An Overview and Efficiency Analysis of Dominant Frequencies Estimation Methods with the Application of DIRECT Global Minimization Method

: This report discusses the application of the following methods that are used to calculate maximum frequency of a spectrum: parabolic, modified parabolic by Rife and Vincent, the method according to Quinn from 1994 and the method according to Quinn from 1997. After introducing DIRECT global minimization method, the results obtained using previously mentioned methods are compared with the results obtained using DIRECT global minimization of the Lipschitz function. Determining vibration frequency and comparing to previously collected data or knowledge base data is a common procedure. In engineering practice it is often necessary to determine not one, but several so - called dominant frequencies or local maxima; the procedure is sometimes called multi peak detection and here we propose an algorithm for computerised multi peak detection.


INTRODUCTION
Monitoring vibration levels is an effective method of forecasting and detecting mechanical failures in rotating components of mechanical systems. Measurements are carried out periodically using hand held portable devices or continuously by means of stationary systems. Measured vibration level may indicate irregularities caused by incorrect assembly even before the occurrence of failure [1]. Ageing of system mechanical components, appearance of wear and minor damage will increase the level of amplitude of certain frequencies; therefore first step in fault detection is interpretation of the measured amplitude and frequency of vibration. Measured data can be used to determine the condition of a mechanical component in exploitation, predict the occurrence of damage and identify type of fault based on the analysis of measured vibration. In cases where there are no specially adapted stationary systems that monitor vibrations of equipment, maintenance staff often uses a visual method to compare measured data and knowledge base that contains spectrum forms of vibrations caused by various damages, classified by type of rotating components and the type of damage.
During visual comparison it is necessary to note and determine any dominant frequencies and their amplitudes. The measured data in time domain is converted to the frequency domain, using Discrete Fourier transform i.e. frequency spectrum is obtained from amplitudes measured in a given period of time using the discrete Fourier transform.
Values of the dominant frequency, phase and amplitude are obtained from the spectrum using one of the known algorithms; see for example [2,3].

INITIAL ASSUMPTIONS
Assume measuring vibration of the electro motor housing, which occurs as a result of its operation. Let (tn, y n ), n = 0, …, m be measured data, while t n is time range, (usually equidistant range of values) on y n corresponding measured displacement values of a fixed point on an operating motor housing, directed as one unit vector.
in vibration analysis is often used Hanning window [6], defined as [ ]  From numerical aspect the problem (6) is extremely complex as the function A w has a large number of local extrema, and this number is growing by increasing data quantity m. In the literature, various procedures can be found on local approximation of periodogram A w aiming to find a global maximum of the function A w . In this article overview of some such functions is given and using numerical experiment their efficiency is examined, Furthermore it is shown that function A w is a Lipschitz class function and for the solution of problem (6) can be used DIRECT method for global optimization of Lipschitz functions [7][8][9]. Such obtained estimator is compared to other methods estimators and its efficiency is determined. Also in order to hasten estimation process DIRECT method is combined with a local method to narrow maximization scope and it will be determined if results can be enhanced comparing the DIRECT method with the mentioned local approximation methods. Finally, based on the numerical experiment findings the method for determining dominant frequencies will be proposed.

OVERVIEW OF LOCAL APPROXIMATION METHODS FOR THE DETERMINATION OF DOMINANT FREQUENCIES
Within this section an overview will be given of several methods used to determine dominant frequencies based on periodogram approximation. Assuming given measurement data is (tn, y n ), n = 0, …, m -1, window function, corresponding periodogram A w , let us define function 1 : 0, 2 Function I w will be called modified periodogram.

Discretization of Modified Periodogram
We consider values of modified periodogram ( ) Notice: [ ] According to this approach maximization problem of real function of real variable is reduced to maximization problem of discrete set of values and therefore for approximation of global maximum * w f of function, is adopted Approximation of amplitude and matching phase is obtained from system of Eq. (7) and (8). While it is fair to expect that such an approximation will not be good enough ( Whole class of methods is based on this approach, some of them described in the text below.

Parabola method
In order to determine dominant frequency, parabola method is proposed in articles [11] and [12], described in this chapter.
, function w I is approximated through three points: where it is: namely 0 a < . Thereby quadratic function P shall reach its maximum in point Approximation of global maximum using parabola method is shown in Fig. 3, parabola was drawn through three points, the maximum of discretized function and two neighbouring points, and approximation * w q is shown.

Figure 3 Parabola method
Approximations for amplitude and matching phase shift are obtained from equation system (7) and (8).

Modified Parabola Method
In works [13] and [14] another global maximum approximation method based on approximation of modified periodogram in case when m is large enough is suggested. Below the method is formulated in analogy to previous chapter.
In proximity of point approximated by parabola through three points, while detecting two cases: , (I ( ) I ( )) , 4 for which applies: Note that 0 a < and the quadratic function P reaches maximum at point where it applies where it is Approximations for amplitude and matching phase shift are obtained from equation system (7) and (8).

Quinn's Period Maximization Method
Two more effective methods for determination of dominant frequencies are suggested in articles [14] and [15]. Without step by step formulation and additional analysis we list these two methods.
Let it be ( ) Approximations for amplitude and matching phase shift are obtained from equation system (7) and (8).

GLOBAL METHOD FOR DOMINANT FREQUENCY ESTIMATION
Function w A maximization problem is equal to the following function minimization problem that it applies (16). Motivated by (18) function instead of (35) we solve minimization problem on smaller domain.

DETERMINATION OF REMAINING DOMINANT FREQUENCIES -MULTIPEAK DETECTION
To determine remaining subsequent dominant frequencies by filtering global maximum in this article is used inverse Fourier transform, can also be used digital filter as shown in [17].
Several methods are described in above sections which can be used to estimate dominant frequency or global maximum, and afterwards using equation system (7) and (8) to obtain amplitude and phase shift.
There are several methods for multiple peak detection, depending on the signal origin and physical meaning eg. [18][19][20]. Also online can be found many solutions for multiple peak detection; some examples are given in [21,22].
Here the authors propose the following algorithm to estimate remaining dominant frequencies and their matching amplitudes: For all methods reconstructed signal will be compared to the original signal. Results of frequency, amplitude and phase are given in Tabs. 1, 2 and 3 for rectangular window function and Tabs. 4, 5, 6 for Hanning window function.     Comparison of results shows that more accurate results are obtained using Hanning window function at larger data dissipation, but for more accurate assessment algorithm should be tested on real measurements data or more realistic data should be obtained such as calculated using finite element method.

Dominant frequencies determination algorithm
Testing input functions has shown that in case when dominant frequencies are sufficiently distant, after isolating dominant frequency from periodogram, algorithm can be effectively used to obtain subsequent dominant frequency. If dominant frequencies are in vicinity then inverse transformation of reduced periodogram gives slightly different result which has larger deviation from real value of maximum.

CONCLUSION
Comparing results of testing methods for vibration dominant frequencies determination, conclusion can be drawn that global method is most accurate in determination of frequencies, amplitude and phase shift in the experiment where rectangular window was used. While using Hanning window DIRECT global method and parabola method give similar results for amplitude but for frequency and phase shift DIRECT global method gives the most accurate results. So overall conclusion that this efficiency analysis shows can be that the global method is more accurate than some previously commonly used methods for frequency estimation.
Method efficiency varies with using different window functions. Further testing shows that algorithm efficiency is questionable at some ratios of amplitude against frequency (when frequency peeks are in vicinity), in some cases Hanning window function causes smoothing of spectra and making frequency isolation more difficult. This problem can be solved by changing sampling ratio and applying rectangular window function.
Proposed algorithm for multi peak detection could be used as part of intelligent fault detection system for rotary machines. In further research algorithm could be used to build more complex computer software and after numerical simulation, results should be tested on experimental rig and real measurement data.