Table 2 shows the 18 analysis models of headstock structure for design optimization using Taguchi method. The SN ratio h and the sensitivity  S  are  expressed  in  Eq.  (3)  and  (4)  respectively. The

thermal displacement result from CAE analysis at the spindle speed of 2000 rpm was used for the calculation.

h    10 log  1 (3)

Ve

n

X

combination with the smallest deviation of thermal displacement at different spindle speed. To verify this combination, the thermal displacement in X direction with spindle speed at 500, 1000, 1500 and 2000 rpm was analyzed respectively. Fig. 5 shows the results: thermal  displacement  in  X  axis  was  —0.0005,  —0.0001      and

—0.0005 mm correspondingly at speed 500, 1000 and 1500 rpm, which confirms the small deviation. Note: No. 11 was selected with control  factors  of  (4,  2,  1,  1,  3,  3,  2)  even  though  the  best

combination was (4, 2, 1, 2, 3, 3, 2) considering the biggest SN ratio. This means the control factor D, thickness of rear wall is 15 mm instead of 25 mm. The reasons behind are two: the SN ratio difference between value 1 and 2 of D was very small as shown in Fig. 4; better thermal conductivity can be gained by using same thickness for the front wall (C) and real wall    (D).

Accordingly, Fig. 6 is the factorial effect graph of sensitivity. As shown in Eq. (4), the sensitivity is calculated as the average of thermal displacement. The effect and the direction of thermal displacement for each control factor were also shown in Fig. 6. It is clear that the Rib II, control factor A with level 3 and 4, has large effect on thermal displacement. As it affects the thermal displacement in negative direction, this is an important factor to minimize the displacement  amount.

S ¼ n

i¼1

xi (4)

where

1

Ve ¼

—

n

1 ðSt — SmÞ

1    n !2

St ¼ Xx2; Sm ¼

i¼1

Xxi

i¼1

Fig. 4. Factorial effect graph of SN ratio.

xi:   thermal   displacement   result   from   CAE   analysis   in   X  axis

direction at each spindle speed  (mm).

Fig. 4 is the factorial effect graph derived from the SN ratio results in Table 2. It can be observed that the smaller the deviation of thermal displacement in X axis direction is, the bigger the SN ratio is. This is because that the SN ratio is the inverse number of the variance Ve shown in Eq. (3). As shown in Fig. 4, each control factor is paired with a biggest SN ratio, for example, level 4 to the control factor A and level 2 to the control factor B. Finally, the analysis model No. 11 of headstock structure is determined as the

Fig. 5. Analysis result of the displacement in X axis direction.

334 M. Mori et al. / CIRP Annals - Manufacturing Technology 58 (2009)   331–334

Fig. 9. Comparison of front and rear wall temperature    change.

Fig. 6. Factorial effect graph of sensitivity.

Fig. 7. Thermal displacement in X axis direction 10 h after spindle start rotation.

Fig. 8. Thermal displacement in Y axis direction 10 h after spindle start rotation.

6.