Analized of thooth wells in the power train.

Pater, Sorin ; Fodor, Dinu ; Mitran, Tudor 等

Abstract: In real functioning conditions, the gearing process has certain deviations versus the ideal conditions. These deviations are provoked both by the execution errors and the other transmission elements of the toothed wheels, and by the assembling errors. The dynamic loads that appear in these conditions can be considerable, in comparison with the static forces, and their being token into consideration at the gearing planning is compulsory.

Key words: frequencies, spectrum, cepstrum, vibrations

1. INTRODUCTION

The objective is to diagnose of gearbox failure with a nondestructive method.

Most of the works in this domain uses spectral analyze, cepstrum analyze and wavelet analyzes to study power trains. We analyzed a five-speed gearbox to diagnose tooth wheel and rolling bearings failures using cepstrum and RMS. By using these methods it is possible to determinate tooth wheel and rolling bearings failures to assure maintenance.

2. VIBRATION PROPAGATION

For the purposes of condition monitoring, we will consider gearboxes (gears, shafts, bearings, and casings) to comprise a linear mechanical system and the gear motion errors to be the sources of vibration. If the gear motion errors are the input signals, then the gearbox can be modelled as a multiple input, single output system. The measured vibration signal on the gearbox casing can be represented by the following summation over N gears and M number of transmission paths for the [k.sup.th] gear. (Andrews, 1979)

[??](t) = [N.summation over (k=1)] [M.usmmation over (n=1)] [h.sub.kn](t)* [s.sub.ek](t) + w(t) (1)

Where [h.sub.kn](t) is the impulse response of the [k.sup.th] input signal via the nth path, [s.sub.ek](t) is the [k.sup.th] gear motion error signal and w(t) is any external noise. Taking the Fourier transform of [??](t) results in the following summation over the same indices where convolution has been replaced by multiplication.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The transfer functions [H.sub.kn](f) can be very complicated structural frequency response functions consisting of the gears, shafts, bearings and casing. Local resonances and time delays due to the propagation time will cause phase changes in the signal. Structural resonances will act as mechanical amplifiers to boost the vibration signal in certain frequency bands. After the vibration propagates through the structure, it is reasonable to expect phase changes, but the underlying frequency components will remain unchanged. It should come as no surprise that good accelerometer placement and mounting are important considerations in gearbox diagnostics. Also, it is important that any automated signal processing techniques used for diagnostics be wideband, since different gearboxes have drastically different structural resonances. Where one narrowband technique may work well centred on a specific frequency, it may fail to give good results on a different class of gearboxes.

3. TOOTH WHEELS DIAGNOSIS

The spectrum of a gearbox signal will usually consist of a number of harmonic families. These harmonic families originate from the different shafts and ball-bearings in the gearbox, and from the tooth meshing of the gears. The gears usually have numbers of teeth equal to prime numbers. This is an advantage as it causes wear to be spread out more evenly on the teeth of the gears, but it is also an advantage from a measurement point of view, as it means that the different harmonic families will usually not overlap. On the other hand, there can often be several harmonic families, and it can be difficult to separate them in the spectrum. Cepstrum is a practical tool that makes it easy to find these different harmonic families, and the individual families can be monitored for changes that might indicate that something is wrong.

The cepstrum and the auto-correlation are closely related. The main difference is that the inverse FFT is performed on the logarithm of the power spectrum, as opposed to the power spectrum itself. The auto-correlation is mainly dominated by the highest values of the spectrum. The logarithm used when computing the cepstrum causes it to take lower level harmonics more into account than does the auto-correlation. It also means that the auto-correlation is strongly influenced by the shape of the time signal, whereas the cepstrum mainly reacts to the harmonics present in the autospectrum and much less to their relative size. Cepstrum analysis has many applications: to machine diagnostics, where its ability to detect periodicities in the spectrum is taken advantage of. The Dual Channel Analyzer

The toothed wheels transmission dynamic is influenced by the following facts:. (Boyes, 1981)

* the rigidity variation of the gearing due to the variable deformations of the teeth in the process of gearing (the load is transmitted by a different number of teeth).

* the technological errors of the gearing

* the rotation speed, especially in those zones that correspond to the resonance phenomenon

The interior sources are represented by the deviations from the tooth-processing precision, especially the error of the measured step on the basis circle, that lends to the appearance of the periodical percussion between the teeth and creates a short term dynamic load and the profile error that creates a permanent dynamic load, as well as the periodic variation of the gearing rigidity, due to the periodic passing of the load from one tooth to two teeth. These sources are of a great interest for the gearing durability. The vibrations generated by these sources and together with them the dynamic forces and the noise become very strong high, especially when the frequency of the perturbation sources which is always in a relation determined by the gearing revolutions superposes on a frequency of it's own--the resonance phenomenon appears.

[FIGURE 1 OMITTED]

The diagrams in fig. 1, have been mode in order to diagrams the gearbox:--the diagrams of the signal acquired in time of the power spectrum in frequency and the cepstrum in time for the faultless gear box, considered as reference.

Amplitude values of the cepstrum were obtained up to 0.55m/[s.sup.2] and the spacing of the side bands corresponds to the frequencies generated by the bearings, gearing and the belt transmission. The RMS value, corresponding to the acquired signal is 1.3761m/[s.sup.2].

For the defect gear box diagnosis, the diagrams in fig.2 have been obtained: the diagram of the signal acquired in time, of the power spectrum in frequency and of the cepstrum in time.

There are peaks equally placed on the cepstrum diagram that correspond to the defect in the gearing. (Dempsey, &, Zakrajsek, 2001)

For the approximate determination through calculation of the signal frequency generated by the gearing defect, the following methodology is used.:

The gearing frequency is determined through:

[f.sub.a]=[f.sub.m]/Na (3)

where: fm represents the rotation frequency of the driving wheel and Na represents the lowest common factor of the teeth number corresponding to the pinion and the toothed wheel. Mathematically, the rotation frequency fm can be expressed through the rotation frequency of the pinion and of it's number of teeth, or though the rotation frequency of the driven wheel and it's number of teeth.

[f.sub.m]=[f.sub.rp] x [Z.sub.p]=[f.sb.rg] x [Z.sub.g] (4)

Where:

* [f.sub.rg]=Rq/60,represents the rotation frequency of the driven wheel expressed in;

* [f.sub.rp]=Rp/60,represents the pinion rotation frequency expressed in Hz.

* [Z.sub.p], represents the pinion number of teeth

* [Z.sub.g], represents the teeth number of the toothed wheel.

The gearing frequency [f.sub.tr], for one tooth of the pinion that comes into gearing with the same tooth of the driver wheel is given by:

[FIGURE 2 OMITTED]

[f.sub.tr]=[f.suyb.mx] [N.sub.a] / [Z.sub.g] x [Z.sub.p] (5)

The gearing frequency will be a low one and it can not be easily detected in spectrum, but it can be easily detected in cepstrum (Bruce, 1983)

For this gear box analyzed in the time domain, the gearing frequency has the value given in the following table:

This obtained values indicates the appearance of frequency peaks on the cepstrum diagrams at an interval of

[DELTA] [t.sub.cp] = 1 / [f.sub.tr] = 1 / 50,94 = 0,0196 [s].

Analyzing the experimental obtained diagrams, we can notice spacing between two peaks in the cepstrum diagram, close to the calculated value of [DELTA][t.sub.mp]=0,0192 [s], so the measured frequency will be [f.sub.m]=1/ [DELTA] [t.sub.mp]=52.083 [Hz].

These peaks and their spacing can be used to discover the gearing defects. In comparison with the defect less gear box, the amplitude in cepstrum is approximately twice two folded.

4. REFERENCE

Andrews, S.A. (1979). Noise and Vibrations of Engines and. Transmissions, MIMechE. University of Waster Australia, Conference Publications, p (47-57).

Boyes, J.D. (1981). Analysis Technique for Gearbox-Diagnosis Using the High Resolution FFT Analyses, Bruel & Kjaer, Application Notes, nr.106,.

Brown, D.N. & Jorgensen, J.C. (1987). Machine Condition Monitoring Using Vibration Analysis, Bruel & Kjaer, Application Note,.

Bruce, B. (1983). Precise control of vibratory stress relief, Poolinght & Production, nov., p.(64-66).

Dempsey, P.J. &, Zakrajsek, J.J; (2001). Minimizing Loan Effects on NA4 Gear Vibration Diagnostic Parameter, NASA/TM-2001-210671,

Table 1. The gearing frequency

 Number Number
Type Speed of wheel of pinion Rotation Gearing
of box step teeth teeth freq. freq.

365 1 42 11 899,97 50,94
 2 38 17 899,98 49,09
 3 34 23 899,98 47,36
 4 33 34 899,96 40,29
 5 31 36 899,93 40,29