In this section, the fuzzy parameters are denoted with tilde (~) as a diacritical mark. The overall functioning of the proposed system comprises the following steps:
Step 1: Given the information on the part, machining operations, machines and tools as input to module A, the setup and machining operation sequences, datum for the setups are obtained. This module is described in detail in Chapter 3.
Step 2: Cutting tool parameters, depth of cut, feed rate and fuzzy specific cutting energy u%s for a machining operation are given as inputs to module B and the values of F F%x, %y and F%z are calculated. u%svaries with undeformed chip thickness t as follows [Shaw, 2005]:
u%s =u t%0 −a% (6.19) where u%0 is the specific cutting energy at 0ο tool rake angle and 0.25 mm undeformed chip thickness and a% is a fuzzy index. The main cutting force in milling is the tangential force component Ft and it is expressed in terms of u%s as follows [Stephenson and Agapiou, 2005]:
F%t =u dt%s (6.20) where d is the depth of cut and t is the undeformed chip thickness. F F%x, %y and F%z can be calculated for different machining cases. As an example, the calculation for an end milling operation is done in the following manner. Undeformed chip thickness t is given by the expression [Stephenson and Agapiou, 2005]:
t= f sinν (6.21) where f is feed per tooth of the cutter and ν is the cutter engagement angle. Figure 6.5 shows the plan view for gradual engagement of the cutter during milling process.
Here b is the radial depth of cut and D is the cutter diameter.
Figure 6.5. Cutter engagement angle ν during milling process
The minimum value of ν is 0ο. The maximum value of the engagement angle can be found from Figure 6.5, as follows:
cos 2 2 D b
ν = D− (6.22) Simplifying Equation (6.22), the maximum value of the engagement angle νm is found as
m cos b ν = − − D
1 1 2 (6.23) Putting the expression of t from Equation (6.21) in Equation (6.20), Ft can be calculated as
F%t =u d f%s sinν (6.24) The radial and axial force components Fr and Fa are found from Ft as [Stephenson and Agapiou, 2005]:
F%r =0 3. F%t (6.25) F%a =F%rtanβ (6.26) where β is the helix angle of the cutter. Now Cartesian components F F%x, %y and F%z can be calculated in terms of F F% %t, r and F%a using Equations (6.1)–(6.3). The maximum value of F%x /F%y/F%z is set as the clamping forceF%clamp.
Step 3: F F F F%x, %y, %z, %clamp and the part weight W are the inputs to module C for locator and clamp position optimization. Objective function and constraints are presented by Equations (6.7)–(6.9). If the feasible solution is not obtained, F%clamp is increased in steps of 5 % until a feasible solution is obtained.
Step 4: The value of F%clamp that provides the positive reaction forces at all locators is the required minimum F%clamp. A factor of safety of value 2 is chosen for F%clamp to take care of the uncertainties associated with dynamic effect of machining forces,
material removal effect and tool wear. The clamp diameter d%clamp is calculated from Equation (6.14) considering clamping torque T%and F%clamp as fuzzy.
Step 5: In this work, a novel strategy is developed to find a range for the value of radius of curvature for the spherical clamp, R%clamp. In Equation (6.11), R Y% %, and E% are fuzzy. As the workpiece surface is planer, R% =R%f =R%clamp. The minimum value of the radius of curvatureR%min is found from Equation (6.11) by replacing the normal load Py with clamping force F%clamp and considering that F%clamp is within elastic limit.
R%min is obtained from Equation (6.11) as min clamp
( . )
F E
R = πY
2 3
6 1 6
% %
% % (6.27) To find the maximum value of the radius of curvatureR%max, the following strategy is developed. Figure 6.6 shows the clamp parameters. Here rclamp is the clamp radius, s is the height of the spherical clamp tip and Rclamp is the radius of curvature of the spherical clamp. From Figure 6.6,
R%clamp2 =(R%clamp−s%)2+rclamp2 (6.28) Neglecting very small terms, Equation (6.28) can be written as
clamp rclamp
R = s
2
% 2
% (6.29) The contacting surface of the workpiece is considered to be a rough surface. With an objective of proper contact between the spherical clamp and workpiece, s must be equal to the peak to valley roughness height Rt of the workpiece surface. Hence, the maximum value of radius of curvature R%max is given by
max clamp
t
R r
= R
2
2
% %
% (6.30)
Figure 6.6. The parameters of a spherical clamp
Step 6: If R%min <R%max, design is proper. If R%min >R%max, there can be two options.
Either the depth of cut/feed can be reduced so that machining forces are lower, or number of clamps can be increased to reduce F%clamp on each clamp. Decision has to be taken depending on the requirements and considering the pros and cons of both the options.
The maximum value of load at the onset of yielding P%y is checked considering planer clamp contact surface with the following relation [Chakrabarty, 1987]
Py rclamp Y π
π
= +
2 1
% % % 2 (6.31) The value of F%clamp should always be lower than P%y calculated from Equation (6.31).
Figure 6.7 shows a typical diagram for two possible cases of membership function ofR%max −R%min. For case (a), all the values of R%max −R%min are positive and the design is proper for all membership grades. For case (b), R%max −R%min is a combination of positive and negative values and T1 and T2 are the positive and negative areas respectively. In this case, the ratio of the positive area T1 to total area (T1+T2) indicates the possibility of the design being successful.
(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.