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, i.e. integration of displacement n times with respect to time, have been indicated. In addition, derivatives where the order is a real number α or a complex number α+βi have been developed. These signals can be utilized 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-meansquare 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 longterm 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.

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Cavitation is harmful to water turbines and may cause operation delays of several weeks. The real-time detection of cavitation risk is increasingly important, and even narrow cavitation-free power ranges can be utilised in load optimisation. Higher derivative signals x (3) and x (4) calculated from acceleration signals are very suitable for detecting impacts. This paper introduces a generalised moment τ σ M p α which is defined by three parameters: the sensitivity of the moment improves when the order p of the moment increases, especially when short sample time τ is used. In this study, sufficently good results were obtained with moments where the order of derivation α =4 ,p ≈ 4, and τ =3s. These moments detect the normal operating conditions, which are free of cavitation, and also provide a clear indication for cavitation risk at an early stage. Sufficiently long signals are required for producing reliable maximum moments and data for analysing short-term cavitation. On-line cavitation monitoring is feasible with this approach since the analysis does not need high frequency ranges and the sample times are very short. The moment can be analysed first, and it is then possible to obtain the cavitation index if the moment value exceeds the threshold. Data compression is very efficient as the detailed analysis only requires the feature values of the appropriate samples.

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