Journal of Mechanical Science and Technology 25 (7) (2011) 1781~1789

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0416-x

Effects of flank deviation on load distributions for helical gear†

Jing Wei1,2,*, Wei Sun1 and Licun Wang3 1School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, China

2The State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China 3School of Mechanical Engineering, Chongqing Technology and Business University, Chongqing 400067, China

(Manuscript Received October 29, 2010; Revised March 10, 2011; Accepted March 22, 2011)

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Abstract Flank deviation is one of the important factors that greatly affect helical gear strength. A qualitative analysis is performed using finite

element method (FEM) for the effects of flank deviation on load distributions for helical gear based on ANSI/AGMA ISO 1328-1 (stan-dard for cylindrical gears - ISO System of Accuracy - Part 1: Definitions and Allowable Values of Deviations Relevant to Corresponding Flanks of Gear Teeth). To analyze the effects of flank deviation, tooth contact analysis (TCA) is developed and load distributions of heli-cal gears with flank deviation are presented. Load distributions in contact lines are derived and compared to each other under grade 5 and 7 after taking five types of flank deviation, including single pitch deviation, profile form deviation, profile slope deviation and helix form deviation as well as helix slope deviation, into account. It is found that the effects of individual flank deviation on load distribution have the superposition property. Flank crowning and tip relief corrections must be carefully regarded in the design process because of the ef-fects of flank deviation on load distributions.

Keywords: Flank deviation; Load distributions; Helical gear; Tooth contact analysis (TCA); Finite element method (FEM) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Flank deviation is one of the factors that greatly affect gear strength. After many studies, Semba Seiso [1] thought that the gear strength calculation equation in every country is still not precise enough; the main reason for this is that the effect of flank deviation is not fully considered. He thought the stan-dard of gear accuracy should be calculated based on the ef-fects of various errors on noise and strength instead of the size of various errors processed in various ways. Therefore, some scholars [2] have put forward the concept of functional equi-valence among the same gear accuracy. The study of gear errors is usually performed through experiments. For example, Velex and Ajmi [3] studied the relationship of the carrying capacity of gears between different errors and transmission ratios, and they pointed out that it is difficult to avoid the ap-pearance of contact stress concentration because of manufac-turing error, installation error, elastic deformation, etc. How-ever, it can be improved greatly through running-in operation. In 2005, Jui-Tang Tseng [4, 5] studied the actual gear form produced by cutting of different hobs and the effect of differ-ent assembly conditions on gear mesh status. There are also

many studies [6] conducted indirectly, for example, by analyz-ing the dynamic incentive generated by error to get the dy-namic load, whereby the strength, noise, vibration, transmis-sion errors, etc., have been studied.

Except for random error, to reduce the partial load, transmis-sion error, dynamic load, noise, etc., as well as to improve the transmission performance of gears, modification techniques to change the shape of the gear flank are also widely used in gear design. Many scholars [7, 8] studied gear modification in order to analyze the effect of different modification methods on gear strength using finite element method. As the effects induced by the errors and modifications on gear are interaction, in order to obtain the best modification effect, both must be considered. In 2001, Litvin et al. [9-11], used gear hobbling when they built gear model, and considered the error and modification parame-ters caused by axial and cross movement. Sun et al. [12] identi-fied principles with modification parameters of manufacturing error gear modification for spur gears. In 2006, Mao [13] built a finite element model for spur gears with axis deviation and studied its effect on the fatigue properties by optimizing the modification parameters to improve fatigue performance. Shut-ing Li [14, 15] has used the finite element and experimental methods to analyze the effect of machining error, assembly error, and modification factors on load distribution, contact strength, bending strength and transmission error of spur gear. It was found that the calculated results agreed well with the

† This paper was recommended for publication in revised form by Associate EditorJeong Sam Han

*Corresponding author. Tel.: +86 411 8470 7435, Fax.: +86 411 8470 8414 E-mail address: weijing@dlut.edu.cn

© KSME & Springer 2011

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measured ones and machining errors, assembly errors and tooth modifications greatly affect the surface contact stress and root bending stress of the gears. Through experiments, the load distribution and bending stress of a helical gear with axis devia-tion and gear modification were investigated by Hotait [16]. The results showed that the gear modification and axis devia-tion are interrelated, and the gear modification can avoid the partial load behavior caused by axis deviation. Most research on gear transmissions have focused on dynamic analysis, in-cluding reducing transmission error, dynamic load and noise and other aspects. However, spur gear is still the main study object. In this study, Tooth contact analysis (TCA) is devel-oped and load distributions are derived. The effects of different deviations on contact stress and load distribution of helical gear are studied and several effects of load distribution by gear flank deviation are deduced.

2. Brief description of gear flank deviation

Several types of individual flank deviations are provided in standard ANSI/AGMA ISO 1328-1. They are single pitch deviation, profile form deviation, profile slope deviation, helix form deviation and helix slope deviation. Other types of inte-grated flank deviations can be combined by these individual types. The gear accuracy used in industrial production is main-ly from grade 5 to 9. It is difficult to study the deviations be-cause of their complex micro-geometry. Therefore, previous studies [17] are mainly laboratory-based. The effect of devia-tions is greatly influenced by processing and laboratory stud-ies cannot accurately obtain the required gear flank deviation. However, they provide an effective way to study flank devia-tions and more accurate flank deviations can be found by nu-merical method. There are some references [18, 19] that have recently introduced related research in this area, but the devia-tion types concerned are not comprehensive.

The two most common types of flank deviations, helix slope and form deviations, are shown in Fig. 1. In Fig. 1(a), the effective length LAE of the profiles is from the gear tip to the top meshing point E of the matching gear. The measured length LAF is defined as the effective length extending to the direction of tooth root about 1mm. The assessment length Lα is from point E to the extended point A1, i. e., the 92% of the length LAE and the tooth profile is divided by the middle MM1. The profile slope deviation fHα is defined as the distance from the two end points in assessment length to the two intersected theory profiles M1N1. The profile form deviation ffα is defined as the distance between the two lines that contains the two actual profiles CC and C1C1.

The helix deviation contains helix form deviation ffβ and he-lix slope deviation fHβ. The assessment scope Lβ on helix de-viation is shown in Fig. 1(b), and b is gear width. Similar to form deviation, the tooth profile is divided by the middle line MM1. The helix form deviation ffβ is defined as the distance from two end points in assessment length to the two inter-sected theoretical spiral M1N1. The helix slope deviation fHβ is defined as the distance between the two lines that contains the

two actual spirals CC and C1C1. The parameters of wheel and pinion used here are listed in

Table 1. Other parameters of wheel and pinion are as follows: Tooth numbers of pinion Z1 and wheel Z2 are 27 and 82, the rotation speed of pinion and wheel are 1731.111 rpm and 570 rpm, respectively. The material of the pinion and wheel in this study is 17CrNiMo6. The material properties of 17CrNiMo6 are listed in Table 2. Table 3 is the flank deviations of the

Table 1. Parameters of the gear and pinion.

Power P (kW) 1200

Normal module mn (mm) 5

Normal pressure angle αn (°) 20

Helix angle β (°) 10

Tooth width B (mm) 100

Center distance A (mm) 280 Table 2. Material properties of 17CrNiMo6.

Young’s modulus (GPa) 206

Poisson’s ratio 0.3

Yield Strength (MPa) 785

UTS (MPa) 930 Table 3. Flank deviations under grades 5 and 7.

Flank deviations Grade 5 Pinion/Wheel

Grade 7 Pinion/Wheel

single pitch deviation ±fpt (um) 7/8 14/16

profile form deviation ffα (um) 8/9 16/18

profile slope deviation ±fHα (um) 6.5/7.5 13/15

helix form deviation ffβ (um) 8.5/9 17/18

helix slope deviation fHβ (um) 8.5/9 17/18

(a) Form deviation

(b) Helix deviation

Fig. 1. Form and helix deviations.

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wheel and pinion under grade 5 and 7 based on ANSI/AGMA ISO 1328-1.

3. Effect of helix slope and form deviations on contact

stress

3.1 Contact stresses caused by flank deviations

Usually, it is difficult to test three dimensional contact stress because of the complex surface of gear tooth surface and time variant characteristics of the contact stress. Therefore, in order to verify the theoretical analysis, a qualitative analysis can be performed using finite element method (FEM) for the effects of flank deviation on load distributions for helical gear. The contact lines in the tooth surface according to the parameters in Table 1 are shown in Fig. 2(a).

Three teeth are engaged at this meshing state, in which con-tact line 1 in tooth pair 1 and contact line 2 in tooth pair 2 pass through the whole tooth surface, while contact line 3 in tooth pair 3 degenerates to a point. Therefore, contact line 3 in tooth pair 3 should be ignored under this analysis when simulating the effects of flank deviation on load distributions. During analysis, node forces and circumferential constraint are ap-plied to the nodes on the circumferential surface of the pinion, while the nodes on the circumferential surface of the wheel are defined as fixed constraint. The boundary conditions of TCA are shown in Fig. 2(b).

As shown in Fig. 3, the contact stress along the contact lines of the two teeth pairs is relatively smooth in most places when considering flank deviations. However, there are significant changes at the start point or end point. The average contact

stress on tooth pair 1 is about 978 MPa and the average con-tact stress on tooth pair 2 is about 1005 MPa. The maximum contact stresses of two teeth pairs were located at the top posi-tion and at the tooth root area, respectively. The maximum contact stress at the contact line is about 2-3 times the average stress in other areas of the tooth surface. This is because the top of the gear experiences a shear stress with the tooth sur-face and stress concentration is relatively serious.

Because the tooth contact stress at the contact line is larger than that in other areas of the gear surface at any meshing state, the stress caused by different flank deviations is given at the contact line in this study.

3.2 Single pitch deviation fpt

The single pitch deviation is defined as the algebraic differ-ence of actual pitch and theoretical pitch on the circle that is close to the central of gear height at the end plane. In this study, the single pitch deviation is obtained by offsetting an equal distance to the theoretical profile flank, and it does not involve other types of individual deviations because of the equal offset.

Among all combinations of deviations, the single pitch de-viation may not be considered when it is both “+” or both “-” in the two gears because of the equal theoretical profiles. Here, in Case A and Case B, the most unfavorable situations to the load distribution are studied, as shown in Table 4. The two cases with single pitch deviation are shown in Fig. 4. The wheel is at the upper position and the pinion is below, the

(a) Contact lines in tooth surface

(b) Boundary conditions of TCA

Fig. 2. Contact lines in tooth surface and boundary conditions.

(a) Tooth pair 1

(b) Tooth pair 2 Fig. 3. Contact stress distributions.

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solid line denotes the theoretical profiles and the dashed line denotes the actual profiles.

Load distributions on the contact line by axial direction are shown in Figs. 5 and 6. Fig. 5 shows the load distributions when input power P = 1200 KW and each curve is almost parallel to each other. It shows that the effect of single pitch deviation on the load distribution is not in the axial direction, but mainly in a different tooth. The load on tooth pair 1 in Case A and tooth pair 2 in Case B increases, and the load in-creasing amplitude under grade 7 is twice that under grade 5. It corresponds to the single pitch deviation of grades 5 and 7. Fig. 6 shows the load distributions when input power P = 420 KW. There is only a small contact area in tooth pair 2 under Case A grade 7, while there is almost no contact for tooth pair 1 in Case B grade 7. It leads to the single gear meshing state because the pitch deviation and the load distribution become very bad. It shows that the effect of single pitch deviation on

load distribution is greater for a smaller load. 3.3 Helix form deviation ffβ and profile form deviation ffα

For profile slope deviation, profile form deviation, helix form deviation and helix slope deviation, the range of the pro-file which is used to evaluate these deviations is 92% of the actual length. The study here mainly considers load distribu-tion affected by the changes of the gear tip. When calculating the load distributions with helix form deviation and profile form deviation, it can be assumed that the effects of the two teeth pairs on load distribution is non-interference. Therefore,

Table 4. Cases with single pitch deviation.

Pinion Wheel Case A － ＋

Case B ＋ －

(a) Case A (b) Case B Fig. 4. Cases with single pitch deviation.

(a) Tooth pair 1

(b) Tooth pair 2

Fig. 5. Load distributions of cases with single pitch deviation P=1200 KW.

(a) Tooth pair 1

(b) Tooth pair 2

Fig. 6. Load distributions of cases with single pitch deviation P=420KW.

(a) Tooth pair 1

(b) Tooth pair 2

Fig. 7. Load distributions of cases with helix form deviation.

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there is only need to study one tooth pair with flank deviations, while the other teeth pairs can use the theoretical involute profiles.

3.4 Profile slope deviation fHα

Profile slope deviation always appears with single pitch de-viation. In order to study profile slope deviation independently, it should be kept at the same pitch circle when building mod-els with profile slope deviation. The combinations of single tooth pair profile slope deviation are shown in Table 5.

The load distribution curves of tooth pair 2 in Cases A and B and tooth pair 1 in Cases C and D are basically parallel. Because there are no deviations on these tooth pairs, the load distributions between the two teeth pairs experiences rela-tively small changes, though these curves does not coincide. The load distribution curves of tooth pair 1 in Cases A and B and tooth pair 2 in Cases C and D have an obvious intersec-tion point close to the pitch circle caused by the same pitch circle when building model. The load distributions with pro-file slope deviation are shown in Figs. 10 and 11.

Figs. 10 and 11 show that in these four Cases (Cases A, B, C and D) with profile slope deviation, the maximum load on tooth pair 2 is significantly higher than that on tooth pair 1. Therefore, the worst load situation should be on tooth pair 2 when these two tooth pairs both have deviations because the load distribution of tooth pair 2 in Cases A and D have a lar-

ger load. When building the worst load distribution model, Case E, the profile slope deviation on tooth pair 1 should be the same as that in Case A, and the profile slope deviation on tooth pair 2 should be same as that in Case D (shown in Table 5).

Fig. 12 shows the comparison of the load distributions in Case E, Case A and Case D. From Fig. 12, the load case of the tooth pair in Case E is the worst condition. Meanwhile, com-

Table 5. Cases with profile slope deviation.

Tooth pair 1 Tooth pair 2

Pinion Wheel Pinion Wheel Case A ＋ － 0 0

Case B － ＋ 0 0

Case C 0 0 ＋ －

Case D 0 0 － ＋

Case E ＋ － － ＋

(a) Tooth pair 1

(b) Tooth pair 2

Fig. 8. Load distributions of cases with profile form deviation.

(a) Case A (b) Case B

(c) Case C (d) Case D

(e) Case E

Fig. 9. Cases with profile slope deviation.

(a) Tooth pair 1

(b) Tooth pair 2

Fig. 10. Load distributions of cases A and B with profile slope devia-tion.

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pared to the incremental load without deviation, the incre-mental load in Case E is approximately equal to the accumula-tion of that in Cases A and D. Thus, the load distribution curves of Case E grade 7 is almost the overlapping of Cases A and D.

3.5 Helix slope deviation fHβ

Similar to profile slope deviation, in order to avoid the in-

fluence of single pitch deviation when analyzing the helix slope deviation, the offset of the axial direction in the middle section of the gear is defined as zero, then an offset of equal distance but in the opposite direction from the middle section to the two cross-sections is performed.

Taking the four Cases with helix slope deviation in Table 6 into account, the load distribution curves are shown in Figs. 13 and 14. These curves are very similar to the curves of profile slope deviation, and the maximum load appears on tooth pair 2 in Case D. However, the amplitude is greater than that of the latter. It shows that the effect of helix slope deviation on the load distribution is larger than that affected by the profile slope deviation under the same accuracy grade. These load distribution curves intersect at the same point located at the middle section in the axial direction.

Cases A and B are studied in Ref. [18] and their load distri-butions are shown in Fig. 15. There are some differences with the results calculated in this paper. It is mainly due to use of a different helix slope deviation in Ref. [18]. The helix slope deviation in one section is zero while in the other it is maxi-mum, which is equivalent to the load distributions affected by the overlapping of the helix slope deviation and single pitch deviation. As the single pitch deviation can lead to some

(a) Tooth pair 1

(b) Tooth pair 2

Fig. 11. Load distributions of cases C and D with profile slope devia-tion.

(a) Tooth pair 1

(b) Tooth pair 2

Fig. 12. Load distributions of case E with profile slope deviation.

Table 6. Cases with helix slope deviation.

Tooth pair 1 Tooth pair 2

Pinion Wheel Pinion Wheel Case A ＋ － － ＋

Case B － ＋ ＋ －

Case C － ＋ － ＋

Case D ＋ － ＋ －

(a) Tooth pair 1

(b) Tooth pair 2

Fig. 13. Load distributions of cases A and B with helix slope deviation.

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changes of load distribution between two teeth pairs, the de-rived curve can be regarded as the overlapping of load distri-butions of the helix slope deviation and single pitch deviation.

4. Profile relief considering effects of flank deviations

The parabolic relief method and the long relief method [22] are used in this paper to calculate the position the start and end

points of engagement according to the parameters of gears. The distributions of contact stress on tooth pairs 1 and 2 are

shown in Fig. 16 after profile relief when considering the ef-fects of flank deviations. Fig. 16 shows that stress concentra-tions have been greatly reduced, while there is still contact stress in addendum. This is caused by the tooth pairs becom-ing thinner, which causes decreased stiffness and increased deformation after relief. However, it does not form a signifi-cant stress concentration and it can reduce the scratches at the top edge and the impact of the engagement and disengage-ment of gear due to the small contact pressure in addendum.

5. Conclusions

The above studies show that flank deviations have great ef-fects on helical gear load distribution, and the following con-clusions can be drawn:

(1) The load increasing amplitude under grade 7 is twice that under grade 5 when considering the effects of flank devia-tions. A lower load will induce a larger effect by flank devia-tions though there is only a small deformation in contact areas.

(2) The effect of single pitch deviation on the load distribu-tion is not in the axial direction, but mainly in a different tooth. There is little effect of load distribution affected by profile form deviation and helix form deviation while it is affected by profile slope deviation and helix slope deviation.

(3) For equal loads, the effect of single pitch deviation on load distribution along the direction of tooth width is overall larger or smaller, while there is a larger effect by the helix deviations. Form deviation can lead to great changes of load distribution along the direction of tooth width, while it is less affected by helix deviations.

(4) According to the comparative analysis of profile slope deviation and helix slope deviation, it is inferred that the ef-

(a) Tooth pair 1

(b) Tooth pair 2

Fig. 14. Load distributions of cases C and D with helix slope deviation.

(a) Tooth pair 1

(b) Tooth pair 2

Fig. 15. Load distributions of cases with helix slope deviation in Ref. [18].

(a) Tooth pair 1

(b) Tooth pair 2 Fig. 16. Contact pressure distributions after profile relief.

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fects of individual deviation on load distribution have the su-perposition property, but whether it is suitable for other flank deviations needs further study.

Acknowledgments

This work is supported by the opening foundation of the State Key Laboratory of Mechanical Transmission, Chongqing University, China (No.: SKLMT-KFKT-200906). The authors would also like to acknowledge grant No.: 20100471443 from China Postdoctoral Science Foundation. The authors would like to take this opportunity to express their sincere apprecia-tion.

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Jing Wei, born in June 1978, received his doctor's degree in engineering from the State Key Laboratory of Mechanical Transmission, Chongqing University, China, in 2008. Dr. Wei is currently a lec-ture in School of Mechanical and Engineer-ing, Dalian University of Technology, China. His research areas of interest include

structural design and optimization, mechanical transmissions, es-pecially in gear transmission.

Wei Sun, born in 1967, is currently a pro-fessor and a Ph.D candidate supervisor in School of Mechanical and Engineering, Dalian University of Technology, China. His main research domains include design and optimization of complex mechanical equipment, mechanical transmission and structure CAD/CAE.

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Licun Wang, born in February 1978, Ph.D., Associate Professor, the director of institute of scroll compressor, Chongqing Technology and Business University, the Assistant Director of Department of Sci-ence and Research. Undertake or partici-pate in national and provincial level project more than 10 items.