Bayesian analysis of rotating machines  A statistical approach to estimate and track the fundamental frequency  Thorkild Find Pedersen
 Abstract  Rotating and reciprocating mechanical machines emit acoustic noise and vibrations when they operate. Typically, the noise and vibrations are concentrated in narrow frequency bands related to the running speed of the machine. The frequency of the running speed is referred to as the fundamental frequency and the related frequencies as orders of the fundamental frequency. When analyzing rotating or reciprocating machines it is important to know the running speed. Usually this requires direct access to the rotating parts in order to mount a dedicated tachometer probe.
In this thesis different frequency estimation techniques are considered for predicting the true fundamental frequency from measured acoustic noise or vibration signal. Among the methods are autocorrelation based methods, subspace methods, interpolated Fourier transform methods, and adaptive filters. A modified version of an adaptive comb filter is derived for tracking nonstationary signals.
The estimation problem is then rephrased in terms of the Bayesian statistical framework. In the Bayesian framework both parameters and observations are considered stochastic processes. The result of the estimation is an expression for the probability density function (PDF) of the parameters conditioned on observation. Considering the fundamental frequency as a parameter and the acoustic and vibration signals as observations, a novel Bayesian frequency estimator is developed. With simulations the new estimator is shown to be superior to any of the previously considered estimation techniques.
Within the Bayesian framework, two schemes for tracking the fundamental frequency are proposed. For both schemes the tracking capability is defined as a PDF of the next frequency estimate conditioned on the previous estimate(s). The first scheme works in realtime and is not guaranteed to find the optimal track, i.e., the track with highest probability. The second scheme is retrospective and requires all observation to be available, but it is guaranteed to find the optimal track.
The Bayesian estimator can also estimate the amplitude of nonstationary frequencies. An example of this is given, where the amplitudes of the orders of the fundamental frequency are estimated. The result is found comparable with existing amplitude order tracking methods.  Type  Ph.D. thesis [Academic thesis]  Year  2003  Publisher  Informatics and Mathematical Modelling, Technical University of Denmark, DTU  Address  Richard Petersens Plads, Building 321, DK2800 Kgs. Lyngby  Series  IMMPHD2003117 (EF 812)  Note  Electronic version available to restricted users  Electronic version(s)  [pdf]  BibTeX data  [bibtex]  IMM Group(s)  Intelligent Signal Processing 
