bearing . Each of the defects is indicated by presence of harmonics around the failing frequencies in a frequency spec- trum corresponding to the vibration spectrum .

FIG 3 shows a flowchart of a method 300 for estimating re- maining useful life of a bearing, in accordance with an em- bodiment of the present invention . The remaining useful life of the bearing is estimated based on defects present in the bearing .

At step 305 , operational data from at least one sensing unit associated with the bearing is received in real-time . The op- erational data comprises a time-domain signal generated by the sensing unit . The operational data may include , but not limited to, vibration signature , current signature and tem- perature signature . In the present embodiment , the operation- al data comprises a vibration signature, hereinafter referred as 'the signal'. In an embodiment, the current signature may be further analysed, using Motor Current Signature Analysis, to determine frequencies corresponding to the vibration sig- nature that may occur due to dynamic eccentricity of the shaft.

At step 310, one or more failure modes associated with the bearing is determined based on one or more features associat- ed with the operational data. In the present embodiment, the failure modes are determined by firstly analyzing the opera- tional data using a plurality of signal processing pipelines. Each of the signal processing pipelines generate a response indicative of the one or more features associated with the operational data. The features may include, for example, peak amplitudes, root mean square values, crest, time duration be- tween events, modified crest, failure frequencies, harmonics, side bands, kurtosis etc. It must be understood by a person skilled in the art that in addition to the above-mentioned features, statistical measures such as standard deviation, mean value etc associated with the operational data may also be determined using the signal processing pipelines.

In an embodiment, each of the signal processing pipelines is associated with at least one signal processing function. The signal processing functions may include, but not limited to, bandpass filtering, envelope analysis, Child Wavelet Trans- form (CWT), Discrete Wavelet Transform (DWT), Fast Fourier Transform (FFT), Short-time Fourier transform (STFT), Spec- tral Kurtosis (SK) and Fast Kurtogram. In addition, the sig- nal processing pipelines may also include signal prepro- cessing functions such as denoising algorithms, bandwidth filtering etc. In the present embodiment, three signal pro- cessing pipelines corresponding to FFT, CWT and DWT are used. FIG 4 shows a diagrammatic representation of processing of a vibrational signature A using a first signal processing pipe- line 410 corresponding to FFT, a second signal processing pipeline 415 corresponding to CWT and a third signal pro- cessing pipeline 420 corresponding to DWT.

In particular, FFT decomposes vibrational signal in time- domain into infinite number of sine or cosine waves in fre- quency domain. Advantageously, FFT helps in determining events from stationary signals. The term 'stationary signal' as used herein refers to a signal wave that has a constant time period and spectral content value. In an embodiment, the FFT helps in determining defect size associated with a defect based on the instances at which a ball associated with the bearing experiences an impact force due to the defect.

Wavelet transforms such as CWT and DWT are used for trans- forming the time-domain signal from the sensing unit into an- other time-domain signal. In other words, wavelet transforms involve breaking up of a signal into shifted and scaled ver- sions of the original signal. When applied to a transient or non-stationary signal, wavelet transforms help in determining features associated with the operational data that vary with time. The wavelet transform is performed by correlating an input signal x(t) with a wavelet basis. For example, wavelet transform of signal x(t) is given by, where W (a, b) represent wavelet coefficients, Ψ(t) is a mother wavelet, a is dilation (scale) parameter and b is translation (position) parameter. The more dilated and trans- lated wavelets are similar to the signal at different levels of resolution, the larger the corresponding wavelet coeffi- cients become . The wavelet coefficients therefore , facilitate extraction of features associated with the signal . The mother wavelet is a waveform of limited duration that has an average value of zero and is represented as :

The dilation ( scale ) parameter 'a ' and the translation (posi- tion) parameter 'b ' play a vital role in wavelet transform . The dilation parameter causes the signal to be dilated by a factor of 'a ' . Dilated wavelets help in capturing slowly var- ying low frequency changes and compressed wavelets help in capturing abrupt , high frequency changes . The translation pa- rameter causes the signal to be time-shifted by a factor of 'b ' along the time-axis . The translation parameter shifts the wavelet in order to align the wavelet with features of inter- est in the signal .

DWT uses filter banks for analysis and synthesis of a signal . Each of the filter banks contains wavelet filters that help in extracting the frequency content of the signal in various sub bands . In an implementation, the DWT is derived by dis- cretization of the CWT of the signal in dyadic scale , fol- lowed by translation . Consequently, the computational time associated with the DWT is reduced . The DWT may be expressed using the following equation :

The original signal x (t ) passes through a set of low pass and high pass filters emerging as low frequency and high frequen- cy signals at a decomposition level 'n' . Therefore , the orig- inal signal x (t ) can be written as : DWT helps in calculating the defect size from the signal. More specifically, the defect size on the outer race of the bearing can be estimated based on duration between entry and exit events corresponding to a ball with respect the defect. The entry event corresponds to the instant at which a ball of the bearing comes in contact with a leading edge of the de- fect. The exit event corresponds to the instant at which the same ball comes in contact with a trailing edge of the bear- ing. The events are extracted from the signal after decompo- sition using DWT. The duration between the events is used to estimate the defect size as follows: where S is the angular velocity, D is the pitch diameter of the bearing, B is the ball diameter, β is the contact angle and At is the duration between the entry and exit events. The contact angle is usually taken as zero for for deep groove ball bearings.

In an implementation, the decomposition is performed by using symlets, such as sym5, as mother wavelets. The scaling fil- ters associated with symlets are near-linear phase filters. Consequently, phase response of the scaling filters is a lin- ear function of frequency. This enables symlets to deal with small discontinuities present in the signal without major loss of information. As a result, symlets may be used in both CWT and DWT.

In the present embodiment, each of the three signal pro- cessing pipelines 410, 415, and 420 generate responses indic- ative of the one or more features. In particular, FFT pro- vides indication of events associated with the ball coming in contact with a defect. CWT provides indication of fault modes such as misalignment as well as cyclicity associated with im- pact force generated when a ball comes in contact with the defect. For example, CWT of a healthy bearing is typically plain, while patterns in the CWT indicate presence of de- fects. DWT provides indication of fault modes similar to CWT.

Similarly, a signal processing pipeline may also include a combination of signal preprocessing and post-processing func- tions as shown in FIG 5.

Further, the one or more features are classified into one or more failure modes using a classification model. The failure modes may be associated with, for example, lubrication- related anomalies, contamination-related anomalies, eccen- tricity, misalignment and loose foot. In the present embodi- ment, the classification model is a convolutional neural net- work (CNN). The structure of a CNN is explained in detail us- ing FIG 7.

At step 315, a best-suited virtual model corresponding to the bearing is identified from a plurality of virtual models. Each of the virtual models indicates a behavior of the bear- ing for a predefined set of conditions. The predefined set of conditions may be associated with, but not limited to, appli- cation (low speed, high speed or medium speed) of the bear- ing, design of the bearing, nature of bearing support, damp- ing and coupler design. In other words, each of the virtual models may correspond to a different behavioral model associ- ated with the bearing.

In the present example, the bearing is mounted on a shaft as- sociated with a motor coupled to a pump. A coupler may be provided between the bearing and the shaft of the motor. How- ever, stiffness of the coupler may vary based on factors in- eluding, but not limited to, application ( low speed, high speed or medium speed) , nature of bearing support , damping and coupler design . Consequently, the bearing may exhibit varying behavior in different scenarios . In an embodiment , each of the virtual models may be digital twins of the bear- ings . Each of the digital twins may be built and calibrated based on experimental data and simulation data corresponding to, for example , different applications , damping, coupler de- signs , faults etc .

The best-suited virtual model is identified by firstly updat- ing the virtual models based on the one or more features and corresponding failure modes . In addition to the one or more features , the virtual model is also updated based on the bearing parameters corresponding to the bearing . Upon updat- ing, simulation instances corresponding to each of the virtu- al models are generated . The simulation instances may be as- sociated with one or more of ID simulation, 2D simulation, 3D simulation, co-simulation, discrete simulation, continuous simulation, stochastic simulations and so on .

The simulation instances are further executed in a suitable simulation environment . Upon execution of the simulation in- stances , simulation results are generated . The simulation re- sults correspond to outputs from each of the virtual models . For example , one or more of the simulation results may indi- cate a maximum impact force experienced by a ball due to a defect . The simulation results may further include a relia- bility coefficient corresponding to each of the virtual mod- els . The reliability coefficient is determined based on vari- ances associated with outputs from each of the virtual models using statistical techniques known in the art . In an embodi- ment , the reliability coefficient is determined using artifi- cial intelligence models trained using test data and simula- tion data for identifying reliability coefficients, along with optimisation models.

Further, the virtual model corresponding to highest reliabil- ity coefficient is selected as the best-suited virtual model.

At step 320, a remaining useful life of the bearing is pre- dicted based on the best-suited virtual model. The term 're- maining useful life' as used herein, refers to a duration be- tween initiation of a detectable failure mode and a function- al failure of the bearing. The remaining useful life is com- puted based on a key performance indicator estimated by the best-suited virtual model. The key performance indicators are characteristic to the application of the bearing and behav- iour of the bearing with respect to the application. For ex- ample, the key performance indicator is a dynamic load on the bearing. In another example, the key performance indicator is a contact force or contact stress associated with a rolling element of the bearing upon coming in contact with the de- fect.

In an example, the remaining useful life may be expressed as number of revolutions before the failure occurs. In another example, the remaining useful life is expressed as number of operating hours at a constant speed before the failure. In a preferred embodiment, the remaining useful life L _RUL for the bearing is determined based on the following rating life mod- el: where, a _ISO is a life modification factor based on systems ap- proach for life calculation given by: where, a1 is a life modification factor for reliability, Cu is fatigue load limit in Newtons, e _c is a contamination factor specific to the defect size, P is dynamic equivalent radial load in Newtons, C is dynamic equivalent radial load rating, K is viscosity ratio.

The life modification factor for reliability is a predefined value specified in ISO 281:2007 for a given value of relia- bility. For example, if the reliability may be considered to be 90%, al is taken as 1. The value of reliability is taken as 90% by default. In an embodiment of the present invention, the value of reliability may be modified by an operator through the client device. The life modification factor for reliability may be further obtained from a first lookup table stored in the database, based on the value of reliability. The fatigue load limit and the dynamic equivalent radial load rating are obtained from the one or more bearing parameters associated with the bearing.

The contamination factor is determined based on the defect size. This is because, defects in the bearing result in re- moval of small, discrete particles of material from the structure of the bearing. These discrete particles increase the concentration of contaminants inside the bearing. The concentration of contaminations further increases with in- crease in the defect size. The defect size is provided as in- put to a trained classification model that classifies the de- fect size into one of a plurality of severity levels. For ex- ample, the plurality of severity levels may correspond to 'normal cleanliness', 'slight to typical contamination', 'se- vere contamination' and 'very severe contamination'. Based on the defect size, the trained classification model outputs a severity level. The severity level thus determined is further used to select an appropriate contamination factor from a second lookup table . The second lookup table may comprise values of contamination factor corresponding to each of the severity levels . It must be understood by a person skilled in the art that the classification model may be trained to clas- sify the defect size into one of any number of severity lev- els .

The viscosity ratio is indicative of a lubrication condition of the bearing during operation . The viscosity ratio is cal- culated as the ratio of an operating viscosity of the lubri- cant to a rated viscosity of the lubricant . The operating viscosity is calculated based on a viscosity grade of the lubricant and an operating temperature . The viscosity grade of the oil may be obtained from a third look up table com- prising viscosity grades corresponding to different types of lubricants . In one embodiment , the operating temperature of the bearing may be obtained from temperature sensors associ- ated with the bearing . In another embodiment , a virtual model of the bearing may be used to determine a thermal profile of the bearing based on the operational data . The rated viscosi- ty is obtained from a fourth look up table based on dimen- sions of the bearing and angular velocity of the bearing .

The equation ( 15 ) for rating life may be rearranged to obtain the remaining useful life model below :

Here , the dynamic equivalent radial load P is the same as the maximum impact force computed by the best-suited virtual model . Therefore , the remaining useful life is a function of the dynamic equivalent radial load P . In an embodiment , the best-suited virtual model determines the maximum impact force based on mathematical model of the bearing, the experimental data and the simulation data corresponding to operational be- haviour of the bearing. Therefore, the remaining useful life of the bearing may be estimated as a function of the mathe- matical model, the experimental data and the simulation data.

The mathematical model of the bearing is of the form: which can be rearranged as : where, X _imax is a maximum deflection of the ball in mm; where v is Poisson' s ratio associated with a material of the bearing, E is Young' s modulus associated with material and E _eq is equivalent stiffness of the material. For example, if the bearing is made of EN31 steel, the Poisson's ratio is 0.3 and

Young's modulus is 210 GPa. where r _ball is the radius of the ball where r _bearing isa radius of the bearing given by: where ω is angular velocity is the angular frequency with which the ball rotates about the edge of the defect.

In the above equation, parameters p1 and p2 are tuned based on the bearing parameters. In an embodiment, the parameters are tuned using a trained machine learning model. The trained machine learning model is an evolutionary algorithm. The mathematical model is further solved for X _imax . Further, the maximum impact force F _max on the ball is calculated from Ximax according to the following equation:

Here F _max represents the value of maximum impact force on the ball calculated mathematically.

The bearing life thus calculated is further outputted as a notification on the output device, notification that indi- cates the remaining useful life of the bearing as a dynami- cally changing parameter based on the best-suited virtual model selected. For example, the notification may include a message 'The remaining useful life of bearing 6319 is 56 hours '.

FIG 5 illustrates a block diagram 500 of a signal processing pipeline in accordance with an exemplary embodiment of the present invention . Here , a vibration signature A is firstly processed using a CWT function 505 . Further, a response from the CWT function 505 is processed using an SK function 510 . The SK function 510 detects presence of transients and their locations in the frequency domain . Further, a passband is se- lected from the output of the SK function 510 using a band- pass filter 515 . The output of the bandpass filter 515 is further processed using a Hilbert Transform function 520 . The Hilbert Transform function 520 helps in obtaining a minimum- phase response from the spectral analysis performed using the SK function 510 . The output of the Hilbert Transform function 520 is further processed using CWT/FFT function 525 . The out- put of the CWT/FFT function 525 may be further analyzed to determine the BPFO of the bearing .

FIG 6 illustrates a workflow 600 representing selection of a best-suited virtual model , in accordance with an embodiment of the present invention . The workflow 600 comprises a plu- rality of signal processing pipelines 610A, 610B . . 610N ( col- lectively called signal processing pipelines 610 ) for pro- cessing a vibration signature A . The responses from each of the signal proces sing pipelines 610 are further processed us- ing an overlaying classification model 615 . Further, a plu- rality of virtual models 620A, 620B . . 620M ( collectively called virtual models 620 ) are updated based on failure modes identified using the classification model 615 and features corresponding to the failure modes . Outputs from the updated virtual models 620A, 620B . . 620M are further analyzed to iden- tify the best-suited virtual model from the virtual models 620 . For example , the best-suited virtual model may be virtu- al model 620B . Further, one or more of key performance indi- cators 625A, 625B . . . 625K ( collectively called key performance indicators 625 ) required for computation of the remaining useful life are identified using the best-suited virtual mod- el.

FIG 7 shows a representation of a convolutional neural net- work 700, in accordance with an embodiment of the present in- vention. Examples of CNN may include, but are not limited to, Fast Region based CNN (Fast R-CNN), Faster Region based CNN (or Faster R-CNN), Mask Region based CNN (or Mask R-CNN) and You Only Look Once (YOLO). In the displayed embodiment, the convolutional neural network 700 comprises an input layer 710, a convolutional layer 711, a pooling layer 712, a fully connected layer 713 and an output layer 714. Alternatively, the convolutional neural network 700 may comprise several convolutional layers 711, several pooling layers 712 and sev- eral fully connected layers 713, as well as other types of layers. The order of the layers may be chosen arbitrarily. In an exemplary implementation, fully connected layers 713 are used as the last layers before the output layer 714.

In the convolutional neural network 700, the nodes 720, ..., 724 of layers 710, ..., 714 may be considered to be arranged as a d-dimensional matrix or as a d-dimensional image. In par- ticular, in the two-dimensional case, the value of the node 720, ..., 724 indexed with i and j in the n-th layer 710, ..., 714 may be denoted as x ⁽ⁿ⁾ [i,j]. However, the arrangement of the nodes 720, ..., 724 of one layer 710, ..., 714 does not have an effect on the calculations executed within the convolu- tional neural network 700 since these are given solely by the structure and the weights of the edges.

In particular, a convolutional layer 711 is characterized by the structure and the weights of the incoming edges forming a convolution operation based on a certain number of kernels. In particular, the structure and the weights of the incoming edges are chosen such that the values x ⁽ⁿ⁾ k of the nodes 721 of the convolutional layer 711 are calculated as a convolu- tion x ⁽ⁿ⁾ k = Kk * x ^{(n-1 )} based on the values x ^{(n-1 )} of the nodes 720 of the preceding layer 710 , where the convolution* is de- fined in the two-dimensional case as :

Here the k-th kernel Kk is a d-dimensional matrix ( in the present embodiment , a two-dimensional matrix) , which is usu- ally small compared to the number of nodes 720 , ..., 724 (e . g . a 3x3 matrix, or a 5x5 matrix) . In particular, this implies that the weights of the incoming edges are not independent but chosen such that they produce said convolution equation . In particular, for a kernel being a 3x3 matrix, there are on- ly 9 independent weights (each entry of the kernel matrix corresponding to one independent weight ) , irrespectively of the number of nodes 720 , ..., 724 in the respective layer 710 , ..., 714 . In particular, for a convolutional layer 711 , the number of nodes 721 in the convolutional layer is equivalent to the number of nodes 720 in the preceding layer 710 multi- plied with the number of kernels .

If the nodes 720 of the preceding layer 710 are arranged as a d-dimensional matrix using a plurality of kernels may be in- terpreted as adding a further dimension (denoted as "depth" dimension) , so that the nodes 721 of the convolutional layer 721 are arranged as a (d+1 ) -dimensional matrix . If the nodes 720 of the preceding layer 710 are already arranged as a (d+1 ) -dimensional matrix comprising a depth dimension using a plurality of kernels may be interpreted as expanding along the depth dimension so that the nodes 721 of the convolution- al layer 721 are arranged also as a (d+1 ) -dimensional matrix . The size of the (d+1 ) -dimensional matrix with respect to the depth dimension is by a factor of the number of kernels larg- er than in the preceding layer 710.

The advantage of using convolutional layers 711 is that spa- tially local correlation of the input data may exploited by enforcing a local connectivity pattern between nodes of adja- cent layers, in particular by each node being connected to only a small region of the nodes of the preceding layer.

In the displayed embodiment, the input layer 710 comprises 36 nodes 720 arranged as a two-dimensional 6x6 matrix. The con- volutional layer 711 comprises 72 nodes 721, arranged as two two-dimensional 6x6 matrices, each of the two matrices being the result of a convolution of the values of the input layer with a kernel. Equivalently, the nodes 721 of the convolu- tional layer 711 may be interpreted as arranges as a three- dimensional 6x6x2 matrix, wherein the last dimension is the depth dimension.

The pooling layer 712 may be characterized by the structure and the weights of the incoming edges and the activation function of its nodes 722 forming a pooling operation based on a non-linear pooling function f. For example, in the two- dimensional case the values x ⁽ⁿ⁾ of the nodes 722 of the pool- ing layer 712 may be calculated based on the values x ^(n-1) of the nodes 721 of the preceding layer 711 as x ⁽ⁿ⁾ [i,j]= f(x(n ^{_ 1} )[id1, jd2], ..., x( ^n-1 )[id1+d1-1, jd2+d2-1]). In other words, the number of nodes 721, 722 may be reduced by using the pooling layer 712, by replacing a number di•d2 of neighboring nodes 721 in the preceding layer 711 with a single node 722 being calculated as a function of the values of said number of neighboring nodes in the pooling layer. In particular, the pooling function f may be the max-function, the average or the L2-Norm. In particular, for the pooling layer 712, the weights of the incoming edges are fixed and are not modified by training .

The advantage of using the pooling layer 712 is that the num- ber of nodes 721 , 722 and the number of parameters is re- duced . This leads to the amount of computation in the convo- lutional neural network 700 being reduced and to a control of overfitting .

In the displayed embodiment , the pooling layer 712 is a max- pooling, replacing four neighboring nodes with only one node , the value being the maximum of the values of the four neigh- boring nodes . The max-pooling is applied to each d- dimensional matrix of the previous layer . In the present em- bodiment , the max-pooling is applied to each of the two two- dimensional matrices , reducing the number of nodes from 72 to 18 .

The fully-connected layer 713 may be characterized by the fact that a majority, in particular, all edges between nodes 722 of the previous layer 712 and the nodes 723 of the fully- connected layer 713 are present , and wherein the weight of each of the edges may be adjusted individually .

In the present embodiment , the nodes 722 of the preceding layer 712 of the fully-connected layer 713 are displayed both as two-dimensional matrices , and additionally as non-related nodes ( indicated as a line of nodes , wherein the number of nodes was reduced for a better presentability) .

In the present embodiment , the number of nodes 723 in the fully connected layer 713 is equal to the number of nodes 722 in the preceding layer 712 . Alternatively, the number of nodes 722 , 723 may differ . Furthermore , in the present embod- iment , the values of the nodes 724 of the output layer 714 are determined by applying the Softmax function onto the val- ues of the nodes 723 of the preceding layer 713 . By applying the Softmax function, the sum of the values of all nodes 724 of the output layer is 1 , and all values of all nodes 724 of the output layer are real numbers between 0 and 1 . In partic- ular, if using the convolutional neural network 700 for cate- gorizing input data, the values of the output layer may be interpreted as the probability of the input data falling into one of the different categories .

The convolutional neural network 700 may also comprise a ReLU (acronym for "rectified linear units" ) layer . In particular, the number of nodes and the structure of the nodes contained in a ReLU layer is equivalent to the number of nodes and the structure of the nodes contained in the preceding layer . In particular, the value of each node in the ReLU layer is cal- culated by applying a rectifying function to the value of the corresponding node of the preceding layer . Examples for rec- tifying functions are f (x) = max ( 0 , x) , the tangent hyperbol- ics function or the sigmoid function .

In particular, convolutional neural networks 700 may be trained based on the back-propagation algorithm . To prevent overfitting, methods of regularization may be used, e . g . dropout of nodes 720 , ..., 724 , stochastic pooling, use of ar- tificial data, weight decay based on the LI or the L2 norm, or max norm constraints .

In the present embodiment , the convolutional neural network 700 is pretrained based on experimental data, simulation data and test data corresponding to the failure modes . Further, each of the features may be correlated with one or more fail- ure modes using training techniques as explained above .

Advantageously, the present invention computes remaining use- ful life of the bearing in real-time based on the best-suited virtual model. Therefore, if additional failure modes occur a new best-suited virtual model is selected and updated to es- timate the real-time key performance indicators. Furthermore, the present invention also facilitates computing remaining useful life of a plurality of bearings by computing key per- formance indicators associated with each of the bearings us- ing corresponding best-suited virtual models.

The present invention is not limited to a particular computer system platform, processing unit, operating system, or net- work. One or more aspects of the present invention may be distributed among one or more computer systems, for example, servers configured to provide one or more services to one or more client computers, or to perform a complete task in a distributed system. For example, one or more aspects of the present invention may be performed on a client-server system that comprises components distributed among one or more serv- er systems that perform multiple functions according to vari- ous embodiments. These components comprise, for example, exe- cutable, intermediate, or interpreted code, which communicate over a network using a communication protocol. The present invention is not limited to be executable on any particular system or group of system, and is not limited to any particu- lar distributed architecture, network, or communication pro- tocol.

While the invention has been illustrated and described in de- tail with the help of a preferred embodiment, the invention is not limited to the disclosed examples. Other variations can be deducted by those skilled in the art without leaving the scope of protection of the claimed invention.