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On the Derivative of Real Number Order and its Application to Condition Monitoring

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This paper examines a derivate x(α), the order of which is α, where α is an arbitrary real number. It is shown with practical examples that x(α) can be used in condition monitoring and that without it we can only obtain an erratic picture of condition in certain cases.
... The motivation of this study arose from a previous result which indicated that the order of derivative can have an effect on the results achieved via enveloping techniques [14]. Along with some earlier studies [15][16][17], it gave an idea that real order derivatives could contribute to recognizing repetitive events in vibration. We found this topic relevant, because no previous research that we know of has focused on the effect of the order of derivative on cyclostationarity analysis performed utilizing spectral correlation or spectral coherence. ...
... In this study, we utilized DFT when obtaining real order derivative using the definition Lahdelma gave in [16]. According to this definition, the real order derivative x (α) of the function x = Xe iωt can be expressed as: ...
... The case here is a fault in a slowly rotating roller bearing of type FAG 23 028-E1A-K-M + H3028. The test rig used for testing is shown in figure 2. We saw this as an interesting case example, because it has some similarities to cases studied before e.g. in [28,58], and these types of faults have been successfully detected using real order derivatives in [16,21,36] for instance. Below we have presented results on detecting the roller bearing fault from an intentionally damaged bearing rotating at 72.86 rpm. ...
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Various methods are used in the field of machine diagnostics for recognizing cyclostationarity in signals. The real order derivatives of vibration signals, however, have been rarely reported from the perspective of their effect on the performance of cyclostationarity detection methods. In this paper, we use real order derivatives together with spectral correlation, spectral coherence and squared envelope. Our results suggest that adjusting the order of derivative can enhance the analysis outcome of spectral correlation and squared envelope in particular. Remarkably, the results also suggest that squared envelope, when used alongside real-order derivatives, may replace spectral correlation and spectral coherence. This approach allows obtaining results with reduced computational power, making it advantageous for applications like industrial edge computing, where cost-effective hardware is crucial.
... The first time derivative of acceleration had been used earlier for assessing the comfort of travelling e.g. in designing lifts. Higher, real and complex order derivatives bring additional methods to signal processing [1,2,3,4,5,6]. Different approaches have been reviewed in [7,8]. ...
... All the other signals in Figure 2 are derived from these acceleration signals. In Figures 2b and 2f both the time signals x (4) are very similar. The reason is obvious, because these signals only differ in the low frequent unbalance component at 30 Hz. ...
... Figure 3 shows time signals from the two other classes: outer race fault and a combination of this with the unbalance. While the signals x (4) and x (2) from the outer race fault have a typical structure with relative constant peaks (Figs. 3a and 3b), the combinations of outer race and unbalance are less structured (Figs. ...
Conference Paper
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Advanced signal processing methods combined with automatic fault detection enable reliable condition monitoring for long periods of continuous operation. Any attempt to detect different types of machine faults reliably at an early stage requires the development of improved signal processing methods. Vibration measurements provide a good basis for condition monitoring. In some cases the simple calculation of root-mean-square and peak values obtained from vibration signals are useful features for detecting various faults. Unbalance, misalignment, bent shaft, mechanical looseness and some electrical faults, for example, can be detected using features of displacement and velocity. Higher order derivatives provide additional possibilities for detecting faults that introduce high-frequency vibrations or impacts. New generalised moments and norms related to lp space have been used for diagnosing faults in a roller contact on a rough surface. This paper extends the field of possible applications from roller bearing fault detection to more complex faults situations where different kinds of fault occur simultaneously. In consequence, feature calculation and signal processing have to be adopted and optimized for each fault type on the basis of one measured signal. The features of x(4) indicate well the intact case and the outer race fault. Velocity x(1) is needed for detecting unbalance. This approach also works for the combined case, outer race fault and unbalance. Derivation reduced the effect of noise by amplifying higher frequency components from bearing faults more than the added noise components.
... In this study, derivatives are first calculated in order to magnify the effects appearing in the signal. This is done according to the definitions by (Lahdelma, 1997) for real order derivatives. The actual vibration features are then extracted from the signal using the generalized norms introduced by (Lahdelma and Juuso, 2008). ...
... The processing was done in the frequency domain by manipulating the sequence of complex numbers resulting from fast Fourier transform (FFT). Lahdelma (Lahdelma, 1997) ...
... The integrals can be calculated similarly by using negative values of α. Furthermore, the value is not limited to integers but any real or complex number can be used (Lahdelma, 1997;Lahdelma and Kotila, 2005). Signals are filtered by multiplying the unwanted frequency components by zero. ...
... The spectrum on the right shows the high frequent vibration in 40 -60 kHz range shortly after the excitement. The severity of faults can be assessed by comparing signals and features in different orders of derivation [3,4,15]. For sinusoidal signals ...
... The measurements were performed in the frequency range 3-2000 Hz, and the rotation frequency was 2 Hz. For signals with a constant amplitude X, the derivation results increase with about the selection of the order of derivation is available in[3,4,5,15]. ...
Conference Paper
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Advanced signal processing methods combined with automatic fault detection enable reliable condition monitoring for long periods of continuous operation. Root-mean-square and peak values obtained from vibration signals are useful features for detecting various faults. Unbalance, misalignment, bent shaft, mechanical looseness and some electrical faults, for example, can be detected using features of displacement and velocity. Higher order derivatives provide additional possibilities for detecting faults that introduce high-frequency vibrations or impacts. Real order derivatives increase the number of signal alternatives. New generalised moments and norms related to lp space have been used for diagnosing faults in a roller contact on a rough surface. Kurtosis provides a strong indication if the order of derivation, α, is at least 4. For peak values, the change is smaller but already starts at α = 3. The generalised moment and norm can be defined by the order of derivation, the order of the moment, p, and sample time, τ. Reliable results can be obtained by relative norms if α and p are in the range between 4 and 6. For the impact of a small scratch, sensitivity is further improved with short sample times, but several sequential samples are required to guarantee the detection of impacts. Then the order p can also be reduced.
... This is due to the fact that 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) . Use of derivative x (α) allows stepless differentiation [3], [1], which means that it will be possible to move from the acceleration signal x (2) , for example, to signal x (4) via a number of intermediate stages [5], [6]. ...
... Sensitivity increases first with increasing order of the differentiation. After feature specific threshold sensitivity starts to decrease [5], [6]. High sensitivity is beneficial for early detection of faults. ...
Conference Paper
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Machine condition monitoring enables reliable and economical way of action for maintenance operations in modern industrial plants. Increasing number of measurement points and more demanding problems require automatic fault detection. Advanced signal processing methods exposed failures earlier and then it's possible to plan more operating time and less shutdowns. Intelligent methods have been increasingly used in model based fault diagnosis and intelligent analysers. Intelligent methods provide various techniques for combining a large number of features. A test rig was used to simulate different fault types and changes in operating conditions. Linguistic equation (LE) models were developed for the normal operation and nine fault cases including rotor unbalance, bent shaft, misalignment and bearing faults. Classification is based on the degrees of membership developed for each case from the fuzziness of the LE models. The classification results of the experimental cases are very good and logical. As even very small faults are detected by a slight increase of membership, the results are very promising for early detection of faults. Together with the compact implementation and the operability of the normal model, this makes the extension to real world problems feasible.
... Here we apply the definition in (11) for an α order derivative of an exponential function where α ∈ R. If we consider displacement x(t), which is usually represented by discrete series {x r } in computer applications (12) , the α order derivative of displacement can be calculated in three following steps (13) : ...
Conference Paper
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Periodically repeating shocks are a quite common indication of certain defect in machinery. Detecting these shocks in early stage, before the defect is severe enough to cause failure, can provide a huge advantage in maintenance planning. The earliest possible warning of a defect may be highly important, especially in targets where failure can lead to a vast loss of production or a safety risk. Shock-like vibrations are, however, usually rather faint when the defect is a minor one. This simply means that the shocks may often be too low in magnitude to be easily detected. In this paper, we use different techniques based on real order derivative to detect gear defects. Higher real order derivative, discrete Fourier transform, and Hilbert transform are discussed.
... where X is a constant representing amplitude, real number α is the order of derivative, ω is the angular frequency, e is the Napier's constant, i is the imaginary unit and t is a time variable. This definition was introduced by Lahdelma in (5) . ...
Conference Paper
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Roller bearing faults are a common fault type in machines. There are several condition monitoring handbooks which describe ways of detecting these faults by means of vibration measurements. Handbooks on neither condition monitoring nor vibration analysis usually mention real order derivatives or cyclostationary technique. These methods have rarely been applied to vibration analysis in industrial condition monitoring , although plenty of studies on both of them have been published. In this study, these methods are applied on the fault detection of a roller bearing.
Conference Paper
Fault diagnosis can utilise signal processing in various ways: informative time domain signals are separated or transformed from the source signals and used in feature extraction to find useful fault indicators. Frequency domain analyses focus on identifying interesting signal components. This research integrates different solutions together: differentiation is used for modifying the source signal and feature extraction is based on the generalised spectral norms which have been studied by the authors before. The concept of these norms is intended to give a method of spectral analysis with less human labour and, potentially, even entirely without it. We discuss the possibility of utilising the spectral norms along with real order derivatives. The solution was tested for bearing and misalignment faults induced in two test grids. The potential challenges of selecting parameters and interpreting results are discussed. Results show that when used with a properly selected set of indices, the presented method can be an effective tool for condition monitoring. Moreover, this technique provides an opportunity for automated analytics also for frequency domain analyses.KeywordsFault detectionGeneralised spectral normReal order derivative
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The papers published in these proceedings are presented in the Second International Seminar on Maintenance, Condition Monitoring and Diagnostics, to be arranged in Oulu, Finland, in 28th – 29th September, 2005. Arranged by the University of Oulu and POHTO – The Institute for Management and Technological Training, the present seminar is supported by a variety of Finnish industrial enterprises.
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