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Modeling and predicting of surface roughness for generating grinding gear
Journal of Materials Processing Technology 213 (2013) 717–721
Contents lists available at SciVerse ScienceDirect
Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Modeling and predicting of surface roughness for generating grinding gear Chen Haifeng, Tang Jinyuan ∗ , Zhou Wei State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha, Hunan 410083, China
a r t i c l e
i n f o
Article history: Received 23 February 2012 Received in revised form 15 October 2012 Accepted 23 November 2012 Available online 1 December 2012 Keywords: Gear grinding Surface roughness Modeling
a b s t r a c t Gear surface has a significant effect on the load-carrying capacity and plays a crucial role for wear, friction and lubrication properties of the contact. However, most mathematical model of a generating gear grinding process trends to focus on grinding forces and temperature distribution in gear tooth. A new method for predicting gear surface roughness with the generating grinding process is developed in this paper. An algorithm for geometrical analysis of the grooves on the gear surface left by idea conic grains is given. To determine the final gear surface roughness produced by thousands of grinding wheel grains with randomly distributed protrusion heights, a search technique is proposed to systematically solve the gear surface roughness, which starts from the addendum circle along the involutes direction and ends at the dedendum circle. The numerical calculated results show that the surface of gear root part is smoother than that of gear top part. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Gear generating grinding is one of the most popular finishing manufacturing processes for hardened gears (kruszinski, 1995). A great deal of research work has been carried out in order to develop models that deal with the characteristics of the gear grinding process, such as force, power, temperature, and surface integrity. However, there are few researches focusing on the forecast for gear surface roughness. Since a gear is usually a highly loaded machine component and works under complex friction as well as high stress condition, unfavorable surface roughness may have a severe negative effect on the functional properties of the gear and thus decreases. Based on various assumptions and different approaches, extensive researches have been done to predict the surface roughness for grinding. On the available literatures, theoretical methods of surface roughness prediction can be classified into virtual simulation and analytical methods. In the virtual simulation method, Gong et al. (2002) applied virtual reality technology to predicate the ground surface roughness by creating the model of the virtual grinding wheel and simulating the grinding process. Kumar and Choudhury (2007) applied a three layer feed forward artificial neural network model to predict the surface roughness and the wheel wear rate which was comprised with experimental investigation during electro-discharge diamond grinding process, and established a second order regression model to represent the relationship between process parameters, such as pulse current, duty
∗ Corresponding author. Tel.: +86 731 88877746; fax: +86 731 88877746. E-mail address: [email protected] (J. Tang). 0924-0136/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2012.11.017
ratio, wheel speed, grit number and output responses, namely, wheel wear and surface roughness. Aguiar et al. (2007) proposed a prediction model of surface roughness based on the neural network which through its mathematical logical system interpreted the acoustic emission and electrical power signals in grinding SAE 1020 steel. Iwai et al. (2009) applied a 3D-CAD model to simulation of virtual ground surface in helical scan grinding, the influence of grit arrangement, protrusion height, apex angle and inclination angle randomly to a real wheel was taken into account and validated with the homologous experiment. Chang et al. (2010) used the Adaptive Network based Fuzzy Inference to establish a grinding model that was trained by experimental data from grinding the SKD11 steel for predicting surface roughness and hardness with respect to grinding parameters, such as wheel speed, feed rate and depth of grinding. Huang et al. (2010) applied the Least Square Support Vector Machine to predict the surface roughness in cylindrical longitudinal grinding. In the analytical methods, a kinematic analysis model, which takes into consideration the random distribution of the grinding wheel grain protrusion heights, is proposed by Zhou and Xi (2002) to simulate the kinematic interaction between the grinding wheel grains and the workpiece. Based on this model, the surface roughness of the workpiece is also predicted. The intersecting points of the trajectories of multiple grains are determined in a sequential manner, which starts from the highest and goes down to the lowest. A probabilistic undeformed chip thickness model is presented by Hecker and Liang (2003) to forecast the arithmetic mean surface roughness, which expresses the ground finish as a function of the wheel microstructure, the process kinematic condition and the material property. An analytical probabilistic model is developed by Stepien (2009) for the grinding process, where the random arrangement of the grain vertices at the wheel active
718
H. Chen et al. / Journal of Materials Processing Technology 213 (2013) 717–721
K
j
ns
C
L
E h
D
A
W
W
B
rbs
v
S
O
r o Fig. 1. The principle of generating-gear grinding machine.
Fig. 2. Trajectory of a single grain.
surface is considered, and where the process of shaping the ground surface roughness, the probability of contact between the grains and the workpiece as well as the undeformed chip thickness are also described in the grinding zone. Another analytical surface roughness model put forward by Agarwal and Rao (2010) is on the base of the stochastic nature of the grinding process, which is mainly governed by the random geometry and the random distribution of cutting edges on the wheel surface with randomly distributed grain protrusion heights. A simple relationship between the surface roughness and the chip thickness is obtained. Most of these models discussed focus on modeling and predicting the plane grinding surface roughness, and the research on modeling and predicting the surface roughness of the gear grinding has not been touched yet. The modeling and predicting of the surface roughness for generating grinding gear is investigated in this paper. First, the generating method of gear grinding is introduced. Then, the mechanism of grinding wheel grain-workpiece interaction is given and the intersection of any two grains with different heights is studied, and a search algorithm is developed to achieve the final gear surface roughness. Finally, the simulation results are given and verified in part of characteristic in comparison with experimental data. 2. Generating method of gear grinding The kinematics of the generating gear grinding is shown in Fig. 1. The main kinematical movements are as follows (kruszinski, 1995): Strictly related linear (v) and rotational (ω) movements of the workpiece, both of which give a generating movement. Rotational (ns ) movement of the grinding wheel. ˛j is the incline angle of the grinding wheel, and L is the grain interval. After completing the grinding of one tooth space an indexing movement of workpiece is performed to start the machining of the subsequent tooth profile. 3. Grinding kinematics The grinding process is very complex. For the convenience of analyzing the grinding kinematics, simplification is needed, which expresses as the following assumptions:
(a) The material of the gear surface in contact with the cutting edges of the wheel is cut off when the wheel is fed into the gear surface (Zhou and Xi, 2002). (b) The diameter of the wheel is much larger than the width of the tooth, and then, the trajectory of grain can be approximated as line. (c) The idea conic grain is uniformly distributed on the wheel. 3.1. Grain trajectory equation According to the stated assumptions, the grooves left on the gear cross section is an isosceles triangle, whose height is equal to the grain protrusion height and whose vertex angle amounts the conic vertex angle as shown in Fig. 2. The wheel meshes with the involutes gear at the point K. The coordinates of point K can be described as
x = rbs [sin(s ) − s cos(s )] y = rbs [cos(s ) + s sin(s )]
(1)
where rbs is the gear base radius, s is the angle parameter of involutes. The line segment length of KE and KC can be described as |KE| = |KC| = h · tan ˛
(2)
The grain protrusion height h is found following the Gaussian distribution with mean value and standard deviation . For a given grit number M, the mean value can be determined as (Hwang et al., 2000) = 68M −1.4
(3)
The standard deviation can be gained as (Hwang et al., 2000) =
15.2M −1 − 68M −1.4 3
(4)
Then the coordinates of the vertex of the triangle can be calculated as
xC = x − |KC| sin ˇ = x − h tan ˛ sin ˇ yC = y − |KC| cos ˇ = y − h tan ˛ cos ˇ
(5)
H. Chen et al. / Journal of Materials Processing Technology 213 (2013) 717–721
1
E2
C1
K2
2
K1
h2
C2
E1 D1
719
According to Fig. 1, the time difference between the two grinding wheel grains can be expressed as
h1
t =
D2
L sin ˛j
(12)
v
where L can be determined as (Hwang et al., 2000)
B2
A
B1
L = 137.9M
−1.4 3
32 − S
(13)
where S is the structural number. Then the angle difference between the two adjacent grains can be determined as
1
= ω · t = ω
O
L sin ˛j
(14)
v
The wheel meshes with the involutes gear at the second point K2 . The coordinates of point K2 can be described as
⎧ L sin ˛j L sin ˛j L sin ˛j ⎪ − 1 − ω cos 1 − ω ⎨ x2 = rbs sin 1 − ω v v v
⎪ ⎩ y2 = rbs cos 1 − ω L sin ˛j + 1 − ω L sin ˛j sin 1 − ω L sin ˛j v
v
v
Fig. 3. Trajectory of two grains.
xD = x + h cos ˇ
(15) (6)
yD = y − h sin ˇ xE = x + |KE| sin ˇ = x + h tan ˛ sin ˇ yE = y + |KE| cos ˇ = y + h tan ˛ cos ˇ
|K2 E2 | = |K2 C2 | = h2 · tan ˛ (7)
where ˇ is the angle between the normal of involutes and the axis x. 3.2. Intersecting point of the trajectories of two grinding wheel grains
xC1 = x1 − h1 tan ˛ sin ˇ1
where subscript 2 indicates the second grain, and h2 denotes its protrusion heights. Then the coordinates of the vertex of the second triangle can be described as
xD1 = x1 + h1 cos ˇ1
(9)
yD1 = y1 − h1 sin ˇ1 xE1 = x1 + h1 tan ˛ sin ˇ1
y1 = rbs [cos(1 ) + 1 sin(1 )]
(11)
(18)
xE2 = x2 + K2 E2 sin ˇ2 = x2 + h2 tan ˛ sin ˇ2
(19)
xCi = xi − hi tan ˛ sin ˇi
(20)
yCi = yi − hi tan ˛ cos ˇi
x1 = rbs [sin(1 ) − 1 cos(1 )]
(17)
Similarly, for the ith grain, its coordinates of the vertex of the triangle can be described as
where subscript 1 indicates the first grain, h1 denotes its protrusion heights, and x1 , y1 can be obtained from Eq. (1) as
xD2 = x2 + h2 cos ˇ2
yE2 = y2 + K2 E2 cos ˇ2 = y2 + h2 tan ˛ cos ˇ2
(10)
yE1 = y1 + h1 tan ˛ cos ˇ1
xC2 = x2 − |K2 C2 | sin ˇ2 = x2 − h2 tan ˛ sin ˇ2
yD2 = y2 − h2 sin ˇ2
(8)
yC1 = y1 − h1 tan ˛ cos ˇ1
(16)
yC2 = y2 − |K2 C2 | cos ˇ2 = y2 − h2 tan ˛ cos ˇ2
Two adjacent grinding wheel grains are considered in Fig. 3. The triangle vertex coordinates of the first grain D1 can be computed according to Eqs. (4)–(7) as
The line segment length of K2 E2 and K2 C2 can be described as
xDi = xi + hi cos ˇi
(21)
yDi = yi − hi sin ˇi
xEi = xi + hi tan ˛ sin ˇi
(22)
yEi = yi + hi tan ˛ cos ˇi where xi , yi can be expressed as
⎧
L sin ˛j L sin ˛j L sin ˛j ⎪ ⎪ x = r sin − (i − 1)ω − − (i − 1)ω cos − (i − 1)ω ⎪ 1 1 1 ⎨ i bs v v v
⎪ L sin ˛j L sin ˛j L sin ˛j ⎪ ⎪ + 1 − (i − 1)ω sin 1 − (i − 1)ω ⎩ yi = rbs cos 1 − (i − 1)ω v
v
v
(23)
720
H. Chen et al. / Journal of Materials Processing Technology 213 (2013) 717–721 Table 1 Grinding wheel and grinding process parameters.
4. Surface roughness modeling To give a more realistic prediction of the surface roughness, the actual distribution of the grinding wheel grain protrusion heights must be considered with different heights. Though the model presented in Eqs. (20)–(23) provides a way to get the coordinates of the vertex of the triangle, the remaining problem is how to solve the final gear surface profile in a systematic way given that there are thousands of grains in the wheel. A search algorithm is also proposed in this paper. As gear mesh belongs to point contact, the marks left on the surface of the workpiece are related to the grain at the meshing point. The traces of adjacent grains will intersect with each other. The grain meshing at the addendum position should be considered first. Then, if the second grain does not cut into the trace generated by the first one, the headmost trace will be the final one. On the contrary, if the second grain does cut into the trace generated by the first one, the profile of the first trace will be changed. The process of modifying the surface profile continues with more grains involved in generating order, and it terminates when the grain meshes at the dedendum. Mathematically, the search method is described as follows. The surface profile of the first grain is formed by connecting three points of the first triangle and is denoted as s1 ⇒ K10 (x, y)
(24)
Grains and gear surface profiles are indexed by subscript and superscript respectively. The superscript value is set to be 0 when the profile is generated by a single grain and it is larger than 0 when the profile is generated by more than two grains. Similarly, the surface profile of the second grain is formed by connecting three vertex of the second triangle, which is determined with Eqs. (17)–(19), and can be expressed symbolically as s2 ⇒
K20 (x, y)
Fig. 4. Surface profile.
1 2
(25)
There are no intersections between the first two profiles when K10 (x, y) ∩ K20 (x, y) = 0. It can be indicated that there is too long grain interval to adopt the process and thus a better grit grinding wheel should be chosen. There are intersecting points between K10 (x, y) and K20 (x, y), when K10 (x, y) ∩ K20 (x, y) = / 0, a new geometry K1 is formed by the combined result, which is satisfied by the following condition K 1 (x, y) = min(K10 (x, y), K20 (x, y))|y
(26)
Which means that the showed surface profile is formed by a series of cut-off sections between the two individual profiles. A cutff section selected from one profile verse another depends on the profile value y that is smaller, which represents the marks left on the surface of the workpiece. The third grain with its individual geometry K30 is considered and as the search process continues, a new geometry K2 is formed from the combined result and satisfies the following condition: K 2 (x, y) = min(K 1 (x, y), K30 (x, y))|y
(27)
Kn
that is formed by the following
The final profile is given as term:
Wheel: 99A120K8V Conic vertex angle 2˛ : 141◦ Slant angle of wheel spindle ˛j : 20◦ Linear velocity v : 2.75 mm/s
0 (x, y))|y K n (x, y) = min(K n−1 (x, y), Kn+1
3 4
Fig. 5. Sample method.
Fig. 4 shows the simulation results on gear cross section when the grains have random heights. A gear is gained by the grinding condition as shown in Table 1, and is measured by the roughness measuring instrument HOMMEL WERKE. Because the probe is unable to reach the dedendum, four
(28)
And n is the last grain meshing with the gear at the dedendum. 5. Simulation and prediction of the surface roughness of generating grinding The proposed method is applied to simulate the surface roughness of generating grinding gear, and the grinding data are listed in Table 1.
Fig. 6. Ra results from experiment and simulation.
H. Chen et al. / Journal of Materials Processing Technology 213 (2013) 717–721
721
grinding trace becomes sparse and the arris height enlarges from dedendum to addendum. As a result, the Ra values increase little by little. 6. Conclusions A new method of predicting the gear surface profile for the generating grinding process is developed in this paper. It takes into account the random distribution of the grain protrusion heights. The simulation shows that the root part of surface is smoother than the top part, and the simulated results are consistent with the experiment, proving the validity of the modeling and predicting method of surface roughness for generating grinding gear presented in this paper. Acknowledgement Fig. 7. Rmax results from experiment and simulation.
The authors gratefully acknowledge the support of the Fundamental Research Funds for the Central Universities of Central South University (2012zzts013), the Notional Key Basic Research Program of China (2011CB706800), the National Science Foundation of China (51275530) and Hunan Provincial Innovation Foundation for Postgraduate (CX 2012B050). References
Fig. 8. Rz results from experiment and simulation.
samples which denote 1, 2, 3 and 4 are taken start from the pitch circle along the tooth profile, as shown in Fig. 5. Each sample length is 0.48 mm, cut off is 0.08 mm. The surface roughness results for experiment and simulation are illustrated in Figs. 6–8. The results show that the simulation values are in agreement with the experimental observations. The surface roughness parameter Ra , Rmax and Rz are increasing monotonously along the tooth profile direction. Considering that the space between two grains is equal here, and the feed rate of rack is changeless, the velocities of meshing points increase from dedendum to addendum and the curvature radiuses augment gradually at the same angular velocity. Hence, the moving distances of the meshing points also increase accordingly at the same time, which leads to the result that the
Agarwal, S., Rao, P.V., 2010. Modeling and prediction of surface roughness in ceramic grinding. International Journal of Machine Tools and Manufacture 50, 1065–1076. Aguiar, P.R., Cruz, C.E.D., Paula, W.C.F., et al., 2007. Neural network approach for surface roughness prediction in surface grinding. In: Proceedings of the IASTED International Conference on Artificial Intelligence and Applications, pp. 96–101. Chang, C.H., Tsai, J.C., Chiu, N.H., et al., 2010. Modeling surface roughness and hardness of grinding SKD11 steel using Adaptive Network based Fuzzy Inference. Advanced Materials Research 126–128, 171–176. Gong, Y.D., Wang, B., Wang, W.S., 2002. The simulation of grinding wheels and ground surface roughness based on virtual reality technology. Journal of Materials Processing Technology 129, 123–126. Hecker, R.L., Liang, S.Y., 2003. Predictive modeling of surface roughness in grinding. International Journal of Machine Tools and Manufacture 43, 755–761. Huang, J.D., Wang, L.S., Li, G.F., 2010. Prediction system of surface roughness based on LS-SVM in cylindrical longitudinal grinding. Optics and Precision Engineering 18, 2407–2412 (Language: Chinese). Hwang, T.W., Evens, C.J., Malkin, S., 2000. High speed grinding of silicon nitride with electroplated diamond wheels, part I: wear and wheel life. ASME Journal of Manufacturing Science and Engineering 122, 32–41. Iwai, M., Shiraishi, Y., Ninomiya, S., et al., 2009. Prediction of surface roughness by 3D-CAD model in helical scan grinding and groove grinding. Advanced Materials Research 76–78, 101–106. kruszinski, B.W., 1995. Model of gear-grinding process. Annals of the CIRP 44, 321–324. Kumar, S., Choudhury, S.K., 2007. Prediction of wear and surface roughness in electro-discharge diamond grinding. Journal of Materials Processing Technology 191, 206–209. Stepien, P., 2009. A probabilistic model of the grinding process. Applied Mathematical Modeling 33, 3863–3884. Zhou, X., Xi, F., 2002. Modeling and predicting surface roughness of the grinding process. International Journal of Machine Tools and Manufacture 42, 969–977.
Contents lists available at SciVerse ScienceDirect
Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Modeling and predicting of surface roughness for generating grinding gear Chen Haifeng, Tang Jinyuan ∗ , Zhou Wei State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha, Hunan 410083, China
a r t i c l e
i n f o
Article history: Received 23 February 2012 Received in revised form 15 October 2012 Accepted 23 November 2012 Available online 1 December 2012 Keywords: Gear grinding Surface roughness Modeling
a b s t r a c t Gear surface has a significant effect on the load-carrying capacity and plays a crucial role for wear, friction and lubrication properties of the contact. However, most mathematical model of a generating gear grinding process trends to focus on grinding forces and temperature distribution in gear tooth. A new method for predicting gear surface roughness with the generating grinding process is developed in this paper. An algorithm for geometrical analysis of the grooves on the gear surface left by idea conic grains is given. To determine the final gear surface roughness produced by thousands of grinding wheel grains with randomly distributed protrusion heights, a search technique is proposed to systematically solve the gear surface roughness, which starts from the addendum circle along the involutes direction and ends at the dedendum circle. The numerical calculated results show that the surface of gear root part is smoother than that of gear top part. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Gear generating grinding is one of the most popular finishing manufacturing processes for hardened gears (kruszinski, 1995). A great deal of research work has been carried out in order to develop models that deal with the characteristics of the gear grinding process, such as force, power, temperature, and surface integrity. However, there are few researches focusing on the forecast for gear surface roughness. Since a gear is usually a highly loaded machine component and works under complex friction as well as high stress condition, unfavorable surface roughness may have a severe negative effect on the functional properties of the gear and thus decreases. Based on various assumptions and different approaches, extensive researches have been done to predict the surface roughness for grinding. On the available literatures, theoretical methods of surface roughness prediction can be classified into virtual simulation and analytical methods. In the virtual simulation method, Gong et al. (2002) applied virtual reality technology to predicate the ground surface roughness by creating the model of the virtual grinding wheel and simulating the grinding process. Kumar and Choudhury (2007) applied a three layer feed forward artificial neural network model to predict the surface roughness and the wheel wear rate which was comprised with experimental investigation during electro-discharge diamond grinding process, and established a second order regression model to represent the relationship between process parameters, such as pulse current, duty
∗ Corresponding author. Tel.: +86 731 88877746; fax: +86 731 88877746. E-mail address: [email protected] (J. Tang). 0924-0136/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2012.11.017
ratio, wheel speed, grit number and output responses, namely, wheel wear and surface roughness. Aguiar et al. (2007) proposed a prediction model of surface roughness based on the neural network which through its mathematical logical system interpreted the acoustic emission and electrical power signals in grinding SAE 1020 steel. Iwai et al. (2009) applied a 3D-CAD model to simulation of virtual ground surface in helical scan grinding, the influence of grit arrangement, protrusion height, apex angle and inclination angle randomly to a real wheel was taken into account and validated with the homologous experiment. Chang et al. (2010) used the Adaptive Network based Fuzzy Inference to establish a grinding model that was trained by experimental data from grinding the SKD11 steel for predicting surface roughness and hardness with respect to grinding parameters, such as wheel speed, feed rate and depth of grinding. Huang et al. (2010) applied the Least Square Support Vector Machine to predict the surface roughness in cylindrical longitudinal grinding. In the analytical methods, a kinematic analysis model, which takes into consideration the random distribution of the grinding wheel grain protrusion heights, is proposed by Zhou and Xi (2002) to simulate the kinematic interaction between the grinding wheel grains and the workpiece. Based on this model, the surface roughness of the workpiece is also predicted. The intersecting points of the trajectories of multiple grains are determined in a sequential manner, which starts from the highest and goes down to the lowest. A probabilistic undeformed chip thickness model is presented by Hecker and Liang (2003) to forecast the arithmetic mean surface roughness, which expresses the ground finish as a function of the wheel microstructure, the process kinematic condition and the material property. An analytical probabilistic model is developed by Stepien (2009) for the grinding process, where the random arrangement of the grain vertices at the wheel active
718
H. Chen et al. / Journal of Materials Processing Technology 213 (2013) 717–721
K
j
ns
C
L
E h
D
A
W
W
B
rbs
v
S
O
r o Fig. 1. The principle of generating-gear grinding machine.
Fig. 2. Trajectory of a single grain.
surface is considered, and where the process of shaping the ground surface roughness, the probability of contact between the grains and the workpiece as well as the undeformed chip thickness are also described in the grinding zone. Another analytical surface roughness model put forward by Agarwal and Rao (2010) is on the base of the stochastic nature of the grinding process, which is mainly governed by the random geometry and the random distribution of cutting edges on the wheel surface with randomly distributed grain protrusion heights. A simple relationship between the surface roughness and the chip thickness is obtained. Most of these models discussed focus on modeling and predicting the plane grinding surface roughness, and the research on modeling and predicting the surface roughness of the gear grinding has not been touched yet. The modeling and predicting of the surface roughness for generating grinding gear is investigated in this paper. First, the generating method of gear grinding is introduced. Then, the mechanism of grinding wheel grain-workpiece interaction is given and the intersection of any two grains with different heights is studied, and a search algorithm is developed to achieve the final gear surface roughness. Finally, the simulation results are given and verified in part of characteristic in comparison with experimental data. 2. Generating method of gear grinding The kinematics of the generating gear grinding is shown in Fig. 1. The main kinematical movements are as follows (kruszinski, 1995): Strictly related linear (v) and rotational (ω) movements of the workpiece, both of which give a generating movement. Rotational (ns ) movement of the grinding wheel. ˛j is the incline angle of the grinding wheel, and L is the grain interval. After completing the grinding of one tooth space an indexing movement of workpiece is performed to start the machining of the subsequent tooth profile. 3. Grinding kinematics The grinding process is very complex. For the convenience of analyzing the grinding kinematics, simplification is needed, which expresses as the following assumptions:
(a) The material of the gear surface in contact with the cutting edges of the wheel is cut off when the wheel is fed into the gear surface (Zhou and Xi, 2002). (b) The diameter of the wheel is much larger than the width of the tooth, and then, the trajectory of grain can be approximated as line. (c) The idea conic grain is uniformly distributed on the wheel. 3.1. Grain trajectory equation According to the stated assumptions, the grooves left on the gear cross section is an isosceles triangle, whose height is equal to the grain protrusion height and whose vertex angle amounts the conic vertex angle as shown in Fig. 2. The wheel meshes with the involutes gear at the point K. The coordinates of point K can be described as
x = rbs [sin(s ) − s cos(s )] y = rbs [cos(s ) + s sin(s )]
(1)
where rbs is the gear base radius, s is the angle parameter of involutes. The line segment length of KE and KC can be described as |KE| = |KC| = h · tan ˛
(2)
The grain protrusion height h is found following the Gaussian distribution with mean value and standard deviation . For a given grit number M, the mean value can be determined as (Hwang et al., 2000) = 68M −1.4
(3)
The standard deviation can be gained as (Hwang et al., 2000) =
15.2M −1 − 68M −1.4 3
(4)
Then the coordinates of the vertex of the triangle can be calculated as
xC = x − |KC| sin ˇ = x − h tan ˛ sin ˇ yC = y − |KC| cos ˇ = y − h tan ˛ cos ˇ
(5)
H. Chen et al. / Journal of Materials Processing Technology 213 (2013) 717–721
1
E2
C1
K2
2
K1
h2
C2
E1 D1
719
According to Fig. 1, the time difference between the two grinding wheel grains can be expressed as
h1
t =
D2
L sin ˛j
(12)
v
where L can be determined as (Hwang et al., 2000)
B2
A
B1
L = 137.9M
−1.4 3
32 − S
(13)
where S is the structural number. Then the angle difference between the two adjacent grains can be determined as
1
= ω · t = ω
O
L sin ˛j
(14)
v
The wheel meshes with the involutes gear at the second point K2 . The coordinates of point K2 can be described as
⎧ L sin ˛j L sin ˛j L sin ˛j ⎪ − 1 − ω cos 1 − ω ⎨ x2 = rbs sin 1 − ω v v v
⎪ ⎩ y2 = rbs cos 1 − ω L sin ˛j + 1 − ω L sin ˛j sin 1 − ω L sin ˛j v
v
v
Fig. 3. Trajectory of two grains.
xD = x + h cos ˇ
(15) (6)
yD = y − h sin ˇ xE = x + |KE| sin ˇ = x + h tan ˛ sin ˇ yE = y + |KE| cos ˇ = y + h tan ˛ cos ˇ
|K2 E2 | = |K2 C2 | = h2 · tan ˛ (7)
where ˇ is the angle between the normal of involutes and the axis x. 3.2. Intersecting point of the trajectories of two grinding wheel grains
xC1 = x1 − h1 tan ˛ sin ˇ1
where subscript 2 indicates the second grain, and h2 denotes its protrusion heights. Then the coordinates of the vertex of the second triangle can be described as
xD1 = x1 + h1 cos ˇ1
(9)
yD1 = y1 − h1 sin ˇ1 xE1 = x1 + h1 tan ˛ sin ˇ1
y1 = rbs [cos(1 ) + 1 sin(1 )]
(11)
(18)
xE2 = x2 + K2 E2 sin ˇ2 = x2 + h2 tan ˛ sin ˇ2
(19)
xCi = xi − hi tan ˛ sin ˇi
(20)
yCi = yi − hi tan ˛ cos ˇi
x1 = rbs [sin(1 ) − 1 cos(1 )]
(17)
Similarly, for the ith grain, its coordinates of the vertex of the triangle can be described as
where subscript 1 indicates the first grain, h1 denotes its protrusion heights, and x1 , y1 can be obtained from Eq. (1) as
xD2 = x2 + h2 cos ˇ2
yE2 = y2 + K2 E2 cos ˇ2 = y2 + h2 tan ˛ cos ˇ2
(10)
yE1 = y1 + h1 tan ˛ cos ˇ1
xC2 = x2 − |K2 C2 | sin ˇ2 = x2 − h2 tan ˛ sin ˇ2
yD2 = y2 − h2 sin ˇ2
(8)
yC1 = y1 − h1 tan ˛ cos ˇ1
(16)
yC2 = y2 − |K2 C2 | cos ˇ2 = y2 − h2 tan ˛ cos ˇ2
Two adjacent grinding wheel grains are considered in Fig. 3. The triangle vertex coordinates of the first grain D1 can be computed according to Eqs. (4)–(7) as
The line segment length of K2 E2 and K2 C2 can be described as
xDi = xi + hi cos ˇi
(21)
yDi = yi − hi sin ˇi
xEi = xi + hi tan ˛ sin ˇi
(22)
yEi = yi + hi tan ˛ cos ˇi where xi , yi can be expressed as
⎧
L sin ˛j L sin ˛j L sin ˛j ⎪ ⎪ x = r sin − (i − 1)ω − − (i − 1)ω cos − (i − 1)ω ⎪ 1 1 1 ⎨ i bs v v v
⎪ L sin ˛j L sin ˛j L sin ˛j ⎪ ⎪ + 1 − (i − 1)ω sin 1 − (i − 1)ω ⎩ yi = rbs cos 1 − (i − 1)ω v
v
v
(23)
720
H. Chen et al. / Journal of Materials Processing Technology 213 (2013) 717–721 Table 1 Grinding wheel and grinding process parameters.
4. Surface roughness modeling To give a more realistic prediction of the surface roughness, the actual distribution of the grinding wheel grain protrusion heights must be considered with different heights. Though the model presented in Eqs. (20)–(23) provides a way to get the coordinates of the vertex of the triangle, the remaining problem is how to solve the final gear surface profile in a systematic way given that there are thousands of grains in the wheel. A search algorithm is also proposed in this paper. As gear mesh belongs to point contact, the marks left on the surface of the workpiece are related to the grain at the meshing point. The traces of adjacent grains will intersect with each other. The grain meshing at the addendum position should be considered first. Then, if the second grain does not cut into the trace generated by the first one, the headmost trace will be the final one. On the contrary, if the second grain does cut into the trace generated by the first one, the profile of the first trace will be changed. The process of modifying the surface profile continues with more grains involved in generating order, and it terminates when the grain meshes at the dedendum. Mathematically, the search method is described as follows. The surface profile of the first grain is formed by connecting three points of the first triangle and is denoted as s1 ⇒ K10 (x, y)
(24)
Grains and gear surface profiles are indexed by subscript and superscript respectively. The superscript value is set to be 0 when the profile is generated by a single grain and it is larger than 0 when the profile is generated by more than two grains. Similarly, the surface profile of the second grain is formed by connecting three vertex of the second triangle, which is determined with Eqs. (17)–(19), and can be expressed symbolically as s2 ⇒
K20 (x, y)
Fig. 4. Surface profile.
1 2
(25)
There are no intersections between the first two profiles when K10 (x, y) ∩ K20 (x, y) = 0. It can be indicated that there is too long grain interval to adopt the process and thus a better grit grinding wheel should be chosen. There are intersecting points between K10 (x, y) and K20 (x, y), when K10 (x, y) ∩ K20 (x, y) = / 0, a new geometry K1 is formed by the combined result, which is satisfied by the following condition K 1 (x, y) = min(K10 (x, y), K20 (x, y))|y
(26)
Which means that the showed surface profile is formed by a series of cut-off sections between the two individual profiles. A cutff section selected from one profile verse another depends on the profile value y that is smaller, which represents the marks left on the surface of the workpiece. The third grain with its individual geometry K30 is considered and as the search process continues, a new geometry K2 is formed from the combined result and satisfies the following condition: K 2 (x, y) = min(K 1 (x, y), K30 (x, y))|y
(27)
Kn
that is formed by the following
The final profile is given as term:
Wheel: 99A120K8V Conic vertex angle 2˛ : 141◦ Slant angle of wheel spindle ˛j : 20◦ Linear velocity v : 2.75 mm/s
0 (x, y))|y K n (x, y) = min(K n−1 (x, y), Kn+1
3 4
Fig. 5. Sample method.
Fig. 4 shows the simulation results on gear cross section when the grains have random heights. A gear is gained by the grinding condition as shown in Table 1, and is measured by the roughness measuring instrument HOMMEL WERKE. Because the probe is unable to reach the dedendum, four
(28)
And n is the last grain meshing with the gear at the dedendum. 5. Simulation and prediction of the surface roughness of generating grinding The proposed method is applied to simulate the surface roughness of generating grinding gear, and the grinding data are listed in Table 1.
Fig. 6. Ra results from experiment and simulation.
H. Chen et al. / Journal of Materials Processing Technology 213 (2013) 717–721
721
grinding trace becomes sparse and the arris height enlarges from dedendum to addendum. As a result, the Ra values increase little by little. 6. Conclusions A new method of predicting the gear surface profile for the generating grinding process is developed in this paper. It takes into account the random distribution of the grain protrusion heights. The simulation shows that the root part of surface is smoother than the top part, and the simulated results are consistent with the experiment, proving the validity of the modeling and predicting method of surface roughness for generating grinding gear presented in this paper. Acknowledgement Fig. 7. Rmax results from experiment and simulation.
The authors gratefully acknowledge the support of the Fundamental Research Funds for the Central Universities of Central South University (2012zzts013), the Notional Key Basic Research Program of China (2011CB706800), the National Science Foundation of China (51275530) and Hunan Provincial Innovation Foundation for Postgraduate (CX 2012B050). References
Fig. 8. Rz results from experiment and simulation.
samples which denote 1, 2, 3 and 4 are taken start from the pitch circle along the tooth profile, as shown in Fig. 5. Each sample length is 0.48 mm, cut off is 0.08 mm. The surface roughness results for experiment and simulation are illustrated in Figs. 6–8. The results show that the simulation values are in agreement with the experimental observations. The surface roughness parameter Ra , Rmax and Rz are increasing monotonously along the tooth profile direction. Considering that the space between two grains is equal here, and the feed rate of rack is changeless, the velocities of meshing points increase from dedendum to addendum and the curvature radiuses augment gradually at the same angular velocity. Hence, the moving distances of the meshing points also increase accordingly at the same time, which leads to the result that the
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