Applied Acoustics 77 (2014) 150–158

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Applied Acoustics

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Effects of operating conditions on the Acoustic Emissions (AE)from planetary gearboxes

0003-682X/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.apacoust.2013.04.017

⇑ Tel.: +56 41 2204149.E-mail address: crimolin@udec.cl

Cristián Molina Vicuña ⇑Laboratorio de Vibraciones Mecánicas (LVM), Departamento de Ingeniería Mecánica, Universidad de Concepción, Edmundo Larenas s/n, Concepción, Chile

a r t i c l e i n f o

Keywords:Acoustic emissions

Planetary gearboxVariable load and rotational speed

a b s t r a c t

In the last decade, the use of acoustic emissions has received growing acceptance for its application inmachine condition monitoring. This is because it offers good possibilities to diagnose failures at earlystages and low rotational speeds. The use of acoustic emissions for condition monitoring of gears, how-ever, is still an active field of research, because several questions remain unanswered. One of these ques-tions is the effect of operating conditions on the AE generated during gear meshing. In this work, theresults of experiments carried out on a non-faulty planetary gearbox test bench are presented. A plane-tary gearbox is considered, because of its usual application on machines subjected to variable operatingconditions. The effects of lubricant temperature, load and rotational speed are investigated. The conclu-sions obtained from the experiments are used for the analysis of the AE measured on the planetary gear-box of a bucket wheel excavator.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The use of Acoustic Emissions (AE) for condition monitoring ofmachines has expanded in the last decade. This is because AE canprovide information related to failures at very early stages ofdevelopment. The most promising results have been found in fail-ure diagnosis of bearings [1]; however, its use for the conditionmonitoring of gears has presented some difficulties [2]. The studyof the AE generated in gears has been actively addressed in recentyears by different researchers. Probably the main difficulty for thereliable use of AE for gear failure diagnosis originates in the uncer-tainties still found in relation to the behavior of the AE in differentoperating conditions. Several experiments have been carried out,aiming to find how the AE generated in gears behave under differ-ent operating and fault conditions [1–4]. However, there are stillquestions to answer and even contradictory results [2,3]. Underthese circumstances, results from new experiments and new view-points should be valuable contributions for the discussion of thisproblem.

In this work, results from AE measured on gears under differentconditions of load and rotational speed are presented. Unlike exist-ing works, the study is centered on a planetary gearbox. The workfocuses on the analysis of the non-faulty case, considered essentialfor the further use of the technique for failure diagnosis.

Planetary gearboxes are the type of gear trains typically used forhigh power transmission, usually at low rotational speeds, which

makes failure diagnosis a difficult task. Common applications ofplanetary gearboxes include wind turbines, bucket wheel excava-tors, mining trucks, etc. A large part of these applications involvevariable conditions of load and rotational speed, a situation thatmakes failure diagnosis even more challenging [5–7]. Severalworks have addressed the study of the dynamics of planetary geartransmission, some including the modeling of different types of de-fects [8–10]. Similarly, results from the use of vibration measure-ments for failure diagnosis in planetary gearbox working on non-stationary conditions have been reported [11,12]. Even thoughthe results presented in these works represent major advanceson this subject, the results and methodologies presented there can-not be directly used for the AE due to the different characteristicsbetween vibrations and AE. In relation to the AE generated in plan-etary gearboxes, we have no knowledge of literature other than[13], where results from AE measurements on a planetary gearbox(non-faulty) from a wind turbine were presented, as well as anexplanation about the origin of the observed AE activity, and a dis-cussion about the cyclostationary characteristics of the measuredsignals. Also the need of an analysis of the AE under different work-ing conditions was pointed out in that work, but this was not done.It is the aim of the present work to provide such analysis.

2. Test bench measurements and results

2.1. Description of the test bench

The test bench used for the measurements has an asynchronousmotor (22 kW) driving the sun gear of the planetary gearbox; the

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output of the gearbox is connected to a DC generator. The shaftconnection at the input and output of the gearbox is made throughflexible couplings. The motor and the generator are controlledthrough frequency controllers allowing the adjustment of rota-tional speed and load. The planetary gearbox has three equallyspaced planet gears (spur type), each of them mounted on needlebearings (Fig. 1a and b). The number of teeth of the sun, planetand ring gear are ZS = 18, ZP = 26 and ZR = 72.

During operation, AE are generated on each of the meshingsoccuring on the gearbox. In this case, since the number of planetsis N = 3 and each planet meshes simultaneously with the sun andring gear, the total number of meshings is 6. Previous experimentshave shown that during the meshing of a pair of spur gears a com-bination of continuous and burst-type AE occurs [2,3,14]. Pres-ently, most researchers agree that the continuous part of the AEoriginates in the gear mesh period portion, where a combinationof rolling and sliding contact occurs; on the other hand, the burstis attributed to be generated at the point where pure rolling occurs(i.e. at the pitch point) – although we believe there is no clear evi-dence for the last statement. Nevertheless, the AE generated oneach of the six meshing has the shape of a continuous backgroundnoise with a superimposed train of burst with mean separationequal to the gear mesh period of the gearbox,

Tpg ¼

ZS þ ZR

ZSZRTS ð1Þ

where TS is the rotation period of the sun gear.It is shown in [15] that a phase-shift can exist between all

meshing in a planetary gearbox. The amount of the shift dependson geometric characteristics of the planetary gearbox, specificallyon the parameters ZS, ZR and the angular position of the planetgears in the carrier plate. It is demonstrated in [16] that differentspectral structures can be expected for planetary gearboxesdepending on the phase-shift, and the effect of the variable trans-mission path due to the revolving of the planet gears around thesun gear. This same study proposes a classification for planetarygearboxes according to the spectral structure of the vibrations;conforming to it, the planetary gearbox of the test bench belongsto Group (A): Planetary gearboxes with equally-spaced planet gearsand in-phase gear meshing processes. Planetary gearboxes pertain-ing to this group theoretically produce vibration spectra with linesat the gear mesh frequency and multiples, each with a symmetricalfamily of sidebands spaced at N-times the carrier rotational fre-quency, when measured with a sensor installed on the outer partof the ring gear. A similar structure is expected for the envelopespectra of the AE signals measured in the same manner.

Fig. 1. (a) Internal view of the planetary gear

An AE sensor and an acceleration sensor were mounted on thetop outer part of the ring gear, as shown in Fig. 2. The portion ofthe outer ring where the sensors were installed was previouslymilled to a flat surface to ensure a proper mounting of the AE sen-sor. The AE sensor was installed directly in contact with the ringgear using a film of grease as a coupling medium. The AE sensorwas held in place by using a clamp system constructed for this pur-pose. As can be inferred, acceleration was also measured during thetest; however, we only present the results of the AE in this paper.

2.2. Influence of temperature

It was suggested in [17] that the lubricant temperature, whichaffects the film thickness, influences the amplitude of the AE gen-erated in the meshing of gear teeth. In order to evaluate the effectof temperature, a temperature sensor (PT100) was installed in thelubricant sump. It is important to notice that the temperature mea-sured in the sump differs from the temperature of the lubricant be-tween the meshing teeth, which is actually the temperatureneeded for the evaluation of the film thickness. In [3] results arepresented for the temperature measurements taken in the lubri-cant sump and in one tooth of one of the gearing wheels, wherethe difference can be observed. Notwithstanding the existence ofthe temperature difference, it is concluded that both temperaturesbehave very similarly during the conducted tests. Hence, the tem-perature of the lubricant sump can be taken as a good estimate ofthe behavior of the temperature of the lubricant between meshingteeth, keeping in mind that the latter is always higher.

The evaluation of the effect of the temperature in the AE wasperformed through the joint assessment of the temperature andthe moving RMS of the AE during the warm-up of the gearbox.The moving RMS was calculated from the AE filtered between50 kHz and 200 kHz. The filter was used to enhance the signal-to-noise ratio of the signal. In the tests, the gearbox was run forapproximately 20 min starting from rest condition. Two types oflubricants were used: a typical gear oil (Mobil CLP144) andhydraulic fluid. Table 1 summarizes the performed tests.

Fig. 3a shows the evolution of the AE-RMS and the temperaturefor Test 1. It is observed that from 0 s. to �350 s. the AE-RMS mag-nitude did not change significantly; whereas the temperature in-creased. Differently, from �350 s. to �550 s an increase in theAE-RMS is observed along with the temperature; the increase inthe AE-RMS is followed by a decrease that ends at �600 s. From�600 s on the increase in temperature produces only a mild in-crease in AE-RMS. The expected behavior according to the elasto-hydrodynamic theory [18] is that an increase in the temperature

box. (b) Planet gear and needle bearing.

Fig. 2. Sensors mounted on the planetary gearbox.

Table 1Tests performed for the evaluation of the effect of temperature.

Test number Motor speed (rpm) Torque (Nm) Lubricant

1 1000 50 Oil Mobil CLP1442 1200 50 Oil Mobil CLP1443 1200 50 Hydraulic fluid4 1200 50 Oil Mobil CLP144

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of the lubricant produces a decrease in its film thickness betweenthe meshing teeth; in turn, this produces an increase in magnitudeand (possibly) rate of asperity contact between the mating toothsurfaces. The results of Test 1 suggest that at the beginning ofthe test, the lubricant film thickness is larger than the asperities

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Fig. 3. Evolution of temperature and

of the tooth surfaces, such that little or no contact between themoccur; hence the AE-RMS is not influenced by the temperature.As temperature keeps rising, a certain lubricant film thickness isreached where the asperity contact begins to increase, thus pro-ducing the strong increase in the AE-RMS. The decrease in AE-RMS that follows could be explained by a reduction of the asperi-ties due to a smoothening of the asperities. The mild increase ob-served afterwards should be due to further reduction of thelubricant film thickness along with the increase in temperature,under the new condition of asperities achieved previously.

Fig. 3b shows the evolution of the AE-RMS and temperature forTest 2. As in Test 1, at first the increase in the temperature does notproduce important changes in the AE-RMS (from 0 s to �350 s).Afterwards, an increase in the AE-RMS is observed along with theincrease in the temperature, but only until a certain time (from�350 s to �700 s). After this point, and unlike the previous test,a sustained decrease in the AE-RMS is observed while the temper-ature remains increasing. It should be noted that the temperaturein the final part of Test 2 is higher than in Test 1. This temperaturedifference could explain the different observed behavior. At highertemperatures the lubricant film is thinner, which favors asperitycontact. As mentioned, this has two contrary effects: (i) higherasperity contact produces more AE activity, and (ii) more asperitycontact produces smoothening of surfaces, which results in lessAE-RMS levels. The total behavior of the AE-RMS will depend onwhich of these effects is predominant. It is reasonable to think that,under constant load and speed conditions, the first effect woulddominate at the beginning and then the second effect prevails.

In Test 3, the gearbox was run with the same speed and load asin Test 2, but this time hydraulic fluid was used instead of oil as lu-bricant. The hydraulic fluid builds a thinner film between themeshing teeth than the oil does, so that different behavior is ex-pected for the AE-RMS. Fig. 3c shows the results of Test 3. An unde-

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(d) Test 4 (Oil Mobil CLP144)

AE-RMS for the tests carried out.

Table 2Tests performed for the evaluation of the effect of load and rotational speed.

Test number Motor speed (rpm) Torque (Nm) Lubricant

5 600; 1200; 1800 15; 40; 65 Oil Mobil CLP1446 600; 1200; 1500; 1800 15; 27.5; 40; 65 Oil Mobil CLP1447 600; 1200; 1500; 1800 15; 27.5; 40; 65 Hydraulic fluid8 1500; 1800 15; 27.5; 40; 65 Oil Mobil CLP144

C.M. Vicuña / Applied Acoustics 77 (2014) 150–158 153

tected error occurred in the measurement chain of the PT100, sothat no temperature signal is available for this test. Only sporadictemperature measurements were taken with a laser probe on theouter part of the ring gear; these are presented as discrete pointson the graph. From these points, an incremental tendency similarto one the observed in the previous tests can be seen, althoughin this test the temperatures were higher. The magnitude of theAE-RMS shows an increase in time, whose rate of change lowersas the time progress. Unfortunately, the test was stopped justwhere the AE-RMS started to stabilize, so is not possible to say ifthe AE-RMS held in this value or started to decrease. It is also inter-esting to note that the behavior of the AE is much more stable thanin tests performed with oil (i.e. the curve is cleaner). This speaks ofa steadier AE generation process when hydraulic fluid was used.The tendency of the AE-RMS observed in this case seems also to re-spond to the two contrary effects previously mentioned.

Fig. 3d shows the results of Test 4, which is the repetition of theTest 2, but after performing a set of measurements with thehydraulic fluid acting as lubricant. Here a slightly different pictureis observed: the AE-RMS starts with higher values and decreases,whereas the temperature increases progressively (from 0 s to

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Fig. 4. Test 5: DFT magnitude spectra o

�450 s). Between �450 s and �750 s, an increase–decrease–in-crease situation is recognized in the AE-RMS. Afterwards, a de-crease in the AE-RMS is observed untill the end of themeasurement, similar to the final part of the tendency observedin Test 2. The end magnitude of the AE-RMS is also very similarto the magnitude at the end of Test 2. One could believe that thesituation from Test 4 would have not differed much from the situ-ation observed in Test 2, if the starting magnitude of the AE-RMSwould have been lower (0.020 V in Test 4 vs. 0.013 V in Test 2).The higher starting magnitude of the AE-RMS could be explainedby a less favorable tooth surface roughness condition at the begin-ning of Test 4, caused by the intermediate tests performed withhydraulic fluid (not only the warm-up test was conducted, but aset of other experiments, as presented in Section 2.3). The situationobserved between �450 s and �750 s suggests the existence of atransition status of the AE-RMS, with some degree of instability.

2.3. Influence of load and speed

It has been reported that load and rotational speed are param-eters that strongly influence the AE resulting from gear meshing[2,3,14,19]. Even though it is clear that these parameters do affectthe AE, their effects on the AE seems not as clear. For example, theresults presented in [3] suggest that load has a significant influencein the AE-RMS (AE-RMS increases along with load;) whereas theresults presented in [2] shows there is no influence from load inthe resulting AE-RMS. We aim to contribute to the discussion ofthe influence of load and rotational speed in the AE, by presentingthe results obtained in our tests, and their corresponding analysis.

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154 C.M. Vicuña / Applied Acoustics 77 (2014) 150–158

Acoustic emissions were measured under different conditionsof load and rotational speed on the outer part of the ring gear ofthe planetary gearbox. Table 2 summarizes the load and rotationalspeed conditions used in the different tests. All measurementswere performed after the warm-up periods described in the previ-ous section. In this case, the magnitude of the envelope spectrumlines calculated from the AE signal filtered between 50 kHz and200 kHz, which are related to the mesh of gears, are taken as anevaluation parameter for the influence of load and rotationalspeed.

The envelope spectrum for the measurements of each test arepresented in Figs. 4–7. In each figure, the temperature of the lubri-cant measured in the sump is also presented (when it was avail-able.) The following observations can be made from these figures:

� The results of all tests show a clear influence of rotational speedon the AE generated in the gear meshing. For example, in Fig. 4,the magnitude of the lines related to the gear meshing at1800 rpm are roughly ten times larger than the lines for themeasurement at 1200 rpm for the same load.� Load did not show a clear influence in the tests where Mobil

CLP144 oil was used as lubricant and where the rotationalspeed was higher than or equal to 1200 rpm. On the contrary,an increase in the load produced a clear increase in the magni-tude of the spectral lines for the measurements at 600 rpm (seeFigs. 4a–c and 5a–d.) However, the observed influence of theload was, in percentage, much lower than the influence of rota-tional speed.

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Fig. 5. Test 6: DFT magnitude spectra o

� When the hydraulic fluid was used, the influence of the loadwas also observed for higher rotational speeds. Like in the othercases, here the influence of the load in the magnitude of thespectral lines was higher at lower rotational speeds.� Interestingly, the magnitudes of the spectral lines when the

hydraulic fluid was used were lower than the observed whenthe Mobil oil was used. Due to the low viscosity of the hydraulicfluid, the film between the meshing teeth is thinner comparedto the film created when using the oil. Hence, an increase inthe asperity contact – and, therefore in the AE magnitudes –was expected for the measurements taken with hydraulic fluid.

Regarding this last observation, analysis of the shape of the AEtime-history suggests that the predominant mechanism of AE gen-eration differs depending on if the hydraulic fluid or the oil wasused. This is depicted in Fig. 8a and b, both presenting a portionof the AE signal for 1800 rpm, 40 Nm; the first figure is when oilwas used, and the second when hydraulic fluid was used. The pres-ence of the burst-type activity only in the former figure means thatthe mechanism that generates them is predominant (the genera-tion of the burst is usually attributed to the rolling portion of thegear mesh period). In the second figure, the AE is predominantlyof continuous type, suggesting that the main mechanism of AEgeneration is the asperity contact during the gear mesh period por-tion where a combination of rolling and sliding occurs. Note that itis possible to do this analysis in this case only because the plane-tary gearbox used belongs to Group (A), according to [16] (i.e.has no phase-shift between the different gear meshes).

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3. Measurements on the planetary gearbox of a bucket wheelexcavator

Acoustic emissions were measured on the planetary gearbox ofa Bucket Wheel Excavator (BWE) used for lignite extraction. Twoplanetary gearboxes are connected in series and constitute the lastreduction stages in the drive system of the bucket wheel. Fig. 9shows a general view of the BWE. The sensor was installed onthe top outer part of the ring gear of the second planetary stage.The measurement system was installed at a distance of about20 m from the planetary gearbox, and was controlled wirelesslyfrom the operator’s room. Several measurements were conductedduring a period of approximate 1 h. All data was stored locallyon the hard drive of the measurement system. Once the systemwas dismounted, the data was retrieved for processing andanalysis.

3.1. Description of the gearbox

The gearbox consists of a fixed-shaft, spur bevel reductionstage, followed by two fixed-shaft spur gear stages. The output ofthe latter stage is connected to the sun gear of the first planetarystage. The carrier plate of the first planetary stage is connectedto the sun gear of the second planetary stage. Finally, the carrierplate of the second planetary stage is coupled to the bucket wheel.Fig. 10 shows a scheme of the gearbox, whose total ratio is 349.1:1.The gearbox is driven by an electrical motor whose nominal speed

Fig. 6. Test 7: DFT magnitude spectra

is 990 rpm; at this rotational speed, the bucket wheel rotates at2.83 rpm.

Both planetary stages have three equally-spaced planet gears;all gears are spur gears. The number of teeth of the first planetarystage are ZS1 ¼ 21, ZP1 ¼ 64 and ZR1 ¼ 150, and of the second stageare ZS2 ¼ 27, ZP2 ¼ 31 and ZR2 ¼ 90. It can be demonstrated that allgear meshing of the first stage are in-phase; whereas in the secondstage, all planet-ring meshing are in phase, all planet-sun meshingare in phase, but there is a phase difference between the planet-ring and planet-sun meshing [16].

3.2. Operating conditions

The BWE is used to retrieve material from the overburden. Asthe bucket wheel rotates, the buckets dig into the overburden,are loaded with material and then transfer the load onto a con-veyor belt at the top part of the bucket wheel. This process is per-formed continuously.

Evidently, the forces resulting from the digging process de-scribed are highly variable, so that the load acting on the differentgearbox stages is also variable. Furthermore, the variable load pro-duces changes in the rotational speed of the complete drive. It iswell known that variations in the rotational speed produce smear-ing of the lines in the magnitude spectrum, which complicates theanalysis. To overcome this problem, the AE signals were analyzedin the angular domain. The angular resampling was possiblethanks to a voltage signal proportional to the rotational speed ofthe motor. This signal was the only available one and is not ideal

of AE envelope (Hydraulic fluid).

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Fig. 7. Test 8: DFT magnitude spectra of AE envelope (Oil Mobil CLP144).

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for the conversion of signals from time domain to angle domain,since some noise will be introduced into the signal in the process.Still, the results obtained were significantly superior that the ob-tained when working with the signals in time domain.

3.3. Results

Fig. 11 shows a portion of the AE signal after angular resam-pling, where a strong amplitude modulation is observed. The signalis composed by a series of bursts spaced apart at the gear meshperiod of the first planetary stage (although they do not strictly re-peat in a periodic fashion) with an amplitude modulation due tothe load variation caused by the repetitive digging of the buckets.The amplitude modulation reveals the existing influence of the

load on the AE generated in the meshing of gears. It is also interest-ing to highlight that the influence of load is important (i.e. theamplitude modulation is strong). Considering the low rotationalspeed of the planetary stage (mean sun rotational speed is100 rpm), this behavior is in accordance with the observationsmade from the test bench results. Fig. 12 shows a joint plot ofthe envelope of the AE and the rotational frequency of the sun gearof the first planetary stage. Note the second signal is plotted withinverted ordinate. The rotational frequency plotted in this manneris used as a rough estimation of the shape of the load acting on thetransmission. This is based on the fact that load variations producespeed variations in rotating mechanical systems. We have ob-served that in systems undergoing continuous load/speed varia-tions, load and speed time signals present very similar shapes,but in opposite direction (i.e. speed reduces when the load in-creases, and vice versas). This situation is also observed in Fig. 3from [12]. Please note that this estimation was only necessary be-cause no signal related to load was available for this measurement.Considering this, Fig. 12 clearly shows the influence of load in theAE measured in the BWE.

It was presented in [13] that acoustic emissions from gears arenon-stationary signals, with high second-order cyclostationary en-ergy content. Fig. 13 shows a portion of the Spectral Coherence(SCoh) [20] of the signal around the gear mesh cyclic order of thefirst stage. A discrete component along the a-axis is recognizedat the gear mesh cyclic order of the first planetary stage ðap

g1Þ.

Although barely recognizable, some sidebands around this compo-nent spaced at three times the carrier plate cyclic order of the firststage ðaC1 Þ can also be identified. They should become clearerimproving the cyclic frequency resolution; however, this was notdone due to the high computational demands needed.

The SCoh also shows that the AE activity related to the gearmeshing of the first stage is concentrated above �50 kOrd. In thefrequency domain, this is equivalent to a high-pass filter with cut-off frequency of �80 kHz. Accordingly, the envelope order spec-trum from the filtered AE signal was calculated and a portionaround the gear mesh cyclic order of the first stage is presentedin Fig. 14. Since the computational demands for calculating theenvelope are significantly lower than the demands for calculatingthe SCoh, it is possible to handle a longer AE signal with the conse-quence of finer cyclic frequency resolution in the envelope spec-trum – note, however, the enhancement of the envelopetechnique due to the optimum filter information obtained fromthe results of the SCoh previously calculated. The finer resolutionallows to clearly identify the components spaced at 3� aC1 aroundap

g1. It was also expected to observe sidebands spaced at aC1 ,

Fig. 9. General view of the bucket wheel excavator.

27 Motor

Bucket wheel

62

198382

2164150

2709 13

Fig. 10. Scheme of the gearbox. The numbers indicate the tooth number of eachgear wheel.

Fig. 11. BWE measurement. AE signal.

Fig. 12. BWE measurement. AE envelope signal and rotational frequency of

Fig. 13. BWE measurement. Spectral Coherence of the angular resampled AE signal(portion around the gear mesh cyclic order).

C.M. Vicuña / Applied Acoustics 77 (2014) 150–158 157

although this was not the case. Even though we have found similarenvelope spectrum structures in other planetary gearboxes, we donot have an explanation for this yet.

Interestingly, no AE activity related to the gear meshing of thesecond planetary stage was recognized in the measurements. TheAE activity was found to come predominantly from the gear mesh-

the sun gear of the first planetary stage (plotted with inverted y-axis).

Fig. 14. BWE measurement. DFT magnitude spectra of AE envelope (portion aroundthe gear mesh cyclic order).

158 C.M. Vicuña / Applied Acoustics 77 (2014) 150–158

ings of the first stage, even though the sensor was placed on thering of the second stage. The conclusions obtained from the resultsof the test-bench experiments allow to attribute this situation tothe much lower rotational speed of the second stage (mean sunrotational speed is 12.3 rpm).

4. Conclusions

Temperature affects the thickness of the lubricant film,which determines the amount of asperity contact that occursbetween the meshing teeth. The decrease in the film thicknesshas two contrary effects on the AE. The relation betweenthem is dynamic and determines the global behavior of theAE-RMS.

The results from the experiments with different loads and rota-tional speeds showed that rotational speed is the most importantparameter influencing the AE generated on the planetary gearbox.At lower rotational speeds, the influence of the load in the resultingAE can be significant; at higher rotational speeds, the influence ofthe load is masked by the influence of the rotational speed. Theseconclusions are in agreement with the behavior of the acousticemissions measured on the planetary gearbox of the bucket wheelexcavator.

Although these results were obtained from experiments carriedout on a planetary gearbox, they could reasonably be extrapolatedto fixed-shaft spur gear transmissions.

Acknowledgment

The author thanks FONDECYT for the support of the Project11110017.

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