multi-objective optimization of process parameters for · pdf filemulti-objective optimization...

Journal of Mechanical Science and Technology 25 (6) (2011) 1519~1527

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0430-z

Multi-objective optimization of process parameters for the helical gear precision

forging by using Taguchi method Wei Feng1 and Lin Hua2,*

1School of Materials Science and Engineering, Wuhan University of Technology, China 2School of Automotive Engineering, Hubei Key Laboratory of Advanced Technology of Automotive Parts, Wuhan University of Technology,

Wuhan 430070, China

(Manuscript Received June 24, 2010; Revised March 10, 2011; Accepted March 20, 2011)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract Precision forging of the helical gear is a complex metal forming process under coupled effects with multi-factors. The various process

parameters such as deformation temperature, punch velocity and friction conditions affect the forming process differently, thus the opti-mization design of process parameters is necessary to obtain a good product. In this paper, an optimization method for the helical gear precision forging is proposed based on the finite element method (FEM) and Taguchi method with multi-objective design. The maximum forging force and the die-fill quality are considered as the optimal objectives. The optimal parameters combination is obtained through S/N analysis and the analysis of variance (ANOVA). It is shown that, for helical gears precision forging, the most significant parameters affecting the maximum forging force and the die-fill quality are deformation temperature and friction coefficient. The verified experi-mental result agrees with the predictive value well, which demonstrates the effectiveness of the proposed optimization method.

Keywords: Helical gear precision forging; Multi-objective optimization; Taguchi method; Finite element method; Orthogonal design ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Helical gears are widely applied as an important transmis-sion component in most of the mechanical and automotive industry. In recent years, there has been an increased interest in the production of gears by the precision forming technique. This is because of their inherent advantages compared with conventional cutting methods. The advantages include the excellent mechanical properties, less raw material, good toler-ance, high productivity and cost savings [1, 2].

The precision forging of the helical gear is a very compli-cated metal forming process under coupled effects with multi-factors. The various process parameters such as deformation temperature, punch velocity and friction affect the forming quality differently, thus the reasonable process parameter de-sign is very important. Actually, for lack of theoretical instruc-tion, the process parameters of gears precision forging were determined by repeated experiments with artificial experience, which consume a large amount of materials and time. As a result, the optimization design of process parameters is sig-nificant to obtain the desired goals such as achieving good die-

fill quality, reducing the forging force, increasing the die life, obtaining favorable grain size.

In recent years, many scholars have made a lot of research on the forming of helical gears. Samanta [3] proposed a proc-ess for cold extrusion of helical gears. Choi et al. [2, 4, 5] de-veloped a new method of cold extrusion for helical gears and analyzed it by using the upper-bound method. Lange et al. [6, 7] made a deformation analysis for the cold forging of helical gears by 3D FEM and analyzed the elastic deformation of the die by 3D-BEM. Yang [8] investigated the clamping-type forging of helical gears through experiments and analysis by FEM. Jung [9] proposed the extrusion of helical gears by two-step process to reducing the forming load. However, previous methods mainly focused on the formability of helical gears by cold extrusion. Optimization technique with multi-objective design of process parameters for a helical gear warm precision forging has not been reported.

Due to the inherent complexity of forming processes, the helical gear forging is of high forging pressure which causes failure, plastic deformation and wear of die. Moreover, it is difficult to fill the teeth corner because of the helical shape. The forging pressure and filling conditions can be predicted by rigid-plastic finite element simulation. This is performed by combining the FEM with an optimization technique allow-ing the adjustment of process parameters in order to meet the

† This paper was recommended for publication in revised form by Editor Dae-Eun Kim

*Corresponding author. Tel.: +86 27 87168391, Fax.: +86 27 87168391 E-mail address: [email protected]; [email protected]

© KSME & Springer 2011

1520 W. Feng and L. Hua / Journal of Mechanical Science and Technology 25 (6) (2011) 1519~1527

specified demand. In this way, the control of process parame-ters is possible and allows us to obtain products with the de-sired demand, such as the shape and so on. Thus, many re-searchers have paid attention to combining the FEM with an optimization technique to optimize process parameters. Guo-qun Zhao and Xinhai Zhao [11, 12] used a finite element-based sensitivity analysis method to optimize the preform die shape for metal forming processes to minimize the difference between the realized and the desired final forging shapes and to minimize the effective strain variation within the final forg-ing, respectively. X. Chen [13] proposed a process parameters optimization method for the hot forging process of gear based on FEM and Taguchi method to minimize the forging force during gear hot forging process. Y.K. Lee [14] simulated the bevel gear forging process by three dimensional FEM based on rigid-plastic material modeling and found a defect-free forging process to improve the product quality and to secure the effective material flow by optimizing the die. Xiaoming He [15] proposed a robust parameters control methodology based on Taguchi method and numerical simulation to control the microstructure of heavy forgings, uniform and small mi-crostructure of the final forging was considered as the objec-tive function.

In this paper, the multi-objective optimization of the helical gear warm precision forging process is studied using a weight-ing factor in the signal-to-noise ratio of the Taguchi method. The multi-objective design includes smaller forging force and better filling condition of gear teeth. Taguchi method deter-mines the optimal values of deformation temperature, punch velocity, friction factor for a given helical gear to minimize forging force and to improve filling condition of gear teeth. The helical gear precision forming process is studied using a FEM to provide the solutions of the deformation, the stress and the internal variable fields under different forming proc-esses. The analysis of variance is also investigated for the multi-objective design parameters. In this study, we propose a method to analyze the effects of the process parameters on precision forging helical gear by combining finite element analysis with the Taguchi method.

2. Description of Taguchi method

Taguchi method [16] was developed by Taguchi, it is util-ized widely in designing and analysis of experimental method to optimize the performance characteristics through the setting of process parameters. It provides an integrated approach to determine the best range of designs simply and efficiently for quality, performance, and cost [17, 18].

In Taguchi method, three-stages such as system design, pa-rameter design, and tolerance design are employed. Parameter design is the key stage, which used to obtain the optimum levels of process parameters for developing the quality charac-teristics and to determine the product parameter values de-pending optimum process parameter values [19]. Based on orthogonal arrays, the number of experiments which may lead

to the increasing of the time and cost can be reduced by using Taguchi method. It employs a special design of orthogonal arrays to learn the whole parameters space with the least ex-periments only. Taguchi method employs the S/N ratio to identify the quality characteristics applied for engineering design problems. Usually, the S/N ratio characteristics can be divided into three types: the-lower-the-better, the-higher-the-better, and the-nominal-the-better [16]. A statistical analysis of variance (ANOVA) can be utilized to present the influence of process parameters on forging force and filling condition. In this way, the optimum levels of process parameters can be predicted.

3. Optimal design problem of the helical gear preci-

sion forging

3.1 FE modeling for precision forging of the helical gear

In this study, the helical gear product used as a component of the automobile transmission was formed by the clamping-type closed-die forging. The specification and dimensions of the adopted helical gear are shown in Table 1. The authors have built a 3D-FE model of isothermal precision forging of the helical gear in DEFORM-3D software as shown in Fig. 1. The process conditions in the FE-simulation are: (1) due to high temperature and large deformation in the process, elastic deformation is negligible and all the dies are regarded as the rigid body and the billet is the deformed body; (2) the friction at the billet-dies interfaces was assumed to be of shear type; (3) the temperature of environment is assumed as the room temperature, 20°C; (4) the initial tetrahedral solid elements of the billet were around 100,000 and automatic remeshing tech-

Table 1. Dimension of the helical gear.

Number of teeth 24 Normal Module 1.745 Normal pressure angle 20° Helix angle 30.3° Modification coefficients 0.228 Width 30mm

.

Fig. 1. 3D-FE model of helical gear warm forging.

W. Feng and L. Hua / Journal of Mechanical Science and Technology 25 (6) (2011) 1519~1527 1521

nique was adopted during simulation; (5) for the complexity of forming processes, the mesh density around the teeth is considered higher to improve the deformation accuracy in these area; (6) the billet material used in the model is the AISI-4120; (7) the governing equations for the solution of the mechanics of rigid-plastic deformation do not consider the volume force and satisfy with the equilibrium equation, the geometrical equation, volume constancy, and the material obeying the Mises yield criterion; (8) using the penalty func-tion method to handle the condition of volume constancy.

3.2 Selection of design parameter based on the Taguchi

method

The goal of the parameter design is to optimize the process parameter values for improving objective functions so as to obtain the desired high quality component without increasing cost under the optimal process parameter. During warm forg-ing processes of helical gear, process parameters such as de-formation temperatures of the billet, interface friction, punch velocity, punch stroke, and flow stress of the billet material have a great influence on the mechanical and metallurgical properties of the final products. Deformation temperature (T) greatly affects the flow and deformation pressure of material, performance and surface quality of the final forging product. Many materials are very sensitive to strain rate, and the distri-bution of strain rate in the forging process greatly affects the deformation behavior. The control of strain rate is usually realized by controlling punch velocity (v). The friction coeffi-cient (µ) of the interface between billet and tools affect tool wear and deformation pressure. Therefore, deformation tem-perature, punch velocity and interface friction coefficient are chosen as optimal process parameters design variables in this study.

3.3 Objective functions

In the helical gear forging process, the high forging force significantly cause failure, plastic deformation and wear of dies. It is one of the most essential factors to take into account when choosing forging equipment. For the same forging, if the maximum forging force can be lessened, small tonnage equipment can be used. This will help to lengthen the life span, and reduce the cost of forging. Therefore, a smaller forging force is one objective that should be looked forward to in forg-ing technology and the process of die design. Hence the maximum forging force Fmax is selected as the optimal objec-tive.

The maximum forging force can be defined as the sum of the forces of all the element nodes that have contacted with the upper die along the Z direction. It is expressed as:

max1

n

izi

F f=

=∑ (1)

where fiz is the maximum Z direction force of the element node i that is in contact with the upper die. n is the total node number that is in contact with the upper die.

In addition, the die-fill condition is also selected as another optimal objective in this work. Incompletely filling of gear teeth is one of the main forging defects. The primary shape requirement in forging design is to make the billet to fill the die cavity adequately and obtain the forging product with less or no flash. The quality of the gear teeth will directly affect the accuracy of dimension and geometry for the final forging part. There are many different evaluation criteria for predicting the filling of gear teeth in gear forging processes, such as the shape difference of the desired final forging and the actual final forging, and the volume difference between them. The minimum distance Dmin is used here as a measure of the filling condition, and it can be defined as:

( ) ( )2 2min ib jd ib jdD x x y y= − + − (2)

where Dmin displays the minimum distance from the surface of the workpiece to the nearest tool, xib, yib is the element node i coordinate values on the surface of the workpiece, xjd, yjd is the element node j coordinate values on the tool surface. The smaller the value of Dmin is, the more adequate the filling of gear teeth. 3.4 Construction of orthogonal array experiment

A large number of experiments need to be carried out when the number of the process parameters increases. To solve this problem, the Taguchi method uses a special design of or-thogonal arrays to study the entire parameter space with only a small number of experiments.

In the process parameter design of the helical gear, four lev-els of the process parameters were selected, as shown in Table 2. Because the interaction between the deformation tempera-ture and velocity is considered, the interaction of two factors can be treated as a new factor according to Taguchi’s sugges-tion. For four factors with four levels, the experimental layout of an L16 (45) orthogonal array is selected for the present re-search. Table 3 shows the L16 orthogonal array in which the 16 rows are carried out to investigate the effects of the four fac-

(a) (b) Fig. 2. The meshed FE model (a) initial billet; (b) deformed block.


tors. The FEM software Deform-3D is used to simulate the gear warm forging process and to calculate the optimal objec-tive function value. Table 4 shows the simulation results of maximum forging force Fmax and minimum distance Dmin.

4. The Taguchi method with multi-objective design

4.1 Signal-to-noise ratio (S/N ratio)

Taguchi method utilizes the S/N ratio approach instead of the average value to transform the FEM results into a value for the evaluation characteristic in the optimum parameter analysis.

This value can estimate the main effect of each factor and their levels on the optimization object. The S/N ratio η is ex-pressed in dB units and it can be defined as below:

10log( )MSDη = − (3)

where MSD is the mean-square deviation for the output char-acteristic. Usually, there are three types of quality characteris-tics in the analysis of the S/N ratio: the nominal-the-better, the smaller-the-better, and the higher-the-better.

To obtain optimal process parameters, the smaller-the-better quality characteristics for the maximum forging force Fmax and the minimum distance Dmin should be taken. The MSD for the smaller-the-better quality characteristic can be expressed as below:

2

1

1MSDn

ii

Sn

=

= ∑ (4)

where Si is the value of maximum forging force Fmax and minimum distance Dmin for the ith experiment and n is the number of tests of data points in an experiment. After con-ducting S/N analysis, S/N ratio values of the maximum forg-ing force Fmax and the minimum distance Dmin are listed in Table 4.

4.2 Taguchi multiple objective optimization based on weight-

ing method

It is suitable for the optimization of only single object to use Taguchi method, so it must be modified to solve multi-objective optimization problems. A weighting method is made use of to determine the importance of each optimum object in this paper. Then the optimization of multiple objective is con-verted into that of single objective by weighting method.

The multi-objective S/N ratio ηj in the jth experiment is de-fined as follows:

2

j kj1

η ηkk

ω=

=∑ (5)

2

1

1kk

ω=

=∑ (6)

where ηj is the multi-objective S/N ratio in the jth experiment, ηjk is the kth single objective S/N ratio for the jth experiment. ωk is the weighting factor in the kth single objective S/N ratio and k is the number of optimization objective. Table 5 shows the multi-objective S/N ratio with different combinations of the weighting factors.

4.3 S/N and ANOVA analysis

The larger the multi-objective S/N ratio is, the smaller the

Table 2. Process parameters and their levels.

Symbol Process parameters

Level 1

Level 2

Level 3

Level 4

A B C

T (℃) v (mm/sec)

µ

750 30

0.05

800 50 0.1

850 80

0.25

900 100 0.45

Table 3. Experimental L16 orthogonal array.

Exp. No. Process parameters A B A×B C

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 1 1 1 1 2 2 2 1 3 3 3 1 4 4 4 2 1 2 3 2 2 1 4 2 3 4 1 2 4 3 2 3 1 3 4 3 2 4 3 3 3 1 2 3 4 2 1 4 1 4 2 4 2 3 1 4 3 2 4 4 4 1 3

Table 4. Simulation results for maximum forging force Fmax and mini-mum distance Dmin.

Exp. No.

Fmax (KN) Dmin (mm) S/N S/N for Fmax(dB) for Dmin(dB)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2370 0.991 -67.49 0.079 2640 1.000 -68.43 0

2850 0.991 -69.00 0.079 3370 0.988 -70.55 0.105 2000 0.997 -66.02 0.026 2770 1.000 -68.85 0 2150 0.983 -66.65 0.149 2270 1.000 -67.12 0 2410 1.010 -67.64 0.086 2120 0.992 -66.53 0.07 1970 1.000 -65.89 0 1970 0.992 -65.89 0.07 1680 0.996 -64.51 0.035 1650 0.990 -64.35 0.086 2540 0.996 -68.00 0.035 2110 1.000 -66.49 0


variance of performance characteristics around the desired value is. The multi-objective S/N ratio for case 1 to 3 for each parameter at different levels is plotted in Figs. 3, 4 and 5, re-spectively. From Figs. 3-5, it can be easily find that the opti-mum factor level combination is A4B1C1for each case, that is the maximum forging force Fmax and minimum distance Dmin are minimum at forth level of deformation temperature (A4), first level of punch velocity (B1), first level of friction coeffi-cient (C1).

In order to investigate the effects of the process parameters on the optimization objective quantitatively, analysis of vari-ance (ANOVA) is carried out. It utilizes the total sum of squares SST , which is a deviation of the multi- objective S/N ratio from the total average S/N ratio η , to evaluate the sig-nificance of process parameters on optimization objective. SST can be calculated as below:

2

1

( )m

T jj

SS η η=

= −∑ (7)

2

1

1 ( )m

jj

mη η

=

= ∑ (8)

where m is the number of experiments in the orthogonal array (m=16 in this study) and ηj is the multi-objective S/N ratio in the jth experiment.

The sum of squares due to the variation from the average S/N ratio for factor p is given by

2

2

1 1

( ) 1 ( )kpr m

p jk j

SSS

r mη

η= =

= −∑ ∑ (9)

where p represents one of the process parameters, k is the level

number of this parameter p, r is the repetition of each level of the parameter p, and

ipSη is the sum of the multi-response S/N

ratio involving this parameter p and level k. The sum of squares for error SSe is given by

( )e T A B C A BSS SS SS SS SS SS ×= − + + + (10)

where SSA, SSB, SSC, SSA×B is sum of squares for factor A,B,C,A×B, respectively.

Sum of squares divided by corresponding degree of free-dom (DOF) can give the mean square. Mean square for each factor MSp and error mean square are given respectively by

Table 5. Multi-objective S/N ratio with different weighting factors.

Multi-objective S/N ratio(dB)

Case 1 Case 2 Case 3 Exp. No. ω1=0.5 ω1=0.4 ω1=0.6 ω2=0.5 ω2=0.6 ω2=0.4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

-33.71 -26.95 -40.46 -34.22 -27.37 -41.06 -34.46 -27.55 -41.37 -35.22 -28.16 -42.29 -33.00 -26.39 -39.60 -34.43 -27.54 -41.31 -33.25 -26.57 -39.93 -33.56 -26.85 -40.27 -33.78 -27.00 -40.55 -33.23 -26.57 -39.89 -32.95 -26.36 -39.53 -32.91 -26.31 -39.51 -32.24 -25.78 -38.69 -32.13 -25.69 -38.58 -33.98 -27.18 -40.79 -33.25 -26.60 -39.89

Fig. 3. Average values of S/N rations for case 1.




pp

p

SSMS df= (11)

ee

e

SSMS df= (12)

where dfp, dfe represents degree of freedom for a factor p and error, respectively.

The F-ratio value for each process parameter is the ratio of mean square for a factor p to error mean square, it can be cal-culated as

.pP

e

MSF

MS= (13)

The percentage contribution Cp for a factor p can be calcu-

lated as

.p p ep

T

SS df MSC

SS− ×

= (14)

Tables 6, 7 and 8 show the results of ANOVA for cases 1, 2

and 3 respectively. It can be seen from the F-ratio value and the percentage contribution result that the significant parame-ters influencing multi-objective optimization design are de-

formation temperature and friction coefficient. The effect of punch velocity is very small compared to that of deformation temperature and friction coefficient. The effect of interaction between deformation temperature and punch velocity in each case is very small, so they could be ignored.

5. Confirmation test

Based on the S/N ratio and ANOVA analysis, the optimal process parameters combination for cases 1, 2 and 3 are A4B1C1. However, the optimum factor level combination is A4B2C1 according to visual analysis from Table 5. Therefore, a confirmation test should be carried out to evaluate the opti-mal combination.

Confirmation test was carried out to predict and verify the improvement of optimization objective, after the optimal level of the design parameters has been selected. The estimated optimum value of S/N ratio ηopt of A4B1C1 for each case can be calculated as

4 1 1 ( 1)opt A B C Pη η η η η= + + − − × (15)

where 4Aη is the average S/N ratio for factor A at level 4 obtained from Table 5, 1Bη is the average S/N ratio for fac-tor B at level 1 obtained from Table 5, 1Cη is the average S/N ratio for factor C at level 1 obtained from Table 5, η is the mean of S/N ratio obtained from Table 3. Executing forg-ing FEM simulation under the optimum setting condition, that is deformation temperature is 900℃, punch velocity is

Table 6. Results of ANOVA for case 1.

Source Sum of squares DOF Mean

square F-

ratio Contribution(％)

A B C

A×B Error Total

5.02 0.72 4.16 0.02 0.25 10.17

3 3 3 3 3 15

1.67 0.24 1.39 0.01 0.08

20.88 3

17.38 0.13

47.00 4.72 38.55 0.00 9.73

100.00



square F-


A B C

A×B Error Total

3.19 0.48 2.68 0.02 0.14 6.51

3 3 3 3 3 15

1.06 0.16 0.89 0.01 0.05

21.2 3.2 17.8 0.2

46.70 5.07 38.86 0.00 9.68

100.00



square F-


A B C

A×B Error Total

7.27 1.05 6.00 0.02 0.35 14.69

3 3 3 3 3 15

2.42 0.35 2.00 0.01 0.12

20.17 2.92 16.67 0.08

47.04 4.70 38.39

0 9.87

100.00

Table 9. Results of the confirmation test for case 1.

Optimal parameters (A4B1C1)

Initial parameters (A4B2C1) Prediction Experiment

T (℃) v (mm/sec)

µ Fmax (kN) Dmin (mm)

S/N ration(dB) Relative error of S/N Improvement of S/N

900 50

0.05 1650 0.990 -32.13

0.28 dB

-32.05 0.63%

900 30

0.05 1550 0.986 -31.85

Table 10. Results of the confirmation test for case 2.

Optimal process Parameters (A4B1C1)

Initial parameters (A4B2C1) Prediction Experiment

T (℃) v (mm/sec)

µ Fmax (kN) Dmin (mm)

S/N ration (dB) Relative error of S/N Improvement of S/N

900 50

0.05 1650 0.990 -25.69

0.23 dB

-25.61 0.59%

900 30

0.05 1550 0.986 -25.46


30mm/s , friction coefficient is 0.05, the optimum result can be obtained. Tables 9, 10 and 11 show the results of the con-firmation test.

The obtained maximum forging force Fmax is 1550kN and its S/N ratio is -63.81, the obtained minimum distance Dmin is 0.986 and its S/N ratio is 0.114. Compared with the results obtained in the initial process parameters A4B2C1, i.e., de-formation temperature is 900℃, punch velocity is 50mm/s, and friction coefficient is 0.05, it shows that both the maxi-mum forging force Fmax and the minimum distance Dmin are improved under the optimal setting of the process parameters which is determined by the approach presented in this study. The increase of the multi-objective S/N ratio from the initial process parameters to the optimal process parameter is 0.28 dB, 0.23dB and 0.34dB for case1 to 3, respectively. Based on the result of the confirmation test, the relative error between the estimated optimum value and the optimum experiment value of the multi-objective S/N ratio for case1 to 3 is 0.63%, 0.59% and 0.6%, respectively, they are within the engineering requirement. This indicates that the optimal settings of process parameters obtained by modified Taguchi approach are reli-able, i.e., deformation temperature is 900℃, punch velocity is 30mm/s, and friction coefficient is 0.05. The minimum dis-tance Dmin distribution and the forging Load-Stroke curve after optimization with before are shown in Figs. 6, 7 and 8, respec-tively.

Fig. 6(a) shows the percentage of the nodes with the same minimum distance Dmin of the total nodes, and Fig. 6(b) shows the Dmin distribution in the deformed block before optimiza-

tion. It can be seen from Fig. 6(a) that the range of the same Dmin is 0~0.02mm and their percentage of nodes is 50.337%, moreover, the filling condition isn’t good in the bottom of the forged gear, as shown in Fig. 6(b).

Fig. 7(a) shows the percentage of the nodes with the same minimum distance Dmin of the total nodes, and Fig. 7(b) shows the Dmin distribution in the deformed block after optimization. It can be seen from Fig. 7(a) that the range of the same Dmin is 0~0.015mm and their percentage of nodes is 50.934%, more-over, the filling condition is very good in the whole forged gear, as shown in Fig. 7(b).

Fig. 8 gives the comparison of the forging Load-Stroke curve for the initial and the optimized parameters. It can be observed from Fig. 8 that the forging load after optimization is hardly less than before during forging. Furthermore, after op-timization, the maximum forging force Fmax is 1550kN in heli-cal gear forming which is 6% lower than before.

6. Conclusions

Multi-objective optimization design based on the Taguchi method and the finite element method has been implemented for the helical gear warm forging process in this paper. In the light of S/N ratio analysis, variance analysis and FEM simula-tion results, the following conclusions can be drawn:

(1) The significant forming parameters affecting the helical gear warm forging process such as deformation temperature, friction coefficient, and punch velocity can be easily recog-nized by performing the experiments which are designed based on the orthogonal array of the Taguchi method.

(2) Deformation temperature and friction coefficient affect the helical gear warm forging process greatly regardless of case 1 to 3, they contribute about 86% together, while punch velocity does not affect the helical gear warm forging process too much, it only contributes about 5%, and interaction be-tween deformation temperature and punch velocity has little effect on the helical gear warm forging process.

(3) For a given helical gear forming process, the optimal combination of process parameters can be determined through the modified Taguchi method to obtain the minimum values of the maximum forging force Fmax and the minimum distance

(a) (b) Fig. 6. The minimum distance Dmin distribution before optimization (a)isoregion distribution; (b) distribution of deformed block.

(a) (b) Fig. 7. The minimum distance Dmin distribution after optimization (a)isoregion distribution; (b) distribution of deformed block.

Fig. 8. Comparing of the forging Load-Stroke curve after optimizationwith before.


Dmin. That is when forming process parameters deformation temperature is 900℃, friction coefficient is 0.05, and punch velocity is 30mm/s, the maximum forging force Fmax and the minimum distance Dmin are the smallest simultaneously.

(4) The optimal process parameters were confirmed with confirmatory experiments.

Acknowledgment

This work has been supported by the Natural Science Foun-dation of China for Distinguished Young Scholars (No. 50725517).

Nomenclature------------------------------------------------------------------------

T : Deformation temperature (℃) v : Punch velocity(mm/s) µ : Friction coefficient Fmax : Maximum forging force (kN) x, y, z : Cartesian coordinates fiz : z direction force of the element node i Dmin : The minimum distance (mm) xib, yib : Coordinate value of node i on the workpiece surface xjd, yjd : Coordinate value of node j on the tool surface η, ηj : Signal-to-noise ratio(dB), multi-objective S/N ratio in

the jth experiment(dB) η : The total average S/N ratio(dB) ηopt : The estimated optimum value of S/N ratio MSD : Mean square deviation Si : The ith experiment value of optimal objective ωk : Weighting factor in the kth single objective S/N ratio SST : The total sum of squares SSp : The sum of squares for factor p SSe : The sum of squares for error MSp : Mean square for factor p MSe : Mean square for error dfp : Degree of freedom for factor p dfe : Degree of freedom for error Fp : F-ratio value for factor p Cp : Percentage contribution for factor p

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Wei Feng is a Ph.D candidate and is also a lecturer of Material processing Engineering at Wuhan University of Technology. She received her M.S. degree in Pressure Processing from Wuhan University of Technology, China, in 2002. Her research areas include advanced forming and its

applications.

Lin Hua received his M.S. degree in Pressure Processing from Wuhan Uni-versity of Technology, China, in 1985. He then received his Ph.D. degree in Mechanical Engineering from Xi’an Jiaotong University, China, in 2000. Dr. Hua is currently a professor at the School of Automotive Engineering,

Hubei Key Laboratory of Advanced Technology of Automo-tive Parts at Wuhan University of Technology, China. Dr. Hua’s research interests include advanced forming and equipment technology.

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