Table 6.5 shows that at VA=1.04, the two different friction situations are providing same curvature, however, they provide different curvatures at VA =1.02. Similarly, Table 6.6 shows the results for a different case. Here also, at VA =1.06, two different friction situations are providing same curvature. At a different speed ratio i.e. VA =1.10, these two friction situations provide different curvature. Thus, it becomes easier to check the friction coefficient at the upper and lower rolls by carrying out the neural network based predictions at two or more speed ratios. Thus, the methodology proposed by Salunkhe [2006] for the estimation of coefficient(s) of friction in rolling can be effectively used with the assistance of a well-trained neural network. The neural network model can also be used for the estimation of curvature for controlling the curvature or imparting the desired curvature.
6.5.1 Fuzzy Modelling of Parameters
The inherent variation in the experimental investigation causes scatter in the experimental data. It causes large deviation with the analytical results also. Due to this, Dixit and Dixit [1996] considered the material and the process parameters as fuzzy parameters. Material and process parameters i.e. flow stress of the materialσY, strain hardening coefficients b and n, and coefficient of friction f are treated as fuzzy parameters which are characterized by their membership functions.
The details of this fuzzy modelling of parameters are explained in Subsection 5.4.1.
In this process of determining the fuzzy parameters, parameters are divided into three subsets such as low (optimistic estimate), most likely estimate and high (pessimistic estimate). It is shown in Fig. 6.12 that at -cut of 0.5, we obtain two values of the corresponding parameters i.e., low and high. Thus, instead of a single curve, band bounded by upper and lower estimation of curvature is generated.
Figure 6.12. Membership functions of , ,b n σY and f
6.5.2 Simulations with Fuzzy Material and Process Parameters
In the process of predicting the curvature by considering fuzzy material and process parameters, three estimates of curvature are obtained as described in Subsection 6.5.1. In order to model the varying frictional conditions, different frictional conditions at upper and lower roll-work interfaces are considered. At -cut of 0.5, friction at upper roll-work interface is considered as 0.085 where a friction at lower roll-work interface is assumed as 0.135. According to adopted sign convention, a positive radius of curvature indicates that the strip curls towards the upper roll whereas a radius of curvature with negative sign indicated that strip curls towards lower roll. It is observed that in the case of asymmetry due to friction, strip curls towards the roll having higher coefficient of friction at roll-work interface. In the past, Richelsen [1997] also observed this trend. Table 6.7 shows the results for optimistic, most likely and pessimistic estimates of radius of curvature. Similar trend was observed for all three set of estimates.
Table 6.7. Radius of curvature obtained due to friction mismatch at -cut of 0.5 (fu =0.085, fl =0.135,VA=1.0)
Radius of curvature, (m) / 1
R h r
Optimistic estimate σY=152.91 MPa, b=0.06,n=0.208
Most likely estimate σY=169.9 MPa, b=0.05,n=0.26
Pessimistic estimate σY=186.89 MPa, b=0.04,n=0.286
50 10 -17.275 -8.6184 -5.5282
65 10 -5.2957 -3.5924 -2.6855
80 10 -2.8012 -2.0805 -1.6470
100 10 -1.6446 -1.2828 -1.0487
50 20 -1.3060 -0.9680 -0.7806
65 20 -0.7655 -0.5873 -0.4815
80 20 -0.5114 -0.4045 -0.3408
100 20 -0.3453 -0.2827 -0.2442
Simulations are also carried out for curvature prediction by considering variation in flow stress only. In this result, material hardening parameters b and n are kept fixed as 0.05 and 0.26 respectively. It is observed from Table 6.8 that variation of
±10%in the flow stress does not change the curvature drastically. Thus, it is observed that the curvature mainly depends on the plastic strains only. However,
type of material does play its role, because roll deformation is affected by the material. Change in roll deformation pattern influences the roll of friction and redundant deformation.
Table 6.8. Comparison of radius of curvature due to variation in flow stress at -cut of 0.5 (fu =0.085, fl =0.135, =0.05, =0.26,b n VA =1.0)
Radius of curvature, (m) / 1
R h r for
σY=152.91 MPa
for
σY=186.89 MPa
50 10 -10.106 -8.1023
65 10 -3.9534 -3.4810
80 10 -2.2344 -2.0314
100 10 -1.3621 -1.2564
50 20 -1.0194 -0.9597
65 20 -0.6175 -0.5805
80 20 -0.4234 -0.4006
100 20 -0.2941 -0.2798
Results are generated for another set of uncertain frictional conditions. In this case, -cut of 0.75 is created. At upper roll-work interface, coefficient of friction is considered as 0.0975 where coefficient of friction at lower roll-work interface is assumed as 0.1225. At the reduced frictional asymmetry, reduction in the curvature is observed for the same sets of flow stress, material parameters. Table 6.9 shows the results for optimistic, most likely and pessimistic estimates of radius of curvature at higher -cut of 0.75.
Table 6.9. Radius of curvature obtained due to friction mismatch at -cut of 0.75 (fu =0.0975, fl =0.1225,VA=1.0)
Radius of curvature, (m) / 1
R h r
Optimistic estimate σY=152.91 MPa, b=0.06, n=0.208
Most likely estimate σY=169.9 MPa, b=0.05, n=0.26
Pessimistic estimate σY=186.89 MPa, b=0.04, n=0.286
50 10 -32.762 -16.645 -10.875
65 10 -10.7550 -7.2410 -5.4235
80 10 -5.6767 -4.1882 -3.3210
100 10 -3.3084 -2.5703 -2.1077
50 20 -2.6168 -1.9361 -1.5685
65 20 -1.5316 -1.1750 -0.9672
80 20 -1.0238 -0.8080 -0.6809
100 20 -0.6882 -0.5608 -0.4846
6.5.3 Control of Undesired Curvature
In order to minimize the undesired curvature, the strategy proposed in the present work may be adopted. In this approach, the friction at upper roll-work interface is treated as lesser than the friction at lower roll-work interface. This asymmetry causes the different locations of neutral points as shown in Fig. 6.1. Now, along with this frictional asymmetry, if we operate the lower roll with the speed higher that that of the upper roll, the width of the shear zone (Zone II) decreases. This leads to reduce the effect of frictional asymmetry and results in the minimization of curvature of the rolled strip. In the present work, optimum speed ratio VA is found out by using exhaustive search method. However, other search methods like golden section search method can also be implemented. Table 6.10 shows the results for the control of curvature with optimum speed mismatch. Column 4 and 5 represents the radius of curvature at optimum speed ratio VA and at VA=1.0. It is observed that with optimum speed ratio, curvature can be minimized up to 90%. Thus, operating the rolls at different speeds helps to minimize the curvature that causes due to uncertain frictional conditions.
Table 6.10. Roll speed adjustment to minimize the curvature that causes due to friction mismatch ( fu =0.085, fl =0.135,σY=169.9 MPa, = 0.05, = 0.26b n )
/ 1
R h r Optimum
VA
Radius of curvature (m)
at VA=1.0
Radius of curvature (m) at optimum VA
deviation %
50 10 1.006 -8.6184 -54.548 84.20
65 10 1.007 -3.5924 -38.125 90.57
80 10 1.009 -2.0805 -20.436 88.87
100 10 1.01 -1.2828 -5.6515 77.30
50 20 1.02 -0.9680 -3.2912 70.58
65 20 1.03 -0.5873 -2.9435 80.04
80 20 1.04 -0.4045 -1.8899 78.59
100 20 1.04 -0.2827 -0.6804 58.45