Computer Aided Constrained Optimisation of Cutting Conditions in Drilling Operations on a Cnc Lathe by Using Geometric Programming

This paper discusses the use of the geometric programming method to determine the optimum values of cutting speed and feed rate which yield minimum cost in a drilling operation that is performed on a CNC lathe, During the formulation of the problem a number of constraints are considered. The determination of economically optimal cutting conditions, i.e. cutting speed and feed rate, is an essential step in computer aided process planning activities. A survey of the literature on the optimisation of cutting conditions indicates that a few number of researchers have studied the optimisation of cutting conditions in drilling operations In the cases especially where HSS drills are used, drilling operation may have considerable influence on the machining time and therefore optimisation of cutting conditions might be necessary. Ermer and Shah [1] considered the problem of optimising cutting conditions in drilling. They used both minimum cost and maximum production rate as optimisation criteria. Arsecularatne [2] and Filiz, Sonmez, Baykasoglu, Dereli [3] studied the constrained optimisation of cutting conditions in drilling by using minimum cost as optimisation criteria. They used the torque available from machine too~ drill buckling, drill strength, axial-circumferential slips in chuck as the constraints. In the above mentioned analyses, cost per operation is expressed in terms of cutting speed and feed rate. One of these variables is found by using partial differentiation of the expression with respect to the variable of concern and the other variable is found as the value which satisfY the above mentioned constraints. In these approaches both independent variables could not be treated simultaneously. In this study, the constrained optimisation of cutting conditions on a CNC drilling operation is successfully and easily treated by the application of a non-linear programming technique, namely, Geometric Programming (GP). In the solution of constrained GP problem Lagrange Multipliers method is used as an additional tool. Minimum cost is used as the objective function and the following restrictions are considered in this work; Maximum machine torque, Limiting torque for the dria Circumferential slip in the chuck, Axial slip in the chuck, Drill buckling, Maximum and minimum speeds available from machine too~ Maximum and minimum feed rates available from machine tool. A computer program is written in QBASIC and implemented on an ffiM compatible computer for automating the calculations in the optimisation procedure.

The determination of economically optimal cutting conditions, i.e. cutting speed and feed rate, is an essential step in computer aided process planning activities.
A survey of the literature on the optimisation of cutting conditions indicates that a few number of researchers have studied the optimisation of cutting conditions in drilling operations In the cases especially where HSS drills are used, drilling operation may have considerable influence on the machining time and therefore optimisation of cutting conditions might be necessary.
Ermer and Shah [1] considered the problem of optimising cutting conditions in drilling.They used both minimum cost and maximum production rate as optimisation criteria.
Arsecularatne [2] and Filiz, Sonmez, Baykasoglu, Dereli [3] studied the constrained optimisation of cutting conditions in drilling by using minimum cost as optimisation criteria.They used the torque available from machine too~drill buckling, drill strength, axialcircumferential slips in chuck as the constraints.
In the above mentioned analyses, cost per operation is expressed in terms of cutting speed and feed rate.One of these variables is found by using partial differentiation of the expression with respect to the variable of concern and the other variable is found as the value which satisfY the above mentioned constraints.In these approaches both independent variables could not be treated simultaneously.
In this study, the constrained optimisation of cutting conditions on a CNC drilling operation is successfully and easily treated by the application of a non-linear programming technique, namely, Geometric Programming (GP).In the solution of constrained GP problem Lagrange Multipliers method is used as an additional tool.Minimum cost is used as the objective function and the following restrictions are considered in this work; Maximum machine torque, Limiting torque for the dria Circumferential slip in the chuck, Axial slip in the chuck, Drill buckling, Maximum and minimum speeds available from machine too~Maximum and minimum feed rates available from machine tool.
A computer program is written in QBASIC and implemented on an ffiM compatible computer for automating the calculations in the optimisation procedure.
In Geometric Programming the objective function is written in the following form: is a minimum subject to; where qo: number of terms in the objective function, Coj: coefficients of the objective function, tk:denotes variables, r: number of variables, p: number of constraint functions, qi: number of terms in the i'th constraint function.
Duffin, Zener and Peterson [4] showed that the dual of the above stated problem (primal programme) is given by; where u(o) is the dual function and Oij, denotes dual vectors.
To solve the problem, the optimum value of the dual vectors 0\ ,which make the dual objective function maximum, should be found from dual constraint equations.
The There are qo equations and r variables.The variables !Ieare then found by solving these equations simultaneously.
If (no.of.equations = r +1), then (s-r-l) is termed the degree of difficulty of the problem, where s is the number of terms in the objective and constraint functions.This represents the number by which the independent variables exceed the number of equations in the system of linear simultaneous equations given by normality and orthogonality conditions.
The basic model describing the cost of a drilling operation, as given by many authors, is expressed as follows; where X is machining cost rate (cost/min), Y: tool cost per cutting edge (in carbide inserts) or drill depreciation cost plus drill resharpening cost (in HSS tools), Tm: machining time (min), Td: tool change time (min), T: tool life (min).
Machining time for a drilling operation can be written as; Tm = 1tDL/(lOOOYf) 8 where D is drill diameter (mm), L: length of cut (mm), f: feed rate (mm/rev), V: cutting speed (m/min).
Taylor's expanded tool life equation for drills has the following form [5]; Then the cost equation can be written as; This equation is the objective function which will be optimised according to the minimum cost criterion.
There are several constraints which effect the cutting conditions in a drilling operation.The source of these constraints may be machine tool, cutting tool and workpiece specifications.One must keep in mind that the larger the number of constraints, the harder the optimisation problem is to solve.

Constraints
Following constraints are considered in this work; 1) Maximum machine torque : Maximum torque which can be provided by a machine is; M1 ~M (Constraint 1) 15 where, Pmaxis maximum power available from machine (W), Nbrcak1 is the break speed of the motor after which tlle power becomes constant (maximum) (rpm).
2) Limiting Torque of The Drill: Limiting torque that the drill can withstand is calculated by the formula; where, De is the equivalent diameter for the drill which is equal to 0.7D (mm), fsl is factor of safety , and 't is the shear strength of the drill shank.material (MPa).3) Drill Buckling : The maximum load to avoid drill buckling can be calculated by using the formula; where; E is the modulus of elasticity of the drill material (MP a), fs2 is factor of safety.

4) Axial Slip in Chuck:
The maximum allowable thrust to avoid axial slip in the chuck can be calculated by using the expression; where, ,u a is coefficient of friction of jaw in axial direction, Fco is clamping force at zero speed (N), mj is mass of chuck jaws (kg.), rj is radial distance of jaws (mm.), Wmin is minimum spindle speed (rad/sec.).So; Fa2 ~Fy (Constraint 4) 21

5) Circumferential Slip in The Chuck:
To avoid circumferenctial slip in the chuck, the torque developed in the cutting operation must be less than the frictional torque (M3) in the chuck which can be calculated by using the formula; M3= ~g (Fco+ L(lIljrj)(wmin)2] 22 where, r ll is component gripped radius (mm), ~c is coefficient of friction of jaw in the direction of spindle rotation

6) Maximum-Minimum Rotational Speeds of the Machine Tool:
The rotational speed can be calculated by using the following equation; f ~tnax (Constraint 7) Where; f max is maximum feed rate of the machine.
The objective function and constraints functions" can be written m the geometric programming formats as explained in section 2; Objective function g,,(t) = C01tl The primal objective function and the constraint functions have been developed in the previous sections.It is seen that the objective function has two terms and there are seven constraint functions.All of the constraint functions have single terms.So the dual objective function turns out to be; t) = L (Cij II tte8 ijk ) :::;; 1 j=l k=1 optimum value of the original objective function g•o(t) is obtained from the dual program after finding the optimum values of the dual vector o•ij.According to the definition of the geometric programming, O•ij are the weight of the terms in the primal objective function, I.e.r C 3oJ"I.<_ •• oj II !Ie -0 oj.g o(t) k=1 m,xv,yv are constants.The substitution of tool life (T) and machining time (Tm)expressions into equation (7) Crl X7tLD)v1 ["1 +(XT +YX 1tLD(l-mx v )) V(m-l)f(myv-l) 1000 d 1000(Cv)m , For convenience, define; n = m.xv,A = (~t ,z = m Yv