(a) (b)
Figure 6.7. Membership function for R%max −R%min
Step 7: The following strategy is developed to find the parameters of the locator.
The deflection δ of the locator on the primary datum under the part weight and other external forces is given by the relation
L L
L
P l
δ = AE (6.32) where PL is the total load on the locator, lL is the locator diameter, A is the cross- sectional area of the locator and EL is the Young’s modulus of elasticity of the locator material. Locator diameter (lL) is calculated by putting A = πlL2/4 in Equation (6.32). Spherical locator button diameter DL is calculated from Equation (6.18).
Radius of curvature of the spherical locator button, RL is found considering the onset of yielding in the workpiece material. Minimum radius of curvature is found from Equation (6.27) where Fclamp is replaced by the maximum locator reaction force.
Maximum radius of curvature can be found using Equation (6.30) assuming proper contact between the workpiece and locators.
is perpendicular to both the primary and secondary datum. The workpiece is a prismatic block of dimensions 70×60×50 mm3 and the workpiece material is AISI 1018 steel. Taking the density of AISI 1018 steel as 7.87 gm/cc, the weight of the workpiece is found to be 16.5 N. It is fixtured with three locators L1, L2 and L3 on the primary datum (x-y plane), two locators L4 and L5 on the secondary datum (x-z plane) and one locator L6 on the tertiary datum (y-z plane). Clamp C1 is placed opposite to the locator on the tertiary datum and C2 is placed opposite to the locators on the secondary datum. Spherical locator and clamp contact surfaces are used in this work for proper contact with rough workpiece surface. Screw clamps made of 2340 medium carbon alloy steel are selected. Locator material is water hardening steel W1 with 0.6 % carbon content. Workpiece, clamp and locator material properties are given in Table 6.1. A 20 mm diameter helical end mill with four flutes and 30ο helix angle is used for the milling operation. A torque of 2000 N-mm is applied at the head of the clamp screw with one hand operation [Rai and Xirouchakis, 2008]. The central line average (CLA) surface roughness height of the workpiece contact surface is considered as 50 µm (N12). The entire tool path is discretized into 170 steps along x and y-axis for applying the computed machining forces.
Figure 6.8. The end milling of the example part
The specific cutting energy, yield stress and Young’s modulus of elasticity of the workpiece material u%s, Y%and E%w, Young’s modulus of elasticity of the clamp and locator materials E%cand E%L, clamping torque T%, index a% in Equation (6.19) and peak to valley roughness height of the workpiece contact surface R%t are treated as fuzzy numbers. The low (l), most likely (m) and high (h) estimates of these parameters are given in Table 6.1. The basis of these estimates is that the variations in specific cutting energy and yield stress of the workpiece material may go up to
±30% and ±10% respectively. The Young’s moduli of elasticity of the workpiece, clamp and locator materials may vary by ±5%. The variations in clamping torque and peak to valley roughness height of the workpiece contact surface are considered as ±10% and index a varies by ±20%. Linear triangular fuzzy membership functions are assumed for these parameters. A linear triangular membership function is constructed by taking the membership grade as 1.0 at most likely (m) and 0.5 at low (l) and high (h) estimates of a parameter. A typical triangular membership function is shown in Figure 4.1, Chapter 4
Table 6.1. The values of the fuzzy parameters
Parameters Low (l) Most likely
(m) High (h) Specific cutting energy u%s
for AISI 1018 steel (J/mm3) 1.8186 2.8856 4.1665
Index a% 0.2 0.3 0.4
Yield stress Y%for AISI
1018 Steel (N/mm2) 347.40 386 424.60
Young’s modulus of elasticity
E%wfor AISI 1018 Steel (N/mm2) 190000 200000 210000 Young’s modulus of elasticity
E%cfor 2340 alloy Steel (N/mm2) 190000 200000 210000 Young’s modulus of elasticity E%Lfor W1
water hardening Steel (N/mm2)
190000 200000 210000
Clamping torque T% (N-mm) 1800 2000 2200
Peak to valley roughness height R%t (mm) 0.200 0.225 0.250
The machining and clamping forces are considered as fuzzy numbers. Figure 6.9 shows the fuzzy machining forces Fx, Fy and Fz and clamping force Fclamp at different membership grades for the end milling operation at 0.5 mm depth of cut and 0.1 mm/tooth feed. Standard 3-2-1 location with one clamp each on secondary and tertiary datum is followed in this case. From Figure 6.9 (a), the high (h) estimates of Fx, Fy and Fz at 0.5 membership grade are 139.89 N, 99.76 N and 28.50 N respectively. From Figure 6.9 (b), the most likely (m) value of clamping force at membership grade 1.0 is 445.68 N and the low (l) and high (h) estimates at 0.5 membership grade are 280.88 N and 643.50 N respectively. Designing for the worst case condition, high estimate of clamping force 643.50 N at membership grade 0.5 is considered. Radius of the clamp rclamp (dclamp/2) is found as 7 mm from Equation (6.14). The value of rclamp is greater than the minimum value of rclamp (4.82 mm) found from Equation (6.15). For finding the radius of curvature Rclamp for spherical clamp, minimum and maximum radius of curvature, Rmin and Rmax as obtained from Equations (6.27) and (6.30) are made equal. The value of Rclamp comes to be 98 mm at 0.5 mm depth of cut. Figure 6.10 shows the variation of machining forces Fx, Fy
and Fz with cutter engagement angle ν for most likely (m) value of machining forces at membership grade 1.0 for one revolution of the cutter.
Considering a very small deflection of 0.001 mm of the locator under the part weight and other external forces, locator diameter lL is found as 12 mm from Equation (6.32). Spherical locator button diameter DL is calculated as 16 mm from Equation (6.18). Radius of curvature of the spherical locator button, RL comes to be 24 mm from Equation (6.17). However, in the proposed design, RL is found considering the onset of yielding in the workpiece material. The minimum value of RL is calculated as 87 mm from Equation (6.27) where Fclamp is replaced by the maximum locator reaction force 503.62 N for worst case machining and clamping forces. Height of the locator button, H and height of the locator, L are found as 8 mm and 12 mm respectively from Equations (6.16) and (6.18). H is considered half of the button diameter DL
.
(a)
(b)
Figure 6.9. Membership function for (a) machining forces (b) clamping forces at 0.5 mm depth of cut and 0.1 mm/tooth feed
0 50 100 150 200 250 300 350 400 -20
0 20 40 60 80 100
Cutter engagement angle (degrees)
Machining force components (N)
Fx Fy Fz
Figure 6.10. Variation of machining forces with cutter engagement angle at 0.5 mm depth of cut, 0.1 mm/tooth feed at membership grade 1.0
Figure 6.11 shows the upper bound of depth of cut at membership grades 0.5 and above for worst case design condition. High estimates of Fx, Fy, Fz and Fclamp at 0.5 membership grade are considered. Upper bound for depth of cut at 0.5 membership grade is 0.5 mm and at 1.0 membership grade, it is 0.722 mm.
Figure 6.12 shows the range of radius of curvature Rmax−Rmin and Rmin for spherical clamp at 0.5 mm, 0.7 mm and 0.8 mm depths of cut. From Figure 6.12 (a), the ratio of positive area T1 to total area (T1+T2) is found as 0.9795, 0.9519 and 0.8575 for 0.5 mm, 0.7 mm and 0.8 mm depths of cut respectively. It indicates that there is 97.95 %, 95.19 % and 85.75 % possibility of the design being successful at 0.5 mm, 0.7 mm and 0.8 mm depths of cut. The extreme low and extreme high cases of machining and clamping forces are also considered for calculating the ratio of T1 to (T1+T2). It is observed that with increase in depth of cut, there is a greater chance of Rmax−Rmin being negative thus reducing the possibility of successful design. In this situation, the options are to either reduce the depth of cut/feed or to
increase the number of clamps. The approximate relation between depth of cut d, feed f and cutting force Fc can be expressed by the following expression
Fc=k f d (6.33)
where k is the proportionality constant. Variable bounds for feed can be calculated using Equation (6.33) by the proposed method.
Figure 6.11. Membership function for upper bound of depth of cut for single clamp design