Graphical Abstract Figure
Graphical Abstract Figure
Close modal

Abstract

This study presents experimental work to analyze spur gear pair with geometrical and operating parameters. The spur gear pair accommodates the correction in tooth addendum, gear backlash, and linear tip-relief profile modification with three levels. As per Taguchi L9 orthogonal array, nine test spur gear pairs are precisely manufactured to analyze the gear dynamics. Other basic gear design parameters and operating parameters were held at a constant level. Root-mean-square (RMS) acceleration in the vertical direction is used to quantify the dynamic response of test gear pairs. Experimental data are analyzed by using the Taguchi method to investigate the rank of influencing parameters and the optimum level of parameters to minimize vibration response. Gear backlash, compared to tooth addendum and linear tip-relief tooth profile modification, emerges as the most influential parameter. The optimal combination is addendum 3.3 mm, backlash 0.05–0.075 mm, and linear tip-relief tooth profile modification, yielding the lowest vibration generation. Finally, the confirmation test was performed by using a simulation study to validate the experimental results. The RMS acceleration value of the simulation study is 0.070 g, which is approximately the lowest of the experimental response values. Similarly, 20 experiments were conducted with different speed and load combinations to check the effect of operating conditions on gear dynamics. From these experimental studies, it is observed that the rank of influencing parameters and optimum level of geometrical are varied with respect to operating conditions. It may be concluded that operating speed and loading conditions play very important roles in the design of a quiet gear system. The optimized performance of spur gear pairs may vary across different operating conditions. It indicates that typically optimized spur gear pair operated at one particular combination of load and speed may not show good performance at other operating conditions. Therefore, this study suggests that the realistic range of operating conditions should be considered while selecting suitable levels of geometrical parameters.

1 Introduction

Geared systems are famous for generating large amounts of mesh and bearing forces under certain operating conditions. Forces acting on gear teeth amplified under resonance conditions result in a larger amplitude of vibration levels that decrease gears' fatigue life. In some constant-speed applications, it is possible to avoid resonance. Although many gear systems work across a wide speed range, some resonances are unavoided. Moreover, changes in natural frequency to avoid resonance are not possible due to the design constraints of the gear pair system. In addition, gear tooth geometry is an essential factor affecting noise and vibration [1,2]. Researchers have developed various mathematical models to investigate gear noise and vibration [3,4]. Some of them suggest methods to control the vibration level of gear systems. However, the relationship between the geometrical parameters of the gear tooth surface and gear vibration needs to be clearly understood when the gear system operates over a wide range of dynamic conditions.

Therefore, it is necessary to investigate the spur gear pairs' dynamic characteristics with variations in geometrical parameters and operating conditions. This study aims to develop a suitable experimental method that simulates the influence of geometrical and operating parameters. The novelty of this study is that this method can help gear designers to select suitable values of gear geometrical parameters when the gear pair systems are required to operate at a wide range of speeds and loading conditions.

Harris [5] examined mesh stiffness variation as a periodical stimulation capable of causing gear vibrations, although the gears lacked manufacturing defects. Mesh stiffness variation in engagement is crucial in analyzing gear pair dynamics even when the gears run at a constant speed and load conditions [68]. Benton and Seireg [7] experimentally simulated instabilities under parametric excitations. Due to limitations in the manufacturing process and design constraints, it is not practically possible to manufacture gears with an ideal involute profile; also, when the gears are operating at nominal speed and torque, the teeth elastically deform. The effect of all this is to deviate actual contact between meshing teeth from its theoretical plane of action, which causes transmission error in the motion of the gear pair. Transmission error under load is widely used for assessing the quality of gear noise and vibration [3,9]. Tooth profile modification is an effective way to minimize gear vibration and noise[10,11]. In 1988, Lin et al. [12,13] developed a simple spur gear transmission model and studied the effect of various factors on its dynamic behavior. Moreover, they concluded that the vibration response depends upon the operating speed and applied torque. Besides, in 1989, Lin et al. [14,15] were the first to publish that an alteration to the tooth profile had successfully decreased load in gear analysis [16]. The best value of the amount and extent of linear tooth profile modification depends on the applied torque carried by a geared system [17].

Cai and Hayashi [18] studied the gear vibration based on the effect of profile error and contact ratio. They further claim that rotational vibration is reduced to zero by optimizing the tooth profiles of spur gears. Yoon and Rao [19] investigated the potential for reducing the maximum dynamic load by modifying teeth using the cubic spline profile. Liou et al. [20] investigated analytically the effect of the involute contact ratio of spur gear pairs without profile modification. Kahraman and Blankenship [21] evaluated a gear pair with external driving excitation, clearance, and parametric in a series of experiments. Also, they experimentally analyzed the response of tip relief and contact ratio on the dynamic response of spur gear pairs [22,23].

Geared systems are always bound to have some backlash caused by manufacturing error or wear, or it may be due to being designed to provide better lubrication or avoid interference. Recently, researchers paid more attention to the geared systems with backlash as the excitation due to backlash may cause tooth separation. Wang [24,25] investigated experimentally that displacement excitation caused by static transmission error and impact excitation caused by excessive backlash may create sub-harmonic components in the response of a geared system. Singh and Comparin [26,27] proposed rattle criteria and demonstrated their application. They also discussed critical issues related to the geared systems. The importance of consideration of manufacturing errors and gear backlash in gear dynamics [28]. Wang et al. [29] discovered nonlinear vibration in geared systems with two nonlinear factors: changing mesh stiffness and backlash. Bonari and Pellicano [30] explored the existence of manufacturing flaws that increase the vibration's amplitude and cause chaotic motion throughout a wide range of rotating speeds with minimal torque. Further, more advanced dynamic models have been developed to study the effect of various geometrical factors on gear pairs.

Ratanasumawong [31] illustrated different tooth forms to analyze the vibration characteristics of helical gears. Surface geometry like bias-in-modification, pressure angle, and lead crowing evaluated the relation among meshing components and surface geometries of helical gears. Moradi and Salarieh [32] investigated the backlash nonlinearity due to the nonlinear oscillation of spur gear pairs based on the multiscale method. Ghosh and Chakraborty [33] optimize analytical methods to study the profile error and pitch effect on optimal modification.

Lin and He [34] studied the static transmission error of gear pairs with assembly errors, machining errors, and tooth modifications. Liu et al. [35] studied the transmission stationarity of the spur gear pair, which is impacted by the pitch deviation induced by the machining error. The meshing features of the spur gear pair with the pitch deviation are analyzed to examine the system's motion characteristics and time-varying contact ratio. Gao et al. [36] established a tooth profile model and optimization method for the involute gear.

From the literature review analysis, it is clear that operating parameters and geometrical factors potentially affect the overall dynamic performance of the gear pair system. However, the interaction between the spur gear pair's operating parameters and geometrical features must be clarified, consistent, and complete. In short, the operating parameters' role in selecting the best value of geometrical parameters is not clearly understood in designing a gear pair system for minimum vibration level. Still, several issues require clarification and further investigation. There is an extension to examine the impact of geometrical parameters resulting in a transition of gear tooth surface geometric features, viz., addendum, backlash, and linear tip-relief tooth profile modification and, further, to investigate the effect of operating parameters on optimizing geometrical parameters of gear pair that resulted in minimum vibration level.

Therefore, this experimental study aims to systematically analyze operating conditions' influence on optimizing the geometrical factors for minimum vibration level. The present paper attempts to clarify the role of operating conditions in optimizing the geometrical factors of spur gear pairs.

2 Experimental Test Setup

A novel dynamic test setup was designed and constructed, as shown in Fig. 1(a). The setup consists of an induction motor (Crompton and Grieves, Power: 5 HP), single-stage gearbox, and rope dynamometer. The flexible coupling is used to connect the motor shaft to the input shaft of the gearbox. Variable frequency drive (VFD) controls and sets the gearbox's operating speed to any desired value. The split-type gearbox facilitates easy mounting and replacing of the gear pair during performing tests. A simple rope brake dynamometer is used to apply the torque on the gearbox's output shaft. Each test spur gear was rigidly mounted on the input and output shaft of the gearbox. A bearing block supported each shaft. Two identical size deep groove ball bearings (SKF 6310) were used for each shaft. Bearing caps were used to fix the bearing position in the bearing block and the shaft position in the gearbox casing. At the end of the output shaft, a rope drum pulley is mounted to apply the load on the gearbox.

Fig. 1
(a) Experimental test rig and (b) accelerometer in vertical position
Fig. 1
(a) Experimental test rig and (b) accelerometer in vertical position
Close modal

The drive motor, split-type gearbox, and rope brake dynamometer are mounted on the ground, and the foundation plate is perfectly aligned with alignment equipment. Variations in vibratory response can be attained only by varying geometrical parameters and operating parameters in this type of test setup. A sweep test was performed beforehand to set the accelerometer's position during testing. Bearing block number 4 was the final location along the vertical direction selected for mounting the accelerometer on the casing, as shown in Fig. 1(b).

2.1 Data Acquisition and Instrumentation.

The vibration signals, particularly in frequency and time domain, are captured and analyzed in real-time using a Fast Fourier Transform (FFT) analyzer. The frequency spectra were examined through an FFT. The vibration signals were measured at a sampling frequency of 4096 Hz, stored in a tape recorder, and consequently analyzed in an FFT analyzer. To at least catch the first tooth mesh frequency and side bands, the frequency range was 1600 Hz. The sidebands of interest are distinguished, as well as any sideband leakage and bin errors, using a standard constant bandwidth of 1 Hz.

The length of the signal was set to one second. The frequency range and the quantity of analyzer lines employed are vital factors. In this experiment, 1600 analyzer lines were used to display a 4096-point transform. After filtering, the root-mean-square (RMS) value of the raw observed vibration signal is used to determine the amplitude of the reference signal. The frequency spectra were averaged using four linear averages, and the time-domain vibration signal was converted to the frequency-domain using narrow-band filters.

3 Methodology to Conduct Experiments

From the literature analysis, it is clear that many studies discussed the nonlinear vibration of gear pair systems with single gear parameters such as contact ratio or backlash. However, in actual practice, gear systems consist of many nonlinear parameters, and their combined effects are not negligible. This experimental study analyzed the combined effect of these nonlinear parameters and operating conditions using the Taguchi method. Further, this study also investigates the combined effect of these parameters with varying operating conditions.

3.1 Taguchi Methodology (Design of Experiments).

Three selected parameters with their three levels are shown in Table 1. These parameters focus on difficulties in maintaining the entire contact surfaces of the tooth profile of mating gears as an involute profile to fulfill the condition of the law of gearing. Maintaining contact between only the involute portion of the profile is achieved by adjusting either addendum or tooth profile modifications as non-standard gears to avoid interference-related problems. These three parameters represent different types of excitation, such as tooth addendum representing parametric excitation, backlash representing impact excitation, and liner tip relief representing displacement excitation. It will be interesting to check the combined effect of these excitations on the dynamic behavior of the gear pair system.

Table 1

Three parameters with three levels

LevelsParameters
(A) Addendum (mm)(B) Backlash (mm)(C) Linear tip-relief profile modification
Amount-(Δ), µmExtent-(L), mm
First2.60.051–0.0740L = 0 (ZR)
Second3.10.101–0.12415–20L = Ln/2 (SR)
Third3.40.151–0.17615–20L = Ln (LR)
LevelsParameters
(A) Addendum (mm)(B) Backlash (mm)(C) Linear tip-relief profile modification
Amount-(Δ), µmExtent-(L), mm
First2.60.051–0.0740L = 0 (ZR)
Second3.10.101–0.12415–20L = Ln/2 (SR)
Third3.40.151–0.17615–20L = Ln (LR)

Note: Ln is normalized tooth profile modification; ZR, SR, and LR refer zero, short, and long relieves.

Tolerance of 25 µm for backlash and 5 µm for linear tip amount is provided due to manufacturing limitations. The levels of parameters for both pinion and gear are kept the same. Based on several factors and their levels, an L9 orthogonal array is selected to design experiments, as shown in Table 2.

Table 2

Orthogonal array

Trial numberParameters
(A) Tooth addendum(B) Backlash(C) Linear tip-relief tooth profile modification
1111
2122
3133
4212
5223
6231
7313
8321
9332
Trial numberParameters
(A) Tooth addendum(B) Backlash(C) Linear tip-relief tooth profile modification
1111
2122
3133
4212
5223
6231
7313
8321
9332

3.2 Design of Test Spur Gear Pairs.

Nine test spur gear pairs are designed for the trials using the Taguchi L9 orthogonal array. Standard design parameters of test spur gear pairs of unity ratio design used for the experimental study are shown in Table 3.

Table 3

Fundamental variables of spur gear pair

S. No.VariablesUnitsValue
1Modulemm3
2Teeth in number50
3Pressure angledeg.21
4Face widthmm21
5Centre distancemm152
6Root-circle diametermm142.320
7Base-circle diametermm142.421
8Pitch-circle diametermm152
10AddendummmVariable
11BacklashmmVariable
12Linear tip-relief profile modificationVariable
S. No.VariablesUnitsValue
1Modulemm3
2Teeth in number50
3Pressure angledeg.21
4Face widthmm21
5Centre distancemm152
6Root-circle diametermm142.320
7Base-circle diametermm142.421
8Pitch-circle diametermm152
10AddendummmVariable
11BacklashmmVariable
12Linear tip-relief profile modificationVariable

Using the values of design parameters given in Table 3 and Taguchi's L9 orthogonal array (OA) shown in Table 2, nine spur gear pairs are designed. Nine test gear pairs are specified and designed by three letters (A, B, and C) and three suffix numbers (1, 2, and 3), respectively.

3.3 Production of Nine Test Spur Gear Pairs.

Test gear pairs produced in quality grade 6 using a gear hobbing machine. The material for the test gear pair was SAE 8620. Variations in geometrical features such as addendum, backlash, and linear tip-relief profile modifications were used for testing gear pairs. Modification in tooth addendum is achieved by the addendum ratio method. In this method, the length of the tooth addendum is controlled by the outside diameter of the test gears. This method does not require a change in the center distance of the gear pair. Modifications in gear backlash are achieved by controlling gear pairs' tooth thickness. Modifications in tooth profiles are obtained by using a unique form of the abrasive wheel on the gear grinding machine. All the dimensions of test gear pairs were measured precisely. Finally, the quality of the gear pair tooth profile was checked by plotting the K-chart shown in Fig. 2 with profile tester. Nine test spur gear pairs are shown in Fig. 3.

Fig. 3
Nine test spur gear pairs
Fig. 3
Nine test spur gear pairs
Close modal

3.4 Test Matrix.

According to Taguchi L9 orthogonal array (OA), nine trials (tests) were performed at constant torque and speed conditions to investigate the effect of selected factors on the dynamic behavior of the spur gear pair system. Also, in order to investigate the effect of operating parameters (speed and torque) on gear pair dynamic behavior, nine sets of spur gear are tested at different operating conditions. For that purpose, totally 20 experiments were performed, with different combinations of load and speed as shown in Table 4. For experimentation zero torque is selected, because the operation of an unloaded gear drive is not a gentle push along one direction. It is an impact type of drive, with impulse trains acting along two opposite directions [24].

Table 4

Operating parameters for conducting 20 experiments

Torque (T), N-m04.95.65.957.0
Speed, (N), rpm900120015001800
Torque (T), N-m04.95.65.957.0
Speed, (N), rpm900120015001800

4 Test Results and Data Analysis

4.1 Taguchi L9 Orthogonal Array (OA) Experiments.

In experiment No. 1, all nine trials were performed for constant torque T = 0 N-m with operational speed N = 900 rpm conditions. RMS acceleration was used to assess the nine gear pairs' dynamic response. Each gear combination is tested three times to increase measurement accuracy. The sample frequency–response plot measured by an accelerometer for gear pair number 1 (A1B1C1) is shown in Fig. 4. In a similar manner, nine experiments are conducted, and their results are compiled in Table 5.

Fig. 4
Sample frequency plot for gear pair 1 (A1B1C1)
Fig. 4
Sample frequency plot for gear pair 1 (A1B1C1)
Close modal
Table 5

Experimental results of the Taguchi L9 orthogonal array (OA) experiments

Trial No.Gear pair designationParametersAcceleration RMS (g)Mean RMS (g)
ABCR1R2R3
1A1B1C12.70.05–0.075ZR0.0770.0780.0800.078
2A1B2C22.70.100–0.125SR0.0870.0870.0880.087
3A1B3C32.70.150–0.175LR0.0750.0770.0790.077
4A2B1C23.00.05–0.075SR0.0620.0610.0600.061
5A2B2C33.00.100–0.125LR0.0830.0840.0850.084
6A2B3C13.00.150–0.175ZR0.0880.0870.0890.088
7A3B1C33.30.05–0.075LR0.0720.0740.0730.073
8A3B2C13.30.100–0.125ZR0.0730.0720.0720.072
9A3B3C23.30.150–0.175SR0.0700.0740.0720.072
Trial No.Gear pair designationParametersAcceleration RMS (g)Mean RMS (g)
ABCR1R2R3
1A1B1C12.70.05–0.075ZR0.0770.0780.0800.078
2A1B2C22.70.100–0.125SR0.0870.0870.0880.087
3A1B3C32.70.150–0.175LR0.0750.0770.0790.077
4A2B1C23.00.05–0.075SR0.0620.0610.0600.061
5A2B2C33.00.100–0.125LR0.0830.0840.0850.084
6A2B3C13.00.150–0.175ZR0.0880.0870.0890.088
7A3B1C33.30.05–0.075LR0.0720.0740.0730.073
8A3B2C13.30.100–0.125ZR0.0730.0720.0720.072
9A3B3C23.30.150–0.175SR0.0700.0740.0720.072

4.2 Experimental Data Analysis.

The Taguchi method expresses the scatter around a target value using the signal-to-noise ratio (S/N). When noise elements are present, the S/N ratio measures the fluctuation within the experiment [37,38]. Taguchi effectively used this technique to determine the best condition from the studies. The better the S/N ratio, regardless of whether the quality characteristic is smaller-the-better or higher-the-better, relates to the minor variation of the output characteristics within the target value. As a result, the experiment's goal is to have the most significant feasible S/N ratio [39].

This research aims to reduce the vibration level of spur gear pairs; therefore, the “smaller- the – better” characteristic is selected, which is the logarithmic function given as
(SN)SB=10Log10(1Ri=1RYi2)
(1)

Table 6 displays the S/N ratios for trials 1 through 9 calculated using the minitab software.

Table 6

Experimental results, mean, and S/N ratios

Trial No.ParametersRMS (g)Mean RMS (g)S/N η (dB)
ABCR1R2R3
11110.0770.0780.0800.07822.1581
21220.0870.0870.0880.08721.2096
31330.0750.0770.0790.07722.2702
42120.0620.0610.0600.06124.2934
52230.0830.0840.0850.08421.5144
62310.0880.0870.0890.08821.1103
73130.0720.0740.0730.07322.7335
83210.0730.0720.0720.07222.8534
93320.0700.0740.0720.07222.8534
Trial No.ParametersRMS (g)Mean RMS (g)S/N η (dB)
ABCR1R2R3
11110.0770.0780.0800.07822.1581
21220.0870.0870.0880.08721.2096
31330.0750.0770.0790.07722.2702
42120.0620.0610.0600.06124.2934
52230.0830.0840.0850.08421.5144
62310.0880.0870.0890.08821.1103
73130.0720.0740.0730.07322.7335
83210.0730.0720.0720.07222.8534
93320.0700.0740.0720.07222.8534

The delta (Δ) rate presented in response Tables 7 and 8 were used to establish the rank. The maximum value of delta indicates a more significant contribution of that factor on the dynamics of spur gear pairs.

Table 7

Response table for means

DesignationParameterMean (g)Delta (Δ)Rank
Level-1Level-2Level-3
ATooth addendum0.080670.077670.072330.008332
BBacklash0.070670.081000.079000.010331
CLinear tip-relief profile modification0.079330.073330.078000.006003
DesignationParameterMean (g)Delta (Δ)Rank
Level-1Level-2Level-3
ATooth addendum0.080670.077670.072330.008332
BBacklash0.070670.081000.079000.010331
CLinear tip-relief profile modification0.079330.073330.078000.006003
Table 8

Response table for (S/N) ratios

DesignationParameterS/N ratios (dB)Delta (Δ)Rank
Level-1Level-2Level-3
ATooth addendum21.8822.3122.810.932
BBacklash23..0621.8622.081.201
CLinear tip-relief profile modification22.0422.7922.170.743
DesignationParameterS/N ratios (dB)Delta (Δ)Rank
Level-1Level-2Level-3
ATooth addendum21.8822.3122.810.932
BBacklash23..0621.8622.081.201
CLinear tip-relief profile modification22.0422.7922.170.743

Based on the delta (Δ) rate, the backlash is the most influencing parameter among the three significant parameters selected for the present study that affect the dynamic behavior of spur gear pairs.

In Figs. 5 and 6, the main effect plot depicts the distinction in mean response (g) and S/N ratio with different factors and levels. The smaller, the better criteria used to optimize the parameters for mean effect plots of means response (g). The S/N ratio represents the scatter of a target value. A large S/N ratio shows that the signal significantly exceeds the random noise sources. The third, first, and second levels, respectively, should be maintained for the backlash (B), tooth addendum (A), and linear tip-relief tooth profile alteration (C). For optimal setting, it is recommended that the selected value should have a maximum S/N ratio [39]. As a result, A3B1C2 is the best combination of spur gear pairs for achieving the lowest dynamic response in terms of RMS. Table 9 shows the optimum levels of control parameters.

Fig. 5
Main effects plot for means
Fig. 5
Main effects plot for means
Close modal
Fig. 6
Main effects plot for S/N ratios
Fig. 6
Main effects plot for S/N ratios
Close modal
Table 9

Optimum level and parameters

DesignationParameterOptimal levelOptimal level (mm)Optimal designation
ATooth addendum33A3
BBacklash10.151–0.176B1
CLinear tip-relief profile modification2“0” reliefC2
DesignationParameterOptimal levelOptimal level (mm)Optimal designation
ATooth addendum33A3
BBacklash10.151–0.176B1
CLinear tip-relief profile modification2“0” reliefC2

In this research investigation, a simulation experiment was carried out as a confirmation experiment employing a combination of the previously assessed elements and levels. A discrete model's set of algebraic equations is created from a set of differential equations using the finite element method. After imposing boundary constraints, these algebraic equations solved for the solution values at the mesh points. The equation solver resolves the discrete equations related to the finite element mesh. For modal analysis, Block Lanczos solver and random vibration solver are used for analysis. The RMS acceleration (g) value is output from the random vibration solver. The test is conducted for analysis at constant torque T = 0 N-m and speed N = 900 rpm.

Figures 7(a) and 7(b) show the frequency plot attained by performing a simulation experiment. The RMS value of the simulation study is 0.070 g and 0.20 mm/s, which is approximately the lowest compared to the response values shown in Table 6. The results obtained using the simulation study are slightly higher because the actual working conditions are somewhat different from those of a simulation study. From this, the experimental results are very similar to those of a simulation study.

Fig. 7
(a) Gear pair simulation and (b) frequency plot of gear pair (A3B1C2)
Fig. 7
(a) Gear pair simulation and (b) frequency plot of gear pair (A3B1C2)
Close modal

This experiment shows that the selected factor for this experimental research plays a significant role in the dynamic response of the gear pair system. Furthermore, it is interesting to investigate the effect of operating conditions on its dynamic behavior. For that purpose, 20 Taguchi L9 orthogonal array experiments are conducted for different speed and loading conditions combinations. For analysis purposes, each experiment was conducted at constant operating conditions. The results obtained by performing 20 experiments are summarized and given in Table 10.

Table 10

Results obtained from 20 different combinations of speed and torque values

Taguchi L9 Expt. No.Operating condition (torque, N-m, speed, rpm)Rank of influencing factorsOptimum combination
1T = 0, N = 900Tooth addendum = 2A3B1C2
Backlash = 1
Profile modification = 3
2T = 0, N = 1200Tooth addendum = 2A2B1C3
Backlash = 1
Profile modification = 3
3T = 0, N = 1500Tooth addendum = 2A2B2C2
Backlash = 3
Profile modification = 1
4T = 0, N = 1800Tooth addendum = 1A2B1C3
Backlash = 2
Profile modification = 3
5T = 5.0, N = 900Tooth addendum = 1A2B3C1
Backlash = 3
Profile modification = 2
6T = 5.0, N = 1200Tooth addendum = 2A2B1C1
Backlash = 1
Profile modification = 3
7T = 5.0, N = 1500Tooth addendum = 2A2B2C1
Backlash = 1
Profile modification = 3
8T = 5.0, N = 1800Tooth addendum = 3A2B1C1
Backlash = 2
Profile modification = 1
9T = 5.5, N = 900Tooth addendum = 1A2B3C1
Backlash = 3
Profile modification = 2
10T = 5.5, N = 1200Tooth addendum = 2A2B3C1
Backlash = 3
Profile modification = 1
11T = 5.5, N = 1500Tooth addendum = 2A1B2C1
Backlash = 3
Profile modification = 1
12T = 5.5, N = 1800Tooth addendum = 3A2B1C1
Backlash = 2
Profile modification = 1
13T = 6.0, N = 900Tooth addendum = 1A2B3C1
Backlash = 3
Profile modification = 2
14T = 6.0, N = 1200Tooth addendum = 3A2B3C1
Backlash = 2
Profile modification = 1
15T = 6.0, N = 1500Tooth addendum = 3A2B3C1
Backlash = 2
Profile modification = 1
16T = 6.0, N = 1800Tooth addendum = 2A1B3C1
Backlash = 3
Profile modification = 1
17T = 7.0, N = 900Tooth addendum = 1A1B2C1
Backlash = 3
Profile modification = 2
18T = 7.0, N = 1200Tooth addendum = 1A2B3C1
Backlash = 3
Profile Modification = 2
19T = 7.0, N = 1500Tooth addendum = 3A2B2C1
Backlash = 2
Profile modification = 1
20T = 7.0, N = 1800Tooth addendum = 1A1B2C1
Backlash = 2
Profile modification = 3
Taguchi L9 Expt. No.Operating condition (torque, N-m, speed, rpm)Rank of influencing factorsOptimum combination
1T = 0, N = 900Tooth addendum = 2A3B1C2
Backlash = 1
Profile modification = 3
2T = 0, N = 1200Tooth addendum = 2A2B1C3
Backlash = 1
Profile modification = 3
3T = 0, N = 1500Tooth addendum = 2A2B2C2
Backlash = 3
Profile modification = 1
4T = 0, N = 1800Tooth addendum = 1A2B1C3
Backlash = 2
Profile modification = 3
5T = 5.0, N = 900Tooth addendum = 1A2B3C1
Backlash = 3
Profile modification = 2
6T = 5.0, N = 1200Tooth addendum = 2A2B1C1
Backlash = 1
Profile modification = 3
7T = 5.0, N = 1500Tooth addendum = 2A2B2C1
Backlash = 1
Profile modification = 3
8T = 5.0, N = 1800Tooth addendum = 3A2B1C1
Backlash = 2
Profile modification = 1
9T = 5.5, N = 900Tooth addendum = 1A2B3C1
Backlash = 3
Profile modification = 2
10T = 5.5, N = 1200Tooth addendum = 2A2B3C1
Backlash = 3
Profile modification = 1
11T = 5.5, N = 1500Tooth addendum = 2A1B2C1
Backlash = 3
Profile modification = 1
12T = 5.5, N = 1800Tooth addendum = 3A2B1C1
Backlash = 2
Profile modification = 1
13T = 6.0, N = 900Tooth addendum = 1A2B3C1
Backlash = 3
Profile modification = 2
14T = 6.0, N = 1200Tooth addendum = 3A2B3C1
Backlash = 2
Profile modification = 1
15T = 6.0, N = 1500Tooth addendum = 3A2B3C1
Backlash = 2
Profile modification = 1
16T = 6.0, N = 1800Tooth addendum = 2A1B3C1
Backlash = 3
Profile modification = 1
17T = 7.0, N = 900Tooth addendum = 1A1B2C1
Backlash = 3
Profile modification = 2
18T = 7.0, N = 1200Tooth addendum = 1A2B3C1
Backlash = 3
Profile Modification = 2
19T = 7.0, N = 1500Tooth addendum = 3A2B2C1
Backlash = 2
Profile modification = 1
20T = 7.0, N = 1800Tooth addendum = 1A1B2C1
Backlash = 2
Profile modification = 3

Table 11 shows that researchers developed a variety of models, methods, and factors considered to predict the dynamic response of gear pairs. Some of them used experimental methods to investigate geometrical parameters' effects on the gear pair system's dynamic behavior.

Table 11

Comparisons spur gear pair dynamic responses

Sr. No.ResearchersModels proposedEffects considerInfluence factorsSolution techniquesConclusions
1Lin and Parker [40]Two- and Multi-mesh gear system modelsParametric instability.Mesh stiffness, Contact ratio, Mesh phasingPerturbation method and numerical integration techniquesInstability regions reduced by selecting contact ratio
2Li [41]FEMMachining errors (ME), assembly errors (AE), and tooth profile modification (TPM) on loading capacity, load-sharing ratio, and transmission errorME, AE, and TPMFinite element analysisME changes waveform and amplitude of TE; however, AE and TPM change only amplitude of TE
3Maliha et. al [42]Nonlinear dynamic modelDynamic model and combined influences of several individual effects.Shaft and bearing external static torque, mesh stiffness variation, gear backlash, and TPMMulti-harmonic balance methodShaft and bearing flexibility should be considered in gear dynamic study for accurate prediction of dynamic loads
4Bruyere and Velex [43]3D Gear modelOutlined optimal transformations in light of transmission error oscillationsTransmission error, center distance variation, linear tip reliefPerturbation methodProfile modification is more suitable for precise ground gears. Effect of center distance significantly reduces the performance of tip relief
5Wang et. al [44]FEMInvestigated effect of backlash caused by gear eccentricity and variation in load on dynamic transmission error (DTE)Load, speed, gear backlashLS-DYNA 3D softwareChange in direction of load causes jumps in DTE curves
6Li et. al [45]FEMDynamic model built using gear compatibility conditionsLoad, speed, lead crown modificationFEMThe suggested model more accurately predicts the gear system characteristics
7Proposed studyExperimental methodStudied the effect of operating conditions for optimization of geometric parameters of spur gear pairAddendum, backlash, tip relief PM, speed, torqueTaguchi methodOptimum levels of geometric parameters of spur gear pair changes with the change in operating conditions
Sr. No.ResearchersModels proposedEffects considerInfluence factorsSolution techniquesConclusions
1Lin and Parker [40]Two- and Multi-mesh gear system modelsParametric instability.Mesh stiffness, Contact ratio, Mesh phasingPerturbation method and numerical integration techniquesInstability regions reduced by selecting contact ratio
2Li [41]FEMMachining errors (ME), assembly errors (AE), and tooth profile modification (TPM) on loading capacity, load-sharing ratio, and transmission errorME, AE, and TPMFinite element analysisME changes waveform and amplitude of TE; however, AE and TPM change only amplitude of TE
3Maliha et. al [42]Nonlinear dynamic modelDynamic model and combined influences of several individual effects.Shaft and bearing external static torque, mesh stiffness variation, gear backlash, and TPMMulti-harmonic balance methodShaft and bearing flexibility should be considered in gear dynamic study for accurate prediction of dynamic loads
4Bruyere and Velex [43]3D Gear modelOutlined optimal transformations in light of transmission error oscillationsTransmission error, center distance variation, linear tip reliefPerturbation methodProfile modification is more suitable for precise ground gears. Effect of center distance significantly reduces the performance of tip relief
5Wang et. al [44]FEMInvestigated effect of backlash caused by gear eccentricity and variation in load on dynamic transmission error (DTE)Load, speed, gear backlashLS-DYNA 3D softwareChange in direction of load causes jumps in DTE curves
6Li et. al [45]FEMDynamic model built using gear compatibility conditionsLoad, speed, lead crown modificationFEMThe suggested model more accurately predicts the gear system characteristics
7Proposed studyExperimental methodStudied the effect of operating conditions for optimization of geometric parameters of spur gear pairAddendum, backlash, tip relief PM, speed, torqueTaguchi methodOptimum levels of geometric parameters of spur gear pair changes with the change in operating conditions

5 Results and Discussion

Using the Taguchi approach, the mutual impact of these geometric parameters on dynamic responsiveness is experimentally investigated. The dynamic response of a spur gear pair is measured using the Taguchi method using means and S/N ratio to account for different combinations of geometrical parameters. A higher S/N ratio implies that the spur gear pair has less dynamic responsiveness. The optimal level is determined by analyzing the means and the S/N ratio. The means and S/N ratio response tables are applied to rate the influencing factors. The response tables for means and S/N ratio and main effects plot for means and S/N ratio are shown in Tables 7 and 8 and Figs. 5 and 6, respectively, for experiment No. 1. The results obtained from such 20 experiments are shown in Table 10.

From the response Tables 7 and 8 for means and S/N ratio, it can be seen that the rank of influencing parameters for experiment No. 1 is as follows: addendum has second rank, backlash first rank, and profile modification has third rank. Also, from the main effects plot for means and S/N ratio (Figs. 5 and 6), it can be seen that the level of vibration response is lower at level 3 of the first-factor addendum, in the case of second-factor backlash level 1 is minimum, and for third-factor profile modification, level 2 is lowest. Hence, an optimal sequence of factors and their levels is A3B1C2. The results obtained from means and S/N ratio analysis provide an optimal combination of spur gear parameters for minimum vibration response in terms of the optimal sequence of factors and their levels as A3B1C2. The optimal sequence of factors and their levels are as the higher value of tooth addendum 1.1 times module = 3.3 mm, the lowest value of backlash 0.050–0.075 mm, and tooth with short profile correction, resulting in the finest sequence to get the minimum RMS (g) level for selected operating condition 900 rpm and zero loads.

Unfortunately, this optimum sequence of factors and levels did not coincide with trials 1–9 in OA in Table 6. Manufacturing spur gear pairs with such a combination of factors and levels is impossible. Therefore, a confirmation test was conducted by using a simulation study for the same operating condition speed of 900 rpm and zero load. The vibration response obtained from this simulation study is 0.07 g, with the lowest among the nine sets tested. Therefore, experimental results are promising in agreement with the simulation study. Response Tables 7 and 8 show that gear backlash is the first dominant factor; this may be due to the presence of backlash causing tooth separation and impacts in the case of unloaded or lightly loaded gear systems. Such impacts result in higher vibration levels and large dynamic loads [24,28,32]. Tooth addendum is the second dominant factor, as it is directly related to the contact ratio and mesh stiffness variation during the mesh cycle. Profile modification has less effect as compared with backlash and tooth addendum.

The rank of influencing parameters and the optimal sequence of factors and their levels change when the nine gear sets are operated with various speed and loading conditions combinations. The results obtained from 20 Taguchi experiments are shown in Table 10. These results show that operating condition plays a significant role in gear vibration response. These 20 trials reveal that tooth profile modification, as opposed to tooth addendum and backlash, is the most influential parameter. An optimum combination of factors and their levels is A2B3C1.

6 Conclusion

In this study, the impact of operating conditions and several factors – including backlash, profile modification and addendum – on the geometrical profile of the gear pair's vibration response is investigated. In this regard, 20 Taguchi L9 experiments are conducted at various combinations of operating speed and loading conditions.

The first experiment was conducted at a speed of 900 rpm and zero load conditions. From Taguchi's analysis, it is seen that the rank of influencing parameters is as follows: backlash first rank, addendum second rank, and profile modification third rank. Moreover, the optimum combination of factors and their levels is A3B1C2. From this experiment, it is observed that geometrical parameters significantly influence the gear pair's vibration response. Similarly, 20 Taguchi L9 experiments were conducted at various operating conditions, and the results are compiled in Table 10. From these results, it is observed that the rank of influencing parameters and optimum levels of parameters for lower vibration response are varied to operating conditions. This is because operating parameters may affect gear tooth deflection and deformation, resulting in variations of contact points from the theoretical line of action. Furthermore, this increases transmission error and affects the vibratory response of the gear system. From these 20 experiments, the rank of influencing parameters are as follows: profile modification first rank, addendum second rank, and backlash third rank. Also, the optimum combination of factors and their level is A2B3C1. From this study, operating speed and loading conditions play significant roles in the design of quiet gear systems. It is only possible to compare gear designs if the operating conditions are varied over their realistic operating ranges.

The future scope is revealed by developing a nonlinear dynamic model that considers time-varying features of a gear pair. The gear diagnosis process can be used to determine the prognosis and anticipated remaining life of the gear. Furthermore, this approach can be developed to incorporate new soft computing methods for gear fault detection and experimental investigation in rotating systems.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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