ACTA TECHNICA CORVINIENSIS - Bulletin of Engineering

ACTA TECHNICA CORVINIENSIS – BULLETIN of ENGINEERINGIn the next stage of tooth damage modelscreation the vibration signals achieved with theuse of five filters were put under the influence ofcontinuous wavelet transform (CWT).Wavelet analysis consists of signaldecomposition and presenting it in the form oflinear combination of basic functions, calledwavelets. The feature which distinguishes thismethod of signal analysis from other methods isthe multi-stage signal decomposition, variableresolution in time and frequency domain andpossibility to use basic functions other thanharmonic functions. Continuous wavelettransform is defined as follows:1+∞⎛ t − b ⎞CWT(a,b) = ∫ ψ⎜⎟⋅xa −∞ ⎝ a ⎠where:ψ t – the analyzing wavelet,a, b()()dt t(1)a – the scale parameter, a ∈ R , a ≠ 0 ,b – the time parameter, b ∈ R .Wavelet frequency is regulated by the parameterof scale a, whereas using parameter b one cantest the local properties of the time runs.This method, due to the possibility of adjustmentof the window width to the analysed frequencyrange enables testing the non-stationary signals.For the long-term variation runs the windowexpands in time domain, whereas with highfrequencies it narrows preserving a constantsurface area. The authors in paper [8] states thatthe proper choice of the basic wavelet and thearray of the scale value determine thecorrectness of the diagnosis process of theobject technical condition with the use ofwavelet analysis.In the preliminary test the usefulness of 83 basicwavelets was checked, which belonged to series:haar, daubechies, biorthogonal, coiflets, symlets,morlet, mexican hat, meyer, reversebiorthogonal, gaussian, complex gaussian,shannon, frequency b-spline, complex morlet,discrete approximation of meyer.In order to describe the change character of thesignal amplitude under the influence of CWT for20 scale analyses, chosen in preliminary stage, ameasure was marked. This measure describedthe change course of the amplitude CWTdistribution in time domain. In the tests theusefulness of 35 diagnostic measures commonlyused in literature was checked (variationcoefficient, peak coefficient, clearance+coefficient, shape coefficient, impulsing andasymmetry coefficient, quarter and averagedeviation; arithmetic, geometric and harmonicaverages; quartiles, non-dimensionaldiscriminants, central moments, cumulants,signal energy, efficiency values, inter-peakvalues, maximum and minimum, variance,positional variation coefficient) [5,6].As a result of tests 14525 sets of models werebuilt. Each set of models had the size of 971x20.This size reflected the number of cases (numberof measured on FZG stand vibration signals) andthe number of network inputs (number of scales,in which the measure was marked). Each optionof model sets was divided into halves and it thisway teaching and testing data were achieved.In preliminary investigations were chosenfollowing wavelets [5,6]:• Daubechies Wavelet 9 (wavelet no. = 9),• Morlet Wavelet (wavelet no. = 37),• Reverse Biorthogonal Wavelet 3.7 (waveletno. = 50),and estimators [5,6]:• impulse ratio (estimator no. = 5),• root mean square (estimator no. = 14),• value of range between maximum andminimum (estimator no. = 25).In the tests performed in order to build adiagnostic classifier the artificial neural networksof multilayer perceptron type (MLP) were used.In case of application of such neural networks,one needs to define the architecture, the methodof network teaching and the type of neuronsused in hidden layers of the network [7,12,16]. Itshould be remembered, however, that the stagesshould not be divided, because they are directlyconnected. At the same time literature shows thelack of rules connected with the choice of theparameters and points out the necessity of theirsetting with the use of experimental method –trial and-error method [7,12,16].In the conducted experiments for each set ofmodels the best option of network architecturewas chosen. By the selection of networkarchitecture the usefulness of classifiers build ofone and of two hidden layers was checked. Onthe basis of preliminary tests [5,6] it was assumedthat for each hidden layer there is a possibility ofexisting of 5, 10, 15, 20, 25 and 30 neurons.Simultaneously, an attempt was made to check,if for an analysed diagnostic task an optimumarchitecture can be defined, independently fromthe way models are built.2010/Fascicule 2/April­June /Tome III 17