A STUDY OF GEAR NOISE AND VIBRATION

M. Åkerblom* and M. Pärssinen♣

*Volvo CE Components AB, SE–631 85 Eskilstuna. *Department of Machine Design, KTH, SE–100 44 Stockholm.

♣MWL, Department of Vehicle Engineering, KTH, SE–100 44 Stockholm.1

Abstract

The influence of gear finishing method and gear deviations on gearbox noise is investigated in this mainly experimental study. Eleven different test gear pairs were manufactured using three different finishing methods as well as different gear tooth modifications and deviations. The surface finish and geometry of the gear tooth flanks were measured. Transmission error, which is considered to be an important excitation mechanism for gear noise, was predicted and measured. LDP software from Ohio State University was used for the transmission error computations. A specially built test rig was used to measure gearbox noise and vibration for the different test gear pairs. The measurements show that disas-sembly and reassembly of the gearbox with the same gear pair can change the levels of measured noise and vibration considerably. The rebuild variations are sometimes in the same order of magnitude as the differences between different tested gear pairs, indicating that other factors besides the gears affect gear noise. Most of the experimental results can be understood and explained in terms of measured and predicted transmission error. However, it does not seem possible to find one single parameter, such as measured peak to peak transmission error, that can be related directly to measured noise and vibration. Shaved gears do not seem to be noisier than ground gears even if their gear tooth deviations are larger. Factors that do seem to reduce gear noise, when compared with profile ground reference gears, are threaded wheel grinding, in-creased face-width, decreased lead crowning, increased pitch errors and de-creased lead twist. Factors that seem to increase noise are a rougher surface fin-ish, increased lead crowning and helix angle error.

Keywords: gear, gearbox, noise, vibration, transmission error.

1 Currently at Scania CV AB, SE-151 87 Södertälje

CONTENTS 1 INTRODUCTION.............................................................................................. 1 2 TEST RIG ........................................................................................................... 2

2.1 Description of the test rig ............................................................................. 2 2.2 Test cycle...................................................................................................... 3

3 TEST GEARS..................................................................................................... 4

3.1 Description of the test gears ......................................................................... 4 3.2 Different gear finishing methods.................................................................. 5 3.3 Test gears with different modifications or errors ......................................... 6

4 GEAR MEASUREMENTS ............................................................................... 8

4.1 Measurement of tooth deviations ................................................................ 8 4.2 Surface finish measurements ...................................................................... 10 4.3 Transmission error measurements .............................................................. 13

5 TRANSMISSION ERROR PREDICTIONS ................................................ 17

5.1 Computation of transmission error ............................................................. 17 5.2 Influence of torque level on predicted transmission error.......................... 21 5.3 Comparison between predicted and measured transmission error ............. 24

6 NOISE AND VIBRATION MEASUREMENTS .......................................... 26

6.1 Instrumentation........................................................................................... 26 6.2 Measurement repeatability ......................................................................... 27 6.3 Order analysis ............................................................................................. 28

7 RESULTS OF THE NOISE AND VIBRATION MEASUREMENTS ....... 29

7.1 Repeatability after reassembling of the gearbox ........................................ 29 7.2 Results of the measurements ...................................................................... 30

8 DISCUSSION AND CONCLUSIONS ........................................................... 38

8.1 Conclusions for gear pairs A–K ................................................................. 38 8.2 General conclusions.................................................................................... 40

ACKNOWLEDGEMENTS................................................................................ 44 REFERENCES.................................................................................................... 44

1

1 INTRODUCTION Legal regulations and customer demands arising from an increased focus on environmental and quality issues can result in requirements to reduce the gear-induced noise from gearboxes. Such requirements can apply to automobiles [1], trucks [2], and off-highway vehicles such as wheel loaders and articulated haulers. Gear researchers and gear-industry experts agree that transmission error is an important excitation mechanism for gear noise, although not the only one [3]. Welbourn [4] defined transmission error as ‘The difference between the actual posi-tion of the output gear and the position it would occupy if the gear drive were perfectly conju-gate.’ One aim of this work is to experimentally investigate the influence of different gear finishing methods and gear tooth deviations on noise from a gearbox. Eleven different test gear pairs were manufactured using different finishing methods and with different deliberately created deviations as well as different surface finishes. A specially built test rig was used for noise testing of the different gear pairs. Noise was measured with 3 microphones and vibration was measured using 3 accelerometers attached to the gearbox housing. A further aim is to investigate the relationship between transmission error and gearbox noise. Accordingly, transmission error was measured as well as computed for the different test gear pairs and the transmission error values were compared to the results of the noise and vibration measurements in the test rig.

2

2 TEST RIG 2.1 Description of the test rig The test rig is described in detail in reference [5], and therefore the description here is very brief. The rig is of the recirculating power type and consists of two identical gearboxes, con-nected to each other with two universal joint shafts. Torque is applied by tilting one of the gearboxes around one of its axles. This tilting is made possible by bearings between the gear-box and the supporting brackets. A hydraulic cylinder creates the tilting force. The torque is measured with a load sensor placed between the cylinder and the gearbox. The test rig princi-ple is shown in figure 2.1.1.

Figure 2.1.1 Sketch of test rig. In order to include the influence of the housing in the investigations, the test gearbox was de-signed to be as similar as possible to a wheel-loader transmission. This was achieved by using gears, shafts and bearings from an existing gearbox and making the housing of the same mate-rial (nodular iron) and of a similar thickness to the housing of a wheel-loader transmission. The test gearbox is shown in figure 2.1.2.

Figure 2.1.2 CAD model of test gearbox with part of housing cut away.

The test gearbox and microphones are shielded from ambient noise by a box made of sound-absorbing material as initial measurements showed that the noise from the electric motor was louder than the gear noise, at least for low RPM.

Hydraulic Cylinder

Test Gearbox Microphone

Accelerometer Articulated Attachment

Electric Motor

Slave or Master Gearbox

Load Sensor

3

2.2 Test cycle The noise and vibration measurements were carried out at three different torque levels, 140, 500 and 1000 Nm. For each torque level, measurements were made for 50 seconds each at constant speeds of 1000, 1500 and 2000 RPM and as speed uniformly (linearly) increased from 500 to 2550 RPM. The test cycle is shown in figure 2.2.1. All speeds and torque levels are for the pinion.

0

500

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2000

2500

0 100 200 300 400 500 600 700 800 900

Time [s]

RPM

[min

-1]

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1100

Torq

ue [N

m]

RPMTorque

Figure 2.2.1 Test cycle for the noise and vibration measurements.

The oil used in the gearbox was SAE 10W–30 engine oil and the temperature was 60°C at the beginning of the test and approximately 80°C by the end of the test. The gearbox was filled with oil to the centre of the gears. Before each measurement, the rig was run at 500 Nm and 1000 RPM for 5 to 10 minutes in order to increase the temperature to 60°C and allow a short running in of the test gears. In a wheel-loader transmission, similar gears would be subject to a maximum torque of ap-proximately 5000 Nm, but at this torque the rotational speed is very low and no noise is cre-ated. At speeds when gear noise can be heard, the torque is typically 100–500 Nm. The maximum rotational speed in a wheel loader is over 3000 RPM, but the test rig is limited to 2550 due to the limits of the electric motor.

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3 TEST GEARS 3.1 Description of the test gears The test gears were chosen to be representative of gears in a wheel-loader transmission. Gear data for the test gears are shown in table 3.1.1. Tolerances and modifications of the test gears are described in table 3.1.2. pinion gear Number of teeth 49 55 Normal module [mm] 3.5 3.5 Pressure angle [ º] 20 20 Helix angle [ º] –20 20 Face width [mm] 35 33 Profile shift coefficient +0.038 –0.529Tip diameter [mm] 191 209 Centre distance [mm] 191.91 Transverse contact ratio εα 1.78 Overlap ratio εβ 1.03

Table 3.1.1 Gear data for the test gears.

[μm] pinion gear Lead crowning 10–18 10–18 Involute alignment dev. 10 10 Involute form deviation 8 8 Lead deviation 10 10 Lead form deviation 8 8 Tip relief (short) 5–10 5–10 Involute crowning – 1–5 Radial run out 50 50

Table 3.1.2 Tolerances and modifications for the test gears.

All test gears were manufactured within these tolerances, unless otherwise stated. For exam-ple, the shaved gears show considerable deviations from the specified tolerances, especially regarding involute alignment deviation, lead deviation and radial run out. The material in all test gears is case-hardening steel V-2525-94 in accordance with Volvo Corporate Standard STD 1125,251 [6]. Table 3.1.3 gives an overview of the different test gear pairs. A more extensive description of each gear pair is given in sections 3.2 and 3.3.

Gear pair Description Finishing method A Reference gears Profile grinding (KAPP) B Shaved Shaving C Gleason ground Threaded wheel grinding (Gleason) D Rougher surface Profile grinding (KAPP) ‘B126’ E Increased face-width Profile grinding (KAPP) F Pitch errors Profile grinding (KAPP) G Increased lead crowning Profile grinding (KAPP) H Decreased lead crowning Profile grinding (KAPP) I Involute alignment error Profile grinding (KAPP) J Helix angle error Profile grinding (KAPP) K Decreased lead twist Profile grinding (KAPP), single flank

Table 3.1.3 Overview of the different test gear pairs.

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3.2 Different gear finishing methods The test gears were manufactured using three different finishing methods, namely, shaving (before case-hardening), profile grinding with CBN-coated steel grinding wheels (KAPP) or threaded wheel grinding (GLEASON TAG 400). Test gears A, ground (KAPP)

The reference gears in this test are identical to the production gears used in wheel-loader transmissions. The gear manufacturing process is hobbing, case-hardening and gear grinding with CBN-coated steel grinding wheels. The finishing grinding wheel is B 91, which means that the average grain size is 91 μm. In this grinding process, one space of tooth is ground, the gear is indexed, and then the next space of tooth is ground. Test gears B, shaved

Shaved gears are finish-machined before hardening. The disadvantage of this inexpensive finishing method is that the case-hardening causes distortions to the gear teeth. Some of these distortions are systematic and can be compensated for when shaving, but others seem to be random or non-symmetrical and are impossible to compensate for. The tooth deviations after case-hardening of the shaved gears are shown in table 3.2.1.

[μm] pinion gear Lead crowning 8–18 7–10 Involute alignment deviation 19 10 Involute form deviation 8 8 Lead deviation 25 35 Lead form deviation 8 8 Tip relief (short) 10–14 5–13 Involute crowning – 10 Radial run out 80 50

Table 3.2.1 Tooth deviations and modifications for test gears B (shaved), to be compared with table 3.1.2.

Test gears C, ground (Gleason)

Test gears C were ground using threaded wheel grinding, which is a continuous generating grinding method. This means that the involute profile is generated by a grinding-wheel with a basic rack profile thread. The gear manufacturing process is hobbing, case-hardening and gear grinding. The test gears were manufactured within the tolerances specified in table 3.1.2, ex-cept in regard to the involute alignment deviation and the lead deviation, which exceeded specified values by a few microns.

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3.3 Test gears with different modifications or errors Test gears were also manufactured with different modifications or errors. All these gears (D–K) were finished using profile grinding with CBN-coated steel grinding wheels (KAPP). Test gears D, rougher surface

Those gears were manufactured in the same way as the A gears except that the finishing grinding wheel was B 126, which means that the average grain size is 126 μm, creating a rougher surface. Test gears E, increased face-width

In this pair, the face-width is 60 mm for the pinion and 58 mm for the gear, giving an overlap ratio (εβ) of 1.80 compared to 1.03 for all other test gear pairs. The amount of lead crowning is 10–18 μm, which is the same as for gear pair A. Test gears F, pitch errors

Those gears are similar to A except for pitch errors deliberately created when grinding the gears. Small increases in the in-feed of the grinding wheel create a wider tooth space, and hence decrease tooth thickness, making it possible to create the desired pitch errors. Because the intention was to imitate pitch errors of shaved gears, pitch errors were created according to table 3.3.1.

Periodicity Amplitude [μm] Once per rev. 15 Twice per rev. 5 Three times per rev. 0 Four times per rev. 10 Random 6

Table 3.3.1 Created pitch errors. Test gears F are within the tolerances specified in table 3.1.2 except for radial run out, which is about 70 μm due to the pitch errors. In table 3.3.2, values of pitch errors for gear pair F are compared with typical values of pitch errors for the test gear pairs manufactured with differ-ent finishing methods.

Transverse pitch deviation fpt [μm]

Transverse tooth to tooth pitch deviation

Δfpt [μm]

Total cumulative pitch deviation

Fp [μm] F (Ground, pitch errors) 8 10 70 A (Ground KAPP) 3 4 25 B (Shaved) 9 8 85 C (Ground Gleason) 6 5 45

Table 3.3.2 Typical values of measured pitch errors.

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Test gears G, increased lead crowning

The lead crowning of the gear is 35 μm (15 μm for A). The pinion is the same as for gear pair A, with lead crowning 15 μm. Test gears H, decreased lead crowning

The lead crowning of the gear is 0 μm. The pinion is the same as for gear pair A, with lead crowning 15 μm. Test gears I, involute alignment error

The involute alignment deviation of the gear is –20 μm (material missing from the top of the teeth). The pinion is the same as for gear pair A, with involute alignment of nominally 0 μm. Test gears J, helix angle error

The lead deviation is 37 μm for the gear. The pinion is the same as for gear pair A, with lead deviation of nominally 0 μm. Test gears K, decreased lead twist

The gears are identical to gear pair A except for the lead twist, which is reduced. Lead twist is a deviation from the desired shape of the teeth. This deviation can be specified as the differ-ence between two lead measurements, one near the root and one near the tip of the gear tooth. Alternatively, it can be specified as involute alignment difference, which is the difference between two involute measurements at each end of the gear tooth. When grinding gears, the method (generating- or profile-grinding) and the amount of lead crowning will cause a certain amount of lead twist. When shaving, the geometry of the shav-ing cutter is the most important factor, but the lead twist will also be affected by whether the shaving method is diagonal, parallel or plunge-shaving. Of course, case-hardening also causes distortions that affect the lead twist. The sign convention is that the lead twist is positive if the helix angle increases at the top of the gear tooth and negative if the helix angle decreases at the top of the gear tooth. The test gears with decreased lead twist were ground in a different way than test gears A, us-ing a specially designed grinding wheel. Instead of grinding one space of tooth (two flanks) at the same time, only one flank was ground at a time. The lead crowning was created by small rotational movements of the gear instead of by varying the in-feed of the grinding wheel. The drawback of this method is increased grinding time. Typical values for measured lead twist for four of the test gear pairs are shown in table 3.3.3.

Lead twist [μm] pinion gear

K (Decreased lead twist) 0 0 A (Ground, KAPP) +29 +21 B (Shaved) –30 –16 C (Ground, Gleason) –22 –26

Table 3.3.3 Typical values of measured lead twist.

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4 GEAR MEASUREMENTS 4.1 Measurement of tooth deviations The test gears were measured using a Höfler ZP 630 gear-measuring machine. Gear tooth deviations were evaluated according to Volvo Corporate Standard STD 5082,81 [7]. The measurements were carried out as cross measurements, which means that the involute (pro-file) is measured at the centre of the teeth face-width and the helix angle is measured at the middle of the controlled profile. An example of a cross measurement is shown in figure 4.1.1, and the results of the measurements of all the test gears are shown in table 4.1.1.

Figure 4.1.1 Example of a cross measurement.

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[μm] In

volu

te a

lignm

ent d

evia

tion

f gα

Invo

lute

form

dev

iatio

n f fα

Invo

lute

cro

wni

ng C

h

Tip

relie

f (sh

ort)

Ca

Lead

dev

iatio

n f H

β

Lead

form

dev

iatio

n f fβ

Lead

cro

wni

ng C

b

Lead

twis

t Vβ

Rad

ial r

un o

ut F

r

Tra

nsve

rse

pitc

h

dev

iatio

n f p

t

Tran

sver

se to

oth

to to

oth

pitc

h de

viat

ion

Δfpt

To

tal c

umul

ativ

e pi

tch

de

viat

ion

F p

A pinion –8 5 – 10 –4 3 15 +29 22 3 3 21 A gear –8 5 2 9 –3 2 13 +21 26 3 4 32 B pinion 19 8 – 13 25 6 14 –30 80 12 11 103 B gear 7 9 9 10 40 7 8 –16 42 6 10 31 C pinion 6 5 – 7 11 1 14 –22 42 5 4 43 C gear –13 4 3 5 –22 1 14 –26 41 6 4 54 D pinion –7 4 – 11 3 3 13 +21 10 2 2 12 D gear –3 4 3 10 –4 2 14 +21 16 3 4 29 E pinion –5 4 – 11 –3 3 11 +9 19 3 2 22 E gear –10 4 2 8 3 2 11 +8 18 3 3 27 F pinion –7 4 – 10 6 3 14 +22 66 7 10 31 F gear –6 5 2 10 1 2 14 +18 54 8 8 43 G pinion Same as A pinion G gear –6 4 3 11 3 2 35 +51 17 2 2 13 H pinion Same as A pinion H gear –5 4 3 10 –2 2 0 +2 20 3 4 24 I pinion Same as A pinion I gear –20 4 2 10 3 2 13 +24 9 3 5 14 J pinion Same as A pinion J gear –10 4 2 9 37 2 13 +20 25 4 4 37 K pinion –5 5 – 11 –7 6 11 0 31 6 9 42 K gear –10 2 5 7 8 4 13 0 27 4 5 37

Table 4.1.1 Gear deviations and modifications. In addition to the cross measurement, the topography of the gear flank was measured in order to obtain information about the teeth geometry in areas not covered by the cross measurement. A topographical measurement gives information about gear flank deviations from a theoreti-cally perfect involute and helix angle, and the result is presented as the deviations in 49 points (7 x 7) per flank. The results of the topographical measurements are used as input to the transmission error computations in section 5.1. An example of a topographical measurement is shown in figure 4.1.2.

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Figure 4.1.2 Example of result of a topographical measurement.

4.2 Surface finish measurements Two-dimensional (2-D) and three-dimensional (3-D) surface finish measurements were car-ried out on the test gears manufactured with different finishing methods (A, B and C) and on the test gears with the rougher surface finish (D). On the gears with the rougher surface finish, the measurement was made before as well as after the noise tests in order to investigate whether the surface finish was affected by wear as the gears ran against each other. The surface finish measurements were made on plastic replicas of the gear flank. The plastic replicas were made of a cold-curing resin with a methylmethacrylate base, as described in Flodin [8]. A Taylor Hobson Form Talysurf MK 1 was used for the 2-D measurements, and the stylus radius was 2 μm. The position for the 2-D profile measurements is shown in figure 4.2.1. A seventh order polynomial was used for form removal of the involute shape and no filter was used. The measured profiles are shown in figure 4.2.2 and the corresponding Ra and Rq values are shown in table 4.2.1.

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Figure 4.2.1 Position on the tooth flank for the 2-D surface finish measurements.

0 1 2 3 4 5 6−0.03

−0.02

−0.01

0

0.01

0.02

0.03

a)

b)

c)

d)

e)

x [mm]

[z [m

m]

Figure 4.2.2 Results of the 2-D surface finish measurements: a) Gear A (KAPP ground), b) Gear B (shaved), c) Gear C (Gleason ground), d) Gear D (KAPP with rougher surface), e) Gear D after noise test. X = 0 corresponds to the tip of the tooth and x = 6 mm corresponds to the root of the tooth.

Gear pair Ra Rq A KAPP ground 0.57 0.71 B Shaved 1.05 1.67 C Gleason ground 0.54 0.68 D KAPP rougher surface 0.86 1.16 D After noise test 0.75 1.01

Table 4.2.1 Ra and Rq values for the gears manu-factured with different finishing methods.

The Ra and Rq values decreased slightly after the noise test. By comparing profiles d) and e) it can be seen that this is due to wear, for some of the highest peaks of profile d) are lower in profile e) (same gear after noise test). Figure 4.2.2 also shows that the surface finish of the shaved gear b) is rougher near the tip than at the middle and near the root.

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For the 3-D surface finish measurements, an UBM 3-D device with a 5 μm stylus radius was used. Measurements were carried out within an area of 1.2 mm x 1.2 mm, positioned ap-proximately at the middle of the gear flank, as shown in figure 4.2.3. A third order polynomial was used for form removal in the profile direction and a second order polynomial was used for form removal in the lead direction. Figure 4.2.4 shows the results of the 3-D surface measurements.

Figure 4.2.3 Area for 3-D surface finish measurements.

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y [um]

z [u

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Gear D (rougher surface KAPP)

Figure 4.2.4 3-D surface measurements showing the differences in surface structure resulting from the different finishing methods. Sampling length is 10 μm in both directions. Involute direction is y.

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4.3 Transmission error measurements Transmission error was measured for the test gear pairs using a Klingelnberg single flank gear tester, equipped with electronic measuring system PEW 03. Examples of result from the transmission error measurements are shown in figures 4.3.1–4.3.6. The total transmission error, for both right- and left-hand rotations of the pinion, is shown in figure 4.3.1. Right-hand rotation corresponds to the pinion driving the gear, and the direction of rotation is the same as when the vehicle is moving forwards. Left-hand rotation corre-sponds to the pinion driving the gear as when the vehicle is moving backwards. All noise and vibration measurements, as well as the transmission error predictions in section 5, are made for right-hand rotation.

Figure 4.3.1 Composite (total) transmission error for gear pair B (shaved), right-hand rota-tion below (r) and left-hand rotation above (l).

Figure 4.3.2 Long wave transmission error for gear pair B (shaved), right-hand rotation be-low (r) and left-hand rotation above (l).

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Long-wave transmission error is shown in figure 4.3.2. Long wave means that the compo-nents with wavelengths equal to or shorter than the gear mesh wavelength are filtered out, and the remaining transmission error is mainly due to run-out and pitch errors. Figure 4.3.3 gives an example of measured short-wave transmission error. Short wave means that components with wavelengths longer than the gear mesh wavelength are filtered out, and the remaining transmission error is mainly due to tooth to tooth transmission error. To obtain information about mean or representative tooth engagement, averaging on FFT-basis was used, meaning that the Fourier coefficients for the tooth mesh frequency and its harmonics were used to plot a mean tooth engagement curve. Three (identical) tooth engage-ments are shown in figure 4.3.4. The curve is computed from the first six harmonics of the tooth mesh frequency.

Figure 4.3.3 Short-wave transmission error for gear pair B (shaved), right-hand rotation (r).

Figure 4.3.4 Average tooth engagement (transmission error) for gear pair B (shaved), right-hand rotation (r).

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FFT analysis was also employed to plot a curve of the average tooth acceleration, computed for a speed of 1000 RPM at the pinion. As with the average tooth engagement, the curve in figure 4.3.5 was computed using the first six harmonics of the tooth mesh frequency. The spectrum obtained from the FFT-analysis is shown in figure 4.3.6. The peak at 55 periods per revolutions corresponds to the tooth mesh frequency.

Figure 4.3.5 Average tooth acceleration for gear pair B (shaved), right-hand rotation (r).

Figure 4.3.6 Spectrum for the deviations (transmission error) for gear pair B (shaved), right-hand rotation (r).

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In order to compare the different test gear pairs, four parameters were chosen from the trans-mission error measurements: • Short-wave transmission error mean peak to peak, f′k_mean, see figure 4.3.3. • Average tooth engagement peak to peak, from FFT analysis, see figure 4.3.4. • Amplitude of the tooth mesh frequency from FFT analysis, see figure 4.3.6. • Average tooth acceleration at 1000 RPM max–min from FFT analysis, see figure 4.3.5. The values of these parameters are shown in table 4.3.1.

Gear pair

Short-wave transmission error mean

peak to peak (f′k_mean)

[μm]

Average tooth engagement peak to peak

from FFT analysis

[μm]

Amplitude of the tooth mesh fre-

quency, from FFT analysis

[μm]

Average tooth acceleration at

1000 RPM max–min from FFT analysis [m/s2]

A 4.7 2.8 1.1 368 B 5.5 3.5 1.7 119 C 2.9 1.5 0.6 276 D 4.9 3.7 1.5 258 E 3.4 1.5 0.7 78 F 4.0 1.1 0.4 244 G 3.6 2.0 0.8 570 H 2.7 1.3 0.6 48 I 5.4 3.2 1.6 365 J 3.4 2.0 0.8 225 K 3.8 2.0 0.7 146

Table 4.3.1 Results of the transmission error measurements, right-hand rotation (r).

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5 TRANSMISSION ERROR PREDICTIONS 5.1 Computation of transmission error Transmission error was computed using LDP software from Ohio State University [9]. Input to the computations was obtained from the gear geometry measurements described in section 4.1. Input files to LDP were created, in which the tooth geometry was described by one lead measurement and seven involute measurements equally spaced over the face width. This means that the geometry of a tooth flank was described by 49 points, as can be seen in figure 4.1.2. Because the teeth of the ground gears were very similar, the topography of only one tooth was measured for each gear. However, the possibility of considerable variations between the teeth of shaved and case-hardened gears meant that for these gears six teeth on each gear were measured. Figures 5.1.1–5.1.3. show examples of results of the transmission error computations.

Figure 5.1.1 Predicted transmission error for gear pair B at 50 Nm, three mesh cycles shown.

Figure 5.1.2 Amplitude of predicted transmission error har-monics of the gear mesh frequency, for gear pair B at 50 Nm.

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Figure 5.1.3 Predicted contact stress [MPa] for gear pair B at torque level 50 Nm. The predicted transmission error values are shown in tables 5.1.1 and 5.1.2. Computations were made for all gear pairs and for five different torque levels, 10, 50, 140, 500 and 1000 Nm. The computations at 10 and 50 Nm were made for comparison with measured transmis-sion error, because the transmission error was measured at a very low torque level. The com-putations at 140, 500 and 1000 Nm were made to allow comparison with the noise and vibra-tion measurements described in sections 6 and 7.

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Predicted transmission error for gear pair A to D

Gear pair Torque [Nm]

Average tooth en-gagement

peak to peak [μm]

Amplitude of the tooth mesh fre-quency [μm]

Amplitude of 2 x tooth mesh fre-quency [μm]

10 1.29 0.57 0.08 50 1.33 0.39 0.06 140 2.34 0.97 0.26 500 2.52 1.08 0.10

A

1000 1.53 0.55 0.32 10 7.44 3.15 0.19 50 6.53 3.03 0.43 140 5.30 2.42 0.55 500 1.28 0.46 0.16

B

1000 1.69 0.73 0.23 10 3.70 1.67 0.22 50 2.23 1.05 0.05 140 0.70 0.11 0.21 500 1.55 0.66 0.04

C

1000 1.65 0.63 0.23 10 2.43 1.13 0.07 50 1.59 0.67 0.20 140 1.59 0.57 0.24 500 1.93 0.94 0.10

D

1000 1.22 0.32 0.29

Table 5.1.1 Results of the transmission error computations for gear pairs A to D.

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Predicted transmission error for gear pair E to K

Gear pair Torque [Nm]

Average tooth en-gagement

peak to peak [μm]

Amplitude of the tooth mesh fre-quency [μm]

Amplitude of 2 x tooth mesh fre-quency [μm]

10 1.69 0.72 0.16 50 1.31 0.57 0.10 140 0.92 0.40 0.03 500 0.96 0.33 0.15

E

1000 0.87 0.37 0.05 10 3.15 1.20 0.32 50 1.83 0.61 0.25 140 2.26 0.97 0.14 500 1.80 0.84 0.18

F

1000 0.83 0.27 0.21 10 2.35 1.14 0.07 50 2.30 0.88 0.17 140 2.55 1.07 0.14 500 2.79 1.23 0.07

G

1000 1.87 0.64 0.39 10 2.15 0.67 0.50 50 1.98 0.79 0.18 140 2.04 0.91 0.07 500 1.49 0.62 0.13

H

1000 0.59 0.15 0.18 10 2.31 0.80 0.31 50 1.38 0.50 0.21 140 1.62 0.71 0.06 500 2.08 0.93 0.02

I

1000 1.38 0.40 0.33 10 2.23 0.92 0.32 50 1.55 0.56 0.10 140 2.08 0.80 0.13 500 2.90 1.30 0.16

J

1000 1.52 0.63 0.20 10 1.63 0.66 0.20 50 1.75 0.78 0.13 140 2.65 1.16 0.13 500 2.95 1.42 0.07

K

1000 1.99 0.91 0.22

Table 5.1.2 Results of the transmission error computations for gear pairs E to K.

21

5.2 Influence of torque level on predicted transmission error Due to deformations, the transmission error depends on the torque level. Predicted peak to peak transmission errors for the different test gear pairs are shown in figures 5.2.1–5.2.3. The results for gear pair A are shown in all figures as a reference. The following are some interest-ing observations: • The shaved gears B show the highest value of the transmission error at low torque levels,

but at 500 and 1000 Nm their transmission error values are among the lowest. This is probably due to the involute crowning, which is largest for this gear pair.

• The gear pair with increased face width (E) seems to be the best because its transmission error does not vary much with torque and the values are low.

• Many gear pairs show decreased transmission error when the torque level is increased from 500 to 1000 Nm. This behaviour is probably due to deformations, which increase the total length of the lines of contact and thereby the effective contact ratio.

• Increased lead crowning (G) increases transmission error. • Decreased lead crowning (H) decreases transmission error, at least at high torque levels. • Involute alignment error (I) and helix angle error (J) do not seem to increase the transmis-

sion error, at least not for errors up to the levels chosen for these test gears. • The gear pair with decreased lead twist (K) shows values of transmission error that are

comparable to or slightly larger than the transmission error values obtained for the refer-ence gears (A).

Figure 5.2.1 Predicted peak to peak transmission error for gear pair A (KAPP), B (shaved), C (Gleason) and D (KAPP rougher surface).

22

Figure 5.2.2 Predicted peak to peak transmission error for gear pair A (KAPP), E (wider), F (pitch errors), G (increased lead crowning) and H (decreased lead crowning).

Figure 5.2.3 Predicted peak to peak transmission error for gear pair A (KAPP), I (involute alignment error), J (helix angle error) and K (decreased lead twist). In an attempt to assign each gear pair one relevant value for each torque level, a transmission error index was computed by adding the amplitude of the gear mesh frequency and four times the amplitude of the second harmonic.

23

( )421 ×+=− amplitudeharmonicndamplitudeharmonicstindexTE

The reason for choosing four times the second harmonic is that experience has shown that the 2nd harmonic will dominate the noise when its amplitude exceeds approximately one quarter of the first harmonic amplitude, possibly because acceleration is the critical parameter rather than displacement. If the displacement of the first harmonic of the gear mesh frequency is

tA ωsin and the displacement of the second harmonic of the gear mesh frequency is

tB ω2sin

A and B are amplitudes (see figure 5.1.2), t = time and ω = 2π f, where f is the tooth mesh frequency. Differentiating the displacement twice gives the acceleration:

tAharmonicstofonAccelerati ωω sin1 2−= tBharmonicndofonAccelerati ωω 2sin42 2−=

Consequently the acceleration is four times as high for the second harmonic compared to the first harmonic, if the displacement amplitudes are equal. This index is computed for each of the tested gear pairs and for torque levels of 140, 500 and 1000 Nm. The values of the index are shown in figure 5.2.4. The shaved gear pair (B) at 140 Nm is the worst with an index of 4.6, and the wider gear pair (E) is the best with indexes close to 0.5 for all three torque levels.

TE-index

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

A B C D E F G H I J K

Gear Pair

TE-in

dex

[um

]

TE-Index 140 Nm

TE-Index 500 Nm

TE-Index 1000 Nm

Figure 5.2.4 Computed transmission error index.

24

5.3 Comparison between predicted and measured transmission error In figure 5.3.1, values of measured short-wave transmission error are compared with predicted values of peak to peak transmission error. In figure 5.3.2, measured peak to peak transmission error values from the FFT analysis are compared with predicted peak to peak transmission error values. Measured and predicted values of the amplitude of the gear mesh harmonic of the transmission error are compared in figure 5.3.3.

Peak to Peak Transmission Error

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

A B C D E F G H I J K

Gear Pair

TE [u

m]

P-P Measured short waveP-P Computed (50Nm)

Fig 5.3.1 Comparison between measured short-wave mean transmission error and predicted transmission error at 50 Nm.

Peak to Peak Transmission Error

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

A B C D E F G H I J K

Test Gear Pair

TE [u

m]

P-P Measured (FFT)P-P Computed (50Nm)

Fig 5.3.2 Comparison between measured short-wave transmission error from FFT analysis and predicted transmission error at 50 Nm.

25

1st Harmonic of Transmission Error

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

A B C D E F G H I J K

Gear Pair

TE [u

m]

1st Harmonic Measured (FFT)1st Harmonic Computed (50Nm)

Fig 5.3.3 Comparison between measured first harmonic amplitude and predicted first harmonic amplitude at 50 Nm.

As can be seen in the above figures, there are considerable differences between the predicted and measured transmission error for some of the gear pairs, while for others the correspon-dence is good. There are several possible reasons for the discrepancy:

• Only one tooth of each gear was measured and used as input for the predictions, while the measurement includes all teeth.

• For the predictions, the shape of a tooth flank is described by 49 points, which might not be sufficient for a fully accurate description.

• Run-out and pitch errors are included in the measurements but not in the computation of transmission error.

• The transmission error is relatively small, typically 1–2 microns and sometimes even less, meaning that it may be in the same order of magnitude as the accuracy of the gear geome-try measurement.

• The computations are made at torque level 50 Nm, while the measurements are made at a torque level close to zero.

26

6 NOISE AND VIBRATION MEASUREMENTS 6.1 Instrumentation The instrumentation consists of one optical tachometer, three microphones and three acceler-ometers. The shaft rotational speed is estimated by attaching a piece of reflecting tape to the part of shaft 1 in front of the gearbox (see figure 6.1.1), so that one pulse is registered per revolution of this shaft. The three microphones are positioned in front of the gearbox, as shown in figure 6.1.1. Additionally, the three accelerometers are attached to the front of the gearbox as shown in figure 6.1.2. Accelerometer 1 registers vibrations in an axial direction; accelerometer 2 registers vibrations in a radial direction, at an angle corresponding to the di-rection of the gear mesh contact force; and accelerometer 3 registers vibrations at a right angle to the direction of accelerometer 2.

Figure 6.1.1 Gearbox shown from above, with tachometer and microphone positions.

Mic. 1

Mic. 2Mic. 3

20 cm29 cm

40.5 cm

26.5 cm 37.5 cm

Vertical positions:Microphone 1: 30 cm above table.Microphone 2: 45 cm above table.Microphone 3: 74 cm above table.

Microphone horisontal positions:

Tachometer

Gearbox

Shaft no. 1 Shaft no. 2

27

Figure 6.1.2 Gearbox shown from front, with accelerometers attached. The arrows denote positive directions of accelerometers 2 and 3. Accel-erometer 1 registers vibrations in the axial direction.

6.2 Measurement repeatability Measurement repeatability was estimated by carrying out a standard test according to the pro-cedure described in section 2.2 and then removing all instrumentation (accelerometers and microphones) from the gearbox. After two hours the instrumentation was remounted and the test repeated. The differences in sound and acceleration levels between these tests give an indication of measurement repeatability. Table 6.2.1 shows the mean deviation in acceleration level and sound pressure level between the two tests. The first step in obtaining the mean deviation is to interpolate the results from the first test so that levels are obtained for the RPM values associated with the second test. This step is necessary because two subsequent tests will not measure levels at precisely the same RPM values. For each separate RPM the relative difference between the measured lev-els is computed. If the relative difference is smaller than 1, its inverse is computed. Otherwise, when computing the mean relative deviation, values smaller than 1 will counteract values larger than 1. After computing the mean deviation value, the corresponding mean level differ-ence in dB is calculated by taking the logarithm of the mean deviation and multiplying by 20. The resulting mean level differences are presented in table 6.2.1.

Instruments Accelerometers Microphones

Load [Nm] no. 1 no. 2 no. 3 no. 1 no. 2 no. 3 140 1.71 2.66 2.63 1.22 1.44 1.26 500 0.56 0.41 0.40 0.85 0.97 0.83 1000 0.49 0.38 0.41 0.98 1.06 0.90

Table 6.2.1 Mean level differences [dB] between two tests. The mean level differences are quite small, although slightly larger for the low load condition.

Acc. 1

Acc. 2

Acc. 3

Shaft 1. Shaft 2.

28

6.3 Order analysis At the test site the signals were recorded on DAT tape as the rotational speed of shaft 1 was increased uniformly from 550 to 2550 RPM over 105 seconds. Signal analysis was carried out in the laboratory by feeding the signal from the DAT recorder to an HP-VXI system con-trolled by I-DEAS software.2 The maximum order of 150 corresponds to slightly above three times the gear meshing frequency. The data acquisition involved synchronous sampling. The current shaft RPM (revolutions per minute) was estimated from the tachometer pulse and the same number of samples were taken per revolution of shaft 1, independent of the shaft speed. The relevant parameters are summa-rised in table 6.3.1.

Number of tachometer pulses per revolution 1 Min and max RPM 550, 2000 Number of samples per revolution 768 Frame size 4096 Maximum order 150.0 Order resolution 0.1875 Window Hanning Broad Frame event delta RPM (20.0)

Table 6.3.1 Order tracking conditions. The frame size of the data acquisition corresponds to 4096 samples; that is, 4096 / 768 = 5.33 rotations of shaft 1. Since the rotational speed is not constant during data acquisition, there is some uncertainty in determining order amplitudes. The lower the shaft rotational speed, the longer a single acquisition will take. At the test site the signals were recorded on DAT tape as the rotational speed of shaft 1 was increased uniformly from 450 to 2550 RPM over 105 sec-onds. The rate R& of the rotational speed increase is thus

RPM/s.20105

4502550 =−

New data are taken for every 20 RPM increase in the rotational speed. The lowest shaft rota-tional speed for acquisition is 550 RPM (see table 6.3.1). At this rotational speed, a single acquisition takes approximately 0.6 seconds. Thus, during the acquisition the rotational speed will vary by 0.6 × 20 = 12 RPM. This variation limits resolution as order amplitudes are pre-sented with respect to actual rotational speed. However, the higher the rotational speed, the better the resolution. At the highest shaft rotational speed used for acquisition – 2000 RPM – the rotational speed varies by only 3.2 RPM during a single acquisition. In section 7, root mean square (rms) amplitudes of the orders are presented. These are ob-tained by integrating the signal components in the vicinity of the order of interest. The inter-val of integration corresponds to ±0.5 orders.

2 I-DEAS © Structural Dynamics Research Corp.

29

7 RESULTS OF THE NOISE AND VIBRATION MEASUREMENTS 7.1 Repeatability after reassembling of the gearbox In addition to the investigation of the repeatability of the noise and vibration measurements, discussed in section 6.2, the repeatability after disassembling and reassembling of the gearbox with the same parts was also investigated. This investigation was done because it was neces-sary to disassemble the gearbox in order to change the gears. The gearbox was disassembled and reassembled with the same gear pair (D), shafts, bearings and housing. The overall sound pressure level for three different measurements is shown in figure 7.1.1. As can be seen, the differences are considerable. For example the peak at 1100 RPM differs by about 7 dB in magnitude and the peak at 1350 RPM is present in only one of the three meas-urements. These figures may indicate that it is not only the excitation from the gear mesh that varies, but also the dynamic properties of the gearbox or the test rig.

Figure 7.1.1 Results of three different measurements of the sound pres-sure level with microphone M1 at torque level 500 Nm. The gearbox was disassembled and reassembled with the same gears (D), shafts, bearings and housing. (Pref =2*10E-5 Pa).

The reassembly variations are in the same order of magnitude as variations reported by Os-wald et al. [10] who investigated the influence of gear design on gearbox radiated noise. In their study, different spur and helical gear designs were tested in a gear noise test rig. One of their conclusions was that ‘In noise reduction tests, variations due to unintended effects, such as testing different part specimens or even re-assembly with the same parts, may be of the same order of magnitude as the effect of deliberate design changes.’

30

7.2 Results of the measurements To get a qualitative conception of the measured noise and vibration, it is often appropriate to study a waterfall plot in which the measured quantity is plotted in a 3-dimensional diagram as a function of frequency and rotational speed. Figure 7.2.1 shows measured sound pressure level for microphone M1 at torque level 500 Nm for gear pair A. It can be seen that the gear mesh frequency and its second and third harmonics are dominating.

Figure 7.2.1 A waterfall plot of measured sound pressure level for microphone M1 at torque level 500 Nm for gear pair A. The gear mesh frequency and its second and third harmonics dominate.

An alternative way of showing the same information is an order plot, in which the measured quantity is plotted as a function of order and rotational speed. An example of such a plot is shown in figure 7.2.2, using the same data as in figure 7.2.1. The gear mesh frequency at or-der 49 (due to 49 teeth at the pinion) dominates, but the second harmonic at order 98 and the third harmonic at order 147 can also be seen. Overall sound pressure level as a function of rotational speed is plotted in figure 7.2.3. The sound pressure level for the gear mesh frequency and for its second and third harmonics are also plotted in the same diagram. It can be seen that the gear mesh frequency determines the overall level, except at 800 RPM and 1800 RPM, where the second harmonic is higher. The conclusion, after studying a number of waterfall plots, is that the gear mesh harmonics determine the overall sound pressure level, making it appropriate to use the overall level as a measure of the gear-related noise. In other words, in the test rig the noise from the electric motor and hydraulic system is always considerably below the gear-induced noise level and does not contribute to the overall noise level.

31

Figure 7.2.2 An order plot of the measured sound pressure level for micro-phone M1 at torque level 500 Nm for gear pair A. The gear mesh frequency at order 49 dominates but the second harmonic at order 98 and third harmonic at order 147 can also be seen.

Figure 7.2.3 Overall sound pressure level, measured with microphone M1 at torque level 500 Nm, for gear pair A, plotted together with the sound pressure level of the gear mesh frequency and its second and third harmonics (Pref =2*10E-5 Pa).

32

The results of the measurements are summarised in table 7.2.1 for torque level 140 Nm, in table 7.2.2 for torque level 500 Nm and in table 7.2.3 for torque level 1000 Nm. The speed is divided into two intervals, 600–1300 RPM (‘Low’) and 1300–2000 RPM (‘High’). The values in the tables are obtained by taking the highest value of the measured quantity in the respec-tive speed interval for each of the six sensors. For example, the peak at 1150 RPM in figure 7.2.3 gives the value 101 dB in table 7.2.2 for gear pair A 17/8 and M1 Low and the peak at 1600 RPM gives the value 101 dB for M1 High. For the gear pairs tested more than once, namely A, B and D, the measurement dates are used to distinguish different tests, and the mean values of the ‘max’ dB values were calculated. For gear pair B 8/8, microphones M2 and M3 were not used because this was the first measure-ment and it had not yet been decided where to place all the microphones. The results for gear pair D 14/12 and D 14/12r show the measurements done to investigate measurement repeat-ability by reattaching accelerometers and microphones but without disassembling and reas-sembling the gearbox. In an attempt to assign each gear pair a few relevant values for their noise ‘activity’, the mean value of the six maximum dB-values (M1 Low, M1 High, M2 Low, M2 High, M3 Low and M3 High) was calculated, and this value is called the ‘mean sound dB’. In the same way a ‘mean vibration dB’ was calculated using A1 Low, A1 High, A2 Low, A2 High, A3 Low and A3 High. These values were calculated for each torque level: 140 Nm, 500 Nm and 1000 Nm. For the gear pairs tested more than once, the mean dB values from the different tests were used. The mean sound dB and the mean vibration dB for the different test gear pairs are shown in figures 7.2.3–7.2.8.

[dB] (Pref =2*10E-5 Pa) [dB] (Aref =10E-5 m/s2) 140 Nm M1 M2 M3 A1 A2 A3 RPM Lo. Hi. Lo. Hi. Lo. Hi. Lo. Hi. Lo. Hi. Lo. Hi. A 17/8 93 93 88 87 86 89 123 123 120 124 117 122 A 16/10 98 93 91 90 90 86 122 124 121 123 116 122 A mean 95.5 93 89.5 88.5 88 87.5 122 124 120 124 116 122

B 8/8 103 93 – – – – 123 124 122 125 118 120 B 15/8 91 93 86 89 85 88 115 121 115 121 111 118 B 25/1 101 95 95 93 93 88 122 128 128 126 119 123 B 30/1 91 96 87 95 86 92 117 122 119 122 114 118

B mean 96.5 94.2 89.3 92.3 88 89.3 119 124 121 124 116 120 C 96 90 91 87 88 85 123 119 113 118 110 114

D 14/8 96 93 89 86 87 87 120 120 121 126 116 120 D 3/10 94 93 94 91 89 88 116 121 120 127 113 119

D 14/12 93 100 91 91 91 92 121 127 127 135 123 130 D 14/12r 94 99 90 91 89 90 120 126 125 134 122 129 D mean 94.2 96.2 91 89.8 89 89.2 119 124 123 130 118 124

E 93 92 88 92 85 87 118 123 113 118 109 117 F 94 93 89 89 86 86 118 119 118 123 112 118 G 96 94 87 92 88 91 119 126 120 127 114 122 H 94 88 87 87 87 83 118 120 117 122 114 118 I 91 96 91 90 86 91 117 126 117 128 116 125 J 98 98 88 99 87 94 122 125 119 128 113 121 K 88 87 88 88 85 83 115 117 113 119 112 116

Table 7.2.1 Results of the noise and vibration measurements at torque level 140 Nm.

33

[dB] (Pref =2*10E-5 Pa) [dB] (Aref =10E-5 m/s2) 500 Nm M1 M2 M3 A1 A2 A3 RPM Lo. Hi. Lo. Hi. Lo. Hi. Lo. Hi. Lo. Hi. Lo. Hi. A 17/8 101 101 94 95 92 96 126 128 123 130 117 131 A 16/10 99 100 96 100 96 91 128 131 126 132 124 126 A mean 100 100 95 97.5 94 93.5 127 130 124 131 120 128

B 8/8 100 95 – – – – 121 125 123 127 119 122 B 15/8 98 92 90 89 92 92 122 126 122 124 117 124 B 25/1 95 101 92 96 93 95 124 127 126 129 119 125 B 30/1 95 101 91 97 90 96 124 127 123 124 119 123

B mean 97 97.2 91 94 91.7 94.3 123 126 124 126 118 124 C 99 95 94 92 92 94 128 129 120 127 118 123

D 14/8 104 100 98 96 97 99 127 127 123 132 124 125 D 3/10 97 98 94 102 96 94 128 130 127 132 121 127

D 14/12 100 99 94 94 94 95 128 135 127 135 124 131 D 14/12r 98 99 93 95 92 94 128 134 127 135 123 132 D mean 99.8 99 94.8 96.8 94.8 95.5 128 132 126 134 123 129

E 92 96 90 97 86 91 120 128 117 123 115 123 F 95 98 91 96 92 94 124 127 118 129 120 124 G 99 100 95 100 93 98 127 133 125 133 121 128 H 100 94 95 97 94 91 126 129 126 128 122 124 I 97 98 92 93 93 92 125 132 123 132 120 130 J 99 105 93 104 93 101 127 132 124 131 120 126 K 93 94 89 92 90 90 123 126 121 123 118 126

Table 7.2.2 Results of the noise and vibration measurements at torque level 500 Nm.

[dB] (Pref =2*10E-5 Pa) [dB] (Aref =10E-5 m/s2) 1000 Nm M1 M2 M3 A1 A2 A3 RPM Lo. Hi. Lo. Hi. Lo. Hi. Lo. Hi. Lo. Hi. Lo. Hi. A 17/8 99 103 93 96 93 96 123 131 125 132 124 132 A 16/10 101 101 97 102 96 92 122 131 126 133 125 127 A mean 100 102 95 99 94 94 122 131 126 132 124 130

B 8/8 101 98 – – – – 123 130 124 129 122 126 B 15/8 101 99 94 91 92 93 121 127 124 127 124 127 B 25/1 99 99 96 97 96 92 126 127 128 133 122 128 B 30/1 96 103 93 97 90 97 126 129 125 126 120 128

B mean 99.2 99.8 94.3 95 92.7 94 124 128 125 129 122 127 C 101 100 94 95 90 94 125 131 122 129 119 128

D 14/8 107 100 99 97 98 99 123 132 125 133 124 127 D 3/10 98 99 93 102 95 94 124 133 125 133 123 127

D 14/12 103 100 94 94 94 90 126 134 127 133 125 132 D 14/12r 104 101 94 95 95 89 126 134 128 133 125 131 D mean 103 100 95 97 95.5 93 125 133 126 133 124 129

E 95 94 92 94 87 90 118 123 120 128 114 122 F 98 99 96 99 93 94 119 130 122 132 122 125 G 98 103 94 104 92 96 122 134 121 135 121 128 H 99 101 95 97 95 92 122 130 125 127 122 126 I 100 98 94 93 92 91 125 132 126 127 123 128 J 100 102 93 102 93 101 126 134 126 132 120 128 K 100 95 93 90 91 89 119 124 118 125 120 124

Table 7.2.3 Results of the noise and vibration measurements at torque level 1000 Nm.

34

As an example, the value for the mean sound dB at 140 Nm for gear pair A in figure 7.2.4 was calculated as the mean of the six dB values for A mean: M1 Low, M1 High, M2 Low, M2 High, M3 Low, and M3 High, in table 7.2.1. Those values are themselves mean values of the maximum dB values measured for gear pair A 17/8 and A 16/10. To get an idea of the rebuild variation of the mean sound dB and the mean vibration dB, those values were calculated for each of the separate tests of gear pair D at 500 Nm and compared to the values for D mean. It was found that the maximum deviation for an individual meas-urement from the mean value was 2 dB and the variation between the four different measure-ments was approximately 4 dB, both for the mean sound dB and for the mean vibration dB. This means that the rebuild variations are in the same order of magnitude as the measured differences between different test gear pairs. Especially for the gear pairs that were only tested once, the uncertainty is considerable and it is necessary to be aware of this when comparing the results for the different gear pairs. Four measurements were made for gear pair B and D, two measurements were made for gear pair A, and one measurement was made for each of the other gear pairs. Calculation of 95% confidence intervals for the mean value of the mean vibration dB and for the mean sound dB were carried out for gear pairs B and D at torque level 500 Nm. The cal-culations were made in accordance with the Six Sigma Guidebook [11].

⎥⎦

⎤⎢⎣

⎡ +−=n

stmn

stmIC xx

xx 2/2/ ;.. αα

C.I. = Confidence interval mx = mean value of the samples t α/2 = value from t-table, for chosen risk, α/2 sx = standard deviation n = number of samples

95 % Confidence intervals for the mean values Gear pair B Gear pair D

Mean Vibration dB 121.7–125.2 [dB] 125.6–131.3 [dB] Mean Sound dB 91.4–96.9 [dB] 94.1–99.4 [dB]

Table 7.2.4 95 % confidence intervals for the mean value of mean vibration dB and the mean sound dB for gear pairs B and D at 500 Nm.

The calculation of confidence intervals showed with more than 95% confidence that the mean value of the mean vibration dB for gear pair B is lower than the mean value of the mean vi-bration dB for gear pair D, because the confidence intervals do not overlap. However, because the 95% confidence intervals for the mean values of the mean sound dB do overlap, it is not possible to say which of gear pairs B or D is the best, or at least not with more than 95% con-fidence. Figures 7.2.5 and 7.2.6 show the confidence intervals for gear pairs B and D. It seems that the rebuild variations are smaller for vibrations than for sound pressure level.

35

"Mean Vibration dB" at 140 Nm

114

115

116

117

118

119

120

121

122

123

124

A B C D E F G H I J K

Gear Pair

[dB

] ref

. 10E

-5 [m

/s2 ]

Figure 7.2.3 Measured mean vibration dB for the different test gear pairs at torque level 140 Nm.

"Mean Sound dB" at 140 Nm

85

86

87

88

89

90

91

92

93

94

95

A B C D E F G H I J K

Gear Pair

[dB

] re

f 2*1

0E-5

[Pa]

Figure 7.2.4 Measured mean sound dB for the different test gear pairs at torque level 140 Nm.

36

"Mean Vibration dB" at 500 Nm

120

121

122

123

124

125

126

127

128

129

130

131

132

A B C D E F G H I J K

Gear Pair

[dB

] ref

. 10E

-5 [m

/s2]

Figure 7.2.5 Measured mean vibration dB for the different test gear pairs at torque level 500 Nm. 95% confidence interval for the mean value shown for gear pair B and D.

"Mean Sound dB" at 500 Nm

90

91

92

93

94

95

96

97

98

99

100

A B C D E F G H I J K

Gear Pair

[dB

] re

f. 2*

10E

-5 [P

a]

Figure 7.2.6 Measured mean sound dB for the different test gear pairs at torque level 500 Nm. 95% confidence interval for the mean value shown for gear pairs B and D.

37

"Mean Vibration dB" at 1000 Nm

119

120

121

122

123

124

125

126

127

128

129

A B C D E F G H I J K

Gear Pair

[dB

] ref

. 10E

-5 [m

/s2]

Figure 7.2.7 Measured mean vibration dB for the different test gear pairs at torque level 1000 Nm.

"Mean Sound dB" at 1000 Nm

90

91

92

93

94

95

96

97

98

99

100

A B C D E F G H I J K

Gear Pair

[dB

] re

f. 2*

10E

-5 [P

a]

Figure 7.2.8 Measured mean sound dB for the different test gear pairs at torque level 1000 Nm.

38

8 DISCUSSION AND CONCLUSIONS 8.1 Conclusions for gear pairs A–K A, ground (KAPP)

Gear pair A is the reference set in this test. The noise and vibration measurements show val-ues that are relatively high, although not the highest. Both the measured and predicted trans-mission error values are among the highest, except for the predicted transmission errors at torque levels 10 and 50 Nm, which are among the lowest. Consequently the difference be-tween measured and predicted transmission error is considerable. B, shaved

The most characteristic deviation from the other tested gear pairs is the high values of the transmission error, measured as well as predicted, at low torque levels. However, the pre-dicted transmission error decreases considerably with increased torque, and its value is among the lowest at 500 Nm. This tendency is also apparent in the noise and vibration measurements where gear pair B is comparable to gear pair A at 140 Nm, while at 500 Nm it is better than gear pair A. Excessive involute crowning may explain this behaviour. C, ground (Gleason)

Predicted transmission error corresponds well to measured transmission error. The tendency shown in figure 5.2.1, with a minimum at 140 Nm and slightly increased transmission error at 500 and 1000 Nm, can also be seen in the noise and vibration measurements where gear pair C is among the best at 140 Nm. D, rougher surface

Both measured and predicted transmission errors at low torque levels are equivalent to or slightly higher than corresponding quantities for gear pair A. At torque levels 140, 500 and 1000 Nm, the predicted transmission error for gear pair D is comparable to or slightly lower than the predicted transmission error for gear pair A. As regards measured noise and vibration levels, the differences between A and D are quite small, but there is a tendency for D to be noisier than A, especially at low torque levels. This is reasonable and may be explained by comparing the width of the contact ellipse to the surface profile amplitude variations with respect to the surface profile length. The computed size of the contact ellipse for different torque levels is shown in table 8.1.1. The computations were made in accordance with K. L. Johnson [12]. The description of the geometry of the gear teeth is somewhat simplified by assuming equivalent spur gears and that all load is carried by one tooth. Of course, the size of the contact ellipse cannot exceed the width of the teeth, but it can be seen as the total length of the lines of contact. The width of the contact ellipse is 0.3 to 0.5 mm for torque levels be-tween 140 and 1000 Nm. When comparing this size with the measured surface profiles in figure 4.2.2, it seems reasonable that surface finish might influence the noise and vibration at low torque levels, but do so less at higher torque levels.

39

Size of contact ellipse [mm] Torque [Nm] 2a 2b Contact stress [MPa]

10 0.11 9.7 206 50 0.19 16.6 353 140 0.27 23.5 497 500 0.41 35.9 760 1000 0.52 45.2 957

Table 8.1.1 Computed size of contact ellipse for gear pair D.

E, increased face-width

At low torque levels, 10 and 50 Nm, the predicted transmission error is equivalent to the pre-dicted transmission error for gear pair A. Measured transmission error values for gear pair E are slightly lower than measured transmission error values for gear pair A, but are not among the lowest. On the other hand, at torque level 500 and 1000 Nm, gear pair E is the best as re-gards predicted transmission error as well as measured noise and vibration. It may be that an increased face-width increases the contact ratio. Other favourable factors may be less defor-mation of the teeth and preserved crowning on wider gears, which results in a larger crowning radius and less lead twist. It is also possible that the dynamic properties of the gearbox and test rig are affected, possibly advantageously, by the heavier and stiffer gears with larger mo-ments of inertia. The disadvantage of this gear pair is its increased cost and weight. F, pitch errors

Measured values of transmission error for gear pair F are comparable to or slightly lower than measured transmission error values for gear pair A. Predicted transmission error values for torque levels 140 to 1000 Nm are also comparable to or slightly lower than the corresponding quantity for A. In the noise and vibration measurements, the gears with pitch errors exhibited lower levels than the reference gears (A). The reason could be that the pitch errors bring about lower amplitudes of the gear mesh harmonics at the expense of more side-bands. The influ-ence of pitch errors on transmission error is discussed in Kohler [13] and Wellbourn [14]. Order plots from the vibration measurements are shown in figure 8.1.1.

Figure 8.1.1 Order plot for gear pair A (left) and gear pair F (right). Vibration measure-ments with accelerometer 3 at torque level 500 Nm.

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G, increased lead crowning

Measured values of transmission error are slightly lower than the values measured for A, but predicted transmission error values are somewhat higher than the values for A. The noise and vibration measurements suggest that increased lead crowning gives equivalent or moderately higher noise and vibration levels compared to A. H, decreased lead crowning

The gear pair with decreased lead crowning shows lower measured transmission error values compared to values for gear pair A. Predicted transmission error values at torque levels 140 to 1000 Nm are lower than values for A. The noise and vibration measurements also indicate that this could be an improvement compared to A. I, involute alignment error

When comparing measured transmission error, the gear pair with involute alignment error on the gear is equivalent to gear pair A. The predicted transmission error values at torque levels 140 to 1000 Nm are lower than the values for A. In the noise and vibration measurements, gear pair I is comparable to gear pair A, except for the noise measurements at 500 and 1000 Nm, where gear pair I is better than A. J, helix angle error

Measured transmission error values are slightly lower than the values for A, while the pre-dicted transmission error values are comparable. The noise and vibration measurements show similar values to those for gear pair A, except for the noise measurements at 140 and 500 Nm, where gear pair J is noisier than gear pair A. K, decreased lead twist

Compared to A, the measured transmission error values are slightly lower for gear pair K. The predicted transmission error values for gear pair K are comparable to the corresponding val-ues computed for gear pair A. However, gear pair K is the best (together with E) in the noise and vibration measurements. Of course, the measurement uncertainty is considerable, espe-cially for gear pairs that were tested only once, but it does not seem unrealistic that gear pair K could be better than gear pair A since the lead twist is an undesired geometric deviation. 8.2 General conclusions Different gear finishing methods produce different surface finishes and structures as well as different geometries and deviations of the gear flanks, all of which influence the transmission error and thereby the noise from a gearbox. It seems that most of the experimental results can be understood and explained by means of measured and predicted transmission error. The relationship between predicted peak to peak transmission error and measured noise at 500 Nm is shown in figure 8.2.1. With the exception of gear pair K, it seems as if there is a strong cor-relation between computed transmission error and noise. However, this breaks down when we look at figure 8.2.2, which shows the relationship between predicted peak to peak transmis-sion error and measured noise at 140 Nm. The conclusion is that it does not seem possible to find one single parameter, such as peak to peak transmission error, and relate it directly to measured noise and vibration.

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This finding is probably a consequence of the fact that two transmission error curves can have different shapes but the same peak to peak value. It might be more relevant to use the trans-mission error ‘acceleration’, i.e. the second derivative of the displacement curve, as a measure of a gear pair’s noise quality. Measured unloaded transmission error acceleration is shown in figure 8.2.3. This figure shows the values in table 4.3.1, plotted as dB, for comparison with the measured vibrations in figure 7.2.3, 7.2.5 and 7.2.7. However, as discussed below, the different torque conditions mean that no direct correlation should be expected. The transmission error measurements were made at no load while the noise and vibration measurements were made at torque levels that considerably influence the transmission error. This means that a direct correlation between measured transmission error and measured noise should not be expected. Measurements of loaded static and dynamic transmission error in the test rig could be an interesting possibility for future research. This would allow comparison between measured transmission error values and noise levels at the same torque levels. The influence of torque level on noise and vibration can be seen in figures 7.2.3 to 7.2.8. As the torque increases from 140 Nm to 500 Nm, the sound pressure level as well as the vibration level (acceleration) increase by approximately 5 dB, but when the torque increases from 500 Nm to 1000 Nm, the noise and vibration levels do not increase further.

Correlation TE–Noise at 500 Nm

90

91

92

93

94

95

96

97

98

99

100

0 0.5 1 1.5 2 2.5 3 3.5

Computed p–p tramsmission error [um]

Mea

sure

d m

ean

dB n

oise

( )

Figure 8.2.1 Relationship between measured mean dB noise and computed peak to peak transmission error for the different test gear pairs at 500 Nm. Line adapted to points by method of least squares. Gear pair K excluded.

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Correlation TE–Noise at 140 Nm

86

87

88

89

90

91

92

93

94

95

0 1 2 3 4 5 6

Computed p–p transmission error [um]

Mea

sure

d m

ean

dB N

oise

Figure 8.2.2 Relationship between measured mean dB noise and computed peak to peak transmission error for the different test gear pairs at 140 Nm.

Measured TE-acceleration

130

135

140

145

150

155

A B C D E F G H I J K

Gear Pair

[dB

] ref

. 10E

-5 m

/s2

Figure 8.2.3 Measured unloaded transmission error acceleration.

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The results of the noise and vibration measurements showed considerable rebuild variation, which remains to be explained. Some possible causes of the rebuild variation are interference from the slave-gearbox, variations in bearing pre-load, or different dynamic properties of the gearbox housing after reassembly. This is a topic for future research. The rebuild variation obviously make it hazardous to draw conclusions from the noise and vibration measurements, but there are some indications that the following conclusions may be warranted: • Shaved gears do not seem to be noisier than ground gears, even if they show considerable

gear tooth deviations. • Gears ground with threaded wheel grinding may be a little less noisy than profile ground

gears. • A rougher surface finish may increase noise and vibration somewhere with the range of 1

to 2 dB, especially at low torque levels. • Wider gears, with overlap ratio εβ=1.8, decrease both noise and vibration by approximately

5 dB. • Pitch errors seem to decrease the gear mesh harmonics and thereby decrease the overall

noise and vibration level by about 2 to 3 dB. • Increased lead crowning increases noise and vibration levels by 1 dB. • Decreased lead crowning decreases noise and vibration levels by between 1 and 3 dB. • Involute alignment errors, up to the magnitude used in this test, do not seem to affect noise

and vibration levels. • Helix angle error (37 μm) increases noise level by 1 to 3 dB. • Decreased lead twist decreases noise and vibration levels by 3 to 5 dB.

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ACKNOWLEDGEMENTS This work was supported by the Swedish Agency for Innovation Systems – VINNOVA. All contributions to this work by colleagues at Volvo Construction Equipment are gratefully ap-preciated. Scania CV AB is acknowledged for carrying out the transmission error measure-ments. Dr. Stefan Björklund is thanked for performing the surface finish measurements. The guidance of my supervisor, Professor Sören Andersson, is gratefully acknowledged. REFERENCES 1. Amini N. ‘Gear Surface Machining for Noise Suppression’, Chalmers University of

Technology, Doctoral thesis, 1999, ISSN 1100-7524.

2. MackAldener M. ‘Tooth Interior Fatigue Fracture & Robustness of Gears’, Royal Insti-tute of Technology, Stockholm, Doctoral thesis, 2001, ISSN 1400-1179.

3. Åkerblom M. ‘Gear Noise and Vibration – A Literature Survey’, TRITA-MMK 2001:11 / ISSN 1400-1179 / ISRN/KTH/MMK/R-01/11-SE, Stockholm 2001.

4. Welbourn D. B. ‘Fundamental Knowledge of Gear Noise – A Survey’ Proc. Noise & Vib. of Eng. and Trans., I Mech E., Cranfield, UK, July 1979, pp. 9–14.

5. Åkerblom M. ‘Gear Test Rig for Noise and Vibration Testing of Cylindrical Gears’, Pro-ceedings OST-99 Symposium on Machine Design, Stockholm 1999, pp. 183–189, ISSN 1400-1179.

6. Volvo Corporate Standard STD 1125,251, http://www.tech.volvo.se/standard/

7. Volvo Corporate Standard STD 5082,81, http://www.tech.volvo.se/standard/

8. Flodin A. ‘Wear of Spur and Helical Gears’, Royal Institute of Technology, Stockholm, Doctoral thesis, 2000, ISSN 1400-1179.

9. LDP, Load Distribution Program v. 10.8, Ohio State University, 2000, http://gearlab.eng.ohio-state.edu/

10. Oswald F. B. et al. ‘Influence of Gear Design on Gearbox Radiated Noise’, Gear Tech-nology, January / February 1998, pp. 10–15.

11. Modig K., Johansson O. ‘Six Sigma Guidebook’, ISBN 91-630-5948-7, 1997.

12. Johnson K. L. ‘Contact Mechanics’, Cambridge University Press, pp. 95–102, 1996.

13. Kohler K., Regan R. ‘The Derivation of Gear Transmission Error from Pitch Error Re-cords’, 61/85 IMechE 1985.

14. Wellbourn D. B. ‘Discussion’ (The Derivation of Gear Transmission Error from Pitch Error Records), IMechE 1986.