A calculation method of thermal deformation for doublehelical gearCheng Wang*
School of Mechanical Engineering, University of Jinan, Jinan 250022, PR China
* e-mail: m
Received: 19 October 2018 / Accepted: 23 January 2019
Abstract. Thermal deformation caused by gear transmission is an important factor causing gear impact,vibration and partial load. Gear modification can effectively improve the effects caused by thermal deformation.The calculation of thermal deformation is the first problem to be solved before gear modification. This papertakes double helical gear as the research object and a calculation method of thermal deformation is proposed.Firstly, temperature of instantaneous meshing points on the tooth surface are measured and these discretetemperature values are fitted by the linear interpolation. Calculation formula of temperature distribution alonggear radial direction is introduced. Combining both, tooth surface temperature field is obtained. Secondly,equation of tooth surface for double helical gear before and after thermal deformation is derived according to thetooth surface temperature field. Finally, an example is given. Compared with the given modification of thermaldeformation, the calculated thermal deformation is almost equal to theoretical value. On this basis, the thermaldeformation of double helical gear considering the installation error and machining error is calculated, whichprovides a theoretical basis for thermal deformation modification of double helical gear.
Keywords: Thermal deformation / double helical gear / temperature field / modification / error
1 Introduction
Thermal deformation caused by gear transmission is animportant factor causing gear impact, vibration and partialload. Gear modification is an effective way to improve itstransmission performance [1–4]. The calculation of thermaldeformation provides a basis for gear modification.
When calculating thermal deformation of gear, thetemperature field should be determined first. It consists oftwo parts: the gear body temperature and the instanta-neous surface temperature (it is also known as flashtemperature). The instantaneous surface temperaturedepends on the body temperature. The traditionalcalculation of gear body temperature has the experienceformula of ISO flash temperature criterion and integraltemperature criterion, but the calculation results cannotdetermine the temperature distribution of gear teeth, andcannot meet the requirements of deformation calculation.Bobach [5] analyzed the regularities of distribution oftemperature field inside the gear teeth. Qiu Liangheng [6]used finite element method to calculate and analyze thetemperature and thermal deformation of gear. Thecalculated results can be used as the basis for analyzingand calculating the lubricating oil film thickness and gear
tooth profile modification. Gong Xiansheng [7] applied thetheories of Hertz contact, tribology, gear engagement andheat transfer, established the finite element analysis modelof the temperature field of gear body, and obtained thesteady-state temperature field of the planetary gear tooth.Patir [8] and Wang [9] used finite element method toestimate the temperature field of spur gear tooth, andmadetheoretical analysis and numerical calculation on dynamicload and oil film thickness. Li [10] proposed the concept ofnon-involute characteristics.When the temperature of gearis changed, the theoretical tooth profile and the practicaltooth profile are not superposition. On this basis,Wang [11]proposed a method for calculating the thermal deformationof helical gear.
Present experimental measurement technology can getthe meshing point temperature value of gear. Reference [1]gives a simplified formula for temperature distributionalong gear radial direction, which can be applied inengineering. Based on this, the specific process of the paperis shown in Figure 1.
2 Determination of tooth surfacetemperature field
Figure 2 showed the schematic diagram of determiningtooth surface temperature distribution. The temperature
Fig. 1. Flow diagram for calculation of thermal deformation ofdouble helical gear.
Fig. 3. Coordinate of standard tooth profile.
Fig. 2. Schematic diagram of determining tooth surfacetemperature distribution.
2 C. Wang: Mechanics & Industry 20, 612 (2019)
values of instantaneous meshing point 1, 2, …, n are firstlymeasured and fitted. Together with the given temperatureformula along gear radial direction, temperature field isdetermined.
2.1 Measurement and fitting of temperature ofinstantaneous meshing points on the tooth surface
Using the infrared technology, the miniature thermocoupleautomatic recorder and the simulated heat source infraredtechnology to measure the temperature of instantaneousmeshing points on the tooth surface. The temperaturecurve in the direction of the meshing line along the toothwidth is obtained by the linear interpolation.
2.2 Temperature distribution along gear radialdirection
Reference [1] gives a simplified formula for temperaturedistribution along gear radial direction, which can beapplied in engineering. The formula can be expressed as
tk ¼ tc þ ðta � tcÞr2k=r2a; ð1Þwhere rk is the radius of gear arbitrary circle k, itstemperature is tk; tc is the temperature of gear center; ra isthe radius of addendum circle, its temperature is ta; Theunit of temperature in the formula is °C.
2.3 Determination of tooth surface temperature field
According to the fitting curve along meshing points andformula (1), the temperature ta of addendum circle alongtooth surface can be obtained. And then through formula(1), the temperature of arbitrary point can be calculated,i.e., the tooth surface temperature field is determined.
3 Determination of thermal deformationof double helical gears tooth surface
According to the structural characteristics of double helicalgear, the standard equation for tooth surface is derived. Onthe basis, the thermal deformation equation of toothsurface is derived according to the tooth surface tempera-ture field.
3.1 Standard tooth surface equation of doublehelical gear3.1.1 Standard tooth profile equation of double helical gear
Take one end tooth surface as an example, the coordinate ofstandard tooth profile is shown in Figure 3. Two toothprofile equation are expressed as
xk1 ¼ rb1 sinmk1 � rb1mk1 cosmk1
yk1 ¼ rb1 cosmk1 þ rb1mk1 sinmk1
�ð2Þ
Fig. 5. Helix diagram of right tooth surface.
Fig. 4. Helix diagram of left tooth surface.
Fig. 6. Left tooth profile after thermal deformation.
C. Wang: Mechanics & Industry 20, 612 (2019) 3
xk2 ¼ rb2 sinmk2 � rb2mk2 cosmk2
yk2 ¼ rb2 cosmk2 þ rb2mk2 sinmk2
�ð3Þ
Formulas (2) and (3) are the profile equation of lefttooth and right tooth, respectively.Where rbi(i=1, 2) is theradius of base circle, mki (i=1, 2) is the roll angle of point kin the involute.
3.1.2 Standard tooth surface equation
The tooth surfaces of double helical gear is involutehelicoid. Figures 4 and 5 are its left and right tooth surfacehelix diagram, respectively.
The left tooth surface equation is expressed as
xk1 ¼ rb1 sinmk1 � rb1mk1 cosmk1 þ rk1 sin uk1yk1 ¼ rb1 cosmk1 þ rb1mk1 sinmk1 þ rk1 cos uk1 � rb1zk1 ¼ rk1uk1 tan gk1
8<: ð4Þ
The right tooth surface equation is expressed as
xk2 ¼ rb2 sinmk2 � rb2mk2 cosmk2 þ rk2 sin uk2yk2 ¼ rb2 cosmk2 þ rb2mk2 sinmk2 þ rk2 cos uk2 � rb2zk2 ¼ rk2uk2 tan gk2
:
8<: ð5Þ
Equations (4) and (5) are all the equation of involutehelicoid. Where, rbi (i=1, 2) is the radius of base circle,mki (i=1, 2) is the roll angle of point k in the involute,rki (i=1,2) is the radiusofpoint k in the involute, uki (i=1,2)is theexpansionangleofpoint k in the involute,gki (i=1,2) isthe helix angle of ascent of point k in the involute,bki (i=1, 2) is the helix angle of point k in the involute.
3.2 Tooth surface equation of double helical gearsafter thermal deformation3.2.1 Variation of tooth profile equation caused by thermaldeformation
Still take the left tooth surface as an example, the left toothprofile after thermal deformation is shown in Figure 6. Thetooth profile equation after thermal deformation can beexpressed as
x0k ¼ r
0b sinm
0k � r
0bm
0k cosm
0k
y0k ¼ r
0b cosm
0k þ r
0bm
0k sinm
0k
(ð6Þ
where r0b ¼ rbð1þ DtlÞ, m0
k ¼ tana0k, a
0k ¼ arcos
r0b
r0k
� �, r
0b is
the radius of base circle after the thermal distortion, m0k is
the roll angle of point k in the involute after the thermaldistortion.
3.2.2 The tooth surface equation after thermal deformation
The helix diagram of left tooth surface and right surfaceafter thermal deformation is shown in Figures 7 and 8,respectively.
The left tooth surface equation after thermal deforma-tion is expressed as
x0k1 ¼ r
0b1 sinm
0k1 � r
0b1m
0k1 cosm
0k1 þ r
0k1 sin u
0k1
y0k1 ¼ r
0b1 cosm
0k1 þ r
0b1m
0k1 sinm
0k1 þ r
0k1cos u
0k1 � r
0b1
z0k1 ¼ r
0k1u
0k1tan g
0k1
:
8><>: ð7Þ
Fig. 7. Helix diagram of left tooth surface after deformation.
Fig. 8. Helix diagram of left tooth surface after deformation.
Table 1. Parameters of double helical gear.
z 65mn 3an 20°b 12°Face width 130 � 2 mmHelical direction Left rightLinear velocity of meshing 130m/s
Fig. 9. Temperature of instantaneous meshing points.
4 C. Wang: Mechanics & Industry 20, 612 (2019)
The right tooth surface equation after thermaldeformation is expressed as
x0k2 ¼ r
0b2 sinm
0k2 � r
0b2m
0k2 cosm
0k2 þ r
0k2 sin u
0k2
y0k2 ¼ r
0b2 cosm
0k2 þ r
0b2m
0k2 sinm
0k2 þ r
0k2 cos u
0k2 � r
0b2
z0k2 ¼ r
0k2u
0k2 tan g
0k2
8><>: ð8Þ
where u0ki ¼ b
0ki cosb
0ki sinb
0ki
r0ki
, b0ki ¼ ar tan
r0ki
ritanbi
� �,
g0ki ¼ p
2 � b0ki, r
0bi (i=1, 2) is the radius of base circle after the
thermal distortion, m0ki (i=1, 2) is the roll angle of point k in
the involute after the thermal distortion, r0ki (i=1, 2) is the
radius of point k in the involute after the thermal distortion,u0ki (i=1, 2) is the expansion angle of point k in the involuteafter the thermal distortion, g
0ki (i=1, 2) is the helix angle of
ascent of point k in the involute after the thermal distortion,bki
’(i=1,2) is thehelixangleofpointk in the involuteafter thethermal distortion, b
0ki is the tooth face width of point k in the
involuteafterthethermaldistortion,ri(i=1,2) is theradiusofreference circle after the thermal distortion.
4 Example
Taking the measured temperature gear as an example [1],the material of gear is 45 steel and the coefficient of linearexpansion l=11.6e�6c�1, room temperature t0= 25 °C.Table 1 shows other parameters of the gear. The toothsurface temperature distribution measured by ZhengzhouInstitute of machinery [1] is shown in Figure 9. Because theinstallation error andmachining error of double helical gearare not considered, the gear tooth temperature distributionof two tooth surfaces is the same.
According to Figure 9, the temperature equation alongreference circle axial with a linear velocity of 130m/s isdeduced.
0 � b � 8 t ¼ 68:58 < b � 33 t ¼ 0:16bþ 67:2233 < b � 53 t ¼ 0:2bþ 65:953 < b � 73 t ¼ 0:1bþ 71:273 < b � 93 t ¼ 0:25bþ 60:2593 < b � 108 t ¼ 0:24bþ 61:18108 < b � 123 t ¼ 0:1bþ 76:3123 < b t ¼ 88:6
8>>>>>>>>>><>>>>>>>>>>:
ð9Þ
Fig. 10. Tooth surface 3D drawing before thermal deformation(left tooth surface).
Fig. 11. Tooth surface 3D drawing before the thermal deforma-tion (right tooth surface).
Fig. 12. Tooth surface 3D drawing after the thermal deforma-tion (left tooth surface).
Fig. 13. Tooth surface 3D drawing after the thermal deforma-tion (right tooth surface).
C. Wang: Mechanics & Industry 20, 612 (2019) 5
According to formulas (4), (5), (7), (8). the toothsurface before and after thermal deformation are shown inFigures 10–13 (where, the x axis describes longitudinaldirection, the y axis describes tooth thickness direction andthe z axis describes radial direction).
Because the installation error and machining error ofdouble helical gear are not considered, the thermaldeformation of Figures 12 and 13 is the same. Take lefttooth surface as an example, the difference of the toothsurface coordinates before and after thermal deformation isshown in Figure 14, the mean value of the difference of thetooth surface coordinates before and after thermaldeformation is shown in Table 2. According to reference [1],Table 3 showed the modification of thermal deformation,which corresponds to the deformation of tooth thickness.In this paper, the diameter of reference circle is 199.36mm,which is close to 200mm in Table 4. Therefore, the
modification can be taken as 0.009mm. In Table 3, themean value of the difference of the tooth surface coordinatesalong tooth thickness direction is 0.009627mm. The error isone order of magnitude smaller than them.
When the installation/machining error of gear exists,the meshing path of two tooth surfaces change. Reference[12] gave the meshing path of left and right tooth surfacesunder installation/machining error of gear (Fig. 15). Thetemperature of instantaneous contact points on the left/right tooth surface is assumed that still satisfies formula(9). The thermal deformation of two tooth surfaces isrecalculated, the mean value of the difference of tooththickness direction without and with considering installa-tion error and machining error of gear is shown in Table 4.
Fig. 14. Difference of the tooth surface coordinates before andafter thermal deformation.
Table 2. Mean value of the difference of thermaldeformation.
Direction Longitudinaldirection
Tooththicknessdirection
Radialdirection
Mean value of thedifference (mm)
0.0346 0.009627 0.059
Table 3. Modification of thermal deformation.
Linearvelocity/(m/s)
Modification of thermal deformation
Gear diameter/mm100 150 200 250 300
130 0.005 0.007 0.009 0.012 0.015
Table 4. Mean value of the difference of tooth thickness direction without and with considering installation error andmachining error of gear.
Tooth thicknessdirection(left tooth surface)
Tooth thicknessdirection(right tooth surface)
Mean value of the difference without consideringinstallation error and machining error of gear (mm)
0.009627
Mean value of the difference under installationerror and machining error of gear (mm)
0.00944 0.0098
Fig. 15. Path of tooth surface contact under installation error and machining error of gear.
6 C. Wang: Mechanics & Industry 20, 612 (2019)
It can be found that thermal deformation of left and righttooth surfaces is difference and different modifications ofthermal deformation are needed.
5 Conclusion
The calculation of gear thermal deformation is one of thekeys to gear modification. Double helical gear is taken asthe research object and a calculation method of thermaldeformation is proposed. The temperature field isdetermined according the measured temperature ofinstantaneous meshing point and radial temperatureformula. According to the structural characteristics ofdouble helical gear, the standard equation for toothsurface is derived. On the basis, the thermal deformationequation of tooth surface is derived according to the tooth
surface temperature field. Taking a temperature measur-ing gears as an example, the tooth surface has changedobviously before and after thermal deformation. Com-pared with the relevant data given in reference [1], theresult is reasonable which verifies the feasibility of theproposed method. By calculating the thermal deformationof two surfaces when the installation error/machiningerror of gear exists, it can be found that the deformation oftwo tooth surfaces is different. That is to say, the left andright tooth surfaces should be modified with differentamounts.
Acknowledgments. The research work is supported by NationalNatural Science Foundation of China (Grant No. 51475210), AProject of Shandong Province Higher Educational Science andTechnology Program (Grant No. J17KA027) and major researchproject of Shandong province (Grant No. 2018GGX103035).
C. Wang: Mechanics & Industry 20, 612 (2019) 7
References
[1] The Directed committee of Gear notebook, Gear notebooksecond edition, China Machine PRESS, 2002, pp. 215
[2] S. Baglioni, F. Cianetti, L. Landi, Influence of the addendummodification on spur gear efficiency, Mech. Mach. Theory49, 216–233 (2012)
[3] V.V. Simon, Influence of tooth modifications on toothcontact in face-hobbed spiral bevel gears, Mech. Mach.Theory 46, 1980–1998 (2011)
[4] C.Wang, Optimization of tooth profile modification based ondynamic characteristics of helical gear pair, Iran J. Sci.Technol. Trans. Mech. Eng. 43, 5631–5639 (2019)
[5] L. Bobach, R. Beilicke, D. Bartel et al., Thermal elastohydrodynamic simulation of involute spur gears incorporatingmixed friction, Tribol. Int. 48, 191–206 (2012)
[6] Q. Liang Heng, X. Yi Xing, W. Tong, et al., A calculation ofbulk temperature and thermal deflection of gear tooth aboutprofile modification, J. Shanghai Jiaotong Univ. 29, 79–86(1995)
[7] G. Xiansheng, W. Huanhuan, Z. Ganqing et al., Analysis ofbulk temperature field and flash temperature for planet geartooth, Trans. Chin. Soc. Agric. Mach. 42, 209–216 (2011)
[8] N. Patir, H.S. Cheng, Prediction of the bulk temperature inspur gear based on finite element temperature analysis,Tribol. Trans. 22, 25–36 (1979)
[9] K.L. Wang, H.S. Cheng, A numerical solution to thedynamic load, film thickness, and surface temperatures inspur gears, ASME J. Mech. Des. 103, 177–194 (1981)
[10] L. Guihua, F. Yetai, Research of the non-involutioncharacteristic of thermal deformation gear, J. Harbin Inst.Technol. 38, 123–125 (2006)
[11] C. Wang, H. Yong Cui, Q. Ping Zhang, The derivation oftransformationmatrix before and after thermal distortion formodification, Proc. Inst. Mech. Eng. C 229, 1686–1692(2015)
[12] C. Wang, H. Yong Cui, Q. Ping Zhang et al., Contact modeland tooth contact analysis of double helical gears withparallel-axis, crossed-axis and modification, Aust. J. Mech.Eng. 13, 1–8 (2015)
Cite this article as: C. Wang, A calculation method of thermal deformation for double helical gear, Mechanics & Industry20, 612 (2019)
