Prediction of optimal stability states in inward-turning operation using genetic algorithms.
Kotaiah, K. Rama ; Srinivas, J. ; Sekar, M. 等
Introduction
The most detrimental phenomenon to productivity is unstable
cutting. This reduces tool--life and surface quality of workpiece. Many
theoretical investigations are available in literature for prediction of
stable and unstable cutting states in orthogonal cutting. Most of the
cases, the stability lobe diagram is generated from an analytical linear
model, by varying one operating parameter at a time. How ever, cutting
processes possess highly nonlinear relationships among the input and
output parameters. In orthogonal turning, it is well known that the
cutting forces depend on the operating variables such as feed, depth of
cut and speed. These variables are often used to control the forces or
machining stability by establishing appropriate regression relations.
Recently, it is found that other parameters such as tool geometry [1],
tool wear [2], variations in shear angle [3] and compliance of work
piece [4-6] have great influence on cutting dynamics. To distinguish
stability states of cutting, the output features such as surface
roughness ([7]-[8]) and type of chips [9] can be employed effectively in
addition to cutting force data.
In practice, there are several other operating parameters like tool
overhanging length and type of material, could also alter the critical
operating conditions in parallel. For example, variation of tool over
hang length changes the stiffness of tool holder, which in turn affects
the tool-life under unstable conditions. Likewise, the effects of
cutting fluids on the surface roughness and tool wear have been
predicted [10]. More recently the overall influence of amount of
lubrication along with cutting speed and feed rates on the surface
roughness and specific cutting forces has been studied [11]. In
addition, variables like steam pressure that influence the surface
roughness of the workpiece have also been considered [12].
In this paper, the effects of operating variables in orthogonal
turning operation including tool overhang length on the critical chatter
lengths of work piece and cutting forces on tool are studied. A series
of cutting experiments are carried out using four different work
materials i.e. En24 steel, EN8 steel, Mild steel and Aluminium at
various operating speeds, feeds and depths of cuts. In all cases,
dynamic cutting forces and critical chatter lengths are measured.
Relations between the input and output parameters are established using
radial-basis function (RBF)neural network model and it is further
employed to arrive at optimized machining data within the operating
constraints using genetic algorithms(GA). Brief description of proposed
neural network model and optimization scheme through GA is presented in
section 3 and the numerical results and discussions are given in section
4. The following section briefly presents the experimental analysis to
get parametric data.
Experimental Analysis.
In present analysis, a series of cutting experiments are carried
out on a center lathe in order to find the critical stability state in
inward turning. Fig. 1 shows the experimental setup employed in the
present work. Cutting is performed from the collar end of the workpiece.
[FIGURE 1 OMITTED]
The cutting operation is limited to a short range since the
structural stiffness rapidly increases as the tool advances inwards. The
spindle speed(V),feed rate(f),depth of cut(d) and tool overhang
length(l) are progressively varied to obtain the cutting forces
([F.sub.x], [F.sub.y] and [F.sub.z]) and critical chatter
lengths([C.sub.c]). The ranges are determined through preliminary
experiments from practical feasibility. The parameters and their
associated levels are depicted in Table 1. In all the cases 50mm
diameter work pieces are employed.
An attached tool post strain gauge dynamometer platform is used to
measure the three-dimensional cutting forces. The required feed rate is
chosen from the lathe preset. The flank wear on the tool faces is
measured before and after the experiments by using tool maker's
microscope. Critical chatter cutting length is recognized by a sudden
increase in the cutting force signature. In this way, 84 data points are
recorded for every work material. Fig. 2a and 2b shows the variation of
output parameters as a function of depth of cut and tool over hang for
EN8 steel workpiece operating at three different spindle speeds. It can
be seen that the cutting forces increase, while the critical chatter
lengths decrease with the depth of cut and tool overhang. However, the
changes are not linear or uniform. Thus output parameters are affected
simultaneously by the operating variables as well as tool over hang in
nonlinear fashion.
[FIGURE 2(a) OMITTED]
Proposed Neuro-genetic Approach
Relation ship between several operating variables and the output
parameters is first obtained from the neural network model. For the last
one decade, several works have been reported the use of neural networks
in modeling of turning process. Few recent applications of neural
networks in turning operation include extraction of surface roughness
information [13-16] and prediction of workpiece motions from cutting
tool vibration signals [17]. There are many types of neural networks in
common use. Main advantage of using neural networks is that the entire
experimental data is consolidated into few cutting parameters known as
weights and centers. Fig.3 shows the schematic diagram of the proposed
approach of obtaining the critical operating variables. A function
approximation model is used to obtain the relationship between the input
and output data. After establishing the neural network model, it is
employed to minimize the cutting forces at different ranges of input
variables.
[FIGURE 2(b) OMITTED]
Proposed Neuro-genetic Approach
Relation ship between several operating variables and the output
parameters is first obtained from the neural network model. For the last
one decade, several works have been reported the use of neural networks
in modeling of turning process. Few recent applications of neural
networks in turning operation include extraction of surface roughness
information [13-16] and prediction of workpiece motions from cutting
tool vibration signals [17]. There are many types of neural networks in
common use. Main advantage of using neural networks is that the entire
experimental data is consolidated into few cutting parameters known as
weights and centers. Fig.3 shows the schematic diagram of the proposed
approach of obtaining the critical operating variables. A function
approximation model is used to obtain the relationship between the input
and output data. After establishing the neural network model, it is
employed to minimize the cutting forces at different ranges of input
variables.
[FIGURE 3 OMITTED]
RBF neural network
Of the available architectures, RBF neural network has principal
advantages such as: single hidden layer, training requiring only output
layer and comparatively rapid convergence. RBF model has three layers:
an input layer, a hidden layer of radial basis neurons and an output
layer of linear neurons[18]. The hidden layer consist of an array of
computing units called hidden nodes. Each hidden node contains a center
vector C that is a parameter vector of the same dimension as the input
data vector X and calculates the Euclidean distance between the center
and the network input vector X defined by : [absolute value of X -
[C.sub.1]]. The results are then passed through a nonlinear activation
function (known as radial basis function) [[phi].sub.j], to produce the
output from the hidden nodes. A popular choice of the activation
function is the Gaussian basis function:
[[phi].sub.j] (t) = [exp.sup.[-[[absolute value of
X-[C.sub.j]].sup.2]/2[[??].sup.2.sub.j]], j = 1, 2, 3,......, M (1)
Where [[sigma].sub.j] is a positive scalar called the width and M
is the number of centers. It is often assumed that the number of hidden
units is significantly less than the number of the data points. The
width [sigma] of hidden unit controls the smoothness property of the
activation function. When the width [sigma] is small, the corresponding
area of the representation space becomes small. Hence, a high number of
centers will be required during the process of training. This results in
over-parameterization. On the contrary, the area of the representation
space may be too extensive when the width [sigma] is large. For both
cases, the generalization capabilities of the network will be poor.
Often [[sigma].sub.j] is selected from the relation: [[sigma].sub.j] =
[d.sub.max]/[square root of (2M)], where [d.sub.max] is the maximum
distance between the centers of hidden units. The center vector C is
obtained from the K-means clustering algorithm in which all the input
sets are arranged into clusters whose centers are initially chosen
randomly from all the input sets. The output layer consists of p neurons
and it is fully connected to the middle layer. Each linear output
neurons forms the weighted some of these radial basis functions. In
other words, the network output:
[??] = [[summation].sup.M.sub.[??]=1] [W.sub.[??]]
[[phi].sub.[??]], i = 1, 2, 3, ...... p (2)
where [[phi].sub.j] is the respone of the jth hidden unit resulting
from all input data and [W.sub.[??]] is the connecting weight between
the jth hidden unit and the ith output unit.
In matrix notation, Eq.(2) can be written as
[??] = W[empty set] (3)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
and [phi] = [[phi].sub.1] [phi].sub.2] ... ....
[phi].sub.[??]].sup.T] (5)
at the end of passing all the input sets (known as epoch or cycle),
a mean square error MSE is computed according to:
MSE = 1/2PAT x p [[summation].sup.PAT.sub.k=1]
[[summation].sup.[??].sub.[??]=1][([y.sub.[??]]-[??]).sup.2] (6)
Where PAT refers to the total number of patterns in each cycle and
[y.sub.[??]] is the target value at the ith output layer weights are
updated using recursive least square or gradient descent algorithm
according to
[W.sup.new.sub.[??]] = [W.sup.old.sub.[??]] + [??][[phi].sub.j]
([y.sub.[??]] [y.sub.[??]]) (7)
where [alpha] is learning parameter whose value is chosen between 0
and 1. After the learning phase, the network can be used to obtain the
output to any unknown input pattern.
Genetic Algorithms.
Genetic Algorithms (GA) also known as 'evolution
strategies', are optimization algorithms imitating principles of
biological evolution. GA is a probabilistic search process based on
natural genetic system, it is highly parallel and efficient optimization
strategy and believed to be robust. GA is capable of solving wide range
of complex optimization problems using three genetic operations:
selection/reproduction, crossover and mutation. The only
'fittest' individuals of every generation survive to obtain n
the next generation. GA considers several points in the search space
simultaneously and the chance of convergence to a local optimum is
reduced. GA does not need the knowledge of the gradient of the fitness
functions, which is very suitable for the optimization problems where an
analytical expression for the fitness function is unknown. In Ga binary
coding of the variables is often employed for convenience. Fitness is
computed for every population before selection of the mating pairs. For
selection of mating variables, either roulette wheel method or
tournament selection can be used. Single-point crossover method is
commonly employed. Essentially GA is developed for unconstrained single
objective optimization problems. Further details of GA can be found
elsewhere [19].
Results and Discussion
For each workpiece-material, experimental results of 80 cases are
selected as learning samples to train the neural network. The remaining
4 sets are used as inputs to verify the accuracy of the model. There are
3 input and 4 output nodes. Four optimum hidden nodes are selected from
several trails. The central vectors are obtained from K-means clustering
algorithm. The learning parameter is chosen as 0.4. Maximum number of
cycles is selected as 100. Fig. 4 shows the progress of network training
for [[sigma].sub.j] = 1. As [[sigma].sub.j] is increased the average
pridictions are found to be comparatively poor. The network centers and
weights are stored at this configuration. The predicted outputs
corresponding to all the trained samples are very close to the target
values. Table-2 shows the accuracy of predictions for all four cases of
work material. Even there are significant deviations of outputs in some
cases due to the improper normalization, average accuracy of the model
is found to be good.
[FIGURE 4 OMITTED]
After the neural network model is established correctly, the GA
procedure is employed to determine the optimum operating parameters. For
obtaining the mating pairs, tournament selection approach is employed.
Each substring length is chosen as 10 and a population of 40 individuals
is considered in each case. For the evaluation, the probability of two
random chromosomes mate in the crossover is considered as 98% and the
probability of mutation is taken as 1%. The evolution was continued
until the number of generations reaches 200. The optimum machining
parameters and corresponding maximum and minimum cutting forces as well
as critical chatter lengths are obtained for each work material. Table 3
and 4 shows the some ranges of operating parameters and corresponding
optimized forces, optimum cutting conditions predicted by GA program. It
is observed that an increase in feed and depth of cut along with tool
overhang rise the cutting forces. While the cutting forces in the
experimental range has relatively less influence on the dynamics.
Relatively high cutting forces are observed when mild steel workpiece is
employed. The convergence time for each run to achieve the desired
cycles is 8 seconds on X86 based PC with 3GHz processor. The combined
effect of all parameters could lead to better visualization at the shop
floor level
Conclusions
In this paper, a multivariate model of orthogonal turning operation
has been presented. Using the experimental data for different workpiece
materials, the cutting dynamics is modelled with radial basis function
neural network. Optimum operating variables namely speed, feed, depth of
cut and tool-overhang length are established for minimum value of total
cutting force. The corresponding chatter lengths are also reported. It
is found that compared to speed feed, depth of cut and overhang of tool
have profound influence on the cutting forces and critical chatter
locations. The work can be extended by considering the feed and depth of
cut as simultaneous variables to obtain more practical model.
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K. Rama Kotaiah * (1), Dr. J. Srinivas (2) and M. Sekar (2)
(1) * Associate Professor and Professor in Dept. Of Industrial and
production Engineering, K.L. College of Engineering, Vaddeswaram, Guntur
(Dist), Andhra Pradesh, INDIA-522502.
(2) School of Mechanical Engineering, Kyungpook National
University, Daegu, South Korea.
(1) * Corresponding Author Email: krk_ipe@yahoo.co.in
Table 1: Operating parameters and their levels used in the experiments
(Tool material: HSS S-200)
Operating Work piece material
parameters EN24 steel EN8 Steel Mild Steel Aluminium
Cutting
speed(m/min) 7,14 and 22 7,14 and 22 7,14 and 22 7,14 and 22
Feed 0.1, 0.138, 0.1, 0.138,
rate(mm/rev) 0.1 0.1 0.175, 0.2, 0.175, 0.2,
0.275, 0.35 0.275, 0.35
and 0.5 and 0.5
0.1-0.7
Depth of 0.1-0.7 with with 0.1 0.1 0.1
Cut(mm) 0.1 interval interval
Tool Overhang 54,57,59 and 53,57,60 54,57,59 53,56,58
Length(mm) 61 and 63 and 61 and 62
Length of
Workpiece(mm) 520 510 560 454
Table 2: Comparison of Cutting forces (N) and critical chatter
length [C.sub.c] (mm) (P: Predicted, M : Measured, MS : Mild
Steel, Al : Aluminium)
Material Case-1
[F.sub.x] [F.sub.y] [F.sub.z] [C.sub.c]
E P 40 64 92 18.
N 4 2 3 4
8 M 38 58 93 19.
6 6 4 1
E P 12 34 67 17.
N 0 8 3 5
24 M 12 39 68 18.
6 1 5 0
Al P 55 17 45 7.6
4 6
M 63 20 44 8.0
3 4
M P 93 48 62 17.
S 0 4 1
M 10 52 63 18.
1 2 6 0
Material Case-2
[F.sub.x] [F.sub.y] [F.sub.z] [C.sub.c]
E P 19 389 523 25.
N 3 6
8 M 18 316 514 27.
8 4
E P 39 510 712 14.
N 1 2
24 M 37 484 721 16.
4 0
Al P 76 230 413 14.
9
M 84 232 399 16.
0
M P 72 234 245 7.3
S 8 5 6
M 74 256 246 8.0
6 8 7
Material Case-3
[F.sub.x] [F.sub.y] [F.sub.z] [C.sub.c]
E P 42 561 784 15.
N 8 3
8 M 41 583 786 16.
2 2
E P 30 465 656 16.
N 6 2
24 M 29 455 663 16.
3 8
Al P 42 187 211 28.
8
M 46 174 221 30.
0
M P 55 196 213 9.5
S 0 5 2
M 54 204 214 10.
5 3 7 8
Material Case-4
[F.sub.x] [F.sub.y] [F.sub.z] [C.sub.c]
E P 20 353 523 24.
N 4 9
8 M 21 315 515 26.
5 6
E P 31 452 645 8.6
N 2
24 M 28 430 643 6
9
Al P 93 27 134 14.
2
M 10 35 146 15
1
M P 64 236 265 11.
S 4 1 4 1
M 65 247 267 10
0 3 8
Table 3: Outputs of GA for different ranges of input variables (feed
f = 0.1 mm/rev (Constant))
Material m/min d(mm) l(mm) Fx(N) Fy(N) Fz(N) Cc(mm)
EN8 7-14 0.1-0.4 53.57 377.4 631.5 823.6 16.23
7-14 0.1-0.4 57-63 377.13 631.0 823.8 16.21
7-14 0.4-0.7 53-57 537.22 851.5 946.9 12.27
7-14 0.4-0.7 57-63 539.32 854.6 946.3 12.28
14-22 0.1-0.4 53-57 406.1 660.1 856.7 16.07
14-22 0.1-0.4 57-63 402.9 655.4 857.2 16.12
14-22 0.4-0.7 53-57 579.4 901.5 1024.6 11.71
14-22 0.4-0.7 57-63 567.3 885.0 946.5 12.12
EN24 7-14 0.1-0.4 52-56 241.6 330.4 425.6 15.7
7-14 0.1-0.4 56-62 389.8 530.8 612.5 11.8
7-14 0.4-0.7 52-56 404.0 551.1 634.8 11.2
7-14 0.4-0.7 56-62 484.0 631.8 721.3 11.0
14-22 0.1-0.4 52-56 324.1 415.0 534.7 15.4
14-22 0.1-0.4 56-62 330.3 423.3 488.6 15.2
14-22 0.4-0.7 52-56 488.6 636.1 689.6 11.0
14-22 0.4-0.7 56-62 489.9 637.8 698.7 11.0
Material V do Lo
EN8 7 0.1 56.95
7 0.1 63.00
7 0.4 56.90
7 0.4 62.97
14 0.1 57.00
14 0.1 57.00
14 0.4 56.89
14 0.4 62.95
EN24 7 0.1 55.9
7 0.1 61.9
7 0.4 55.9
7 0.4 62.0
14 0.1 55.9
14 0.1 61.2
14 0.4 55.9
14 0.4 61.5
Table 4: Outputs of GA for different ranges of input variables
(Constant DOC)
Material N(rpm) F(mm) L(mm) Fx(N) Fy(N) Fz(N)
Aluminum 7-14 0.1-0.3 53-56 570.8 690.6 754.6
(d = 0.1 mm) 7-14 0.1-0.3 56-62 592.6 618.1 723.7
7-14 0.3-0.5 53-56 533.5 665.2 717.8
7-14 0.3-0.5 56-62 557.2 676.6 734.5
14-22 0.1-0.3 53-56 616.5 766.0 803.4
14-22 0.1-0.3 56-62 455.6 678.1 722.3
14-22 0.3-0.5 53-56 588.3 755.2 786.5
14-22 0.3-0.5 56-62 586.3 751.9 803.2
Mild steel 7-14 0.1-0.3 54-57 598.6 1988 2034.2
(d = 0.2 mm) 7-14 0.1-0.3 57-61 597.3 1983 2039.7
7-14 0.3-0.5 54-57 620.2 2300 2543.2
7-14 0.3-0.5 57-61 639.0 2231 2442.6
14-22 0.1-0.3 54-57 673.9 2275 2456.8
14-22 0.1-0.3 57-61 673.9 2275 2457.9
14-22 0.3-0.5 54-57 716.7 2524 2756.3
14-22 0.3-0.5 57-61 688.6 2563 2743.4
Material Cc(mm) No do Lo
Aluminum 10.14 7 0.1 56.0
(d = 0.1 mm) 10.06 7 0.1 61.5
10.36 7 0.3 55.9
10.32 7 0.3 61.8
8.83 14 0.1 55.6
8.54 14 0.1 61.9
8.85 14 0.3 55.9
8.89 14 0.3 61.9
Mild steel 9.64 7 0.1 56.9
(d = 0.2 mm) 9.65 7 0.1 61.0
7.73 7 0.3 56.9
9.24 7 0.3 61.0
8.96 14 0.1 56.9
8.97 14 0.1 60.9
8.57 14 0.3 56.8
7.04 14 0.3 60.9
