Real Order Derivatives and Generalised Norms in Condition Monitoring with Noisy Data Sulo Lahdelma 1 , Esko Juuso 2 and Jouni Laurila 1 1 Mechatronics and Machine Diagnostics Laboratory, Department of Mechanical Engineering, P.O.Box 4200, FI-90014 University of Oulu, Finland Phone: +358-8-5532083 E-mail: sulo.lahdelma@oulu.fi 2 Control Engineering Laboratory, Department of Process and Environmental Engineering, P.O.Box 4300, FI-90014 University of Oulu, Finland E-mail: esko.juuso@oulu.fi Abstract Rapid changes in acceleration become emphasised upon the derivation of the acceleration signal ) 2 ( x . Higher order derivatives work very well in the whole range from slowly to very fast rotating rolling bearings. Real order derivatives ) (α x provide additional possibilities, and generalised norms with the real order p are used in feature extraction. The optimum setting of the orders α and p is fault-specific. Added random noise takes the form of additional fault, which in this case makes it more difficult to detect misalignment of a claw clutch by means of acceleration. Derivation reduces the effect of noise by amplifying the higher frequency components of misalignment more than the added noise components. Correspondingly, integration amplifies the lower frequency components. The analysis is fine-tuned with generalised norms. Added noise changes the optimal setting of the orders α and p: high order α combined with high order p operates well in misalignment detection, and negative order α combined with low order p has good sensitivity for detecting unbalance. The results clearly show that an extended analysis with a wide range of orders α and p is needed for the detection of simultaneous faults in order to obtain the best sensitivity for specific faults, also if the measurements are noisy. The power of generalised norms is in selecting the amplified frequency ranges by the order of derivation and in fine-tuning the sensitivity with the order of moment. Keywords: Condition monitoring, real and higher order derivatives, fractional derivatives, feature extraction, generalised norms, noise reduction 1. Introduction Any attempt to detect different types of machine faults reliably at an early stage requires the development of improved signal processing methods. The mathematical background of fractional derivatives has a very long history that goes back to Leibniz, Johann Bernoulli and L’Hospital. The theory of signal processing and ideas for practical problems and more about the history of fractional integrals and derivatives can be found in (1) . Practical condition monitoring application use vibration measurements, which were started by means of mechanical or optical instruments, used displacement The Ninth International Conference on Condition Monitoring and Machinery Failure Prevention Technologies