Author: ShuGen Wang, YehLiang Hsu (20040923);
recommended: YehLiang Hsu (20050118).
Note: This paper is published in Proceedings of the I MECH E Part B
Journal of Engineering Manufacture, Vol. 219, p. 177 – 181.
Onepass milling machining
parameter optimization to achieve mirror surface roughness
Abstract
In this paper,
the possibility of just using general machine centers in onepass milling
process to finish an aluminum plate with the mirror surface roughness is
studied. In particular, how to find the optimal setting of machining parameters
is presented. In this optimization problem, to evaluate whether the surface
roughness meets the average criteria of mirror surface requires real cutting
experiments, and it is desirable to find the optimal machining parameters using
as few experiments as possible. The “Sequential Neural Network Approximation
Method (the SNA method)” was used to find the optimal machining parameters, including
the spindle speed, feed rate, depth of cut, and the number of inserted blades
in the cutter to maximize the metal removal rate while the surface roughness
meets the average criteria of mirror surface.
Keywords: mirror surface roughness, machining parameter
optimization, the sequential neural network method.
1.
Introduction
Mirror surfaces
based on metal matrix intended for application on reflectors and optical parts
have been expected to become true in mass production. The cost of mirror
quality fabrication of metal surface using superprecision machines or
multiloop machining is still high. On the other hand, for cheap and fast mass
production, the possibility of just using general machine centers in onepass
milling process to finish an aluminum plate with the desired mirror surface
roughness is studied. In particular, how to find the optimal setting of
machining parameters for onepass milling of metal mirror surfaces is presented
in this paper.
The effect of
various machining parameters (such as spindle speed, feed rate, depth of cut,
and different types of cutters) on surface roughness has been well studied
[14], but few researchers have paid special attentions on mirror surface
machining with onepass milling simple procedure. This paper describes the
process of optimizing the machining parameters in a onepass milling process by
a general machine center for mass production of 6061T6 aluminum plates. The
purpose is to maximize the metal volume removal rate while the finished surface
will pass the desired average roughness of the mirror surface. The spindle
speed, feed rate, depth of cut, and the number of inserted blades in the
cutter, are the design variables to be decided.
The spindle
speed of the CNC machine center used in this research ranges from 40 to 7100
rpm (cutting speed 20m/min.
to 3560m/min.), with feed
increment of 0.001 mm.
The cutting tool was a MAPAL face miller, 160
mm in diameter, with 1 to 10 diamond face milling blades
inserted. A noncontact, cutoff light type
TaylorHobson microscope with 0.02 mm resolution of pickup diamond probe was used
to measure the cutting surface roughness. The working piece was a 55 mm^{3} aluminum alloy
(AL6061T6) material.
The metal
removal rate Q (in cm^{3}/min) can be calculated as:
_{} (1)
where _{} is the width of
cut in mm, _{} is the depth of
cut in mm, _{} is the feed speed
in mm/min. Moreover,
_{} (2)
where z is the number of blades or cutter
teeth, and _{} is the feed per
cutter tooth in mm/tooth.
In this case, _{}=55 mm,
and the objective is to maximize the metal removal:
Max. _{} (3)
In the meantime,
the surface roughness measured by the TaylorHobson microscope has to pass the criteria
of mirror surface. In general, for a mirror surface, the centerline average
roughness R_{a} as defined in Equation (4) has to be lower
than 0.05 mm.
_{}mm (4)
where y_{i} is the
measured roughness height according to each individual division from the
average centerline, and n is the
total number of divisions.
This is a
typical engineering optimization problem that cannot be solved by directly
applying the existing numerical optimization algorithms. In this optimization
problem, the mirror surface constraint is the socalled “implicit constraint”
[5]. It cannot be expressed as an analytical function in terms of the design
variables. Many factors can affect the surface roughness of a work piece, and
it is hard to analyze the surface roughness and establish a theoretical form
using cutting theory. To evaluate whether the surface
roughness meets the average criteria of mirror surface requires real cutting
experiments. It is desirable to find the optimal machining parameters using as
few experiments as possible.
There has been
considerable interest in the area of nonlinear discrete optimization. Some
review/survey articles on the algorithms for nonlinear optimization problems
with mixed discrete variables have been published [6, 7]. Among these methods,
the branch and bound method, simulated annealing, and genetic algorithm are
suitable implementations for problems with nondifferentiable functions. But
these methods require many function evaluations, which may not be suitable for
engineering optimization problems with implicit constraints.
One important category of numerical optimization algorithms is the sequential approximation
methods. The basic idea of sequential approximation methods is to use a
“simple” subproblem to approximate the hard, exact problem. By “simple”
subproblem, is meant the type of problems that can be readily solved by
existing numerical algorithms. For example, linear programming subproblems are
widely used in sequential approximation methods. The solution point of the
simple subproblem is then used to form a better approximation to the hard,
exact problem for the next iteration. In an iterative manner, it is expected
that the solution point of the simple approximate problem will get closer to
the optimum point of the hard exact problem. One major disadvantage of the
existing sequential approximation methods is that they are usually
derivativebased approximation methods, which require at least the first derivatives
of the constraints with respect to the design variables.
2.
The Sequential Neural Network
Approximation Method
In this paper,
the “Sequential Neural Network Approximation Method (the SNA method)” [8, 9] is
used to solve the problem. In this method, first a backpropagation neural
network is trained to simulate the feasible domain formed by the implicit
constraints using a few representative training data. The “exact optimization
model” Equations (3) and (4) can be approximated below:
min. f(x)
s.t. NN(x)=1 (5)
where f(x)=Q. The binary constraint NN(x)=1
approximates the feasible domain of the implicit surface roughness constraint Equation
(4). If NN(x)=1, the design point x (a
set of values for the machining parameters) is feasible; if NN(x)=0,
the design point x is infeasible, that is, the surface roughness
obtained by the set of values for the machining
parameters x cannot meet the average
criteria of mirror surface.
Table 1 shows the possible discrete
values of the design variables. The set of initial training points should reasonably
represent the whole design domain. Various types of matrices are commonly used
for planning experiments to study several input variables. Orthogonal arrays are
highly popular in industrial applications because they are geometrically
balanced in the coverage of experimental region with just a few representative
experiments. The orthogonal array is also adopted in this research to form the
set of initial training data. A training data
consists of two pieces of information: a design point and whether this design
point is feasible or infeasible. Table 2 shows the set of initial
training points using the L_{9}(3^{4}) orthogonal array. Note
that 4 initial training points are feasible (_{}), and training point No. 3 (2000,
4.5, 0.085, 10) has the maximum metal removal rate.
Table 1. Possible discrete values of machining
parameters for the mirror surface

1

2

3

4

5

6

7

8

9

10

11

12

13

v_{s} (10^{3}rpm)

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7



r_{f} (mm/t)

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5





d_{c} (mm)

.025

.03

.035

.04

.045

.05

.055

.06

.065

.07

0.75

.08

.085

z (tooth)

2

4

6

8

10









Table 2. The set of initial training data using the
L_{9} orthogonal array
no.

L_{9} orthogonal array

v_{s}(rpm)

r_{f}(mm/t)

d_{c}(mm)

z(tooth)

Q(cm^{3}/min)

R_{a}(mm)

Feasible

1

1

1

1

1

2000

0.5

0.025

2

0.00275

0.08627

N

2

1

2

2

2

2000

2.5

0.055

6

0.09075

0.04587

Y

3

1

3

3

3

2000

4.5

0.085

10

0.42075

0.04786

Y

4

2

1

2

3

4500

0.5

0.055

10

0.06806

0.02997

Y

5

2

2

3

1

4500

2.5

0.085

2

0.10519

0.05580

N

6

2

3

1

2

4500

4.5

0.025

6

0.16706

0.05088

N

7

3

1

3

2

7000

0.5

0.085

6

0.09818

0.04744

Y

8

3

2

1

3

7000

2.5

0.025

10

0.24063

0.06227

N

9

3

3

2

1

7000

4.5

0.055

2

0.19058

0.10546

N

Table 3. Iteration history
Iteration

v_{s}(rpm)

r_{f}(mm/t)

d_{c}(mm)

z(tooth)

Q(cm^{3}/min)

R_{a}(mm)

Feasible

Start point

2000

4.5

0.085

10

0.42075

0.04786

Y

1

3500

4.5

0.085

10

0.73631

0.06459

N

2

3500

3.5

0.085

10

0.75269

0.05625

N

3

2500

4.5

0.085

10

0.52594

0.05230

N

4

2500

4.5

0.080

10

0.49500

0.04878

Y

5

3000

4.5

0.080

10

0.59400

0.05391

N

6

2500

4.5

0.080

10

0.49500

0.04878

Y

7 (Restart)

7000

4.5

0.085

10

1.47263

0.14341

N

8

7000

4.5

0.080

10

1.38600

0.13582

N

9

7000

4.0

0.080

10

1.23200

0.12439

N

10

7000

3.0

0.080

10

0.92400

0.10147

N

11

5000

2.0

0.065

10

0.35750

0.04401

Y

12

6000

4.5

0.065

10

0.96525

0.09150

N

13

5500

3.0

0.075

10

0.68063

0.06804

N

14

5000

4.5

0.065

10

0.80438

0.07147

N

15

4500

3.0

0.080

10

0.59400

0.05843

N

16

7000

3.5

0.055

10

0.74113

0.08766

N

17

7000

2.5

0.085

10

0.81813

0.09555

N

18

7000

2.5

0.075

10

0.72188

0.08494

N

19

7000

2.0

0.085

10

0.65450

0.08354

N

20

7000

2.0

0.080

10

0.61600

0.07850

N

21

7000

2.5

0.065

10

0.62563

0.07636

N

22

6000

2.5

0.065

10

0.53625

0.06068

N

23

5500

2.5

0.070

10

0.52938

0.05722

N

24

5000

2.5

0.065

10

0.44688

0.04953

Y

25

5000

2.5

0.080

10

0.55000

0.05773

N

26

5000

2.5

0.075

10

0.51563

0.05449

N

27

5000

2.5

0.065

10

0.446875

0.04953

Y

28 (Restart)

2500

4.5

0.080

10

0.49500

0.04878

Y

29

2500

4.5

0.080

10

0.49500

0.04878

Y

30 (Restart)

2500

4.5

0.080

10

0.49500

0.04878

Y

Opt

2500

4.5

0.080

10

0.49500

0.04878

Y

3.
Conclusion
The size of the
input layer of the threelayer network depends on the number of variables and
the number of discrete values of each variable. Figure
1 shows the representation of training point No. 2 [(2000, 2.5, 0.055, 6),
feasible] in Table 2. In this example, a total of 38 neurons are used in
the input layer. Each neuron represented with a circle or a cross sign in the
input layer has value 1 or 0, respectively, to represent the discrete value in
the sequence corresponding to each variable. There is only one single neuron in
the output layer to represent whether this design point is feasible (the output
neuron has value 1) or not (the output neuron has value 0).
There are 12
neurons in the hidden layer in this example. The transfer functions used in the
hidden and output layers of the network are both logsigmoid functions. The
neuron in the output layer has a range [0, 1]. After the training is completed,
a threshold value 0.25 is applied to the output layer when simulating the
boundary of the feasible domain. In other words, given a discrete design point
in the search domain, the network always output 0 (if output neuron’s value is
less than the given threshold) or 1 (otherwise) to indicate whether this
discrete design point is feasible or not.
The
computational effort required in the neural network training is critical. If
the computation required is larger than that of evaluating the implicit
constraints, then the SNA method will lose its advantage. Here all the training
data are represented in a clear 01 pattern, which make the training process
relatively faster. A quasiNewton algorithm is used for the training. In our
numerical experience, the error goal of 1e6 is usually met within 2000 epochs,
even for cases with many training points.
A search
algorithm then searches for the “optimal point” in the feasible domain
simulated by the neural network, starting from the best feasible design point
in the initial training set [2000, 4.5, 0.085,10], Q=0.421 cm^{3}/min. This search algorithm is specially designed for the SNA
method, and is described in details in [7]. When the “optimal point” is found,
a cutting experiment is performed to evaluate whether this point is feasible,
that is, whether the surface roughness meets the average criteria of mirror
surface. The new training data is then added to the training set. The neural
network is trained again with this added trained data, hoping that the network
will better approximate the boundary of the feasible domain of the exact
optimization model. This process continues in an iterative manner until the
same design point is obtained repeatedly and no new training point is
generated.
Table 3 and
Figure 2 show the iteration history of this problem. The SNA method terminates
after 6 iterations. The final optimal machining parameters are: spindle speed v_{s}=2500 rpm, feed per cutter tooth r_{f} =4.5 mm/tooth, depth of cut d_{c}=0.08mm, and the number of blades or
cutter teeth, z=10. The maximum volume cut off 0.495 cm^{3}/min, increasing 17.6%. Using this set of machining parameters
in the cutting experiment, R_{a}=0.04878mm, which meets the average
criteria of mirror surface. Note that a total of 15 design points out of 11×9×13×5=6435
possible combinatorial combinations (0.23%) were evaluated by real cutting
experiments.
To ensure a
better chance to reach a global optimum, the searching process was restarted
from 3 other feasible points in the set of initial training data in Table 2. As
shown in Table 3 and Figure 2, after a total of 30 iterations, the same optimal
machining parameters were obtained. Note that the final optimal design point is
close to the starting point obtained by the L_{9} orthogonal array, which is commonly used for planning experiments
to study several input variables, while the metal removal rate Q
increases by 17.6%.
4.
References
[1]
Takeuchi,
Y., Kawakita, S., Sawada, K., and Sata, T., “Ultra
precision milling (Sculptured surface generation),” Nippon Kikai Gakkai
Ronbunshu, C Hen/Transactions of the Japan Society of Mechanical Engineers,
Part C, Vol. 59, No 566, 1993, pp. 31933198.
[2] Fuh, K. H., Wu, C. F., “Proposed statistical
model for surface quality prediction in end milling of Al alloys.” Int. J.
of machine tools & manufacture, v35 n8, 1995, pp.11871200.
[3] Nieminen, I., Paro, J.; Kauppinen,
V. “Highspeed milling of advanced materials,” J. of Materials Processing Technology, v56, n 14, 1996, pp. 2436.
[4] Kin, J. D., Kang, Y. H.,
“Highspeed machining of aluminum using diamond endmills.” Int. J. of
Machine Tools and Manufacture, v37 n8, 1997, pp.11551165.
[5] Hsu, Y. L., Sheppard, Sheri D. Wilde, and Douglass J. “Explicit
approximation method for design optimization problems with implicit constraints,”
Engineering Optimization, v 27 n1,
1996, pp. 2142.
[6] Arora, J. S., and Huang, M.W., 1994. “Method for optimization of
nonlinear problems with discrete variables: a review,” Structural
Optimization, Vol. 8, pp. 6985.
[7] Thanedar, P.B., and Vanderplaats, G..N., 1995, “Survey of discrete
variable optimization for structural design,” Journal of Structural
Engineering, Vol. 121, No.2, pp. 301305.
[8] Hsu, Y. L., Dong, Y. H., Hsu, M. S., “A Sequential Approximation
Method Using Neural Networks for Nonlinear Discrete Variable Optimization with
Implicit Constraints,” JSME International Journal, Series C, v 44, n1,
2001, pp. 103~112.
[9] Hsu, Y. L., Wang, S. G., and Yu C. C. “A sequential approximation
method using neural networks for engineering design optimization problems,” Engineering Optimization, v 35 n5, 2003, pp. 489511.