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ANALYSIS OF EPICYCLIC GEARBOX VIBRATION V IBRATION
David Forrester Air Vehicles Division, DSTO David Blunt Air Vehicles Division, DSTO
ABSTRACT
Many aircraft transmissions use epicyclic gear trains, particularly helicopter main rotor gearboxes and propellerr reduction gearboxes. propelle g earboxes. As these th ese gears form a non-redundant non-r edundant critical part of the drive to the the main main rotor, rotor, or propeller, it is important to have advanced techniques and tools to assess the condition of these components. One such tool is vibration analysis. However, epicyclic gear train vibrations are difficult to analyse. Not only are there multiple planet gears producing similar vibrations, but there are multiple and time-varying time-varyi ng vibration vibrati on transmission path s from the gear mesh points po ints to any vibration vibrati on transducer mounted mounted on the gearbox gearb ox housing. These factors factor s combine to reduce r educe the sensitiv s ensitivity ity of conventiona conventi onall fault detec detection tion algori algorithms thms when they are applied to epicyclic gears. This paper outlines the DSTO-develo DSTO-developed ped techniques for analysing epicyclic gear train vibra vibration, tion, based on an algorithm for separa separating ting the meshing vibrations from each planet. The results of applying these techniques to seeded fault tests, using DSTO vibration data, are shown to significantly improve the detection of localised gear faults.
INTRODUCTION
Synchronous signal averaging has proved to be the most useful Synchronous vibration analysis tool for detecting faults in gears. However, there has been a problem in the past in applying the technique to epicyclic gearboxes. An epicyclic gearbox has a number of planet gears which all mesh with the sun and ring gears. The problems encountered when attempting to perform a signal average for components withi n an epicyclic epicycli c gearbox are twofold. twof old. Firstly, Firstly, there there are multiple multi ple tooth contacts, conta cts, with each ea ch planet being simultane simultaneously ously in mesh with both the sun and ring gears, and secondly, the axis of the planets move with respect to both the sun and ring gears. An earlier method of performing selective signal averaging on epicyclic gearboxes was developed and tested at the Defence Science and Technology Organisation (DSTO) [1,2], and proved successfull in detecting successfu detect ing faults on o n individual planet gears. gears. Howeve However, r, this method was tedious to implement, required an excessively long time to perform even a small number of averages and required the selective slicing up of the time signal which proved to introduce discontinuities in the signal average. ©
In this paper, an alternative method for performing signal averaging for epicyclic gearbox components components is presented which overcomes these problems. A mathematical derivation of the method is provided which shows that the averaging can be performed performe d with no loss of information by proportionally proportion ally dividing dividing the vibration data amongst the individual gear meshes. Practical examples are presented which show that the method has far superior performance than conventional signal averaging. The techniques techniq ues presented present ed here are the subject of Australian Patent Patent 40638/95 [3], United Unit ed States Patent 6,298,725 [4], and United Unite d States Provisional Patent [5]. EPICYCLIC GEARBOX VIBRATION
Epicyclic gearboxes are typically used in applications requiring a large reduction in speed (greater than three to one) at high loads, such as the final reduction in the main rotor gearbox of a helicopter. helic opter. A typical typic al epicyclic epicyc lic reduction reduct ion gearbox has three or more more planet gears each meshing with a sun and ring gear, as shown in Figure 1. Drive is provided via the sun gear, the ring gear is
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stationary and the axes of the planet gears are connected to a carrier which rotates in relation to both the sun gear and the ring gear. The planet carrier provides the output of the epicyclic gear train.
mounted on the ring gear of an epicyclic gear train will be the sum of the individual planet gear vibrations multiplied by the planet pass modulations, P x(t ) =
Vibration transducer
∑
α p (t )v p
(t )
(4)
p = 1 where α p(t) is the amplitude modulation due to planet p, and v p(t) is the tooth meshing vibration for planet p.
The amplitude modulation function α p(t) (planet pass modulation) will have the same form for all planets, differing only by a time delay, and will repeat with the planet carrier rotation period 1/f c
p α p (t ) = a t + f P = c
∞
∑()
A m cos 2π mf c t +
2π mp
(5)
P
m=0
where a(t) is the planet pass modulation function and A(m) is its Fourier Transform. Equation (4) can be rewritten in terms of the common planet pass modulation function giving Figure 1: Typical epicyclic gear train.
P
x(t ) =
Tooth meshing frequencies and relative rotations
∑ p =1
Where f c , f p and f s are the rotational frequencies of the planet carrier, planet and sun gear respectively, and there are N r, N p and N s teeth on the ring, planet and sun gears respectively, the meshing frequency of the epicyclic f m is given by: f m = N r f c = N p f p + f c = N s ( f s − f c ) (1) The relative frequencies f p + f c of the planet to the carrier and f s - f c of the sun to the carrier are: f p + f c = f m N p = f c N r N p (2) f s − f c = f m N s = f c ( N r N s )
(3)
Planet pass modulation
The only place in which it is normally feasible to locate a transducer to monitor the vibration of an epicyclic g ear train is on the outside of the ring gear. This gives rise to planet pass modulation due to the relative motion of the planet gears to the transducer location. As each planet approaches the location of the transducer, an increase in the amplitude of th e vibration will be seen, reaching a peak when the planet is adjacent to the transducer then reducing as the planet passes and moves away from the transducer. For an epicyclic gear train with P planets, this will occur P times per revolution of the planet carrier, resulting in an apparent amplitude modulation of the signal at frequency Pf c. Expected epicyclic gear train vibration signal
The expected planet gear vibration signal recorded at a transducer
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a t +
p v p (t ) . f c P
(6)
PLANET SEPARATION TECHNIQUE
An alternate method of extracting representative signal averages for each planet is to incorporate a selective (continuous) time filter into the signal averaging process. The time filter proportionally divides the overall vibration signal into the estimated contributions from each planet. For each planet signal average, the time window, b(t), is centred at the point at which the planet is adjacent to the transducer. Signal averaging of the filtered vibration signal is performed with a period equal to the relative pl anet rotation, 1/(f p+f c), gi ving, where N is the number of averages, N −1
z p (t ) =
1
N
∑ l =0
b t +
p f c P
+
l x t + (7) f p + f c f p + f c l
With careful selection of the time w indow characteristics and the signal averaging parameters, the separation can be performed with minimum ‘leakage’ of vibration from other planets, no loss of vibration data, and no distortion of the signal average. Properties and restrictions
It is shown in Appendix A that where: a) the separation time window is real valued, even and periodic with the planet carrier rotation, 1/f c, b) the signal averaging is performed over the relative planet rotation period, 1/(f p+f c), and the number of
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averages is an integer multiple of the number of teeth on the ring gear, N r , and c) the time window, b(t), is a Fourier series with less than P terms, the time filtered signal average for planet p (7) reduces to P
z p (t ) =
∑
v k (t )c( p − k ) .
(8)
k =1
The separation f unction, c(n), is defined by both the applied time window, b(t), and the planet pass modulation function, a(t),
1 c(n ) = B(0) A(0) + 2
2π mn B(m) A( m) cos P m=0 P−1
∑
(9)
where the applied time window is
SUN GEAR TECHNIQUE
The sun gear technique computes ‘separated’ averages of the sun gear vibration ‘seen’ through each planet, which are then phase shifted so that the beginning of each average starts with the same sun gear tooth in mesh with each planet, and recombined (averaged) to produce a modified sun gear average. Ideally, with complete separation of the vibration from each planet, the technique produces a modified average that represents the average meshing behaviour of the sun gear seen through a single typical, or average, planet. I n contrast, a conventional sun gear average represents the average meshing behaviour of the sun gear seen through all the planets simultaneously (with the accompanying summation and cancellation of certain vibration components).
P −1
b(t ) =
∑()
B m cos(2π mf c t ) .
(10)
m =0
Note that the summation of the time filtered signal averages is equal to the sum of the mean planet vibration signals multiplied by a constant P
P
∑
∑
p =1
k =1
z p (t ) =
P
∑
vk (t )
c( p − k ) = PB(0) A(0)
p =1
P
∑ v (t ) . k
k =1
That is, the separation process is performed using all of the available vibration data. Separation window functions
If the planet pass modulation was known exactly, complete separation could be made by setting the time window coefficients, B(0) = 1/A(0) and B(m≠ 0) = 2/A(m) , giving c(0) = P and c(n≠ 0) = 0. However, this is rarely practical in operational gearboxes.
The procedure for computing the modified sun gear average is: a) Compute the separated sun gear averages. This is the same as computing the separated planet gear averages, except that the averaging period is that of the sun gear instead of the planet gear. The same separation window function is used (i.e., the window function is still based on the planet-carrier position). b) Phase shift the separated sun gear averages so that the mesh points align. For example, in a three planet gear train, Planet 2 must be shifted +120°, and Planet 3 must be shifted -120° (or +240°) to align both with Planet 1, as shown in Figure 2. Note that the sun gear meshes with the planets in the reverse order that they pass the transducer on the ring gear. c) Combine (average) the aligned averages. The theoretical development for this is shown in Appendix A. Vibration transducer
+120°
Two separation windows which have been found to perform well are a cosine window raised to the power of P-1 [3],
1
P −1
, (11) b(t ) = (1 + cos(2π f c t )) which is a tapered function with maximum value when the planet is adjacent to the transducer and a value of zero when the planet is furthest from the transducer, and a window with B(0) = ½ and B(m≠ 0) = 1,
2
-120°
P −1
b(t ) =
1
+
2
∑
cos (2π mf c t ) ,
(12)
m =1
3
which relies upon the planet pass modulation itself to provide the separation. That is, where the time window in (12) is used, the separation function, c(n), becomes P −1
c(n ) =
∑
2π mn A(m) cos . P
Figure 2: Phase shifts for separated sun gear averages.
m =0
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IMPLEMENTATION Angle domain
In the preceding description of the t echnique, it is assumed that the speed of the epicyclic gearbox is constant. In practice, even for a nominally constant speed machine, this is not always the case. To allow for speed fluctuations all analysis is done in the 'angle' domain rather than the time domain; this simply involves the substitution of a angular reference for the time based variable, t . In practice, the conversion from the time domain to the angle domain is done by synchronising the vibration signal sampling with an angular reference on one of the shafts of the gearbox. The synchronisation can be done either using phase-locked frequency multipliers or by digital re-sampling [4, 5].
signal enhancement [6] on the separated signal averages for the faulty planet using: (a) the planet separation technique with the time window defined in (11), and (b) the conventional signal averaging technique. For this example, the analysis time is 75 seconds (10 x 32 revolutions of the planet carrier). The kurtosis of the residual signal is used as a measure of local variation in the tooth meshing behaviour. The kurtosis is defined as the fourth statistical moment normalised by the square of the variance. The residual signal is obtained by removing all the known regular frequency components such as the mesh harmonics. A kurtosis value greater than 4.5 is considered to be a clear indication of a local defect and a value below 3.5 indicates a ‘good’ gear.
Planet carrier positional reference
For the calculation of the time windowed signal averages (7) a planet carrier positional reference is required t o set the starting time (t = 0) to a point at which one of the planets ( p = P) is adjacent to the transducer. The positional reference can be obtained either by using a shaft encoder/tacho on the planet carrier (usually the output of the gearbox) or by software synchronisation to the planet pass modulation signal. In the case where a carrier positional reference is not directly available, the planet carrier position can be estimated by examining the ‘planet pass modulation.’ This involves performing a signal average of planet carrier (ring gear) vibration. As each planet passes the transducer location the vibration level increases, giving an amplitude modulation of the vibration signal. Demodulation of the ring gear si gnal average about the gear mesh vibration [8] is used to determine the modulation peaks as each planet gear passes the transducer location. The point with the maximum amplitude in the demodulated signal average is selected as the zero point for the planet carrier positional reference.
Figure 3: Three-planet epicyclic gearbox. 0.10
) g ( n o i t a r e l e c c A
-0.10
EXAMPLES
0
360 Rotation (degrees)
The following examples are from a recorded vibration signal of the epicyclic gearbox shown in Figure 3 with three p lanet gears each having 32 teeth, a sun gear with 28 teeth, and a ring gear with 95 teeth. Planet Gear
A small fault was implanted on one of the planet gears [2]. Approximately 0.05 mm was ground from the face of one of the teeth to form a narrow flat surface at the pitch line. The gearbox was reassembled so that when under load the damaged tooth face meshed with the ring gear.
(a) Planet separation technique – Planet Gear 2 (Kurtosis = 5.9) 0.10
) g ( n o i t a r e l e c c A
-0.10
0
360 Rotation (degrees)
(b) Conventional signal averaging technique (Kurtosis = 3.1) Figure 4 shows the results obtained by performing a residual
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Figure 4: Planet gear fault.
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The planet separation technique gives a kurtosis of 5.9, clearly indicating the presence of the fault. Over the same analysis period the conventional signal averaging t echnique (kurtosis=3.1) gives no indication of the fault.
averaging technique only gives a kurtosis of 4.9. It is also possible to see that the impacts of the fault with each of the planet gears are distributed at intervals of 120° in the conventional average,but they are aligned in the modified sun gear average, as shown in the figure.
Sun Gear CONCLUSION
A small fault was impl anted on the sun gear [2]. Approximately 0.05 mm was ground from the face of one of the teeth to form a narrow flat surface at the pitch line. The gearbox was reassembled so that when under load the damaged tooth face meshed with the planet gears. Figure 5 shows the results obtained by performing a residual signal enhancement [6] on the signal averages for the faulty sun gear using: (a) the sun gear technique with the time window defined in (11), and (b) the conventional signal averaging technique. For this example, the analysis time is also 75 seconds (10 x 32 revolutions of the planet carrier). 0.40
1, 2, 3
) g ( n o i t a r e l e c c A
-0.40
0
360 Rotation (degrees)
(a) Sun gear technique (Kurtosis = 7.9) 0.40 3 2
1
) g ( n o i t a r e l e c c A
-0.40
0
360 Rotation (degrees)
(b) Conventional signal averaging technique (Kurtosis = 4.9) Figure 5: Sun gear fault.
The sun gear technique gives a kurtosis of 7.9, very clearly indicating the presence of the fault, while the conventional
Techniques for analysing the individual planet and sun gear vibration signatures in a epicyclic gearbox have been developed. It has been shown that these t echniques significantly improve the detection (by up to 90% using t he residual kurtosis method) of a single tooth fault in a planet or sun gear, and consequently have considerable advantages over conventional synchronous signal averaging for condition monitoring of aircraft transmission systems incorporating epicyclic gear trains. REFERENCES
[1] Howard, I.M., “An Investigation of Vibration Signal Averaging of Individual Components in an Epicyclic Gearbox”, Propulsion Report 185, Department of Defence, Aeronautical Research Laboratory, March 1991. [2] McFadden, P.D. and Howard, I.M ., “The Detection of Seeded Faults in an Epicyclic Gearbox by Signal Averaging of the Vibration”, Propulsion Report 183, Department of Defence, Aeronautical Research Laboratory, October 1990. [3] Forrester, B.D. , “Method and Apparatus for Performing Selective Signal Averaging”, Australian Patent 672166 (40638/95). [4] Forrester, B.D., “A Method for the Separation of Epicyclic Planet Gear Vibration Signatures”, United States Patent 6,298,725, October 2001. [5] Blunt, D.M., “Synchronous Averaging of Epicyclic Sun Gear Vibration”, United States Provisional Patent Application, 24 January 2003. [6] Forrester, B.D. , “Advanced Vibration Analysis Techniques for Fault Detection and Diagnosis in Geared Transmission Systems”, PhD Thesis, Sw inburne University of Technology, February 1996. [7] McFadden, P.D. , “A Model for the Extraction of Periodic Waveforms by Time Domain Averaging”, Aero Propulsion Technical Memorandum 435, Department of Defence, Aeronautical Research Laboratory, March 1986. [8] McFadden, P.D. , “Examination of a Technique for the Early Detection of Failure in Gears by Signal Processing of the Time Domain Average of the Meshing Vibration”, Mechanical Systems and Signal Processing, Vol. 1(2), pp. 173-183, 1987.
APPENDIX A: Theoretical Development Planet Separation
Assuming that all vibration which is not synchronous with the relative planet rotation will tend toward zero with the signal averaging
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process [6, 7], the time filtered signal average (7) using the time window, b(t), taken over N periods of the relative planet rotation, 1/(f p+f c), can be expressed as
z p (t ) =
1
N −1
l =0
l x t + + f c + f f p c
p
l
∑ b t + f P + f N c
p
P lN p k l t + + + + a t v k f c N r k =1 f c P f c N r f p + f c P N −1 lN p lN p 1 p k a t + + + vk (t ) b t + N f P f N f P f N c c r c c r k =1 l =0
p = b t + N l =0 f c P 1
N −1
∑
∑
=
lN p
∑
(A.1)
∑
where vk (t ) is the mean vibration for planet k , which repeats with period, 1/(f p+f c), and a(t) is the planet pass modulation (5). Condition a): b(t) is real valued, even and periodic in 1/f c
Under the condition that b(t) is real valued, even and periodic in 1/f c, ∞
b(t ) =
∑ B(m)cos(2 mf t ) π
c
m =0
, and the time filtered signal average becomes
(A.2)
P
z p (t ) =
∑ v (t )
( p, k , t )
(A.3)
Ψ
k
k =1
where ∞
∞
Ψ ( p, k , t ) = ∑∑ B(m ) A(n )
( (
cos 2π (m + n ) f c t + 2 2 l = 0 cos 2π (m − n ) f t + c
N −1 1 2 N
m= 0 n = 0
∑
( mp + nk )
π
P
( mp − nk )
π
P
+ +
)+ ) ) ( ) )− ( ) )+ ( ) )− ( ) )
2π l ( m + n ) N p
N r
2π l (m − n N p
N r
( ( ( (
cos(2π (m + n) f c t + 2 (mp + nk ) )cos 2 l m +n N P N l m 2 + n N 2 (mp + nk ) ∞ ∞ N −1 sin (2π (m + n ) f t + ) sin c P N 1 B(m ) A(n ) 2 N = 2 (mp − nk ) m = 0 n =0 l = 0 cos(2π (m − n ) f t + )cos 2 l m N −n N c P 2 ( mp − nk ) )sin 2 l m N −n N sin (2π (m − n ) f c t + P π
∑
π
p
r
π
π
π
p
r
π
π
∑∑
π
(A.4)
p
r
p
r
Condition b): the number of averages is an integer multiple of N r
If the number of averages, N , is an integer multiple of the number of teeth on the ring gear, N r , then, since 1
iN r 1
iN r we have Ψ
iN r −1
∑ cos(
2π l ( m + n ) N p
N r
) = 1, n = −m
l=0
,
iN r −1
∑ sin (
2π l ( m + n ) N p
N r
(A.5)
) = 0, n ≠ −m
l =0
( p, k , t ) =
∞
[ (m) A(− m) cos( ∑ B 2 1
2π m ( p − k )
P
)+ B(m ) A(m )cos(
m=0
∞ 2 = B(0 ) A(0 ) + B(m) A(m ) cos( 2 m =0
1
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∑
m ( p − k ) ) . P
2π m( p − k )
P
)]
π
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(A.6)
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Condition c): b(t) is a Fourier series with less than P terms
The summation in m is a discrete Fourier series with period P, and, to avoid aliasing, the number of terms in the series needs to be limited to less than P. This is done by setting the limitation that the time window function, b(t), be a Fourier series of less than P terms, P −1
b(t ) =
∑ B(m)cos(2 mf t ) π
(A.7)
c
m =0
and the time filtered signal average over iN r averages with period, 1/(f p+f c ), becomes P
z p (t ) = ∑ vk (t )c( p − k )
(A.8)
k =1
where 1 2π mn c(n ) = B(0) A(0) + B(m) A( m) cos . 2 P m= P −1
∑
0
(A.9)
Sun Gear Technique
A similar result can be found for the separated sun gear averages by substituting relative sun rotation, 1/(f s-f c), instead of 1/(f p+f c), into (A.1) giving the separated sun gear average for planet p, as
z s , p (t ) =
P
∑v
s , k
(t )c( p − k )
(A.10)
k =1
where v s,k (t ) is the mean vibration of the sun gear with planet k , and c(n) is the same as (A.9). The modified sun gear average, z ms (t ) , is then P
z ms (t ) =
P
∑∑ v
s , k
p =1 k =1
2π p t − c( p − k ) P
(A.11)
. gear averages so that the beginning of each average starts with the same sun gear tooth where the delay, 2π p/P, aligns the separated sun in mesh with each planet. If the sun gear vibration with each planet is identical, it repeats with a period of 2π p/P, i.e.,
v s , k (t ) = v s t +
2π p
(A.12)
P
, where vs is the meshing vibration of the sun gear with a planet. Substituting (A.12) into (A.11) gives P
P
∑∑ c( p − k )
z ms (t ) = v s (t )
(A.13)
p =1 k =1
= PB(0 ) A(0 )v s (t ). and the modified sun gear average would thus represent, to within a constant, the average vibration of the sun gear with a single planet. However, since the sun gear vibration will, in practice, not be identical with each planet, the modified average will only tend to averageout the differences between the meshi ng behaviour with each planet. Nevertheless, because the separated aver ages will be aligned at the same sun gear tooth, any localised sun gear defect will always appear at the same angular position, and thus be reinforced. This should lead to an improved ability to detect the fault compared to an ordinary sun gear average, where the influence of the defect will be more distributed.
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