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Coupling misalignment is a fairly common fault in rotating machines and causes excessive loads to bearings and other machine components. In general, misalignment and its severity can be detected reliably using vibration measurements. Misalignment is typically detected on the basis of the velocity spectrum. An increase in amplitude at rotational frequency and its harmonics are an indication of misalignment or a bent shaft. However, the behaviour of a claw clutch (jaw coupling) is very different as compared with a conventional elastic coupling. This paper presents the results of research where the detection of claw clutch misalignment is investigated based on acceleration signals. Real order derivatives and lp norms are calculated from the signals. In a test rig, which is used in this study, different degrees of misalignment are generated by moving a motor in the horizontal direction with steps of 0.1 mm. The tests were carried out at five rotational frequencies. Studies show that after a certain level of misalignment, the claw clutch gives rise to impacts that induce high frequency vibrations.

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Content uploaded by Sulo Lahdelma

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All content in this area was uploaded by Sulo Lahdelma on Jan 27, 2016

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... In references (1,2,3,4) Lahdelma et al. have defined real order derivatives and integrals of function (1) by replacing n ∈ N with a real number α ∈ R ...

... Norm (17) can be generalised with weight factors, which we choose to be 1/N (3,4) . Generalised norm l p is 1 N weighted l p norm ...

... Lahdelma presented in 1992 (19) the measurement index, or M IT index, utilising rms values of displacement and its derivatives and integrals of order n ∈ N. Later it has been generalised to l p norms and real order derivatives and integrals (3,4) . Thus it is formulated as ...

Fractional calculus and generalised norms provide a powerful toolkit for analysing vibration from rotating machines. They have been used effectively in the condition monitoring of immobile machines. A harsh environment and varying operating conditions complicate the reliable condition monitoring of mobile machines in underground mining industry. In this paper, we focus on the condition monitoring of the front axle of a load haul dumper and the numerical calculation of complex order derivatives and integrals. Measurements are performed with accelerometers, which measure horizontal and vertical vibrations near the planetary gearboxes. A tachometer records the rotational speed of the drive shaft, which is essential for recognising different operation stages of the machine. We describe how the mentioned difficulties can be overcome and what real order derivatives and generalised norms can reveal about the condition of the axle. An improved algorithm for the numerical calculation of complex order derivatives and integrals is given.

... The motor and the driven shaft are coupled by means of a claw clutch with a four-tooth elastic element (spider). More information on the test rig can be found in Lahdelma and Laurila, 2012), where the same test rig has been used. ...

... Coupling misalignment of the claw clutch causes impacts and invokes high frequency vibrations as shown in earlier studies (Laurila and Lahdelma, 2013;Lahdelma and Laurila, 2012). Figure 5 shows projections of the signals x (3.4+i) in the z plane, when different rates of coupling misalignment occur. ...

... The weighted l p norm was utilised in the vibration signal analysis in order to evaluate the levels of vibration signals and the stress of the cutter. It is defined by (Lahdelma and Laurila, 2012), ...

The papers published in these proceedings are presented in the International Conference on Maintenance, Condition Monitoring and Diagnostics, and Maintenance Performance Measurement and Management, MCMD 2015 and MPMM 2015, to be arranged in Oulu, Finland, in 30th September – 1st October, 2015.
Arranged by the University of Oulu and POHTO – The Institute for Management and Technological Training, the present conference is supported by a variety of Finnish industrial enterprises.

... More information on the test rig and the original measurements can be found in (31) . We have studied the detection of the offset misalignment of a claw clutch in (32) . Figure 1 shows the relative peak values of the x (3) signals in the case of motor offset and rotational frequency changes. ...

... The relative jerk peak or x(3) p values when the rotational frequency of the motor and offset misalignment changes(32) . ...

The processing of measured signals plays a very important role when information is needed on a physical phenomenon or machine faults. The differentiation of a signal is one of the basic methods used in many signal processing applications. The order of derivative is typically an integer though it can also be any real or complex number. This paper discusses the application of real and complex order fractional differentiation. The aim is to bring out different applications of these derivatives, especially in the field of condition monitoring.

... The motor and the driven shaft are coupled by means of a claw clutch with a four-tooth elastic element (spider). More information on the test rig can be found in (7,8) , where the same test rig has been used. ...

... For example, when the unbalance mass is doubled from 5.5 g to 11 g, the diameter of the circle is doubled too from 200 µm/s 0.4+i to 400 µm/s 0.4+i . Coupling misalignment of the claw clutch causes impacts and invokes high frequency vibrations as shown in earlier studies (4,8) . Figure 5 shows projections of the signals x (3.4+i) in the z plane, when different rates of coupling misalignment occur. ...

There is a wide range of signal processing methods in the field of machine condition monitoring, which are used in feature extraction and fault diagnosis. Derivation and integration are very common and important operations in vibration signal processing. Often a signal that has been measured using an accelerometer is integrated to obtain velocity or displacement signal. These commonly used three signals and an infinite number of other signals can be obtained by means of fractional order derivation. Especially in challenging fault and process cases, signals whose order of derivation is a real or complex number could be clearly more sensitive to fault detection than the commonly used signals. This paper discusses the application of complex order derivation to acceleration signals in order to detect various faults at an early stage.

... The motor and the driven shaft are coupled by means of a claw clutch with a four-tooth elastic element (spider). More information on the test rig can be found in (Lahdelma et al., 2011;Lahdelma and Laurila, 2012), where the same test rig has been used. ...

... Coupling misalignment of the claw clutch causes impacts and invokes high frequency vibrations as shown in earlier studies (Laurila and Lahdelma, 2013;Lahdelma and Laurila, 2012). Figure 5 shows projections of the signals x (3.4+i) in the z plane, when different rates of coupling misalignment occur. ...

The reliable condition monitoring of machines and the early detection of faults play an important role in condition-based maintenance. Information on the condition of machines is needed in different forms but finally it has to be in as simple form as possible so that it can be utilised in the decision-making by the maintenance and management personnel. In the field of machine condition monitoring, there is a wide range of signal processing methods, which are used in feature extraction and fault diagnosis. Differentiation and integration are very common and important operations in vibration signal processing. The order of derivative is typically an integer number but it can also be any fractional number. Especially in the challenging fault and process cases, signals whose order of derivative is a real or complex number could be clearly more sensitive in fault detection than the commonly used signals: displacement, velocity and acceleration. In this paper, the application of complex order differentiation of acceleration and sound signals was discussed. With the help of them, it is possible to find good and sensitive new types of methods and indicators for different fault detection situations.

... The weighted l p norm has been applied for several cases of fault detection. This norm has proved to be an effective tool in condition monitoring and it can be applied to any type of signal (4,9,10,11,12) . It is defined by ...

... Moreover, we end up with (5), when w 1 = w 2 = ... = w N = 1 N . This is the l p norm, which Lahdelma introduced in (12) , and it is defined by ...

In condition monitoring detecting changes in measured signals is often desired.
Another crucial piece of information is the frequency which shows the change. In
industrial applications today, it is quite common to monitor some feature e.g. RMS
value, and for information about the frequency content the frequency spectrum is
visually analysed. The RMS is in fact a special case of the lp norms, which have
shown to be a quite useful way to monitor changes in signals. While effective for
detecting changes, the l p norms calculated from the time domain signal fail to answer
the question in which frequency range the changes are. Calculating spectral norms
is a way to perform both of these tasks simultaneously and effectively.

... Unstable running of an electric motor [26,62,67] Bearing faults [26,[66][67][68][69]72,[84][85][86][87][88][89][90]93,94,98] Bent shaft [88] Misalignment [88,91,92] Cavitation [95][96][97] Poor lubrication [91] A laboratory experiment is examined in which automatic diagnostics were used with acceleration and jerk signals. The outer race of a rolling bearing contained the faults shown in Table 6 [69,87,94] . ...

The time derivatives of acceleration offer a great advantage in detecting impact-causing faults at an early stage in condition monitoring applications. Defective rolling bearings and gears are common faults that cause impacts. This article is based on extensive real-world measurements, through which large-scale machines have been studied. Numerous laboratory experiments provide additional insight into the matter. A practical solution for detecting faults with as few features as possible is to measure the root mean square (RMS) velocity according to the standards in the frequency range from 10 Hz to 1000 Hz and the peak value of the second time derivative of acceleration, ie snap. Measuring snap produces good results even when the upper cut-off frequency is as low as 2 kHz or slightly higher. This is valuable information when planning the mounting of accelerometers.

... where all the weight factors are equal to N 1 , has been proved to be an efficient indicator in condition monitoring Juuso, 2011a, 2011b;Lahdelma et al., 2010;Lahdelma and Laurila, 2012). ...

Epicyclic gearing or planetary gearing is a gear system that consists of one or more outer gears, or planet gears, revolving around a central, or sun gear. Typically, the planet gears are mounted on a movable arm or carrier, which itself may rotate relative to the sun gear. It is challenging to monitor epicyclic gearboxes, due to their complex structure consisting of many rolling elements. A complex structure and versatile components also result in a long stoppage if a failure occurs. Therefore, it is important to detect incipient faults at an early stage. In this paper, vibration analysis is used for the condition monitoring of an epicyclic gearbox at a water power station. There is a distinct difference between vibration quantities: vibration velocity responds very well to vibrations with frequencies less than 1000 Hz; an even better response is obtained when using acceleration and its higher derivatives, which also provide more information on higher frequencies. Because of the quite high rotational speed of the output, vibration velocity is not good enough for the condition monitoring of the gear in question. Acceleration and its higher order derivatives should be used in order to obtain better responsiveness to changes in the condition of the gearbox. Complex models based on mechanisms are needed in order to calculate the vibration components in the frequency range and to identify the possible faulty components. There was also one vibration component in the gear the source of which could not be discovered with certainty.

The jaw coupling with a flexible spacer is frequently used in the torque transmission between shafts with misalignment for machinery. Its torsional stiffness and limit torque closely determine the operational capacity and the dynamic characteristics of the system because the coupling is usually the most flexible link in the driving chain. In this study, the optimal design of a jaw coupling with an elastomeric spacer was investigated using the Taguchi method by considering four design factors: the tightening method of the clamping bolts, the tightening torque of the clamping bolts, the hardness of the thermoplastic polyether ester elastomer spacer and the installation angle between the two end blocks of the coupling. All specimens were tested by using an in-house torsion tester to record the torque-angular deformation responses. The results showed good reliability and repeatability, with a coefficient of variance within 5%. Spacer's hardness was found to be the most significant factor regarding the torsional stiffness, while the magnitude of the clamping torque had the most critical role in the limit torque. The estimation formulae for the torsional stiffness and limit torque of the jaw coupling were obtained by using the statistical regression of the measured data, respectively. Both formulae predicted the performance of the optimal designs within 5% of error compared to the confirmation tests.

Acquiring signals without disturbance is a crucial part of condition monitoring. This paper discusses some observations concerning disturbance in signals caused by two different frequency converters when measuring acoustic emission and acceleration. Signals are acquired by means of two acoustic emission and acceleration sensors. The test rig includes an electric motor and a worm gear. Measurements are performed when the motor is driven by a frequency converter at 594 and 1494 rpm. The results are compared to the motor connected directly to the power line. In acoustic emission signals one frequency converter caused a repeating peak in the spectrum at intervals of 40 kHz. The other frequency converter produced a general rise in amplitudes from about 200 to 400 kHz. When measuring acceleration in the frequency range 0...30 kHz the signals show no significant distortion.

The investigation provides new criteria for use in the evaluation of the results obtained from vibration measurements performed as part of condition monitoring. The author sets out from practical long-term tests which involved paper machines and woodpulp manufacturing machines in a number of mills, pointing out reasons for which displacement time derivates of a higher order than acceleration x(2) should be employed for technical diagnostic purposes. Practical results are presented for the measurement parameters x(2), x(3) and x(4).

Advanced signal processing methods combined with automatic fault detection enable reliable condition monitoring for long periods of continuous operation. The signals x(3) and x(4) are very suitable for the condition monitoring of slowly rotating bearings since rapid changes in acceleration become emphasised upon the differentiation of the signal x(2). Real order derivatives x(α) provide additional possibilities, e.g. in a bearing fault case the sensitivities of some features have been found to reach a maximum when α = 4.75. In earlier cavitation indicators, the rms values were combined with either kurtosis or peak values. Generalised moments τMαp can be defined by the order of derivation (α), the order of the moment (p) and sample time (τ). The moment normalised by standard deviation can be used as kurtosis in the model-based analysis.
This paper introduces a new norm ‖τMαp‖p = (τMαp)1/p = [1/NΣi=1N|xi(α)|p]1/p, where the orders α and p are real numbers. The number of signal values s N =τ N where Ns is the number of samples per second. The new norm has the same dimensions as the corresponding signals x(α ) . The cavitation of a Kaplan water turbine was analysed in the power range 1.5…59.4 MW based on measurements collected with sampling frequency 12800 Hz. The order p was compared in the range from 0.25 to 8 with a step 0.25, and a total of 11 sample times were used: τ = 1, 2,…, 6, 8, 10, 20, 30, and 40 seconds. An optimum order p was detected for each sample time τ. The relative max(‖3^M_4^2.75‖) compared to a cavitation-free case is alone a good indicator for cavitation: strong cavitation and cavitation-free cases are clearly detected, and the power ranges for shortterm cavitation are only slightly wider than in the previously developed knowledge-based cavitation index. Short sample times and relatively small requirements for the frequency ranges make this approach feasible for on-line analysis and power control. The weighted sums of features on a different order of derivation form fault-specific measurement indices MIT, and several indices MIT define the health index SOL. The sensitivity increases with the order of derivation to some limit of α. For faults causing impacts, high MIT values for lower orders of derivation indicate an increase in the severity of the fault.

Any attempt to detect different types of machine faults reliably at an early stage requires advanced signal processing methods. It is well known that vibration measurements provide a good basis for condition monitoring and fault detection. The complexity of signal processing techniques depends on the type of fault. In many cases root mean square and peak values are useful features for fault detection. Unbalance, misalignment, bent shaft, mechanical looseness and some electrical faults, for example, can be detected using features of displacement and velocity. Advanced filter settings and higher order derivatives provide additional possibilities if faults cause high frequency vibrations or impacts. This paper presents results from investigations about various faults, e.g. unbalance, misalignment, resonance, cage fault, absence of lubrication in a ball bearing and their combinations. To detect the faults, also higher order derivatives, lp norms and dimensionless measurement index MIT are utilised.

A combined signal processing and feature extraction approach can be based on generalised moments and norms, which are defined by the order of derivation (α), the order of the moment (p) and sample time (τ), where α and p are real numbers. The analysis can be further improved by taking into account nonlinear effects by monotonously increasing scaling functions. Several features can be combined in condition indices and, in some cases, only one feature is needed if the orders p and α are chosen properly. The aim of the application analysis is to test new features and whether their sensitivity is sufficient to detect faults from the absolute values of the dynamic part of the signals x(α) at an early stage when the faults are still very small. For faults causing impacts, the sensitivity of the features is clearly improved by higher and real order derivation, and low order norms can be used if the order of derivation is sufficient. Correspondingly, subharmonic vibrations can be amplified by integration. A limit of sensitivity is reached on a certain order α, and on each order α an optimal sensitivity is chosen by the order p. Both the orders can be chosen fairly flexibly from the optimal area. Sample time, which connects the features to the control application, is process-specific. The analysis methods allow the use of lower frequency ranges, and a multisensor approach can also be used. The approach is well suited for rotating process equipment with speeds ranging from very slow to very fast. In this article, the types of slow rotating equipment are a lime kiln, a washer and the scraper of a digester, all from pulp mills. A centrifuge and a turbo compressor are examples of very fast rotating machines. Features can also be combined in stress indices, which react to harmful process conditions. For example, strong cavitation, cavitation-free cases and even short-term cavitation are clearly detected in a Kaplan water turbine. Several features, also from different frequency ranges, can be combined in condition and stress indices.

Advanced signal processing methods combined with automatic fault detection enable reliable condition monitoring for long periods of continuous operation. Rapid changes in acceleration become emphasised upon derivation of the signal x(2). Higher order derivatives, especially x(4), work very well in the whole range from slowly to very fast rotating rolling bearings. Real order derivatives x(α) provide additional possibilities. If α < 0, we can also talk about the fractional integration of displacement. It has interesting application potential for faults that occur in frequencies below the rotating frequency. The aim of the analysis is to introduce new features whose sensitivity is sufficient to detect faults from the absolute values of the dynamic part of the signals x(α) at an early stage, ie when the faults are still very small. Generalised moments τMαp and norms, ‖τMαp‖p = (τMαp)1/p = [1/NΣi=1N|xi(α)|p]1/p, can be defined by the order of derivation (α), the order of the moment (p) and sample time (τ), where α and p are real numbers. The number of signal values N = τNs, where Ns is the number of signal values taken in a second. The norm ‖τMαp‖p has the same dimensions as the corresponding signals x(α). Both the order of derivation and the order of the moment improve sensitivity to impacts. There are many alternative ways to normalise the moments: absolute mean deviation is an easy solution and normalisation by standard deviation generalises the moments to standardised variables. Dimensionless vibration indices obtained using different orders α and p can be combined in measurement indices. The analysis can be further improved by taking into account non-linear effects by monotonously increasing the scaling functions. Several features can be combined in condition indices and, in some cases, only one feature is needed if the orders p and α are chosen properly.

Advanced signal processing methods combined with automatic fault detection enable reliable condition monitoring even when long periods of continuous operation are required. The parameters x(3) and x(4) are very suitable for the condition monitoring of slowly rotating bearings, as although the acceleration pulses are weak and occur at long intervals, the changes in acceleration are rapid and become emphasised upon differentiation of the signal x(2). Grounds for the need of x(-n) signals, ie integration of displacement n times with respect to time, have been indicated. In addition, derivatives where the order is a real number or a complex number + i have been developed. These signals can be utilised in process or machine operation by combining the features obtained from the derivatives. The importance of each derivative is defined by weight factors.
Dimensionless indices are obtained by comparing each feature value with the corresponding value in normal operation. These indices provide useful information on different faults, and even more sensitive solutions can be obtained by selecting suitable features. Widely used root-mean-square values are important in many applications, but the importance of the peak values increases in slowly rotating machines. Further details can be introduced by analysing the distributions of the signals. The features are generated directly from the higher order derivatives of the acceleration signals, and the model can be based on data or expertise. The intelligent models extend the idea of dimensionless indices to nonlinear systems. Variation with time can be handled as uncertainty by presenting the indices as time-varying fuzzy numbers. The classification limits can also be considered fuzzy. The reasoning system will produce degrees of membership for different cases. Practical long-term tests have been performed e.g. for fault diagnosis in bearings, cogwheels, gear boxes, electric motors and supporting rolls, and for cavitation in turbines and pumps.