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The frequent impulse collisions and wear between the ball and cage pocket not only affect the bearing stability but also significantly impact the deterioration of the bearing service life. A comprehensive dynamic model for analyzing the stability, skidding degree, ball-cage collision, wear distribution, and wear rate of four types of cage pocket is proposed. A series of tests of cage whirling conducted on self-lubricating bearing test bench using high-speed photographic technology have proven the accuracy of the model. It is found that the cage with a combination of circle and rectangle pockets has the highest stability. The circle pocket has a larger collision area and higher wear rate than rectangle and diamond pockets. The combined rectangle-diamond pocket has a better wear resistance.

The term “over-skidding” indicates that the cage rotational speed ratio exceeds the theoretical value as ball purely rolls on the raceway. Different from the skidding phenomenon that occurs in low-load and high-speed bearing, over-skidding usually occurs in large-size angular contact bearings, and it is still difficult to suppress under high load conditions. The main forms of damage to the raceway by over-skidding are spinning and gyro slip. To further explore the vibration characteristics and thermal effects of this phenomenon, a set of over-skidding tests of an angular contact bearing with a bore diameter of 220 mm were conducted on an industrial-size test bench. Through the experiment, the influence of axial load, rotational speed, and lubrication conditions on the occurrence of over-skidding were determined. Based on a previous dynamics model, the heat generation and thermal network models were integrated in the present study to predict the over-skidding and its thermal behavior. The model was validated in terms of the measured degree of over-skidding and temperature rise. The results showed that the degree of over-skidding reaches up to 12% of the theoretical value, and the friction power loss of the ball-pocket accounts for 30% of the total power loss. The analysis of the vibration signal showed a strong correlation between the bearing vibration characteristics and over-skidding behavior, thereby providing a way to indirectly measure the degree of over-skidding.

Spiral bevel gears are important part of many mechanical transmission systems and are known for their smooth operation and strong load carrying capacity. This type of gear has a high contact ratio, which makes it very difficult to diagnose even serious defects. Therefore, spiral bevel gears have rarely been used as a reference for defect diagnosis techniques. To overcome these challenges, artificial intelligence (AI) techniques are used in this research to diagnose defects in spiral bevel gears. Although Al techniques in the field of fault diagnosis have been very successful, however, these methods largely use the assumption that the training and test data come from the same operating conditions. However, when the operating conditions in which the trained model is deployed for predictions, differ from the operating conditions in which the model was trained, the performance of these approaches might be significantly reduced. Outside the laboratory, in real-world applications, operating conditions significantly vary, and it is difficult to obtain data for all potential operating conditions. To overcome this limitation and to make AI techniques suitable for diagnosing spiral bevel gear faults under different operating conditions, an effort is made to find fault distinguishing features, which are lesser sensitive to operating conditions. Artificial neural network (ANN) and K-nearest neighbors (KNN) are used as classifiers for fault detection. Performance comparison between both classifiers is made to determine their individual capability and suitability for diagnosing defects of spiral bevel gears under different operating conditions.

The rotor thermal sensitivity often affects the dynamic behavior of power unit generators. Owing to this phenomenon, increments of field current and other process parameters that are related to it may cause a shaft thermal bow and significant changes in the synchronous vibration. This symptom can also be caused by many other common malfunctions that affect rotating machines. Therefore, diagnostic techniques aimed at identifying the actual fault are very useful for optimizing maintenance activities. The thermal sensitivity of generator rotors can be deemed as a fault because it is commonly caused by a local deterioration of the winding insulation as well as by jamming phenomena between conductors and rotor slots, caused by friction forces due to the different thermal expansions of these components. This paper shows the results obtained by applying a diagnostic method, based on multiple linear regression models, which has been developed for the analysis of generator vibrations caused by thermal sensitivity. Nevertheless, nonlinear relationships between vibration and process parameters have also been taken into account. The capabilities of this diagnostic technique have been validated using the analysis of experimental data collected in a power plant. The results of this investigation are shown and discussed in the paper.

Hydraulic generating systems are widely modeled in the literature for investigating their stability properties by means of transfer functions representing the dynamic behavior of the reservoir, penstock, surge tank, hydro-turbine, and the generator. Traditionally, in these models the electrical load is assumed constant to simplify the modeling process. This assumption can hide interesting dynamic behaviors caused by fluctuation of the load as actually occurred. Hence, in this study, the electrical load characterized with periodic excitation is introduced into a hydraulic generating system and the responses of the system show a novel dynamic behavior called the fast–slow dynamic phenomenon. To reveal the nature of this phenomenon, the effects of the three parameters (i.e., differential adjustment coefficient, amplitude, and frequency) on the dynamic behaviors of the hydraulic generating system are investigated, and the corresponding change rules are presented. The results show that the intensity of the fast–slow dynamic behaviors varies with the change of each parameter, which provides reference for the quantification of the hydraulic generating system parameters. More importantly, these results not only present rich nonlinear phenomena induced by multi-timescales, but also provide some theoretical bases for maintaining the safe and stable operation of a hydropower station.

This paper addresses the optimization of the tribological characteristics of lubricating oils used in the process industry. In many cases, the machines are organized in several “stands” to form “lines” and are equipped by spindles supported by oil-film journal bearings that are fed by the same oil. The modelling of a steel roll forming line is presented by using a thermo-elasto-hydro-dynamic model for the calculation of the power dissipated in each journal bearing. The optimal lubricating oil characteristics are defined by means of a multivariate optimization on the parameters of viscosity, temperature and thickness of the oil-film. The results of the experimental tests are shown for the oil used in the real plant and the new oil formulated by simulations.

Many industrial rotating machines are equipped with hydrodynamic journal bearings, such as centrifugal compressors, steam turbines, pumps and motors. After some time from the installation, however, the surface of the bearings often presents imperfections and slight damages mainly caused by the presence of harder particles in the lubricant during start-ups and shut-downs, when the hydrodynamic mechanism is not well developed and the mixed lubrication can occur. The presence of scratches on a bearing can lead to variations of the oil film thickness which, in turn, causes significant degradation of the bearing hydrodynamic performance. For example, the reduction of the minimum oil-film thickness can lead to the increase in the local temperature, to local pressure peaks and, finally, to the failure of the bearing. Experimental data relating to scratches on journal bearings are extremely limited in the literature especially for tilting-pad journal bearings (TPJBs). An experimental activity was carried out to study the effect of artificial scratches on pads on the static and dynamic behaviors of a TPJB. The number of scratches, the depth and the axial position have been investigated and the dynamic coefficients have been estimated as well. The experimental results confirmed a degradation of the dynamic performance of the bearing in case of scratches, that it has has been also confirmed by means of numerical simulations.

The vibration caused by an early defect on the rolling element bearing (REB) is very weak and easy to be submerged other signals and noise. Therefore, the performance of a bearing fault diagnosis method mainly depends on two key steps, namely, bearing fault signal component extraction and bearing fault type identification. In this article, authors have proposed a bearing fault diagnosis method that combines the techniques of squared envelop spectrum (SES) analysis and singular value decomposition (SVD). The original vibration signal will be decomposed into several sub-signals through SVD. Then, sub-signals are grouped according to their similarity. Later, the SES of the grouped signal is applied to identify the fault type. The performance of this method is tested through actual vibration signals obtained from the test-rig.

Rotating machines are generally equipped with tilting-pad journal bearings. In general, during the design phase, the aspects that are most taken into account are the dynamic performances in terms of stiffness and damping and the minimum thickness of the oil film to avoid mixed lubrication conditions that can lead to white metal wear phenomena. The thermal aspect, on the other hand, is often underestimated and can be critical for machines operating at high speed and high load. Thermal heating is caused by shear stresses in the lubricating film. In general, an increase in lubricant temperature corresponds to a reduction in oil viscosity followed by a reduction in lubricant film thickness. This has a beneficial effect on bearing stiffness. On the contrary, a high temperature in the oil film is detrimental to the mechanical characteristics of the white metal which is characterized by a low melting temperature and subject to thermal creep phenomena. The article analyzes, by means of numerical simulations, the influence of the bearing geometry on the thermal behavior of the bearing. The bearing geometry will then be optimized to reduce the maximum temperature in the bearing.

The skidding phenomenon, consisting of sliding and spinning may between the rolling elements and the raceways of bearings, can reduce significantly the service life. In order to early predict the arising of the skidding in angular contact bearings, a dynamic model is introduced in this paper. The centrifugal force and the gyroscopic effect are also taken into account to determine the load distribution determined by the rolling elements. The model of the rolling element orbital kinematics considers the elasto-hydrodynamic (EHD) lubrication and its contribution to the friction forces, along with the lubricant oil drag effect, the rolling element gravity and the ball-cage interactions. The fourth order Runge-Kutta integration and Newton-Raphson iteration methods are employed to calculate the load distribution, and the dynamic motion of the rolling elements and cage. The comparison between the results obtained by means of the proposed model and some experimental data shows a general good agreement. The experimental data have been obtained in a test rig equipped by an industry scale angular contact ball bearing under different axial load. Under high load, the experimental speed ratio of the cage versus the bearing inner race exceeds the pure kinematically determined value: this phenomenon is defined as over-skidding behavior and it is also simulated by the proposed model. The pure rolling state of rolling elements are also predicted by the proposed model at high axial load. The mechanism of skidding and over-skidding are also discussed based on the proposed model. In the future, the model presented will be supplemented with a temperature analysis, considering the variation of the lubricant viscosity in a full thermo- elasto-hydrodynamic (TEHD) approach.

The coating materials commonly used in hydrodynamic bearings are the so-called “Babbitt metals” or “white metals”, as defined by ASTM B23-00. Their low Young’s modulus and yield point have encouraged researchers to find new coatings to overcome these limitations. In this paper, the friction and wear of PEEK are studied in a dry sliding environment (without lubrication) using a ball-on-disk tribometer and compared to those of Babbitt metal. Furthermore, the bond strength tests between PEEK and metals/alloys are evaluated. PEEK polymer samples were obtained from cylindrical rods, manufactured by an innovative process for polymer bonding on bearing surfaces, using additive manufacturing technology. The morphologies of the degraded surfaces were examined using a high-resolution metallurgical optical microscope (OM) and a scanning electron microscope (SEM). The coefficients of friction (CoF) were obtained under the alternating ball-on-disk dry tribometer. The results of the experimental activity show that PEEK polymers have CoFs of about 0.22 and 0.16 under the 1 and 5 N applied load, respectively. The CoF and wear volume loss results are reported and compared to the reference Babbitt coating.

The occurrence of skidding and overskidding behaviour is likely to scratch rolling element raceways and lead to a significant temperature rise and thermal deformation. The ball-cage interaction and cage whirling motion are likely to oscillate as this skidding is aggravated. To address the damage problems introduced by skidding behaviour, a kinematic-Hertzian contact thermal-elasto-hydrodynamic (KH-TEHD) model with a 5-layer loop structure is proposed. Various factors are considered in the model, including the dynamic beahviour of bearing components, quasi-static analysis on bearing load distribution and deformation of bearing components, and the miscellaneous forces (unbalanced force, collision tangential friction, drag force) acting on the ball and cage. Ball-raceway lubrication, lubricant temperature rise, thermal expansion of shaft and bearing housings, and hydrodynamic pressure acting between the cage surface and guide-ring surface are established in this paper. A 7307AC bearing experimental study under varying operating speeds, axial loads and two lubricating conditions is conducted to validate the accuracy and effectiveness of the KH-TEHD model. The superiority of the developed model is illustrated by comparison with previous models, and the results suggest that the thermal expansion and cage-guide ring hydrodynamic pressure cannot be ignored. After comparisons with literature results, it is found that the overskidding degree is approximately proportional to the square of the bearing size under the same level of operating speed. The maximum thermal deformation at the bearing raceway under starvation lubrication reaches 4 times the ball-raceway clearance. The cage simulation results indicate that the cage whirling oscillation amplitude increases with increasing operating speed and unbalanced mass and generally decreases with increasing axial load. The proposed model and discussion of the results are useful for mitigating the skidding degree under certain conditions and avoiding cage instability and wear.

The symptoms of the propagation of shaft cracks in rotating machines, which can be brought out of the vibration data collected by condition monitoring systems, are often not very evident, especially when the crack depth is not very high. In fact, shaft cracks commonly propagate rather far from the cross-sections where the vibration probes have been mounted. At these measurement planes, the sensitivity of the machine dynamic behavior to the effects caused by a shaft crack can be fairly low. Therefore, it is very important to improve the capabilities of diagnostic techniques aimed at the early detection of shaft cracks. In this paper, the dynamic behavior of the cracked shaft of a HP-IP steam turbine is shown. The results of investigations on the influence of the periodic and non-periodic breathing of the crack on the rotor response, measured in different machine thermal states and at unusual cross-sections of the shaft, are shown and discussed.

The aim of this paper is to describe the diagnostics of bearing failures in rolling element bearings by means of special processing techniques applied to vibration data. The vibration data have been measured by using a test device, designed in accordance with EN 12082:2017. Two data sets, corresponding to two different bearings, employed for the wheel axle for railway application, have been analyzed, by applying the most suitable algorithms for the identification of the bearing defect and state. Some damage severity indices are also introduced. The results are described in detail in this paper.

Over the last few decades, the increasing demand on efficiency and performance for steam turbines has resulted in OEMs operating machines near critical conditions of their structural and thermal capabilities. In this paper, a new thermo-elasto bulk-flow model for labyrinth seals has been introduced. The model includes the bulk-flow model for estimating the dynamic coefficients, heat transfer model for evaluating the temperature distribution in the rotating and stationary parts and structural-mechanics model for calculating the radial growth. By considering a staggered labyrinth seal installed in the balancing drum of a steam turbine, different operating conditions, such as the boundary pressure, rotational speed and inlet pre-swirl ratio, as well as the stability of the seal are investigated in this study.

In Oil and Gas framework, the need of innovative turbomachineries in terms of efficiency, weight, and flexibility is quickly growing. For these reasons, the new LM9000 is entirely designed with aviation design criteria. It is composed by a gas generator derived from the GE90-115B twin shaft turbofan, coupled to a free power turbine (FPT). Both the gas generator and the FPT use rolling element bearings that contribute to reduce the weight and the complexity of the engine and its auxiliary systems architecture. The free power turbine represents the new element. Due to the FPT dimensions, weight and characteristics, it was necessary to realize a dedicated test bench that is in a class by itself in the Oil and Gas industry. To give an idea of the dimensions, the ball bearing installed has a bore diameter of 320mm with a ball diameter close to 60mm. The test bench was crucial to validate the sump configuration for on-shore and off-shore applications and to define the inputs range in which the FPT will operate properly. The first part of the paper aim at describing the solutions adopted to realize the test bench. The main results are reported in the second part of the paper. Moreover, the test bench provides flexibility to perform complex tests for different configurations thanks to its 20 tons allowable payload, considerable volume for the test article installation and its modular design.

Rotor balancing is probably the most discussed topic in the entire literature about rotor dynamics. It would therefore seem that, from the point of view of theory, this is a problem of little interest, however, balancing is very relevant in the industrial practice and sometimes there are very particular cases that cannot be addressed and solved by traditional methods. Moreover, many papers deal with only simulated results or with small-scale tests-rigs, which can hardly reproduce the behavior of real rotors. The case described in this paper is just one of these and presents what could be defined as “predicting the effect of balancing” at rotational speeds that are higher than those possible on balancing machines. Rotordynamics modeling, identification techniques developed by the authors and the available vibration measurements allow the simulation of the behavior, i.e. the vibrations, of the considered turbine rotor on the balancing machine, even at rotational speeds higher than those are possible to be reached, but that correspond to the trip speed in the plant.

The rolling element bearings, and gears are the main components of rotating machines and are most prone to defects which may result in significant economic loss. The main purpose of this study is an automated diagnosis of rolling element bearings and gears defects using machine learning (ML) technique and statistical features extracted from time domain vibration signal and spectral kurtosis. Extracted features are used to train K- nearest neighbors (KNN) as diagnostic classifier. The significance of segmentation size for time domain raw vibrational signals for the purpose of feature extraction is studied. This analysis is carried out by varying the window/segment length for features extraction and observing its effect on classification accuracy. Importance of feature selection for optimal performance of KNN in defect classification is studied by selecting most important and useful features using Genetic Algorithm (GA). Furthermore, effect of value of K on performance of KNN classifier has been observed by varying the value of K between 1 to 10 with step size of 1. Results show the ability of KNN classifier in combination with GA for correct and confident fault diagnosis of rotating machine elements in case of proper selection of parameters for features extraction.

Because tilting-pad journal bearings are more stable and efficient than conventional bearings, they have been commonly applied to many rotating machinery applications. Most of the studies about steady state and dynamic characteristics of tilting-pad journal bearings are usually evaluated by means of thermo hydrodynamic models assuming nominal dimensions for the bearing. However machining errors could lead to actual bearing geometry and dimensions different from the nominal ones. In particular for tilting-pad journal bearing the asymmetry of the bearing geometry is the principal cause of unexpected behavior. In this paper a theoretical analysis on dynamic characteristics of a five-pad tilting-pad journal bearing is investigated with non-nominal geometry, that is, different thickness for each pad. The dynamic coefficients of a five-pad tilting-pad journal bearing with a nominal diameter of 100 mm, length-to-diameter ratio (L/D) of 0.7 are evaluated versus rotor rotational speed, load direction and static load. Then, the analytical results of the non-nominal bearing are compared to those of a bearing having nominal (i.e. ideal) geometry.

The diagnosis of early-stage defects of rolling element bearings (REBs) using vibration signals is a very difficult task since bearing fault signals are usually weak and masked by shaft rotating signals, gear meshing signals, and strong background noise. Therefore, there are two important main points for bearing fault diagnosis, namely, extracting weak bearing fault signals and identifying the fault types. In this paper, a new bearing fault diagnosis method combining singular value decomposition (SVD) and the squared envelope spectrum (SES) is proposed. An optimal range for the number of rows in the Hankel matrix used in SVD for bearing fault diagnosis is obtained by means of numerical simulation. Then, the sub-signals obtained by SVD are grouped according to their similarity. Eventually, the composite SES is applied to identify the fault type. The performance of the new method is evaluated by means of the actual vibration signals collected from a full-scale test rig of a high-speed vehicle traction system.

Health condition analysis and diagnostics of rotating machinery requires the capability of properly characterizing the information content of sensor signals in order to detect and identify possible fault features. Time–frequency analysis plays a fundamental role, as it allows determining both the existence and the causes of a fault. The separation of components belonging to different time–frequency scales, either associated to healthy or faulty conditions, represents a challenge that motivates the development of effective methodologies for multi-scale signal decomposition. In this framework, the Empirical Mode Decomposition (EMD) is a flexible tool, thanks to its data-driven and adaptive nature. However, the EMD usually yields an over-decomposition of the original signals into a large number of intrinsic mode functions (IMFs). The selection of most relevant IMFs is a challenging task, and the reference literature lacks automated methods to achieve a synthetic decomposition into few physically meaningful modes by avoiding the generation of spurious or meaningless modes. The paper proposes a novel automated approach aimed at generating a decomposition into a minimal number of relevant modes, called Combined Mode Functions (CMFs), each consisting in a sum of adjacent IMFs that share similar properties. The final number of CMFs is selected in a fully data driven way, leading to an enhanced characterization of the signal content without any information loss. A novel criterion to assess the dissimilarity between adjacent CMFs is proposed, based on probability density functions of frequency spectra. The method is suitable to analyze vibration signals that may be periodically acquired within the operating life of rotating machineries. A rolling element bearing fault analysis based on experimental data is presented to demonstrate the performances of the method and the provided benefits.

Electrical discharges in bearings may occur in shaft lines equipped with electrical machines. This effect, known as electrical pitting, consists of the removal of material from the pad surface; this can lead to bearing pad failure mainly because of overheating and reduced load capacity. The case history of a fault caused by electrical discharges that occurred in the thrust bearing of a turbine is described in the first part of the paper. This phenomenon is then simulated by means of a thermal elastohydrodynamic model of the bearing, considering the effect of electrical pitting on the surface of the pad. Finally, the results of the simulations allow the irreversible failure of the bearing caused by the electrical pitting to be shown.

From a kinematic point of view, rolling elements should continuously roll on the raceways of rolling element bearings. When the dynamic behaviour is also considered, pure rolling occurs if the bearing is properly loaded and the system is correctly lubricated. In the absence of these conditions, the rolling elements may slide, or skid, from time to time. In the literature, this behaviour is well documented and occurs generally for low-load roller bearings. During a long-lasting experimental test on a large-scale industrial angular contact ball bearing (ACBB), not only skidding behaviour but also so-called over-skidding behaviour was observed on the rolling elements of the bearing. The term over-skidding, or negative-skidding, means that the cage/rotor speed ratio exceeds the value calculated under pure rolling kinematic conditions. To the best of authors’ knowledge, this phenomenon has not been fully described or analysed before. Therefore, a comprehensive model considering the kinematics of the bearing components, the Hertzian contact between the rolling elements and raceways, the interaction between the rolling elements and cage, the hydro-dynamic lubrication, and the thermal effects is introduced in this paper to study and forecast the over-skidding and skidding mechanisms. The model acronym is KH-THD, that is the kinematic-Hertzian-thermo-hydro-dynamic model. The empirical existence of over-skidding indicates that the use of the theoretical value for the cage/rotor speed ratio is inaccurate for determining whether the bearing rolling elements are slipping or not on the raceways, especially for large-scale industrial bearings. The results of the experimental tests on such kinds of bearings obtained by varying the load and under three different operating rotational speeds and two lubricant supply conditions suggest that the KH-THD model is more accurate than existing models, which neglect the thermal effects due to friction. The analysis of the friction thermal effects due to skidding shows that a considerable temperature gradient forms in the bearing. The increase in the lubricant flow rate can somehow mitigate the increase in temperature even though it can worsen the skidding. The proposed model is useful for determining this trade-off for a given load.

In the broad framework of condition-based maintenance, the final objective of bearing condition monitoring is to evaluate the health state and to estimate the remaining useful life of the bearings. The latter is a particularly challenging task, considering that remaining useful life is inextricably linked to a projection of what will happen in the future. Often, health indices, whose reliability relies on their effectiveness and consistency, are used for bearing condition monitoring. Most of the existing health indices pursue the property of monotonicity, but generally there are no obvious boundaries between the different health states of the bearings. Hence, it is quite difficult to give an objective and independent estimation of the health states, especially in real applications under different operating conditions and in the presence of noise. Furthermore, it is also difficult to set the failure threshold for a given health index when it is employed in different applications. In this paper, a new health index called ‘MAC2PSD’ is proposed, based on the moving average cross-correlation (MAC2) of the power spectral density (PSD) of the vibration signals. An interesting property of MAC2PSD is its capability to track the health condition and to discriminate clearly between the different health states. As shown in the paper, the MAC2PSD can also be used to estimate the remaining useful life by using its values during the defect-propagation phase. The effectiveness of MAC2PSD is shown by means of two different cases of bearing run-to-failure experimental data, from two different test rigs. Additionally, the capability to avoid false positives is evaluated by means of bearing vibration data measured on a locomotive in commercial service.

Tilting-pad journal bearings (TPJB) are widely installed in rotating machines owing to their higher stability given by negligible values of the cross-terms of dynamic coefficients if compared to full circular or elliptical oil-film journal bearings. TPJBs have also some drawbacks, like hot oil carry over, the risk of flutter in the unloaded pads and higher cost. The former leads to the increase in the temperature of the pads due to the shear stresses in the oil-film and can be reduced by adopting suitable oil nozzles and higher oil inlet flow-rates. The increase in the oil-film temperature leads to the reduction in the lubricant viscosity and, therefore, to the reduction in the oil-film thickness. Moreover, the increase in the pad temperature leads to high thermal deformations of the pad that correspond to changes in the oil-film thickness due to the change in the geometry of the pads. In general, the lower the temperature in the oil-film and pads, the higher the load that can be applied. Therefore, the reduction in the temperature of the lubricant is a key aspect for oil-film bearings. For high speed machines the adoption of a cooling system for the pads can reduce the temperature issues in the bearing. The case of a TPJB with pads having internal channels for a forced circulation of an external cooling fluid will be considered and investigated in this paper and the effectiveness of the system proved by means of numerical simulations.

Plain journal bearings are commonly equipping various rotating machineries from small electrical motors to large generators, turbines and pumps. They are preferable to rolling element bearings because of their long life performance, low cost, quiet operation, high damping properties, split design, and smaller outside dimension. In this paper the static and dynamic characteristics of a cylindrical journal bearing working in critical conditions, namely at very low speed and high specific pressure, are described. A bearing with the nominal diameter of 160mm and the length of 145mm was used. The rotational speed was fixed at very low speed of 60rpm while the applied static load in the vertical direction ranges in the Sommerfeld number from 0.0086 to 0.1314. Measurements of static position of the shaft, hydrodynamic pressure, temperature profiles, and dynamic coefficients have been presented and discussed.