外文翻译
Finite Element Modeling and Analysis on Interfacial Contact Temperature of
Gears Based on ANSYS
Li Jie , Zhang Lei, Zhao Qi
State Key Laboratory of Automobile Dynamic Simulation
Jilin University
Jilin, Changchun, 130025, China
5337lei@https://www.wendangku.net/doc/ef17999722.html,
Abstract
The development of modern transmission design leads to scuffing becoming more serious for transmission gears. The finite element method (FEM) is proposed that can be used as the most economical and effective designing tool for predicting the gears temperature and the distribution, determining whether the tooth scuffing or not, and analyzing the thermal stress and the deformation of gears. The thermal finite element model is developed based on ANSYS simulate the heat generation process of the friction in the involute gears engagement, which includes the bulk temperature and the transient temperature (flash temperature), analyze the heat generation on the surface of gear engagement in the different parts and the moving heat loading method. Through comparing with the result in ISO, it has showed that the finite element model that gears temperature field could calculate the bulk and the transient temperature more accurately. While the moving heat load function could not only calculate the transient temperature on the gear contact surface, but also facilitate using the method to the other types of flat or curved surface the issue of finite element moving heat loading, so it has a broader significance in the practice.
Keywords:Scuffing, Finite Element, Transient Temperature, Bulk Temperature
1. Introduction
Due to high-speed rolling and sliding contact surfaces of two gears, the heat is generated by friction, which resulting in the gear bulk temperature rise. The major targets of transmission design today are higher efficiency, higher torque capacity and reduced size. Increasingly smaller transmissions with higher torque lead to increasing operating temperatures. This trend is further intensified by the use of noise abatement
devices and improved aerodynamic body styling that reduces the airflow around transmissions [1, 2]. There is a particularly sever form of gear tooth surface damage in which seizure or welding together of areas of tooth surface occurs, due to absence or breakdown of a lubricant film between the contacting tooth flanks of mating gears, caused by high temperature and high pressure. This form of damage is termed scuffing and most relevant when surface velocities are high.
At present, the authority methods of calculation gears scuffing were: ISO scuffing load capacity of gears and AGMA gears scuffing design guide for helical and spur gears. The two methods are both based on the gear contact temperature which doesn’t exceed the tooth scuffing corresponding temperature as the calculation basis.
Therefore to accurately predict the temperature of gears contact surface is to prevent gear from scuffing damage occurred in the most direct and effective means.
At the fixed speed and torque conditions, the gear transmission system would reach the thermal equilibrium, the temperature at this time known as the bulk temperature. However, the contact interface in the gears engagement would still have a lot of heat, known as the transient rise of the temperature (flash temperature). The interfacial contact temperature is conceived as the sum of two components: the interfacial bulk temperature and the rapidly fluctuating flash temperature.
Many scholars at home and abroad, used different analytical methods and means to study the temperature of the gear contact surface. However, the test methods of the gear temperature was not only the complexity of measuring technology, the need for specialized test equipment and equipment, the use of higher cost; but also there is some error in the tooth surface transient temperature measurement, the accuracy of that is limited. The finite element method was developed in recent years, although it has gradually been applied to
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the calculation of gears temperature field, such as Wang Tong and Long Hui have used finite element method to analysis the bulk temperature of gears, and got a lot of useful the results and conclusions [3, 4].However, it is very difficult to deal with the moving heat load in the finite element model, so the research on the real cause of the scuffing damage by the transient temperature has been very limited.
2. Finite Element Model of the Gear
Transient Temperature
2.1. Moving Heat Load of the Gear Transient Temperature
According to the well-known Block transient temperature calculation model, the width of contact interface is . Due to the relative sliding friction, the heat is generated in the contact area, as shown in Figure 1. While, the transient temperature rise (flash temperature) may be regarded as a heat generated by friction to the speed of movement along the contact plane 12b [5].
At this point, the transient temperature rise (flash temperature) may be regarded as the heat source generated by friction, has a movement along the contact plane in the speed of s q v , solving the temperature distribution of the gear. The distribution of the heat source and contact stresses is the same in the border:
21
max )(1b x
q q s ?1?
Where is instantaneous heat source intensity,
is the maximum heat intensity, is semi-width of Hertzian contact band.
s q max q 1b
1
Figure 1. Heat Distribution on the Contact Area
In theory, the heat intensity that is generated by the relative sliding friction could be determined by the following equation:
D 21max v v Pf q ?2?
Where P is the average unit load on the contact line,is the coefficient of friction, and are the velocity of pinion and wheel,f 1v 2v D is the coefficient of temperature conductivity.
2.2. Average Load on the Contact Line
ca P P 4
S
?3?Where is the maximum load on the contact line, in accordance with the spur gear tooth along the contact line, determination by calculation:
ca P t
a t
n ca b KF L KF P D H cos
?4?Where is the total length of tooth contact line,is the normal tooth load, is the nominal
tangential load at reference circle, L n F t F a H is the contact ratio,is the face width, smaller value of pinion or wheel,b t D is the transverse pressure angle,K is the scuffing load factor, which could be calculated from the following equation:
D E B B V A K K K K K ?5?Where is the application factor,is the dynamic factor, is the scuffing transverse load factor,is the scuffing transverse load factor.
A K V K E
B K D B K 2.3. Friction Coefficient
In general, the coefficient of friction is determined by experimental method in accordance with the specific working conditions. For example, the steel gear measured friction coefficient is about 0.06 ~ 0.08, which includes friction losses in the rolling bearing. If the removal friction losses of bearing, the reduction of the tooth surface friction coefficient is about 1 / 4 to 1 / 5, so the friction coefficient is about 0.045 ~ 0.065. In the calculation of gears bulk temperature, it could be used as the tooth surface average coefficient of friction 05.0 f .
While in the calculation of the transient temperature rise, the actual friction coefficient would be changed with different the gears engagement point on the tooth surface. In order to get the friction heat input accurately, a great number of experiments and calculations should be made.
The friction coefficient varies with the surface roughness of tooth flanks, the sliding velocities, the load, the lubricant used, the properties of gear materials and heat treatment. The average friction coefficient on the different tooth surface engagement
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points obtained from experiments, having approximate relations with a great number of important factors:
25.0)(12.0redi
i M a
t i R w f U X K 6 ?6?
Where is the scuffing specific tooth load, is the arithmetic mean roughness, t w a R i 6X is the sum of tangential velocities at arbitrary point, redi U is the relative radius of curvature at arbitrary point,M K is the dynamic viscosity at bulk temperature.
2.4. Temperature Conductivity Coefficient
In the previous calculations, the heat is seen as loaded only on the surface of one object, while there are two surfaces in friction, so the friction heat would be allocated to the two objects in the contact area. If the conductivity temperature coefficients of the two materials are different, then heat distribution on the two objects would be uneven.
The temperature conductivity coefficient could be calculated from the following equation:
2
22211111
111v c v c v c U O U O U O D
?7?
Where 1O ,2O is the heat conductivity of pinion and wheel,1U ,2U is the density of pinion and wheel material, ,is the specific heat per unit mass of pinion and wheel,,is the Poisson’s ratio of pinion and wheel material. If the material of pinion and wheel are the same, the temperature conductivity coefficient could be determined by the following equation 1c 2c 1v 2v [6]:
1
2
11
v v D ?8?
2.5. Load Sharing Factor Coefficient
The load P is the average load on the contact line, without taking into the load distribution relations along the contact line, and therefore need to introduce a correction factor to calculate the load along the contact line in the corresponding to the different engagement points. The load sharing factor coefficient has some relations with the teeth stiffness, the tip relief of gears and the tooth manufacturing error.
*X The single pair tooth contact area , while two pairs of teeth contact area, would be changed with the stiffness of the tooth and the tip relief of gears. In order to simplify the calculation, the calculation of the
load sharing factor direct use the ISO method, for the no tip relief of gears, determination by calculation:
1 *X *X 3
1
**E A X X ??9?
1 **D B X X 3. Load of the Finite Element Model of Gears Interfacial Contact temperature
3.1. Load of the Gear Bulk Temperature Model
Figure 2. Bulk Temperature Model of Tooth
The two-dimensional model of the single tooth is shown in Figure 2. Taking the calculation of the interfacial contact temperature on the outer point of single pair tooth contact as an example, and simplify the contact area as the Block transient temperature model, which is shown in Figure 3. The boundary conditions of the gear bulk temperature model are as follows:
Tooth engagement area (N region):
q T T n T k t w w )(/0D ?10?
No engagement area of the tooth (t region):
)(/0T T n T k s w w D ?11?
Inside of the tooth (p and q region):
q p T T || ?q p n
T n T ||w w w w ?12?Where is the heat conductivity of gears, k D is the heat coefficient, T the bulk temperature, the environment temperature, n is the outward normal of heat exchange surface, q is the friction heat input of the engagement surface.
0T 3.2. Load of the gear Transient Temperature Model
In order to calculating the corresponding transient temperature at arbitrary point, the boundary conditions of the transient temperature model are as follows: Tooth engagement contact area (A region):
s q n T k w w /?13?
Surrounding areas of gear engagement (and region).Desirable for the gear bulk temperature, but for
B C 670
the purpose of simplify the calculation, determination by calculation:
0 C B T T ?14?
Inside of the tooth (D region):
0/ w w n T ?15
?
Figure 3. Contact Area of Gear Engagement Model
ANSYS couldn’t get the result of moving heat load directly, so first of all, it need to discriminate the moving heat source, assuming that the fixed heat source loads on the contact area in a very short period of time, and then the fixed heat source loads on the other region in the next period of time, and the results of the last time would be treated as the initial conditions in this time.
As the heat load is not a constant, which would change with the location of gear engagement. Therefore it need break down the parabola heat function into the table parameters. If the width of gear engagement contact area is , dividing the region into even ten parts, there are a total of eleven nodes. Firstly, one-dimensional array is defined as follows:
12b 11
,,,ARRAY Qheat DIM
Secondly, the parabola heat source is assigned to the contact area on the corresponding nodes:
max
max max max 2524
,2521,2516,259,0)1(q q q q Qheat 0
,259
,2516,2521,2524,)6(max max max max max q q q q q Qheat It has been known that the speed of the moving heat
is 21v v , even though and is different in different moments, but as the contact width is small, which can be similar to that of the residential area is a constant value. In zero time, the front point of the heat source A is located at some point N on the contact area, while the each time step as follows:
1v 2v 2
11
5v v b t ?16?Each time step after a long, the heat would move
the unit length , to be followed by analogy, as shown in Figure 3, until the moment 5/1b )/(2211v v b t ,the time-step heat source load over.
Figure 4. Moving Heat Load on the Gears Contact Area
4. Analysis on the Finite Element Model of the Gears Interfacial Contact temperature
In order to verify the accuracy of the finite element model of gears interfacial temperature, the gear parameters (see table 1) is provided by the literature [7] and the ISO calculation method of the interfacial temperature as the reference, The results of the finite element model and ISO are compared, as shown in table 2.
Table 1. Structure parameters of the example gears
Parameter value unite
Module()m 3mm Pressure angle(D )20deg Helix angle(E )
29deg Coefficient of addendum ()
a h 1Clearance factor()
c 0.25mm
Number of teeth of pinion()
1z 16Number of teeth of wheel()
2z 92Addendum modification coefficient of pinion()1x 0.5Addendum modification coefficient of wheel()
2x -1.4Facewidth of pinion ()1b 27.85mm Facewidth of wheel()2b 27.85mm Actual center distance()a c 105mm
Arithmetic roughness a R 0.63m
P Modulus of elasticity()E 206GPa Poisson’s ratio(Q )0.3Nominal torque()
1T 49.1
m
N 671
Oil temperature oil
T 65
C
$
The gear bulk temperature distribution is Figure 5 (a), while the block model of the transient temperature corresponds to the outer point of single pair tooth as shown in Figure 5 (b).
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?a ?Bulk temperature ?b ?Transient temperature Figure 5. Finite Element Model of Interfacial
Temperature
In order to facilitate the study the law of the transient temperature on the contact area (flash temperature), as shown in Figure 6.
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Figure 6. Temperature Curve of Gear on Contact Area Table 2. Transient Contact Temperature on the points of
Gears Engagement ???
Interfacial temperature ISO FEM Bulk temperature 8578.397Root of tooth A 14.5205 5.604Inner point of gears contact B 7.1590 4.476Outer point of gears contact D 49.897849.132Transient Temperature on Contact Area
Tooth tip E
22.906527.487Maximum interfacial temperature
134.898
127.529
As shown in Figure 5, the maximum bulk temperature is located at outside of the gear pitch circle, and the temperature is slightly lower around the tooth tip and the root of tooth. Although in practice, the heat input is not uniform along the tooth profile, the heat flow is averaged by the meshing gears in per lap, so the average heat input of bulk temperature is got. It compared the results of the bulk temperature of the gear finite element model with the integral method of the gear scuffing load capacity, which showed that the
finite element model of bulk temperature is in compliance with the law.
Temperature rise curve of the finite element model in Figure 6 has showed that the maximum transient temperature rise (flash temperature) appears in some time 211/6.1v v b t . The integration method and the numerical results are basically consistent with the Block flash temperature model of the transient temperature law, which proved that the finite element model and the moving heat load based on the ANSYS is a proper way. The model and the method could be used for the actual calculation of scuffing load capacity of gears,
Although From the view of the data comparison in Table 2, there are some errors in the calculation of the finite element model of the tooth tip and the root of tooth. This was mainly due to the finite element model of heat load sharing coefficient is derived from the ISO calculation methods and the design of the conclusions, so in order to achieve the engagement of the tooth surface contact area at a more accurate calculation, which depends on the amendment the finite element model for further study in the future.
5. Conclusions
The thermal finite element models are developed based on the ANSYS Parametric Design Language APDL, which include the bulk temperature and the transient temperature (flash temperature). Through the modeling and analysis, it is found that both models could not only to solve gear temperature field separately, but the results could be superimposed to obtain more accurate interfacial contact temperature. While the moving heat loading function could not only calculate the transient temperature on the gear contact surface, but also facilitate using the method to the other types of flat or curved surface the issue of finite element moving heat loading, so it has a broader significance in the practice.
Acknowledgement
This work has been supported by Foundation of National High-tech R&D Program (63 Program 2006AA110101). The authors would like to thank the sponsors of the state key laboratory of automobile dynamic simulation for their financial support and encouragement during the course of this study. The state key laboratory is also to be thanked for supplying access to its facilities for performing the finite element analyses.
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6. References
[1] Christoph Wincierz, Klaus Hedrich. ‘‘Formulation of Multigrade Gear Oils for High Efficiency and Low Operating Temperature,’’ SAE Paper 2002-01-2822
[2] Long H, Lord A A, Gethin D T, et al. ‘‘Operating tempera-tares of oil-lubricated medium-speed gears: Numerical models and experimental results,’’ Journal of Aerospace, 2003, 217(2): 87–106.
[3] Qiu Liangheng, Xin Yixing, Wang Tong, et al. ‘‘A Calculation of Bulk Temperature and Thermal Deflection of Gear Tooth about Profile Modification,’’ Journal of Shanghai Jiaotong University, 1995, 29(2): 79–86. [4] Long Hui, Zhang Guanghui, LuoWenjun. ‘‘Modeling and Analysis of Transient Contact Stress and Temperature of Involute Gears,’’ Chinese Journal of Mechanical Engineering, 2004, 40(8): 24–29.
[5] Zhu Xiaolu, Er Xhongkai, “Analysis of Load Capacity of Gears,” BeiJing: National Ministry of Education Press, 1992 [6] Yang Wenong, Ding Jinyuan, Ma Xiangui. ‘‘Finite Element Analysis of the Flash Temperature,’’ Journal of Northeast University of Technology, 1990, 11 (2): 192–190. [7] Gong Guiyi, Chen shichong, Wang Yongjie , “Involute Cylindrical Gears strength calculation and structural design,” BeiJing:Machinery Industry Press, 1986
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