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 disassembly 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, increased face-width, decreased lead crowning, increased pitch errors and decreased lead twist. Factors that seem to increase noise are a rougher surface finish, increased lead crowning and helix angle error. Keywords: gear, gearbox, noise, vibration, transmission error.

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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 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 position of the output gear and the position it would occupy if the gear drive were perfectly conjugate.’ 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.

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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, connected 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 gearbox 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 principle is shown in figure 2.1.1. Hydraulic Cylinder Slave or Master Gearbox

Load Sensor Test Gearbox

Microphone

Electric Motor

Accelerometer Articulated Attachment Figure 2.1.1 Sketch of test rig. In order to include the influence of the housing in the investigations, the test gearbox was designed 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 material (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 soundabsorbing material as initial measurements showed that the noise from the electric motor was louder than the gear noise, at least for low RPM.

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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.

1100 2500

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800

-1

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300 500

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0 0

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Time [s]

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 approximately 5000 Nm, but at this torque the rotational speed is very low and no noise is created. 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. Number of teeth Normal module [mm] Pressure angle [ º] Helix angle [ º] Face width [mm] Profile shift coefficient Tip diameter [mm] Centre distance [mm] Transverse contact ratio εα Overlap ratio εβ

pinion gear 49 55 3.5 3.5 20 20 –20 20 35 33 +0.038 –0.529 191 209 191.91 1.78 1.03

[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 3.1.1 Gear data for the test gears.

Table 3.1.2 Tolerances and modifications for the test gears.

All test gears were manufactured within these tolerances, unless otherwise stated. For example, 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 A B C D E F G H I J K

Description Reference gears Shaved Gleason ground Rougher surface Increased face-width Pitch errors Increased lead crowning Decreased lead crowning Involute alignment error Helix angle error Decreased lead twist

Finishing method Profile grinding (KAPP) Shaving Threaded wheel grinding (Gleason) Profile grinding (KAPP) ‘B126’ Profile grinding (KAPP) Profile grinding (KAPP) Profile grinding (KAPP) Profile grinding (KAPP) Profile grinding (KAPP) Profile grinding (KAPP) 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 Lead crowning 8–18 Involute alignment deviation 19 Involute form deviation 8 Lead deviation 25 Lead form deviation 8 Tip relief (short) 10–14 Involute crowning – Radial run out 80

gear 7–10 10 8 35 8 5–13 10 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, except 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 Once per rev. Twice per rev. Three times per rev. Four times per rev. Random

Amplitude [μm] 15 5 0 10 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 different finishing methods. Transverse pitch Transverse tooth to Total cumulative deviation tooth pitch deviation pitch deviation fpt [μm] Δfpt [μm] 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 difference 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 shaving 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, using 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.

K (Decreased lead twist) A (Ground, KAPP) B (Shaved) C (Ground, Gleason)

Lead twist [μm] pinion gear 0 0 +29 +21 –30 –16 –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 (profile) 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.

8

11

–5

4

3

10

–20

4

2

10

–10 –5 –10

4 5 2

2 – 5

9 11 7

Total cumulative pitch deviation Fp

3

Transverse tooth to tooth pitch deviation Δfpt

4

Transverse pitch deviation fpt

–6

–4 3 15 +29 –3 2 13 +21 25 6 14 –30 40 7 8 –16 11 1 14 –22 –22 1 14 –26 3 3 13 +21 –4 2 14 +21 –3 3 11 +9 3 2 11 +8 6 3 14 +22 1 2 14 +18 Same as A pinion 3 2 35 +51 Same as A pinion –2 2 0 +2 Same as A pinion 3 2 13 +24 Same as A pinion 37 2 13 +20 –7 6 11 0 8 4 13 0

Radial run out Fr

10 9 13 10 7 5 11 10 11 8 10 10

Lead twist Vβ

– 2 – 9 – 3 – 3 – 2 – 2

Lead crowning Cb

Tip relief (short) Ca

5 5 8 9 5 4 4 4 4 4 4 5

Lead form deviation ffβ

Involute crowning Ch

–8 –8 19 7 6 –13 –7 –3 –5 –10 –7 –6

Lead deviation fHβ

Involute form deviation ffα

A pinion A gear B pinion B gear C pinion C gear D pinion D gear E pinion E gear F pinion F gear G pinion G gear H pinion H gear I pinion I gear J pinion J gear K pinion K gear

Involute alignment deviation fgα

[μm]

22 26 80 42 42 41 10 16 19 18 66 54

3 3 12 6 5 6 2 3 3 3 7 8

3 4 11 10 4 4 2 4 2 3 10 8

21 32 103 31 43 54 12 29 22 27 31 43

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2

2

13

20

3

4

24

9

3

5

14

25 31 27

4 6 4

4 9 5

37 42 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 theoretically 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 carried 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.

[z [mm]

0.03

0.02

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d)

−0.02

e)

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3 x [mm]

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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 A KAPP ground B Shaved C Gleason ground D KAPP rougher surface D After noise test

Ra 0.57 1.05 0.54 0.86 0.75

Rq 0.71 1.67 0.68 1.16 1.01

Table 4.2.1 Ra and Rq values for the gears manufactured 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 approximately 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. A (KAPP)

B (Shaved)

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C (Gleason)

D Rougher surface after noise test (KAPP)

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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. 12

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 corresponds 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 rotation below (r) and left-hand rotation above (l).

Figure 4.3.2 Long wave transmission error for gear pair B (shaved), right-hand rotation below (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 components 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 FFTbasis 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 engagements 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), righthand rotation (r).

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In order to compare the different test gear pairs, four parameters were chosen from the transmission 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]

A B C D E F G H I J K

4.7 5.5 2.9 4.9 3.4 4.0 3.6 2.7 5.4 3.4 3.8

Average tooth Amplitude of Average tooth engagement the tooth acceleration at peak to peak mesh fre1000 RPM from FFT quency, from max–min analysis FFT analysis from FFT analysis [μm] [μm] [m/s2] 2.8 1.1 368 3.5 1.7 119 1.5 0.6 276 3.7 1.5 258 1.5 0.7 78 1.1 0.4 244 2.0 0.8 570 1.3 0.6 48 3.2 1.6 365 2.0 0.8 225 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 harmonics 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 transmission error, because the transmission error was measured at a very low torque level. The computations at 140, 500 and 1000 Nm were made to allow comparison with the noise and vibration measurements described in sections 6 and 7.

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Predicted transmission error for gear pair A to D Average Amplitude of Amplitude of tooth enthe tooth 2 x tooth Torque gagement mesh fremesh freGear pair [Nm] peak to peak quency quency [μm] [μm] [μm] 10 1.29 0.57 0.08 50 1.33 0.39 0.06 A 140 2.34 0.97 0.26 500 2.52 1.08 0.10 1000 1.53 0.55 0.32 10 7.44 3.15 0.19 50 6.53 3.03 0.43 B 140 5.30 2.42 0.55 500 1.28 0.46 0.16 1000 1.69 0.73 0.23 10 3.70 1.67 0.22 50 2.23 1.05 0.05 C 140 0.70 0.11 0.21 500 1.55 0.66 0.04 1000 1.65 0.63 0.23 10 2.43 1.13 0.07 50 1.59 0.67 0.20 D 140 1.59 0.57 0.24 500 1.93 0.94 0.10 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 Average Amplitude of Amplitude tooth enthe tooth of 2 x tooth Torque gagement mesh fremesh freGear pair [Nm] peak to peak quency quency [μm] [μm] [μm] 10 1.69 0.72 0.16 50 1.31 0.57 0.10 E 140 0.92 0.40 0.03 500 0.96 0.33 0.15 1000 0.87 0.37 0.05 10 3.15 1.20 0.32 50 1.83 0.61 0.25 F 140 2.26 0.97 0.14 500 1.80 0.84 0.18 1000 0.83 0.27 0.21 10 2.35 1.14 0.07 50 2.30 0.88 0.17 G 140 2.55 1.07 0.14 500 2.79 1.23 0.07 1000 1.87 0.64 0.39 10 2.15 0.67 0.50 50 1.98 0.79 0.18 H 140 2.04 0.91 0.07 500 1.49 0.62 0.13 1000 0.59 0.15 0.18 10 2.31 0.80 0.31 50 1.38 0.50 0.21 I 140 1.62 0.71 0.06 500 2.08 0.93 0.02 1000 1.38 0.40 0.33 10 2.23 0.92 0.32 50 1.55 0.56 0.10 J 140 2.08 0.80 0.13 500 2.90 1.30 0.16 1000 1.52 0.63 0.20 10 1.63 0.66 0.20 50 1.75 0.78 0.13 K 140 2.65 1.16 0.13 500 2.95 1.42 0.07 1000 1.99 0.91 0.22 Table 5.1.2 Results of the transmission error computations for gear pairs E to K.

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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 interesting 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 transmission 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 reference 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).

21

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.

22

TE − index = 1st harmonic amplitude + (2nd harmonic amplitude × 4 )

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

A sin ω t and the displacement of the second harmonic of the gear mesh frequency is

B sin 2ω t 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: Acceleration of 1st harmonic = −ω 2 A sin ω t Acceleration of 2nd harmonic = − 4ω 2 B sin 2ω t 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

5 TE-Index 140 Nm

4.5

TE-Index 500 Nm

4

TE-Index 1000 Nm

TE-index [um]

3.5 3 2.5 2 1.5 1 0.5 0 A

B

C

D

E

F Gear Pair

Figure 5.2.4 Computed transmission error index.

23

G

H

I

J

K

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 7.00

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

6.00

TE [um]

5.00 4.00 3.00 2.00 1.00 0.00 A

B

C

D

E

F

G

H

I

J

K

Gear Pair

Fig 5.3.1 Comparison between measured short-wave mean transmission error and predicted transmission error at 50 Nm. Peak to Peak Transmission Error 7.00 P-P Measured (FFT) P-P Computed (50Nm)

6.00

TE [um]

5.00 4.00 3.00 2.00 1.00 0.00 A

B

C

D

E

F

G

H

I

J

K

Test Gear Pair

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

24

1st Harmonic of Transmission Error 3.50 1st Harmonic Measured (FFT) 1st Harmonic Computed (50Nm)

3.00

TE [um]

2.50 2.00 1.50 1.00 0.50 0.00 A

B

C

D

E

F

G

H

I

J

K

Gear Pair

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

25

6 NOISE AND VIBRATION MEASUREMENTS 6.1 Instrumentation

The instrumentation consists of one optical tachometer, three microphones and three accelerometers. 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 direction of the gear mesh contact force; and accelerometer 3 registers vibrations at a right angle to the direction of accelerometer 2.

Microphone horisontal positions: Shaft no. 1

Shaft no. 2

Tachometer Gearbox

29 cm

20 cm Mic. 1

40.5 cm

Mic. 2 Mic. 3 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.

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

26

Acc. 2 Acc. 1 Shaft 1.

Shaft 2.

Acc. 3

Figure 6.1.2 Gearbox shown from front, with accelerometers attached. The arrows denote positive directions of accelerometers 2 and 3. Accelerometer 1 registers vibrations in the axial direction. 6.2 Measurement repeatability

Measurement repeatability was estimated by carrying out a standard test according to the procedure 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 levels 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 difference 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 Load [Nm] 140 500 1000

no. 1 1.71 0.56 0.49

Accelerometers no. 2 2.66 0.41 0.38

no. 3 2.63 0.40 0.41

Microphones no. 1 no. 2 no. 3 1.22 1.44 1.26 0.85 0.97 0.83 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.

27

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 controlled 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 summarised in table 6.3.1. Number of tachometer pulses per revolution Min and max RPM Number of samples per revolution Frame size Maximum order Order resolution Window Frame event

1 550, 2000 768 4096 150.0 0.1875 Hanning Broad 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 seconds. The rate R& of the rotational speed increase is thus

2550 − 450 = 20 RPM/s. 105 New data are taken for every 20 RPM increase in the rotational speed. The lowest shaft rotational 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 presented 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 obtained by integrating the signal components in the vicinity of the order of interest. The interval of integration corresponds to ±0.5 orders.

2

I-DEAS © Structural Dynamics Research Corp.

28

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 necessary 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 measurements. 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 pressure 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 Oswald 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.’ 29

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 order 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.

30

Figure 7.2.2 An order plot of the measured sound pressure level for microphone 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).

31

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 respective 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 measurement 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 repeatability by reattaching accelerometers and microphones but without disassembling and reassembling 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. 140 Nm RPM

[dB] (Pref =2*10E-5 Pa) M1 M2 M3 Lo. Hi. Lo. Hi. Lo. Hi.

[dB] (Aref =10E-5 m/s2) A1 A2 A3 Lo. Hi. Lo. Hi. Lo. Hi.

A 17/8 A 16/10

93 98

93 93

88 91

87 90

86 90

89 86

123 122

123 124

120 121

124 123

117 116

122 122

A mean

95.5

93

89.5

88.5

88

87.5

122

124

120

124

116

122

B 8/8 B 15/8 B 25/1 B 30/1

103 91 101 91

93 93 95 96

– 86 95 87

– 89 93 95

– 85 93 86

– 88 88 92

123 115 122 117

124 121 128 122

122 115 128 119

125 121 126 122

118 111 119 114

120 118 123 118

B mean C

96.5 96

94.2 90

89.3 91

92.3 87

88 88

89.3 85

119 123

124 119

121 113

124 118

116 110

120 114

D 14/8 D 3/10 D 14/12 D 14/12r

96 94 93 94

93 93 100 99

89 94 91 90

86 91 91 91

87 89 91 89

87 88 92 90

120 116 121 120

120 121 127 126

121 120 127 125

126 127 135 134

116 113 123 122

120 119 130 129

D mean E F G H I J K

94.2 93 94 96 94 91 98 88

96.2 92 93 94 88 96 98 87

91 88 89 87 87 91 88 88

89.8 92 89 92 87 90 99 88

89 85 86 88 87 86 87 85

89.2 87 86 91 83 91 94 83

119 118 118 119 118 117 122 115

124 123 119 126 120 126 125 117

123 113 118 120 117 117 119 113

130 118 123 127 122 128 128 119

118 109 112 114 114 116 113 112

124 117 118 122 118 125 121 116

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

32

RPM

[dB] (Pref =2*10E-5 Pa) M1 M2 M3 Lo. Hi. Lo. Hi. Lo. Hi.

[dB] (Aref =10E-5 m/s2) A1 A2 A3 Lo. Hi. Lo. Hi. Lo. Hi.

A 17/8 A 16/10

101 99

101 100

94 96

95 100

92 96

96 91

126 128

128 131

123 126

130 132

117 124

131 126

A mean

100

100

95

97.5

94

93.5

127

130

124

131

120

128

B 8/8 B 15/8 B 25/1 B 30/1

100 98 95 95

95 92 101 101

– 90 92 91

– 89 96 97

– 92 93 90

– 92 95 96

121 122 124 124

125 126 127 127

123 122 126 123

127 124 129 124

119 117 119 119

122 124 125 123

B mean C

97 99

97.2 95

91 94

94 92

91.7 92

94.3 94

123 128

126 129

124 120

126 127

118 118

124 123

D 14/8 D 3/10 D 14/12 D 14/12r

104 97 100 98

100 98 99 99

98 94 94 93

96 102 94 95

97 96 94 92

99 94 95 94

127 128 128 128

127 130 135 134

123 127 127 127

132 132 135 135

124 121 124 123

125 127 131 132

D mean E F G H I J K

99.8 92 95 99 100 97 99 93

99 96 98 100 94 98 105 94

94.8 90 91 95 95 92 93 89

96.8 97 96 100 97 93 104 92

94.8 86 92 93 94 93 93 90

95.5 91 94 98 91 92 101 90

128 120 124 127 126 125 127 123

132 128 127 133 129 132 132 126

126 117 118 125 126 123 124 121

134 123 129 133 128 132 131 123

123 115 120 121 122 120 120 118

129 123 124 128 124 130 126 126

500 Nm

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

RPM

[dB] (Pref =2*10E-5 Pa) M1 M2 M3 Lo. Hi. Lo. Hi. Lo. Hi.

[dB] (Aref =10E-5 m/s2) A1 A2 A3 Lo. Hi. Lo. Hi. Lo. Hi.

A 17/8 A 16/10

99 101

103 101

93 97

96 102

93 96

96 92

123 122

131 131

125 126

132 133

124 125

132 127

A mean

100

102

95

99

94

94

122

131

126

132

124

130

B 8/8 B 15/8 B 25/1 B 30/1

101 101 99 96

98 99 99 103

– 94 96 93

– 91 97 97

– 92 96 90

– 93 92 97

123 121 126 126

130 127 127 129

124 124 128 125

129 127 133 126

122 124 122 120

126 127 128 128

B mean C

99.2 101

99.8 100

94.3 94

95 95

92.7 90

94 94

124 125

128 131

125 122

129 129

122 119

127 128

D 14/8 D 3/10 D 14/12 D 14/12r

107 98 103 104

100 99 100 101

99 93 94 94

97 102 94 95

98 95 94 95

99 94 90 89

123 124 126 126

132 133 134 134

125 125 127 128

133 133 133 133

124 123 125 125

127 127 132 131

D mean E F G H I J K

103 95 98 98 99 100 100 100

100 94 99 103 101 98 102 95

95 92 96 94 95 94 93 93

97 94 99 104 97 93 102 90

95.5 87 93 92 95 92 93 91

93 90 94 96 92 91 101 89

125 118 119 122 122 125 126 119

133 123 130 134 130 132 134 124

126 120 122 121 125 126 126 118

133 128 132 135 127 127 132 125

124 114 122 121 122 123 120 120

129 122 125 128 126 128 128 124

1000 Nm

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

33

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 measurement from the mean value was 2 dB and the variation between the four different measurements 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 calculations were made in accordance with the Six Sigma Guidebook [11]. s s ⎤ ⎡ C.I . = ⎢m x − tα / 2 x ; m x + tα / 2 x ⎥ n n⎦ ⎣ 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

Mean Vibration dB Mean Sound dB

95 % Confidence intervals for the mean values Gear pair B Gear pair D 121.7–125.2 [dB] 125.6–131.3 [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 vibration 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% confidence. 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.

34

"Mean Vibration dB" at 140 Nm 124 123

[dB] ref. 10E-5 [m/s2]

122 121 120 119 118 117 116 115 114 A

B

C

D

E

F

G

H

I

J

K

Gear Pair

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 95 94

[dB] ref 2*10E-5 [Pa]

93 92 91 90 89 88 87 86 85 A

B

C

D

E

F

G

H

I

J

K

Gear Pair

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

35

"Mean Vibration dB" at 500 Nm 132 131 130 [dB] ref. 10E-5 [m/s2]

129 128 127 126 125 124 123 122 121 120 A

B

C

D

E

F

G

H

I

J

K

Gear Pair

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 100 99

[dB] ref. 2*10E-5 [Pa]

98 97 96 95 94 93 92 91 90 A

B

C

D

E

F

G

H

I

J

K

Gear Pair

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.

36

"Mean Vibration dB" at 1000 Nm 129 128

[dB] ref. 10E-5 [m/s2]

127 126 125 124 123 122 121 120 119 A

B

C

D

E

F

G

H

I

J

K

Gear Pair

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 100 99

[dB] ref. 2*10E-5 [Pa]

98 97 96 95 94 93 92 91 90 A

B

C

D

E

F

G

H

I

J

K

Gear Pair

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

37

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 values that are relatively high, although not the highest. Both the measured and predicted transmission 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 between 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 predicted 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 between 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.

38

Torque [Nm] 10 50 140 500 1000

Size of contact ellipse [mm] 2a 2b Contact stress [MPa] 0.11 9.7 206 0.19 16.6 353 0.27 23.5 497 0.41 35.9 760 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 predicted 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 regards 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 deformation 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 moments 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 influence 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 measurements 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 predicted 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 values 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, especially 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 correlation 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 transmission 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 transmission 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 100 99

Measured mean dB noise

98 97 96 95 94 93 92 ( )

91 90 0

0.5

1

1.5

2

2.5

3

Computed p–p tramsmission error [um]

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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3.5

Correlation TE–Noise at 140 Nm 95

Measured mean dB Noise

94 93 92 91 90 89 88 87 86 0

1

2

3

4

5

6

Computed p–p transmission error [um]

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 155

[dB] ref. 10E-5 m/s

2

150

145

140

135

130 A

B

C

D

E

F

G

H

I

Gear Pair

Figure 8.2.3 Measured unloaded transmission error acceleration.

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J

K

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 appreciated. Scania CV AB is acknowledged for carrying out the transmission error measurements. 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 Institute 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’, Proceedings 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 Technology, 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 Records’, 61/85 IMechE 1985. 14. Wellbourn D. B. ‘Discussion’ (The Derivation of Gear Transmission Error from Pitch Error Records), IMechE 1986.

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