Analyze of transmission errors in gearboxes.

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

1. INTRODUCTION

Thus far, the discussion has been an idealization that has assumed that the gears are uniformly spaced, perfectly formed and completely rigid. Actual gears deviate slightly from perfect involute surfaces and elastically deform under load. These tiny surface deviations result in an unsteady force component of the torque. The unsteady forces are transmitted as vibration from the gear, through the shaft and bearings to the power transitions casing where it is measured. It is this unavoidable vibration that we exploit to no invasively investigate the operating condition of gearboxes while in operation.

Mathematically, this unsteady forcing function due to the pinion is best described by a complex Fourier series with a fundamental frequency equal to the pinion rotational rate, [f.sub.r].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The Fourier transform of the complex Fourier series is a one-sided pure line spectrum at multiples of the gear rotation rate.

S(f) = [[infinity].summation over (n=0)] [c.sub.n][delta](f - [nf.sub.r]) (2)

We can now develop a mathematical expression for the composite vibration due to both the pinion and the gear. This is accomplished by summing two infinite complex Fourier series. Therefore, if the pinion has N teeth and the gear has M teeth, then equation 3 describes the composite vibration due to both gears.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

In equation 3, [f.sub.r] is the pinion rotational frequency and (N/M)[f.sub.r] is the gear rotational frequency. The Fourier transform of the composite vibration is also a one-sided pure line spectrum.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

It should be clear that the two summations share a common set of frequencies. In the second summation, whenever m equals any integer multiple of M, the summations share the same frequency component. Therefore, equation 4 can be further decomposed as equation 5.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

If we then transform back to the time domain, we arrive at the following equation.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The first summation in equation 1-6 is composed of vibration components from both the pinion and the gear. The first summation's fundamental frequency (N [f.sub.r]) is the gear mesh frequency. We have just shown that the components of vibration due to the gear mesh frequency and its harmonics are due to both the pinion and the gear. This is a fact that is neglected in the literature on gear diagnostics. In the next section, we will find that the summation over " in equation 6 is called the harmonic error signal. The second summation in equation 6 is then due solely to the pinion and the third summation is due solely to the gear. Also in the next section, we will find these two summations are called the residual error signals for the pinion and the gear. Thus, in the frequency domain, it should be possible to separate the vibration produced by different gears. The component of the static transmission error that occurs at multiples of the gear meshing frequency is caused by elastic tooth deformations and the mean deviations of the tooth faces from perfect involute surfaces (Zaveri 1985).The remaining components of the static transmission error that occur at multiples of the gear rotational frequency are caused by the dynamic components of the tooth face deviations. Thus, we have a concrete, physical justification for using the static transmission error for gear diagnostics. This includes, but is not limited to, heavily worn teeth, missing teeth, and cracked or chipped teeth. In the next section, we will show how the residual error signal is defined as the dynamic component of the static transmission error.

2. GEAR MOTION ERROR

The gear motion error is a real part of the static transmission error (Fhay & Perez 1990),. The gear motion error is a real signal, described by an infinite cosine series with fundamental period f r. The Fourier transform is then a two-sided pure line spectrum. Whereas the static transmission error was developed for predicting the amount of vibration produced by meshing gears, the gear motion error was developed for gearbox diagnostics. It should be noted that since we are dealing with real valued signals, the static transmission error and the gear motion error contain the same information and simply differ by a factor of 2. We will use the term gear motion error and its interpretation throughout the remainder of this thesis. The decomposition of the composite gear motion error has three components--the harmonic error component [][][][][s.sub.eh](t), the residual error component due to the pinion [s.sub.er,p](t), and the residual error component due to the gear [][][][][s.sub.er,g](t).

[FIGURE 1 OMITTED]

s(t) = [s.sub.ch](t) + [s.sub.er,p](t) + [s.sub.er,g](t) (7)

Equation 7 is expressed graphically in figure 6. Notice how important good spectral resolution is when separating the vibration produced by the pinion and the gear.

The situation is made increasingly difficult when monitoring multiple gearsets. It should be apparent that several frequency components in the spectrum will be sufficiently close together so that it may not be practical to resolve them with reasonably sized FFTs. In this case, it is possible for information from another gear to leak into a different gear's residual error signal. There is no easy way of dealing with this situation when performing time domain averaging. This is part of the motivation behind developing the Comblet basis function (Dempsey & Zakrajsek 2001).

3. COMB FILTERS

We know that a comb filter is the most natural way to extract fault information from a composite vibration signal. Then we can do away with the adaptive sampling needed to compute the motion error signals. This is accomplished by giving the comb filter "teeth" a small but finite passband. Thus, even though we do not hit the shaft rate frequency bins exactly, we are still able to extract fault information. This approach also allows us to incorporate shaft speed measurement uncertainty into the filter design. This small passband can also capture variability in the shaft speed. Additionally, a modular wavelet basis will also allow us the flexibility to neglect "ghost" frequency components that mask early fault detection in multiple gear systems. Which is to say, there will be frequency bins that are contaminated by information from other vibration sources, which we can selectively neglect in our modular wavelet design? To my knowledge, the Comblet transform is the only signal processing technique for machinery diagnostics that addresses this issue.

An ideal comb filter is a unit amplitude periodically spaced impulse train in the frequency domain (Max 1981). A finite impulse response (FIR) comb filter is designed to increase the signal to noise ratio for periodic signals in broadband noise. The z transform of a FIR comb filter with D unity gain peaks at [[omega].sub.k] = 2k[pi]/D and D zeros at [[omega].sub.k] = (2k + 1)[pi]/D

H.sub.comn](z) = b(1 + [z.sup.-D]/1 - [az.sup.-D]) (8)

where the parameters 'a' and 'b' are determined by the half power bandwidth, [DELTA][omega], from the following design equation

[beta] = tan(D[DELTA][bar.[omega]]/4) (8)

4. CONCLUSION

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.

The toothed wheels transmission dynamic is influenced by the following facts:

* 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 (Randall 1990).

The diagrams in fig.2, 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.

[FIGURE 2 OMITTED]

5. REFERENCES

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

Fhay, K., &Perez, E., (1990), Fast Fourier Transforms and Power Spectra in Lab. VIEW, National Instruments Corporation, Application Note 040.

Max, J. (1981). Methodes et techniques de traitement du signal et applications aux mesures physiques. Methods and techniqes for signal analyse on phisical measurements. Paris, Ed. Masson.

Randall, R.B., (1990), Cepstrum Analysis and Gearbox Fault Diagnosis, Bruel & Kjaer, Application Note, 1990.

Zaveri, K.(1985). Modal Analysis of Large Structures Multiple Exciter Systems. Bruel & Kjaer,