Inspection of Tooth Surface Geometry by Means of Vibration ...

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Inspection of Tooth Surface Geometry by Means of

Vibration Measurement∗(Assessment of Tooth Surface Undulation from Synchronous Averaged

Signal and Application of Frequency Response Function)

Chanat RATANASUMAWONG∗∗, Shigeki MATSUMURA∗∗∗ and Haruo HOUJOH∗∗∗

Tooth surface undulation is one of the important sources of gear noise and vibration.The vibration caused by this source is observed as the occurrence of non-meshing vibrationcomponent or ghost noise on a vibration spectrum. Frequently ghost noise occurs at thesame frequency with natural frequency of a gear pair, consequently its amplitude is amplifiedto the considerable level and lead to unexpected and severe noise and vibration problems.In this paper a method for inspecting tooth surface undulation is proposed and applied to ahelical gear pair. Vibration characteristics of individual gear are extracted from the vibrationsignal of a gear by synchronous averaging technique, then a frequency response functionthat can be determined experimentally is applied to the individual averaged signal to assessthe tooth surface undulation. The undulations are evaluated by applying this method to themeasured vibration signals of the gear pair operated at various speeds and various torques,and show good agreement with each other regardless of operating conditions and also withthe expectation by precise tooth surface measurement, even though the undulation is verysmall in the level of 0.1 µm. These results suggest the ability of this method to assess thetooth surface geometry relevant to vibration.

Key Words: Gear, Vibration, Synchronous Averaging, Frequency Response Function,Tooth Surface Undulation

1. Introduction

It is known that existence of irregularity during gearmanufacturing process leads to various kinds of gear er-rors such as profile error, pressure angle error, lead er-ror, and tooth surface undulation. These errors resultin gear vibration problems at both harmonics of mesh-ing frequency and non-meshing frequencies. Gear vibra-tions at harmonics of meshing frequency are caused byvariation of tooth stiffness during meshing and caused byerrors which are common in all teeth. These vibrationcomponents generally have higher amplitude than non-

∗ Received 18th February, 2005 (No. 05-5017)∗∗ Graduate Student, Department of Precision Machinery

Systems, Tokyo Institute of Technology, 4259 Nagatsuta,Midori-ku, Yokohama 226–8503, Japan.E-mail: [email protected]

∗∗∗ Precision and Intelligence Laboratory, Tokyo Institute ofTechnology, 4259 Nagatsuta, Midori-ku, Yokohama 226–8503, Japan

meshing vibration components, therefore there are a lot ofresearches in the past that studied about meshing compo-nents. Some of them are the works by Kubo and Kiyono(1),Sato et al.(2), Umezawa et al.(3)

Although the non-meshing vibration componentsusually have less amplitudes than the meshing compo-nents, they are likely to occur at various harmonics of shaftfrequency and to excite system natural frequencies at var-ious operating speeds. In this case the amplitude of non-meshing frequencies are magnified by the resonance effectto considerable levels and frequently become larger thanthe meshing components and result in unexpected and se-vere noise problems. Moreover, because the non-meshingcomponents always occur at the frequencies that do notrelate with any nominal tooth geometry of a gear pair thevibration source is hardly detected. And with this reason,the non-meshing vibration components are usually calledas ghost noises.

It is known that the ghost noise is created by a smallperiodic waviness or a cyclic undulation on the gear tooth

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surface continuing from one tooth to the next as mesh-ing proceeds. This means the waviness is smoothly con-tinuous when the tooth profiles are drawn along the lineof action. The source of this cyclic undulation can betraced back to the irregularity in the process of gear man-ufacturing(4). Mark(5) showed theoretically how the peri-odic undulations on tooth surfaces could generate ghostnoises. Matson and Houser(6) showed experimentally thatthe cyclic undulation on the gear tooth surface acts as anexciter to generate ghost noise. In general, it is difficult todetect such undulation because of its small amplitude aswell as small number of cycles per tooth.

Matsumura et al.(7), (8) introduced Maximum EntropyMethod (MEM) and conventional Fourier Analysis (DFT)along with tooth surface measurement to examine the ex-istence of the cyclic undulation on tooth surface that isthe cause of ghost noise. Although both of them can beused to investigate the tooth surface undulation, they re-quire two-dimensional and multi teeth measurement, andhence plenty of time and cost are required to detect by thisway. With this disadvantage, other methods for inspectionof the cyclic undulation on gear tooth surface are required.

One of the alternative techniques that has been ap-plied to detect ghost noise and cyclic undulation ontooth surface is “synchronous averaging technique”. Syn-chronous averaging technique is a time domain averagingtechnique even it is processed in spectrum domain. Eachblock of the measured signal to be averaged is selectedsynchronously with the trigger signal of shaft rotation. Af-ter enough times of averaging, the signal asynchronouswith the trigger signal is eliminated leaving the signal ofinterest that is completely periodic with the trigger signal.This technique has been used to reduce noise and improvethe quality of the measured signal that is periodic(9) – (11),and has also been applied for diagnosis of the failure orearly defect in gears and bearings(12) – (19).

In the previous work by authors(20), synchronous av-eraging technique was used to extract vibration character-istics attributed to driving or driven gear in the measuredsignal, therefore the gear causing the ghost noise can bedetected. However, although this indicates the existenceof surface waviness, the relation between the amplitude ofvibration and the magnitude of cyclic undulation has notbeen known at that time because the vibration characteris-tics of a gear system was not considered.

In this paper, a method to assess quantitatively thetooth surface undulation is proposed, in which the fre-quency response function of a gear system is inverselyapplied to the synchronously averaged signal. Effective-ness of the proposed method even for the waviness of sub-micro metric amplitude is verified by precise tooth surfacemeasurement results.

Fig. 1 Vibration model

2. Basic Formulation of Gear Vibration

Gear vibration comes from the combination of vari-ation of tooth stiffness with time and tooth surface formerrors. The former one is related to parametric excita-tion, which produces vibration at meshing frequency andits harmonics. The latter is related to displacement excita-tion, which potentially produces vibration at any harmon-ics of shaft rotation including meshing frequencies. Un-derstanding of the relation between vibration source andthe resultant vibration is the basic step for gear inspectionbased on vibration measurement.

To simplify consideration, the rotational vibration ofa helical gear pair can be treated as a single degree offreedom system similar to the vibration model used bymany investigators(21) – (25). The vibration model is shownin Fig. 1, where J is the moment of inertia, rb is the baseradius, T is the applied torque, and θ is the rotation angle.Subscript 1 and 2 mean driving and driven gear, respec-tively. From this model, equation of motion for the gearpair can be expressed along the line of action as,

Mx+Cx+K(t)(x−e(t))=W (1)

where

M=J1J2

J1r2b2+ J2r2

b1

, x= rb1 ·θ1−rb2 ·θ2,

W =T1

rb1=

T2

rb2,

M is the equivalent inertia mass of the gear pair alongthe line of action

x is the relative displacement between two gearsalong the line of action

C is the damping coefficientK is the meshing stiffness of the gear paire is the error of meshing teeth of a gear pairW is the static normal transmitting load

By decomposing the meshing stiffness and the relative dis-placement into their time invariant components and fluc-tuating components, the following equations are obtained.

K(t)= K+∆K(t) (2)

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x= x+∆x(t) (3)

W = K · x (4)

Substituting Eqs. (2) – (4) into Eq. (1), and neglectinghigher order term as a primary approximation, yield

M∆x+C∆x+ K∆x= K ·e+∆K ·e−∆K · x (5)

The gear error, e, can be considered as the sum of errorsthat are pressure angle error or any common error of ev-ery gear tooth, e1, fluctuating part from the common errorsuch as pitch error, e2, and the cyclic undulation on toothsurface continuing one tooth to the next, e3. Single toothdefects such as nick damages are also probable, but sincethe existence of these defects can be detected in time do-main by the method proposed in Refs. (12), (14), (17) and(18), and diagnosis of tooth defect is not the objective ofthis paper, this kind of defect is excluded from the argu-ment of this paper.

With above consideration, Eq. (5) can be written inthe form of three kinds of gear errors as expressed inEq. (6). The terms having relatively small amplitude(∆K ·e1, ∆K ·e3) are also neglected.

M∆x+C∆x+ K∆x

= (K ·e1−∆K · x)+ (K+∆K) ·e2+ K ·e3 (6)

Both the variation of meshing stiffness, ∆K, and thecommon error of every gear tooth, e1, vary at the frequen-cies equal to harmonics of meshing frequency, thereforethe first term in the right hand side of Eq. (6) means theexcitation to vibrate at harmonics of meshing frequency.In other words, the vibration components at harmonics ofmeshing frequency are caused by the variation of toothstiffness during meshing and by the common error of ev-ery gear tooth.

The second term in the right hand side of Eq. (6)shows the excitation affected by the error, e2. This kind oferror is caused by pitch error or misalignment, thereforethe variation of e2 always has low frequency, say, sev-eral cycles per one revolution. The vibration excited bythis error will occur at low harmonics of shaft frequency.This kind of error also causes the amplitude and phasemodulation to harmonics of meshing frequency due to theproduct of the error with the variation of meshing stiff-ness (∆K ·e2). The result of this modulation is observed assidebands surrounding meshing vibration components.

In addition to the components presented above, thereare also the occurrences of non-meshing components, orghost noises. These vibrations come from the excitationshown by the third term in the right hand side of Eq. (6).The cause of these components is the existence of cyclicor quasi-cyclic undulation on tooth surface continuing onetooth to the next and represented with e3.

The gear vibration is the result of inputting all excita-tions discussed above into the frequency response function

Fig. 2 Experimental apparatus

of the gear system. In this paper only the tooth surface un-dulations and their relevant vibration components that arenon-meshing components, or ghost noises, are consideredand will be processed further.

3. Experimental Apparatus and Gears

The experiments for assessing the tooth surface undu-lation were performed with an apparatus shown in Fig. 2.A test helical gear pair was driven by a variable speed ACmotor via V-belts. Load torque was applied to the gearpair by a dynamometer connected to a driven shaft. Twodiaphragm couplings mounted between gear shafts and themotor or the dynamometer were used to isolate vibrationattributed to the other parts in the apparatus from the gear-box.

The vibration of the gear pair was measured by a pairof accelerometers attached tangentially at the driven gearto extract only the rotational vibration. Then the vibrationsignal was sent via a slip ring to a signal analyzer. At thesame time, trigger signals, one pulse per revolution, weremeasured simultaneously at both driving and driven shaftby magnetic pick-ups placed close to the shaft surfaces.These signals were used as the triggers for synchronousaveraging.

The driving gears used in this experiment have 30teeth, while the driven gear has 53 teeth, to give primegear ratio. Module is 2.5. Face width is equal to 20 mm.Other parameters of test gears are shown in Table 1.

The identical driven gear, finished by indexed gener-ation method, was used in all experiments, and is calledas the “master gear”. On the other hand, the driving gearwas changed among ones by five different finishing meth-ods to provide various tooth surface conditions and vari-ous characteristics of ghost noise. The details of finishingmethod of test gears are shown in Table 2. It should benoted here that the “master gear” does not mean that thedriven gear is ideal having no error, but that the identicaldriven gear, which has reasonably small errors, was used


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throughout all experiments to keep the measuring condi-tion unchanged. The reason why errors are allowed toexist on the master gear is that they can be distinguishedthrough synchronous averaging process as reported in theprevious paper(20).

Figure 3 shows tooth surface forms of the test gearsinspected by a gear inspection machine. The left hand side

Table 1 Gear parameters

Fig. 3 Tooth surface forms of the test gears

shows tooth profile forms, whereas tooth trace forms areshown in the right hand side. The abscissa of measuredtooth profile result is the base tangent length that is nor-malized with base pitch. On the other hand, the abscissaof measured tooth trace result is axial position. The ordi-nates of all results are tooth surface deviation. From theseresults, only the tooth profile form and the tooth trace formof the gear SV1 are apparently different while another gear

Table 2 Gear finishing methods


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Fig. 4 Synchronous averaging result of the gear pair GTR4

tooth surface forms are similar. Moreover it is impossibleto specify the difference of cyclic surface undulation byviewing these tooth surface forms.

In the experiment, the gear pair was warmed up be-fore measuring vibration signals so that it can be operatedsteadily. Moreover because running time is short, the de-veloping of wear on gear tooth surface is considered to bevery small and reasonable to neglect. This was confirmedwith repeatability test.

4. Processing Method and Synchronous AveragingResults

To assess the tooth surface undulation of individualgear, it is necessary to know the vibration characteristicsof each gear firstly. Even though placing accelerometers atonly one of either driving or driven gear, measured vibra-tion is still the combined effect of both driving and drivengear. This means that the vibration measurement at onegear in the gear pair gives vibration characteristics of bothgears. To extract the vibration characteristics of the indi-vidual gear, synchronous averaging technique is applied tothe measured vibration signal along with the use of drivingor driven shaft trigger signal. The detail of synchronousaveraging technique with this test gear pair is the same asreported in the Ref. (20).

The number of averaging times is one of the impor-tant parameters that affect to the accuracy of the result. Itmust be large enough so that the vibration characteristicsattributed to driving or driven gear can be distinguishedfrom each other thoroughly. Theoretically, if there are noother vibrations than that produced by the gear pair, num-ber of averaging times can be reduced to 53 or 30, which

are the multipliers for the least common multiple of toothnumbers of the test gear pair. However, because measuredvibration always includes the vibration components at-tributed to the other parts such as bearing or random noise,larger number of times of averaging is required. From theprevious report by authors(26), it is found that synchronousaveraging results done by averaging more than 256 timesare almost unchanged, therefore 256 times averaging islarge enough and was used in this experiment.

Figure 4 (a) and (b) show averaged waveforms ofthe gear pair GTR4 at driven shaft rotational speed of1 800 rpm and applied torque 245 Nm (25 kgf ·m) aftersynchronous averaging with driving and driven trigger.The abscissa is time that is presented to fit with the periodof driving and driven shafts individually. The ordinate isthe amplitude of vibration (acceleration).

Because remaining signal after synchronous averag-ing are absolutely periodic with driving or driven shaftperiod depending on trigger source, these synchronousaveraging waveforms conform theoretically line spectracontaining only vibration components having shaft har-monic frequencies. Therefore the window length was setto equal to the period of driving or driven shaft rotation,and rectangular window function (no window) was usedin the discrete frequency analysis process to bring suchideal spectra that have the resolution equal to shaft fre-quency. The line spectra of the gear pair GTR4 processedby this method are shown in Fig. 4 (c) and (d) for drivingand driven gear, respectively.

If frequency analysis was done by general methodwith arbitrary window length, window function such ashanning window is necessary to reduce the leakage effect.


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Fig. 5 Synchronous averaging results

However the use of window function makes the loss ofdata at the beginning and the end of data block inevitably.Moreover the frequency resolution of spectrum analyzedby a general method does not always match with shaftfrequency. Therefore frequency analysis by the generalmethod using hanning window is inaccurate. The spec-tra of the gear GTR4 processed by the general methodare shown in Fig. 4 (e) and (f). By comparing the spec-tra processed by both methods, it is found that the spec-trum processed by the general method has componentswith less amplitude than those of the line spectra as shownin Fig. 4 (c) and (d) and has leakage.

In the synchronous averaged spectra with driving anddriven trigger shown in Fig. 4 (c) and (d), the vibration at-tributed to the gear GTR4 (Fig. 4 (c)) has a series of non-meshing component at every six order of driving shaft ro-tation from 18th to 84th order. On the other hand the mas-ter gear has some peculiar peaks of vibration between thesecond and the third harmonic of meshing frequency (2fz

and 3fz) at about 122nd and 138th order of driven shaftrotation (Fig. 4 (d)).

Synchronous averaged results of the other gear pairswith driving and driven trigger are shown in Fig. 5. Themeshing components that are synchronous with both driv-ing and driven shaft rotation remain at identical levelswithin each gear pair after averaging. On the contraryfor different gear pair, although gear specifications are thesame in all cases, small differences of each driving pair ineither profile form or lead make the amplitude different.

Because the master gear was identical in all experi-

ments, those non-meshing components after synchronousaveraging with driven shaft trigger, right hand side ofFig. 5, are almost the same regardless of the kinds of driv-ing gear. On the other hand the ghost noises in the caseof driving trigger are different according to the tooth sur-face conditions of the test gears. From these results, itis concluded that the use of synchronous averaging tech-nique along with spectrum analysis with the rectangularwindow having the length equal to one period of shaft rev-olution can extract the vibration characteristics attributedto individual gear from the complicated measured signal,and then the order of cyclic undulation can be detected.

5. Assessment of Cyclic Undulation on Tooth Surface

Figure 6 shows the waterfall plots of vibration spectrawithout averaging of the test gear pairs used in this exper-iment. These waterfall plots show the vibration character-istics of each gear pair operated from 550 rpm to 2 300 rpmof driven shaft rotational speed. Applied torque in everycase was 245 Nm at driven shaft. The height of the spec-trum is the amplitude of the rotational vibration (accelera-tion) along the line of action. The abscissa is the frequencyof the vibration in the range 0 to 5 kHz.

Meshing components (fz, 2fz, and 3fz) are seen asthe oblique linear lines. Natural frequencies of this gearsystem are seen as vertical bands and are marked by theblank arrows. Natural frequencies at about 3 600 Hz and4 000 Hz are prominent among them. These are the causeof the relatively high peaks between 2fz and 3fz in the syn-chronous averaged results shown in Figs. 4 and 5. From


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Fig. 6 Waterfall plots of the test gear pairs

these results, it is clear that the spectra obtained by syn-chronous averaging show only the vibration character-istics of an individual gear including orders of existingcyclic undulation. However they do not directly show thegeometrical magnitude of the cyclic undulation on toothsurface.

To assess the tooth surface undulation quantitatively,the equation of motion shown in Eq. (6) is taken into con-sideration. If only the effect of the tooth surface undula-

tion, e3, is considered, Eq. (6) can be reduced to Eq. (7) byneglecting other excitations as expressed below.

M∆x+C∆x+ K∆x= K ·e3 (7)

Equation (7) can be written in the form of spectrumpresentation in terms of the tooth surface undulations,E(ω), the output vibration of non-meshing components,A(ω), and the frequency response function, H(ω) as,

A(ω)=H(ω)×E(ω). (8)

Therefore if the frequency response function isknown, the tooth surface undulation can be inversely as-sessed from the measured vibration of non-meshing com-ponents at each frequency.

5. 1 Determination of frequency response functionIn Eq. (8), the frequency response function is essential

for inverse evaluation of the cyclic undulation. The proc-ess for determining the frequency response function canbe divided into 3 steps; 1) determination of the shape ofthe frequency response function in logarithmic representa-tion, 2) selection of the reference order for amplitude cal-ibration of the frequency response function and 3) findingthe sensitivity of the vibration corresponding to the cyclicundulation at the reference order. In these three steps, firststep is about determination of the response function shape,whereas the second and the third step are about method forcalibrating the amplitude of the frequency response func-tion. The details of each step are described below.

5. 1. 1 Determination of the shape of frequency re-sponse function Because the value of the mean mesh-ing stiffness K is invariable in each driving condition,Eq. (7) indicates that the non-meshing components, or theghost noises, are affected only by the displacement errorclassified as the tooth surface undulation, e3. With thisreason, the shape of frequency response function of thegear pair can be decided by considering the shape of aresponse due to one undulation component at a particu-lar order obtained through varying speed. However, ifone tries to determine the response function precisely, itis difficult to do for wide range of frequency because thelower bound of operating speed exists. Therefore, the sev-eral responses due to undulations existing at several ordersshould be considered and be combined together.

Figure 7 (a) shows the shapes of vibration responseof non-meshing components in the case of the gear pairGTR4 at various shaft orders. Here these responses wereobtained by tracking vibration at each order from the wa-terfall plot. These responses can also be obtained by shaftorder tracking analysis directly. Then each frequency re-sponse in logarithmic scale is vertically adjusted such thatthey conform a unique frequency response function cur-vature. By averaging these collected responses, the shapeof the frequency response function of the gear pair is ob-tained and is shown in Fig. 7 (b).


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Fig. 7 Determination of the shape of the frequency responsefunction

At this moment, the frequency response functionshown in Fig. 7 (b) performs only the shape of the re-sponse. However the relation between the magnitude oftooth surface undulation and the vibration level is still un-known. To calibrate the relation between the magnitudeof surface undulation and the amplitude of resultant vibra-tion, it is essential to know the tooth surface form of onegear of a mating pair. In this research, the master gear isregarded as the reference gear having known tooth surfaceform and is used for calibration as explained next. On theother hand the driving gears are treated as the inspectedgears having unknown tooth surface forms, and will beevaluated further. However, for the verification of the pro-posal, every one of both driving gears and driven gear wasinspected by a high precision gear tooth profile measuringequipment.

5. 1. 2 Selection of the reference order for ampli-tude calibration The reference order is the order ofwhich both the amplitude of vibration and the magnitudeof undulation are known. This order is used for amplitudecalibration of the frequency response function. Theoret-ically, any order of the non-meshing components wheremagnitude of undulation corresponding to this order isknown can be used as the reference order for calibration.But for accurate calibration, order that has relatively largeamplitude of vibration and undulation should be selectedas the reference order. With this criterion, the undulationof tooth surface at 26th order of the master gear and the vi-bration corresponding with this order obtained from syn-chronous averaging spectrum was selected for calibration.

5. 1. 3 Finding the sensitivity of the vibration cor-responding to the cyclic undulation Figure 8 showsthe method for amplitude calibration of the frequency re-sponse function. This figure is drawn in the same formwith waterfall plot shown in Fig. 6, but in this figureonly the spectrum obtained from synchronous averaging

Fig. 8 Amplitude calibration of the frequency responsefunction with known tooth profile

at 1 800 rpm in logarithmic representation is shown. Theoblique aligned curve shown in Fig. 8 is the frequency re-sponse (FR) of the 26th order (reference order). Its shapeis obtained from the method described in the first step.Because this frequency response is the vibration responseof the gear pair when it was excited by the cyclic undu-lation at 26th order, its height is placed such that it inter-sects with the synchronous averaged spectrum at 26th or-der. Therefore once the magnitude of exciting undulationat the reference order is known from tooth surface formmeasurement, the sensitivity of the magnitude of the vi-bration corresponding to the cyclic undulation is decided.

By applying of the frequency response function deter-mined and calibrated by the procedures described above,tooth surface undulation can be inversely evaluated.

5. 2 Assessment of tooth surface undulationThe frequency response function calibrated with the

undulation on the driven gear tooth can also be applied tothe driving gear to assess cyclic undulation on the driv-ing tooth surface because of the linearity of the system.This means the tooth surface undulation of the driving gearcan be evaluated from the synchronous averaged spectra(Figs. 4 and 5) using the calibrated frequency responsefunction.

The cyclic undulations assessed by the proposedmethod are shown in Fig. 9 (a) – (c) in the case of thegear pair GTR4, GTM2, and SV1. Results for the gearsGTO1 and THO1 are omitted in this paper, because theghost noises relevant to undulations were hardly observed.All evaluated results in Fig. 9 are obtained from the syn-chronous averaged results of the gear pair operated at1 800 rpm and applied torque equal to 245 Nm. Sincemeshing components and sidebands neighboring themwere caused by the other vibration sources beside toothsurface undulation, they are not included in the evaluationhere. Comparing these evaluated results with the surfacemeasurement results (whole surfaces were measured andanalyzed by the same method as described in Ref. (20))


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Fig. 9 Evaluated spectra of tooth surface undulation and tooth surface measurement results(omit meshing vibration components)

Fig. 10 Evaluated spectra of tooth surface undulation at variousrotational velocities (constant applied torque 245 Nm)of gear GTR4

shown in Fig. 9 (d) – (f), the evaluated results agree withthe tooth surface measurement results qualitatively. Verysmall undulation in the level of 0.1 µm is also detectedprecisely by this method. This confirms the ability of theproposed method for inspection of the cyclic undulation.

6. Discussion

It should be noted that the cyclic undulation consid-ered here is the undulation which can be recognized view-ing from a point along the line of contact and is continu-ing one tooth to the next to produce continuous harmonictransmission error, or vibration(20). The undulation ob-served from a point in tooth trace direction, which hasonly little effect to the gear vibration, is not detected bythe method proposed here.

Although the evaluated results discussed above wereonly under the operating condition that driven shaft speedis equal to 1 800 rpm and torque 245 Nm, this method isalso valid at other operating conditions. The results eval-uated from the vibration of the gear pair GTR4 operatedat various rotational speeds, but constant applied torque245 Nm are shown in Fig. 10. Each datum is the undula-tion at each dominant order (orders marked by ∇) in thespectrum shown in Fig. 9. Because the meshing stiffness

Fig. 11 Evaluated spectra of tooth surface undulation at variousapplied torques (constant rotational velocity 1 800 rpm)of gear GTR4

of the gear pair operated at a given torque does not changewith operating speed, it is obvious that the frequency re-sponse function in each case is the same from Eqs. (7)and (8). Therefore the frequency response function cali-brated at 1 800 rpm can be used to assess the cyclic undu-lation from the vibration signal measured at any rotationalspeeds. In the results shown in Fig. 10, the cyclic undula-tions evaluated from various operating speeds are almostthe same.

The result in the case of the gear pair GTR4 operatedat the constant rotational velocity 1 800 rpm but differentapplied torques is shown in Fig. 11. In this case, becauseapplied torque changes, meshing stiffness may vary de-pending on surface-to-surface contact condition. For thisreason, it seems to require new frequency response func-tion corresponding to each applied torque every time ap-plied torque changed. From the results shown in Fig. 11,however evaluated cyclic undulations are almost the sameat high torques and agreeable with surface measurementresult. There is significant difference between the eval-uated result and tooth surface measurement only for thelow torque condition. The cause of this difference is thevariation of tooth contact area during operation at different


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torques. At small torque, tooth contact area become nar-row, hence apparent magnitude of undulation when con-tact area is partial are different from the case of high torquethat tooth contact area widens to enclose almost full tootharea.

7. Conclusion

This paper introduced the concept for inspecting geartooth surface geometry by vibration measurement. Mea-sured gear vibration signal is considered separately ac-cording to the kinds of excitations that are the source ofvibration. This paper focuses only the non-meshing vibra-tion components which are relevant to undulations.

First, synchronous averaging technique was appliedto extract the vibration characteristics of individual gearfrom complicated measured signal. And next the fre-quency response function of the gear pair that can be de-termined experimentally was applied to the non-meshingcomponents of the averaged spectrum to evaluate toothsurface undulation.

From the results shown here, it is verified that the pro-posed method can assess the cyclic undulation existing ontooth surface effectively, even though it is very small in thelevel of 0.1 µm. On the contrary, at this amplitude level, itis difficult to detect the cyclic undulation directly by ordi-nary tooth surface measurement. This concludes that thisproposed method has more ability than surface measure-ment method and has strong potential for tooth surface in-spection.

In the practical application of this method for toothsurface inspection, the master gear which may be one ofdriving or driven gear should be prepared such that itstooth surface condition is known and has the number ofteeth prime with that of the inspected gear. Moreover itshould have cyclic undulation at reasonable magnitude toproduce reasonable amplitude of ghost noise, so determi-nation and calibration of the frequency response functioncan be done precisely.

Determination and calibration of the frequency re-sponse function can also be done indirectly by using anominal gear, which has the specific undulation to matewith the master gear. The frequency response function cal-ibrated by the nominal gear will be applied to the vibrationsignal measured from the inspected gear mating with themaster gear to assess the cyclic undulation on tooth sur-face of the inspected gear later.

Operation at reasonably high speed and high torqueis also recommended so that the amplitude of vibration ishigh and amplitude calibration of the frequency responsefunction can be done easily and accurately. Moreover hightorque also gives more accurate magnitudes of evaluatedundulation and can be comparable with real undulation ex-isting on the full tooth surface area. With these prepara-tions and operating conditions, the tooth surface undula-

tion of the inspected gear can be assessed precisely. Fur-ther work for the common tooth form errors and their vari-ation is in progress.

Acknowledgement

The work was conducted under a part of researchproject for gear future technology (RC-184) founded bythe Japan Society of Mechanical Engineers.

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