Estimation of the instantaneous rotation speed using complex shifted Morlet wavelets Konstantinos C. Gryllias n , Ioannis A. Antoniadis Dynamics and Structures Laboratory, Machine Design and Control Systems Section, School of Mechanical Engineering, National Technical University of Athens, Greece article info Article history: Received 20 January 2012 Received in revised form 20 June 2012 Accepted 22 June 2012 Available online 26 July 2012 Keywords: Complex shifted Morlet wavelets Condition monitoring Instantaneous rotation speed estimation abstract The ability of the complex continuous wavelet transform (CCWT) to provide also an estimation of the instantaneous frequency of a signal, parallel to the estimation of the instantaneous amplitude of the signal, is proposed as an approach for the estimation of the instantaneous rotation speed of machinery. Complex shifted Morlet wavelets (CSMW) present a number of advantages. The concept of shifting the Morlet wavelet in the frequency domain allows the simultaneous optimal selection of both the wavelet center frequency and the wavelet bandwidth. In this paper it is shown that the recovery of the signal frequency can be performed accurately, without the requirement that the wavelet center frequency coincides to the signal frequency. Contrarily, the accurate recovery of the signal amplitude requires additionally this last condition. The algorithm is tested on two synthetic signals and four non-stationary experimental vibration signals, in an experimental fault test rig and in a motorcycle engine. The proposed instantaneous frequency estimation approach presents very good results and in comparison to the Hilbert Transform achieves a significantly lower RMSE. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction The estimation of the instantaneous speed of rotating machinery under non-stationary operating conditions (e.g., under time-varying speed and/or load) is fundamental for a plethora of mechanical engineering applications, including condition monitoring, real time control, etc. However, the direct measurement of the rotational speed is not always easy or even possible, either due to difficult ambient conditions or due to shaft accessibility problems. In order to overcome this limitation, a number of signal processing methods can be applied for the extraction of the instantaneous rotating speed indirectly from vibration measurements. These signal processing methods have mainly been developed for the estimation of the instantaneous frequency (IF) of a time signal. The estimation of the IF has been an issue of intensive research [3] due to its importance in a significant number of applications other that machine condition monitoring, such as communications, radars, speech processing, seismography, biomedical applications etc. The IF estimation algorithms can be grouped as phase differencing methods, signal modeling methods, phase modeling methods and time-frequency-representation methods [4]. The authors [3,4] contributed an interpretation of the IF concept as well as a comprehensive review of the literature about IF estimation methods published until then. In general, the IF estimation methods can be classified into five main Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ymssp Mechanical Systems and Signal Processing 0888-3270/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2012.06.026 n Corresponding author. E-mail addresses: kosgryl@central.ntua.gr (K.C. Gryllias), antogian@central.ntua.gr (I.A. Antoniadis). Mechanical Systems and Signal Processing 38 (2013) 78–95