Maritime University of Szczecin
Akademia Morska w Szczecinie
2011, 26(98) pp. 33–37 2011, 26(98) s. 33–37
Toothed gear transmission diagnosis based on optimal
features of vibration signal
Diagnozowanie przekładni zębatej w oparciu o optymalne
cechy sygnału drgań
Łukasz Jedliński, Józef Jonak
Lublin University of Technology, Faculty of Marine Engineering, Department of Machine Design Politechnika Lubelska, Wydział Mechaniczny, Katedra Podstaw Konstrukcji Maszyn
20-618 Lublin, ul. Nadbystrzycka 36, e-mail: l.jedlinski@pollub.pl, j.jonak@pollub.pl
Key words: bevel gear, feature selection, artificial neural network Abstract
The article presents a method for reducing amount of discriminants required to evaluate technical condition of an object and a trial of evaluating it using artificial neural networks as a way of increasing certainty of the obtained results.
Słowa kluczowe: przekładnia stożkowa, selekcja cech, sztuczne sieci neuronowe Abstrakt
W artykule przedstawiono metodę redukcji liczby dyskryminant wymaganych w ocenie stanu technicznego obiektu oraz próbę oceny stanu z użyciem sztucznych sieci neuronowych, jako środek zwiększenia pewności prognozy.
Introduction
In case of very responsible gears (e.g. in avia-tion) typical procedures of evaluating technical condition of their structural components require execution of control tests in latter stages of assem-bling. Correctness of the components and of the assembly is mainly evaluated by verification of the traces of contact between teeth for particular gears. Moreover, exploitation of such objects is realized according to overhaul life, i.e. after certain periods of time the transmission gear is disassembled and its components are evaluated using e.g. wearing measurements. Such procedures are time-consum-ing and expensive, require turntime-consum-ing gear off the exploitation (whole object is then out of order) and do not guarantee that during work between over-hauls a defect will not occur since procedures of condition monitoring (oil temperature and pressure measurements, measurements of filings content in an oil) do not guarantee proper recognition of
changed technical condition (particularly in case of rapid changes) what can lead to a disaster for whole object in the worst case.
Intensive development of vibroacoustic methods of evaluating technical condition allows exploita-tion of the objects according to their actual condi-tion and performing forecasts of the changes during exploitation and determining dates of possible overhauls ensuring safe and reliable exploitation. However, also in this case, one of the problems that has not been solved yet is a problem of selecting discriminants of the measurement signals and re-duction of their amount to the necessary minimal value [1, 2, 3].
Large number of features can cause problems with their interpretation during diagnosing, espe-cially if inconsistent trends of features occur. Level of correlation between a feature and machine condi-tion depends on type of defect and properties of the analyzed object. Certain discriminants can not be sensitive to a particular defect or can provide the
same information (effect of too much data). Selec-tion of the signal features is then a crucial step that affects final result of the evaluation of technical condition and its efficiency.
Features selection can be realized in an automa-tic way. Using different methods can lead to obtain-ing optimal data, e.g. connectobtain-ing statistical methods with artificial intelligence, we obtain reliable and efficient tool for detecting defects of toothed gears. However, in solutions currently used in Poland there are not such procedures hence a need of re-search in that area.
There are no procedures allowing selection of sets of signal measures (discriminants) that would result in evaluation of object’s technical condition with required safety level and optimal in terms of amount of work. Thus, it is necessary to develop algorithms of reducing amount of necessary discri-minants and effective methods of condition evalua-tion, e.g. using artificial neural networks.
Investigated object
Investigated object was a two gear helicopter transmission (reducer) with a first gear being coni-cal transmission with a ratio i1 = 2.7. Second gear
was a planetary gear with ratio i2 = 5.1 (four planets
with spur gears). Total ratio is ic = ~13.89. Maxi-mum continuous power is 365 KM (M1 = 420 Nm).
Starting power is 450 KM (torque M1 = 510 Nm).
Nominal rotational speed of the input shaft is
n1 = 6016 rpm. A diagram of the transmission is
shown in figure 1.
Basic characteristic frequencies (rotational and meshing) of this transmission are presented in table 1.
Table 1. Characteristic frequencies Tabela 1. Charakterystyka częstotliwości
Gear
number Number of teeth
Rotational speed [rpm] Rotational frequency fo(k) [Hz] Meshing frequency fz(k) [Hz] 1 2 3 4 5 6 7 8 9 yoke 23 62 29 43 119 17 23 47 27 6016 2231.7419 2231.7419 752.56411 0 8139.3 6016 2231.7419 3884.8837 437.3 100.267 37.1957 37.1957 12.54 0 135.65 100.267 37.1957 64.75 7.288 2306.1331 2306.1332 1078.6753 539.3 0 2306.1331 2306.1331 1748.1979 1748.1976
The transmission was mounted in the investiga-tion stand located in the line of the transmission final assembly in one of Polish aviation plants.
Vibrations measurements were realized in con-stant periods of time with sampling frequency equal to 26 kHz with assumed work conditions (known load and rotational speed). The measurements were performed using three-axis vibration acceleration amplitude sensors Bruel & Kijaer 4321.
Analysis method
The values of selected diagnostic indicators were computed for the vibration signals registered
Fig. 1. Kinematic diagram of the transmission being investigated (w.we. – input shaft – driving, w.wy. – output shaft, w.went. – fan shaft, w.p.hydr. – hydraulic pump shaft, w.p.1, w.p.2 – shafts of oil pumps no. 1 and no. 2)
Rys. 1. Kinetyczny schemat transmisji poddany analizie (w.we. – wał wchodzący, w.wy. – wał wychodzący, w.went. – wał wentyla-tora, w.p.hydr. – hydrauliczny wał pompy, w.p.1, w.p.2 – wały pompy oleju nr 1 i 2)
during tests. Simple dimension and dimensionless measures were applied, as well as dimensionless discriminants based on torques of higher order. For the purpose of presented analysis a set of discrimi-nants was created to which only a few sample ones were selected.
For the transmission in good and damaged con-ditions six signal features were computed: average value xave, median Me, root-mean-square value xRMS, signal power P, peak value xpeak,peak to peak value
xpp.
N n ave N x n x 1 ) ( 1 (1) 1 2 2 2 1 2 1 , n n n Me x x x Me (2)
N n RMS N x n x 1 2 ) ( 1 (3)
N n n x N P 1 2 ) ( 1 (4) ) ( max nx xpeak (5)
0 max
0 max x n x n xpp (6) where: x(n) – discrete signal,N – number of samples in the signal, σ2 – variance.
Taking into account using neural networks as a condition classifier, it was necessary to apply properly large base of measurement data. Using smaller amount of signal features causes that neural network usually has smaller amount of hidden neu-rons and works faster. Additionally, larger number of features means larger data base required to real-ize network learning process.
To select optimal features the algorithm described in [4, 5, 6] was used. Input data is a three-dimensional matrix (dimensions Mc C J),
where: J – number of signal features, Mc – number of values for each feature, C – number of consi-dered object’s states. pm,c,j means m-th value of j-th measure for c-th state of the object, where:
m = 1, 2, …, Mc; c = 1, 2, …, C; j = 1, 2, …, J. The work recognizes two states of the object (C = 2), for which eight features were computed (J = 8) with two hundred twenty nine values (Mc = 229).
Computation method consists of the following steps:
Stage (1)
Compute average distance between particular measures values in the same condition:
c M l,m l,c,j m,c,j c c c,j M ..., , , l,m p p M M D c 2 1 1 1 1 2
(7)Then compute average distance for C states of the object:
C c j c w j C D D 1 , ) ( 1 (8) Stage (2)Determine and compute (w)
j V in the following way:
c j j c w j D D V , , ) ( min max (9) Stage (3)Compute average distance for all measures val-ues for the same state:
c M m j c m c j c M p a 1 , , , 1 (10) Then obtain average distance between the measures values for different states:
C ... e c a a C C D C e c c,j e,j b j , , 2 , 1 , 1 1 1 , 2
(11) Stage (4)Determine and compute (b)
j
V for the object in different state:
|a a |
c e ... C | a |a V c,j e,j c,j e,j b j , , 1, 2, , min max (12) Stage (5)Determine and compute j as below:
1 max max b j b j w j w j j V V V V λ (13) Stage (6) Compute ratio b j D and (w) j D and determine Ej w j b j j j D D λ E (14)then normalize Ej with maximum value and obtain valuation criterion:
j j j E E E max (15)Last stage of the computations (normalization) results in obtaining numbers between 0 and 1. Val-ues closer to one means that the given feature better represents object’s state (condition). The selection can be made using criterion Ej , where P is P
a selected threshold value for the features.
The algorithm operation results are shown in the graph of measures classification (Fig. 2). Threshold value of the optimal feature was set to 0.8 hence in the further analysis the average, root mean square and signal power features were used (marked on the graph with a frame).
Fig. 2. The graph showing sensitivity level of the measure of object’s condition change
Rys. 2. Wykres poziomu wrażliwości mierzony zmianą stanu obiektu
A Radial Basis Function (RBF) neural network was used as a classifier of transmission state. In our case, a single hidden layer is enough to model arbi-trarily complex nonlinear function what is an ad-vantage of this network comparing to Multilayer Perceptron Network [7]. To find optimal network a number of neurons in the hidden layer was chang-ing and different functions of neurons activations were applied. The best results for three features were obtained using the network with five hidden neurons and for all six features the network had eight hidden neurons.
Results of investigating technical condition of the transmission
Two cases of neural networks were tested. For selected three signal features a neural network number 1 was created and for whole six features
a neural network number 2. Network learning using all features was to verify correctness of features selection. Table 2 presents the results for both net-works.
Table 2. Efficiency of network classification Tabela 2. Skuteczność klasyfikacji sieci
No. Network name quality [%] Learning quality [%] Testing quality [%] Validation 1 RBF 3-5-2 95.10 95.24 95.71 2 RBF 6-8-2 95.10 95.71 95.24
For learning and verification of network opera-tion correctness 1400 cases were used (Table 3). 70% cases took part in the learning process, 15% cases in the testing during learning and 15% to verify the network after learning. Quality of learn-ing, testing and validation is determined in percen-tage values as a number of cases correctly classified per number of all cases in a given test.
Table 3. Operation of investigated neural networks Tabela 3. Działanie badanych sieci neutronowych
No. Good Damaged All cases
1 Together 700 700 1400 Correct 670 663 1333 Incorrect 30 37 67 Correct [%] 95.71 94.71 95.21 Incorrect [%] 4.28 5.28 4.78 2 Together 700 700 1400 Correct 669 664 1333 Incorrect 31 36 67 Correct [%] 95.57 94,85 95 Incorrect [%] 4.42 5.14 4.78
Tested neural networks achieved similar classi-fication efficiency at the level approx. 95%. The difference is number of hidden neurons that is 5 for the network for smaller amount of features and 8 for the second one.
Conclusions
Performed research showed that it is possible to efficiently reduce number of investigated features (discriminants) of the vibration signals (without loss of possibility of correct condition evaluation). It allows for significant reduction of amount of obtained information that has to be saved and stored (required storage capacity) and accelerates the process of evaluation. Using neural networks as the condition classifier significantly accelerates, facilitates and improves reliability and repeatability of the performed analyses.
1 2 3 4 5 6 Measure number Measures classification M ea su re imp ortan ce d eg re e 1 Average 2 Mediana 3 Root mean square 4 Power 5 Peak value 6 Peak-to-peak value
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Recenzent: dr hab. inż. Andrzej Adamkiewicz, prof. AM Akademia Morska w Szczecinie
