96 CHAPTER 5 NON-LINEAR SURROGATE MODEL 5.1 INTRODUCTION As set out in the research methodology and in sequent with the previous section on development of LSM, construction of the non-linear surrogate model is presented in this section. The main aim of this phase of work is to bring out the non-linear behaviour of the processing variables influence on the response sink depth and to include those terms in the prediction model to improve its effectiveness. CCD-RSM design of experiments is used for the construction of NLSM. Results of the experiments and a summary on NLSM are also presented in this chapter. 5.2 NON-LINEAR SURROGATE MODELLING The CCD-RSM designs of experiments require five levels for the variables. The variables and its levels used in this phase are tabulated in the Table 5.1.
97 Table 5.1 Variables and its levels for Central Composite Design Response Surface Methodology Melt Mould Packing Rib-towall Levels Temperature Temperature Pressure ( C) ( C) (MPa) Ratio (%) Un-coded T me T mo P pr R Coded T me T mo P pr R -2 200 20 20 35-1 220 40 23 50 0 240 60 26 65 1 260 80 29 80 2 280 100 32 95 5.2.1 CCD-RSM for NLSM Based on the CCD array, a set of twenty-six experiments were needed inclusive of the sixteen experiments of full factorial design of experiments as discussed in the research methodology. Hence, additional ten experiments were conducted at centre and axial points. The maximum sink mark depths were measured at the intersections of rib and base wall. Recorded sink mark depths along with CCD array for the twenty-six experiments are tabulated in Table 5.2.
98 Table 5.2 Experimental results of Central Composite Design Variables Melt Temperature in C Mould Temperature in C Pack Pressure in MPa Rib-towall ratio in % Sink Depth in mm Trial No. (T me ) (T me ) (P pr ) (R) (S) 1 220 40 23 50 0.0888 2 220 40 23 80 0.0957 3 220 40 29 50 0.0676 4 220 40 29 80 0.0756 5 220 80 23 50 0.0911 6 220 80 23 80 0.0995 7 220 80 29 50 0.0718 8 220 80 29 80 0.0793 9 260 40 23 50 0.0586 10 260 40 23 80 0.0696 11 260 40 29 50 0.0465 12 260 40 29 80 0.0570 13 260 80 23 50 0.0660 14 260 80 23 80 0.0740 15 260 80 29 50 0.0533 16 260 80 29 80 0.0609 17 200 60 26 65 0.0968 18 280 60 26 65 0.0558 19 240 20 26 65 0.0651 20* 240 100 26 65 0.0745 21 240 60 20 65 0.0930 22 240 60 32 65 0.0567 23 240 60 26 35 0.0572 24 240 60 26 95 0.0775 25 240 60 26 65 0.0705 26 240 60 26 65 0.0705 * Due to the higher mold temperature (100 C) for 20 th run, ejection temperature was set at 105 C.
99 Regression analysis was carried out using the data collected as per the CCD array. The NLSM equation (in coded form) for the sink depth (S nl ) was formulated through second-order regression is given by the following Equation 5.1: S nl = (7.050 10-2 ) (1.106 10-2 ) T me (2.303 10-3 ) T mo (8.499 10-3 ) P pr (4.518 10-3 ) R (5.313 10-4 ) T me* T mo (1.894 10-3 ) T me* P pr (3.938 10-4 ) T me (3.063 10-4 ) T mo * R (4.375 10-5 ) T mo * P pr * R (4.375 10-5 ) P pr (1.476 10-3 ) T 2 me (1.479 10-4 ) T 2 mo (1.102 10-3 ) P 2 pr (7.629 10-4 ) R * R 2 (5.1) The Equation 5.1 can be rewritten as follows after removing the insignificant terms and using the uncoded variables: S nl = (74.40 10-2 ) (3.23 10-3 ) T me (1.15 10-4 ) T mo (1.716 10-2 ) P pr (7.03 10-4 ) R (3.16 10-5 ) T me * P pr (3.86 10-6 2 ) T me (1.3 10-4 ) P 2 pr (3.1 10-6 ) R 2 (5.2)
100 Significance tests were carried out to confirm the main effects, their interaction effects and non-linear 2 nd order effects. Three-way and four-way interactions were not included in the regression model building. The significance test results are listed in the Table 5.3. Table 5.3 Significance test results of NLSM Term Coeff SE Coeff T p Constant 0.07050 0.001409 74.23 <0.0001 T -0.01106 0.000288-40.35 <0.0001 me T 0.00230 0.000288 8.40 <0.0001 mo P -0.00850 0.000288-31.00 <0.0001 pr R 0.00452 0.000288 16.48 <0.0001 T me * T 0.00053 0.000352 1.58 0.142 mo T me * P 0.00189 0.000352 5.64 0.000 pr T mo * P pr 0.00004 0.000352 0.13 0.899 T me * T mo * P pr * R 0.00039 0.000352 1.12 0.2897 R -0.00031 0.000352-0.91 0.381 R -0.00004 0.000352-0.13 0.899 T me * T 0.00148 0.000419 4.59 0.001 me T mo * T mo -0.00015 0.000419-0.46 0.654 P pr * P pr 0.00110 0.000419 3.43 0.006 R * R -0.00076 0.000419-2.37 0.037
101 As the p values (refer Table 5.3) of the terms T me, T mo, R, T me, me T 2, P pr 2 and R 2 were found to be less than 0.05, these were considered to have significant contribution on the sink depth at 95% confidence level, whereas all other effects were found insignificant since their p values are more than 0.05. From the above table, it can be seen that all variables but for the mould temperature, which exhibited second order behaviour. From the significance test, it was observed that a similar kind of trend as that of LSM on the main effects and interaction effects. The melt temperature had a major contribution on the response sink. It had 35 % of direct share in controlling the sink. It is followed by the variable pack pressure with 26.5 %. The interactive effect between the melt temperature and pack pressure contributed to an extent of 4.84%. The significant second order terms, combined, contributed to the tune of 8.91% for the response sink depth. Figure 5.1 (a) to 5.1 (f) shows the 3D surface plots for the response of sink depth. Following observations can be drawn from the surface plots: 1. From the surface plots Figure 5.1 (a) 5.1 (c) it is found that increased melt temperature in combination with other factors has a positive influence on reduction of sink depth. This could be due to increased flowablity and longer gate-seal-off time.
102 T me T mo Figure 5.1 (a) Response surface plot for sink mark depth: Melt temperature Vs Mould temperature T me P pr Figure 5.1 (b) Response surface plot for sink mark depth: Melt temperature Vs Pack pressure R T me Figure 5.1 (c) Response surface plot for sink mark depth: Melt temperature Vs Rib-to-Wall ratio T mo P pr Figure 5.1 (d) Response surface plot for sink mark depth: Mould temperature Vs Pack pressure R R T mo P pr Figure 5.1 (e) Response surface plot for sink mark depth: Mould temperature Vs Rib-to-Wall ratio Figure 5.1 (f) Response surface plot for sink mark depth: Pack pressure Vs Ribto-Wall ratio Figure 5.1 3D response surface plots.
103 2. It is observed that from Figures 5.1 (b), 5.1 (d) & 5.1 (f) that, increased pack pressure reduces sink depth. From Figure 6.1 (b), it can be observed that combination of increased melt temperature with increased pack pressure aids rapid reduction in sink depth. It is also observed that, from ANOVA, melt temperature and pack pressure interaction has significant contribution on sink depth. This can be attributed to the fact that increased melt temperature makes the gate sealing time longer thereby effective packing can be achieved through higher pack pressure. 3. It is found that lower rib-to-wall ratio, in general, leads to less sink. This can be due to reduced thermal mass at the intersection. As is seen from the Figure 5.1 (c), lower rib-towall ratio with increased melt temperature reduces sink depth rapidly. This reduction could be due to increased flowablity due to high melt temperature. It is observed from Figure 5.1 (e) that, low rib-to-wall ratio in combination with low mould temperature reduces sink. This can be due to faster heat transfer between melt and mold. 4. It is observed that generally lower mold temperature causes less sink. From Figure 5.1 (d), it is found that lower mould temperature with increased pack pressure has an impact on reduction of sink. From Figure 5.1 (a), it is found that decreased mould temperature in combination with increased melt temperature decreases the sink depth. This could be due
104 to quicker thermal heat transfer during solidification and increased surface rigidity of the part under cooling. From the above observations, it could be concluded that sink depth decreases with higher melt temperature, higher pack pressure, lower mold temperature and lower rib-to-wall ratio. Contribution of processing variables on the sink depth is shown in Figure 5.2, graphically using the pie diagram. Melt Temp Mould Temp Pack Pr R ratio Melt Temp * Pack Pr Melt Temp^2 Pack Pr^2 R Ratio^2 others Figure 5.2 Contribution of variables on sink depth -NLSM ANOVA was performed to test the significance of the variables for the response sink depth. The results of the ANOVA are listed in Table 5.4.
105 Table 5.4 Analysis of variance table for Non-linear Surrogate Model Source SS df Adj SS Adj. MS F Value p-value Prob > F Model 5.46E-03 8 5.43E-03 6.82E-04 402.74 < 0.0001 T 2.94E-03 1 2.94E-03 1734.44 < 0.0001 me T 1.27E-04 1 1.27E-04 75.15 < 0.0001 mo P pr 1.73E-03 1 1.73E-03 1023.93 < 0.0001 R 4.90E-04 1 4.90E-04 289.28 < 0.0001 T me * P 5.74E-05 1 5.74E-05 33.89 < 0.0001 pr T 2 5.24E-05 1 5.24E-05 30.94 < 0.0001 me P 2 pr 3.01E-05 1 3.01E-05 17.77 0.0006 R 2 1.06E-05 1 1.06E-05 6.29 0.0226 Residual 2.88E-05 17 1.69E-06 The model F - value of 402.74 implies that the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise. Prob > F values less than 0.05 indicates that the model terms are significant. It is also inferred from the ANOVA that the model termst me, mo and R 2 are significant model terms. T, P pr, R, T me* P pr, T 2 me, P 2 pr
106 Figure 5.3 shows the normal probability plot of residuals. It shows no abnormality in the methodology adopted (R 2 = 0.9948). The R 2 analysis is listed in Table 5.5. The R 2 prediction of 0.9831 is in reasonable agreement with the R 2 adj of 0.9923. The PRESS and RMSE statistic were also found satisfactory. 99 95 N ormal Probability Plot (response is sink depth) 80 50 20 5 1-0. 002-0. 001 0.000 Re sidual 0.001 0.002 Figure 5.3 Normal Probability Plot for Residuals of Non-Linear Surrogate Model Table 5.5 R 2 Analysis of Non-Linear Surrogate Model Sl. No. Statistic Values 1 R 2 0.9948 2 R 2 adj 0.9923 3 R 2 prediction 0.9831 4 RMSE 1.301 E-03 5 PRESS 9.254 E-05
107 The statistical analysis shows that, the developed NLSM based on the CCD-RSM is also statistically adequate and can be used to navigate the design space. 5.3 SUMMARY By sequentially adding another ten experiments to the full factorial array, NLSM was constructed using the CCD-RSM array. Nonlinear second order behaviour is found with variables such as melt temperature, pack pressure and rib-to-wall ratio. Mould temperature did not exhibit second order behaviour. As highlighted by the LSM, all main effects and an interaction effect between melt temperature and pack pressure was found to be significant with NLSM also. Significance tests and other statistical tests established the adequacy of the model in prediction. The NLSM model can be used for navigating the design space. In order to find the better out of the two surrogate models i.e. LSM and NLSM, prediction performance evaluation was undertaken. In the following chapter, the performance evaluations of the surrogate models are presented.