failure diagnostics of a gear unit - wit press

Failure diagnostics of a gear unit

A. Belšak & J. Flašker University of Maribor, Faculty of Mechanical Engineering, Smetanova 17, 2000 Maribor, Slovenia

Abstract

The influences of various defects of a single-stage gear-unit upon the vibrations they produce are dealt with. Different methods are applied to analyse time signals obtained by experiments. Significant changes in tooth stiffness are caused by a fatigue crack in the tooth root. This leads to a dynamic response which differs from the one in concern to an undamaged tooth. A time signal of a helical single-stage gear-unit has been obtained by means of accelometers and then prepared for further analyses. Statistical methods for analysis are most frequently applied for time signal. Amplitudes of time signal are, by frequency analysis, presented as a function of frequencies f. Digital frequency analysis has been carried out by means of Fourier Fast Transformation (FFT) and Time Frequency Analysis (STFT), which demands sampling of time signal in equal intervals during the measured period.

1 Introduction

Keeping a technical system (gear unit) in the most suitable working condition is the aim of maintenance; its purpose is to discover, to diagnose, to foresee, to prevent and to eliminate damages. The purpose of modern maintenance, however, is not only to eliminate failures but also to define the stage of a potential danger of a sudden failure of system operation. The procedure, which is referred to as diagnostics, includes the definition of the current condition of the system and the location, shape and reason of the damage formation. Incorrect operation, the possibility and the location of damages and the possibility of elimination of those damages are defined by means of the following diagnostic values: different signals, condition parameters and other indirect signs. The form of a damage is determined on the basis of deviations from the expected or wished for values. A hypothesis, i.e. a possible explanation of the damage

Damage and Fracture Mechanics VIII, C. A. Brebbia & A. Varvani-Farahani (Editors)© 2004 WIT Press, www.witpress.com, ISBN 1-85312-707-8

formation and condition, is based upon current data concerning the damage as well as upon knowledge and experience. If a correct hypothesis is selected, this leads to performing the corresponding activity. If the hypothesis is not correct, the whole procedure is repeated. The excitation of a gear directly in the meshing area is caused by the following internal sources: the impact at the beginning of meshing, tooth stiffness, resulting in parametrical excitation, geometrical deviations of teeth and deformation of bearings and shafts [1]. Consequently, a fatigue crack in the tooth root causes significant changes in tooth stiffness, which leads to a dynamic response that differs from the one caused by an undamaged tooth. Cracks (Figure1), their depths being 4,5 mm, have been formed in the tooth roots of pinions by means of a machine for dynamic tests of mechanical elements.

Figure 1: Gear with fatigue crack in the tooth root. The following differences between a fatigue crack and an artificially produced notch can, however, be observed: a notch is flat whereas a crack propagates in the direction of the gradient of maximum stresses; the boundary of a notch is flat, which is usually not the case with cracks; additionally, a notch is thicker than a crack. Vibrations of two pairs of spur gear-units have been measured, one of the pairs being with a fatigue crack and the other one without it. Tests performed under different loads and vibrations have been measured directly by means of accelometers, fixed on the housings; this fixing method yields better results than when using a magnet or special glue.


256 Damage and Fracture Mechanics VIII

2 Time signal analysis

A time signal of a tested gear-unit obtained by accelometers has been, for the purpose of further analyses, conditioned by filtering and windowing. A time signal is usually described by means of the following basic statistical methods: mean square factor, root mean square factor, crest factor and shape factor. Their applicability is, however, rather limited; therefore, performing frequency analysis is required. According to frequency analysis, amplitudes of time signal are presented as a function of frequencies f. Digital frequency analysis has been carried out by means of Fourier Fast Transformation (FFT); during the measured period, it is required to sample time signals in equal intervals. Time signal analysis [3] is one of the fundamental dynamic process analyses. The data obtained in such a way is rarely sufficient to diagnose a certain system. From the comparison between Figure 2a and 2b, it is evident that the changes in the values of amplitudes are only insignificant. Meshing is disturbed by a crack, which results in impulses and, consequently, to a small extent, in a time signal amplitude. Statistical analyses (average values, Crest factor, etc.), time signal with or without a crack, have not produced results clear enough to make it possible to establish the failure (crack) in a gear. Therefore, it is clear that it is almost impossible to foresee a crack only by means of time signal analysis.

3 Analysis of frequency spectrum

The gear-unit has been loaded with different torques and different numbers of rotations. The following equation has been used to calculate characteristic teeth frequencies and high harmonics depending upon the number of speed rotations:

( )60

)1(60

1 2211 znN

znNf N

⋅+=

⋅+= [Hz] (1)

fN – tooth frequency , N – 0, 1, 2, 3,... fundamental frequency and higher harmonics , n1 – the number of rotations of the input shaft [min-1] , n2 – the number of rotations of the output shaft [min-1] , z1 – the number of teeth of a pinion , z2 – the number of teeth of a gear wheel. Various defects in teeth of a gear wheel lead to the formation of sidebands round the fundamental tooth frequency and its higher harmonics [5]. There are different levels of sidebands; however, the more defects a gear wheel has, the larger is the number of sidebands formed; consequently, the spectrum is completer. Therefore, observing sidebands is one of the most important indicators for the condition of a gear wheel. Additionally, a damaged tooth (a crack in the tooth root) does not mesh in the same way as an undamaged one: there is a difference in stiffness; during meshing, the deformation of a damaged tooth is larger than that of other teeth. Furthermore, the frequency spectrum is also influenced. The reduction of tooth stiffness leads to the reduction in the


Damage and Fracture Mechanics VIII 257

contact between the engaging tooth of a gear-unit and the damaged tooth of a pinion; thus, in view of the length of the crack, its contribution to the frequency spectrum is being decreased by factor k (Figure 3). ( )prcb lsfk ,= (2)

Figure 2: Time signal of a gear-unit (a) without and (b) with a crack.

b)

a)



sb lc pr

α

lc

Figure 3: Base thickness sb, diminished by the crack length lc in the pinion tooth root.

The new tooth frequency can be expressed as:

( )

60)1( 111 kzznN

f pnNs

⋅+⋅⋅+= [Hz] (3)

[ ]( )

601)1( 111' +⋅+⋅⋅+

=kzznN

f pnNs [Hz] (4)

fNs, f’Ns – sidebands at fundamental tooth frequency (lower and higher) and its higher harmonics (N – 0, 1, 2, 3, …), z1n – the number of undamaged pinion teeth, z1p – the number of damaged pinion teeth, k – the factor of tooth thickness in the root. The factor k of a tooth thickness in the root is, in case of notch, the function of base thickness sb and of notch length l in the tooth root (Figure 2):

3

−=

b

prcb

sls

k (5)

Factor k has exponent 3 because the inertion moment Ix changes with cube of tooth thickness in the tooth root. Thus, Equations 3 and 4 are transformed into:

60

)1(3

111

−⋅+⋅⋅+

=b

prcbpn

Ns

sls

zznN

f [Hz] (6)



60

2)1(

3

111

'

−⋅⋅+⋅⋅+

=b

prcbpn

Ns

sls

zznN

f [Hz] (7)

The increasing propagation of a notch l in the interior of the tooth leads to the factor k being reduced and approaching the value of 0. In Figure 4b, the increase in sidebands of higher harmonics, resulting from the crack, is evident. The portion of the amplitude reduction of harmonics is, therefore, compensated by increased sidebands primarily in the central part of the frequency spectrum, between the 6th and 7th harmonics of the gear with a crack.

Figure 4: Frequency spectrum of a gear a) without and b) with a crack in the

tooth root (n=1200 min-1, M=30Nm)

b)

a)



Vibration measuring can represent measuring of a dynamic quasi periodical signal as the gear consists of rotating elements (shafts with gears and bearings). This represents complex repeated rotating movements. The information obtained by measuring the rotation speed, making it possible to follow the stability of rotation speed during the measuring, is important for the frequency analysis. During measuring, rotation speed can oscillate, even more so in case of gears in machines, which are under loading. This results in an unreliable frequency spectrum, even more so in case of frequency analysis of higher harmonics; additional sidebands namely appear. By following the rotation speed signal (TTL signal), the beginning and the end of rotations are located and their length and time of duration are measured The oscillation of rotating speed leads to spectrum disfiguration; however, correction of time signal can be performed. Signal can be reconstructed by means of integral interpolation, or by the method of extended value. The decision concerning the selected method depends upon difference criterion, which depends upon sampling, signal length and further analysis. After reconstruction, time signal obtains the character of a periodically changing dynamic system, which is in accordance with the actual rotating speed (Figure 5).

Figure 5: Reconstruction of time signal.



Figure 6: Short Time Frequency Spectrum of a gear with fatigue crack in the

tooth root.

Figure 7: Short Time Frequency Spectrum of a gear without fatigue crack in the

tooth root.



4 Analysis of time-frequency spectrum

In signals of technical diagnostics, some frequencies appear only in some cases. On the basis of classical frequency analysis of such signals, it is impossible to establish the time when particular frequencies appear in the spectrum. Joint Time Frequency Analysis (JFTA) is used to find out how frequencies of nonstationary signals change with time, and to establish the levels of their intensity. The idea of Short Time Fourier Transformation (STFT) is to divide a time signal into short time intervals and then, to carry out frequency analysis of each interval separately. STFT is a linear time frequency transformation. A very simple method for eliminating defects of Fourier Transformation is to compare signals with elementary functions, which are defined in time space and in frequency space.

( ) ( ) ( )∫+∞

∞−

−−⋅= ττγ dettxftSTFT jft*, (8)

Equation 8 is a product of time signal x(t) and elementary function

( ) jftet −−τγ * . Function γ(t) is short, i.e. it is reduced in view of time and presents windowing. Equation 8 is a short Fourier transformation, or windowing Fourier transformation. Analysis depends primarily upon the selection and the size of the field of sampling. Figure 6 and 7 presents JTFA of a signal of a gear without a defect and of a gear with a defect. Primarily the application of a suitable time window and its length leads to a correct identification of changes in the spectrogram; using this it is possible to foresee a defect. Figure 7 (a faultless gear) shows that separate frequencies are of the same intensity through the whole time window. This, however, differs from Figure 6.There it is evident that some frequencies are present only from time to time, i.e. they appear periodically through the whole time axis. By defining the frequency of appearance, it is possible to identify the source.

5 Conclusion

In a frequency and time frequency space, clearer results are obtained by time signal of vibrations, processed on the grounds of dependence upon rotation speed; this leads to improved reliability of defining the damage, i.e. the fault. The measured signals, having some changes which are characteristic in case of the presence of a defect, should, according to the forecast, increase the number of sidebands in a frequency spectrum. However, this is not always the case. Conditions, caused by other defects (primarily, by eccentricity), prevail over the influencing values, caused by the changes in the notch length. Variables generally increase with the notch extension. However, the probability of this is not sufficient to conclude that there is a defect in a gear-unit. The time signal shows a dynamic response of a gear-unit, which is not clear enough also on the



grounds of disturbing values being rather large. Therefore, detecting the defect or its size is not completely reliable.

References

[1] Smith, J.D., Gear noise and vibration, Marcel Dekker, Inc, pp. 10-68, 1999. [2] McConnel, K.G., Vibration Testing – Theory and Practice, John Wiley &

Sons, Inc, New York, 1995. [3] Belšak, A., Ploj, M, Noise analysis of a single-stage gear drive with a notch

in gear tooth root, Proc. of the int. conf. on Advances in Fracture and Damage mechanics 2, ed. M. Guagliano, M. H. Aliabadi, Geneva, pp. 479-484, 2001.

[4] Kolerus, J, Zustandsüberwachung von Maschinen, Expert – Verlag, pp. 84-114, 2000.

[5] Randal R.B., Frequency Analysis, Bruel &˛Kjear, Naerum, Denmark, September 1987.



failure diagnostics of a gear unit - wit press

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