Application of an improved genetic algorithm in the optimization design of NC
machine tool spindle
Zhiwei Xu 1, Yongxian Liu 1, Hualong Xie1 ˈPeng Tian1
1. School of Mechanical Engineering & Automation, Northeastern University, Shenyang 110004, China
E-mail: zhwxu@mail.neu.edu.cn
Abstract: In the optimization design of machine tool spindle, an improved genetic algorithm is proposed in this paper.
Through the analysis of the specific structure of the spindle and the specific force and deformation in the process of
machine tool processing, a mathematical model of the genetic algorithm optimization design of the spindle is established
in accordance with the actual use of the spindle. Genetic algorithm has the advantages of good adaptability and faster
convergence to the optimal solution. The coding methods and genetic operators are improved to compensate for their
limitations. In this paper, a real coded scheme and an improved genetic operator are adopted to optimize the structural
design of the spindle. This paper introduces the main steps and process optimization design, through the example of CNC
machine tool spindle design of Shenyang Machine Tool Co. Ltd. The experimental data show that the method can
quickly find the global optimal solution and verify the feasibility and practicality of the method.
Key Words: genetic algorithmχMachine spindleχoptimal designχCAD system
1 INTRODUCTION
Optimization design is an important research field in
modern mechanical design theory and method.
Optimization design can not only reduce the weight of
mechanical products, decrease the manufacturing cost, but
also improve the quality and performance of the products.
The optimal design method is to use the mathematical
programming method to set the optimal design parameters
on the computer, so that the design index can reach the
optimal value under the given design parameters and
restriction condition.
In the mechanical optimization design, the genetic
algorithm is an adaptive heuristic global search algorithm,
which has the incomparable advantages of the traditional
algorithm, and is easier to solve various optimization
problems. Chakraborthy [1] et al. takes the fatigue life as the
optimization objective function and uses the genetic
algorithm to describe the design optimization of a rolling
bearing with five design parameters. They have obtained the
geometrical parameters inside the bearing from different
boundary dimensions. Due to the limitation of the
traditional genetic algorithm, such as: slow solution in
genetic algorithm, strong reliance on coding scheme and
inferior robustness of the algorithm, etc., some modification
need to be made, which has become a hot issue in genetic
algorithm research.Li Gongfang [2] et al. combined the
global search ability of genetic algorithm with local search
ability of chaos optimization, and then proposed a chaos
genetic algorithm which has fast convergence speed and
high precision. GuanQi Ming [3], with the aim to optimize
the lightest weight of the machine tool spindle, used neural
network instead of finite element structural approximate
This work is supported by Science and Technology Project of Liaoning
Province under Grant No. 2011216010
c
978-1-5386-1243-9/18/$31.00 2018
IEEE
analysis. Genetic algorithm is adopted in his work to solve
the optimal mathematical model, and the structural
optimization design of the spindle is carried out. The
experimental results show that the method is a better method
for optimization.
Spindle is one of the core components of machine tools, and
its performance has a decisive impact on the processing
quality and production efficiency. How to reduce its volume
and mass, how to increase its stiffness and improve its
ability to transfer torque are the main objectives of the
optimization design of the main axis of the machine tool. In
this paper, an improved genetic algorithm is used to
optimize the spindle of CNC machine tool.
2 OPTIMIZATION DESIGN MODEL
SPINDLE OF MACHINE TOOL
OF
A spindle is generally a hollow stepped shaft. In the process
of optimizing the structure parameters of the spindle, the
stepped shaft is simplified into a uniform shaft with
equivalent diameter. The structural analysis and
optimization design is carried out by the discretization
processing and the selecting appropriate optimization
design method
2.1 Design Variable
A design can be represented by a set of basic parameters,
such as the length of the component, the size of the section,
the weight of the component, etc.. The optimal design of a
specific problem can be expressed as a column vector[4]:
X = [ x1 , x 2 , x3 , " , x n ]T
In the optimization design, the n-dimensional vector space
described by the coordinate axis of each design variable is
called the design space, and one point of the design space is
a design scheme.
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According to the above theory, the design variables of the
multi-support machine tool with concentrated force and
torque are designed with the inner diameter dimension G, the
outer diameter dimension D and the length of each section
of the shaft L as the design variable [L L = Q , n is
the number of design variables of the model. As shown in
figure 1.Therefore, the optimum design variable of spindle
is:
In the design of machine tool spindle, according to the
design requirements, and the working condition and load
condition of spindle, maximum stiffness and minimum
volume are the target of design. This can not only meet the
requirements of the spindle drive, but also can reduce
weight, save material and reduce cost. Fig. 2 is a simplified
drawing of the main axle load
If I [ is used as a function to reflect the spindle stiffness,
; = >G '' " ' Q // " /P @7
and I [ is a function that reflects the spindle volume, then
the objective function of the spindle optimization can be
expressed as:
I [ = ωI [ + ωI [
㸦1㸧
where, ω ࠊ ω are weighting factors, which reflect the
importance
of
each
sub
objective
function.
I [
Fig.1 The spindle structure
N =
2.2 Object Function N =
The objective function is the function relation between the
optimal target and the variable that the design problem is to
be achieved. The function relation which can make the
design optimization is also the evaluation criterion of
comparing the merits of the design scheme, so it is also
called
the
evaluation
function,
denoted
as
I [ or ) [ ,which is to emphasize its dependence on
design variables, such as life, weight, price, etc. The
objective function can be written as:
I [ = I [ [ [ " [ Q
For multi-objective functions, n objective functions can be
listed independently:
I [ = I [ [ [ " [ Q
I [ = I [ [ [ " [ Q
#
#
IQ [ = IQ [ [ [ " [ Q
To convers the multi-objective problem into a
single-objective problem a comprehensive objective
function expression can be established
Q
I [ =
ω IM [
¦
M
L
=
N + N )D /
(, D
,
I [
=
¦
π
' L − G
× /L
+
D
)
N D
> + $
+
+
N$
N % /
/
Where, K1 indicates the bending deformation of the spindle
itself at the end of the shaft
K2 represents the displacement of the spindle end caused by
the deformation of the bearing
F is the cutting force
N $ is the stiffness of the front support
N % is the stiffness of the rear support
Therefore, the objective function can be expressed as:
I [ = ω
)D /
)
N D D
+ + > + $ +
+ @
(, D
N$
N% /
/
+ ω ¦
π
'L − G × /L
(2)
In the design of machine tool spindle, according to the
design requirement of spindle, the constraints are divided
into static constraints and dynamic constraints. The static
constraints includes cutting force deflection constraint,
strength constraint, angle constraint, torsion deformation
constraint and design variables boundary constraint;
dynamic constraint includes first order vibration frequency
constraints and so on.
=
where, ωL is the weighting factor of item M 㸪 it is a
nonnegative real number, whose value is chosen according
to the importance of the index in the optimization design.
3 OPTIMIZATION DESIGN PROCEDURES
In accordance with the mathematical model of the optimal
design of the spindle established above, the fitness function
of the genetic algorithm is constructed. According to the
practical problems, the appropriate coding method is
selected to improve the corresponding genetic operators,
and then the optimization design of the spindle is carried out
based on the improved genetic algorithm.
3.1 Chromosome Encoding
Fig.2 The simplified diagram of spindle load
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The standard genetic algorithm uses binary encoding.
Binary encoding is the genetic space that maps the solution
of the original problem into 0 and 1 encoding strings.
Through the calculation of the fitness function the suitable
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solution is found when the fitness of the individual reaches a
given threshold, and then the decoding process is reduced to
the solution of the actual problem. Real number coding is
using real numbers directly to encode the solution of the
problem. It is a direct natural description of the continuous
parameter optimization problem, and there is no encoding
and decoding process. In a real coded case, a real parameter
vector corresponds to a chromosome, a real number
corresponds to a gene, and a real value corresponds to an
allele.
Real coding eliminates the hidden trouble that due to
insufficient coding accuracy possible solutions that have
better fitness values but fail to represent in the search space.
The real coded genetic algorithm has the ability to make use
of the gradient of continuous variable functions. In addition,
real coded encoding can eliminate the Hamming cliff in
binary coding.
3.2 Produce Initial Population
The vector ; .M is used to represent the design variable
vectors corresponding to the gene chain of the j-th
individuals in the K-th generation group, and the random
real numbers are generated by the UDQGYDO functions in
MS VC++6.0 between RAND_MIN and RAND_MAX.
void GA::initialize()
{
double lbound[11];
double ubound[11];
for (i = 0; i < NVARS; i++) // Initialization variable gene
{
for (j = 0; j < POPSIZE; j++)
{
population[j].fitness = 0;
population[j].rfitness = 0;
population[j].cfitness = 0;
population[j].lower[i] = lbound[i];
population[j].upper[i]= ubound[i];
population[j].gene[i] =
randval(population[j].lower[i],population[j].upper[i]) ;
// The initial population is randomly generated in the
interval of design variables
}
}
}
The function UDQGYDO generates [min, max] random
floating numbers which fill individuals gene chains one by
one. The random number generated by UDQGYDO is
uniformly distributed between [min, max], so that the
original population is produced throughout the solution
space.
the next generation, while individuals with lower fitness are
less likely to pass on to the next generation. It is important to
establish an adaptive evaluation function HYDOXDWH [ . The
evaluation of the individual's advantages and disadvantages
is to obtain the function value H M of the adaptive
evaluation function HYDOXDWH [ of each individual. The
evaluation criteria for each individual are as follows: the
greater the fitness value, the better the vector ; M of the
design variable.
In this paper, the penalty function is used to transform the
constrained optimization problem into the penalty function
of unconstrained optimization problem, and the constrained
optimization problem is transformed into an unconstrained
problem. For individuals who do not meet the constraint
conditions, a small fitness value is given, and the
corresponding adaptive evaluation function is:
HYDOXDWH ; = I ; + S ;
Where , S ; is a penalty function.
Penalty function method belongs to the indirect method to
solve the problem of optimization, In genetic algorithms,
the number of infeasible solutions in the iteration is limited
by adding penalty terms to the fitness function, at the same
time the amount of non feasible solutions are allowed in
order to preserve the excellent individuals and the diversity
of the population.
In the static penalty function method, the number of
penalty factor parameters is very large, and the choice of
parameters has a great impact on the search efficiency. At
the same time, the value of the appropriate penalty factor
will change with the evolution process.Therefore, the static
penalty factor can not guarantee that the search is feasible.
This paper chooses the dynamic penalty function method to
deal with the constraint problem.
The dynamic penalty function method can dynamically
adjust the penalty factor as it evolves, and its penalty
function is written as㸸
S ; = FW
α
P
J Mβ ;
¦
M
㸦3㸧
=
Where , t is for evolutionary algebra, usually F = 㸪
α = β = ࠋ
Therefore, the fitness function of the spindle of the machine
tool can be constructed as follows:
HYDOXDWH [ = I [ + S [
= ω
ω
N D D
)D /
)
+ +
>+ $
+
+ @
(, D
N$
N % /
/
π
¦ 'L
− G × /L + FW α
+
(4)
P
J Mβ ;
¦
M
=
3.4 Operation of Genetic Operators
3.3 Construct the Fitness Function
The probability of a genetic algorithm that is proportional to
individual fitness determines how many individuals in the
current population are inherited to the next generation.
Individuals with higher fitness are more likely to pass on to
3.4.1 Selection Operator
Proportional Model is a playback random sampling method.
The basic idea is that the probability that each individual is
selected is proportional to its fitness size. The idea of
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optimal preservation strategy is to copy the individuals with
the highest fitness in the group to the next generation
without pairing and crossing.
The combination of these two methods can ensure that the
optimal individuals are not destroyed by genetic operations
such as crossover and mutation. At the same time, the global
search ability of the algorithm is not deteriorated, so as to
avoid gene loss, improve global convergence and
computational efficiency.
3.4.2 Crossover Operator
In genetic algorithms, individuals need to be paired in a
group before cross computing. The commonly used pairing
strategy is a random pairing, in which M individuals in a
group form M/2 pairs of paired individuals in a random way,
and crossover is performed between two individuals in these
paired individuals.
The way chromosome coding directly affects the design of
crossover operators. For real coded chromosomes, the
arithmetic crossover method can be used. It is a linear
combination of two individuals producing two new
individuals. Therefore, this paper chooses the dynamic
arithmetic crossover method, and its main operation is:
1 to determine the coefficients of a linear combination of
two individuals.
2 to produce new individuals in a linear combination.
If two individuals make arithmetic intersections
between ; LW
and ; MW , the two new individuals generated after the
Initialization
Initial population
generation
Calculating initial
population
Fitness value
Does it satisfy the
stopping criteria?
Yes, decoding
optimum
solution
no
Elite proportional
selection operation
crossover operation are:
Dynamic arithmetic
cross operation
­; W + = α; W + − α
M
° L
®
W +
°̄; M
= α; LW + − α
Uniform mutation
operation
; LW
; MW
Where , α is a variable determined by the number of
evolutionary generations
3.4.3 Mutation Operator
The operation of uniform mutation is to replace the random
numbers in a certain range and replace the original gene
value in each locus of the individual coding string with a
smaller probability. The uniform mutation method makes
the search points can move freely in the entire search space,
so it can increase the diversity of the population; In the
meanwhile, uniform mutation method is generally for the
real number encoding, so the method of uniform mutation is
adopted in this paper.
3.5 The Implementation of Genetic Algorithm
The operation flow of the genetic algorithm is shown in
figure 3. The specific implementation steps are as follows:
First step, the initial population is randomly generated, and
the number of individuals is certain, and each individual is
coded as a real number system of chromosomes;
Second step , calculate the fitness of the individual and
determine whether it conforms to the optimization criteria.
If the output is in line with the optimal individual and the
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optimal solution represented, it will be finished; otherwise,
proceed to the third steps;
Third step 㸪the optimal selection strategy and fitness ratio
selection method are adopted to select the regenerative
individual according to the fitness;
Fourth step, a new individual is generated according to a
certain crossover probability and dynamic arithmetic
crossover;
Fifth step , generate new individuals according to the
probability of variation and the method of uniform
variation;
Sixth step, a new generation of population produced by
crossover and mutation, returns to the second step.
Calculating new
populations
Fitness value
Fig.3 The procedure of Genetic algorithm
4 OPTIMIZATION DESIGN EXAMPLE
4.1 Optimization Calculation of Machine Tools
Taking SSCK40-750 machine tool developed by Shenyang
Machine Tool Co., Ltd. as an example, the optimization
design based on improved genetic algorithm is carried out.
The main motor power SSCK40-750 type machine, the
minimum speed UPLQ = U PLQ , the main cutting force
(perpendicular to the plate) ) = 1 , side cutting
force ) = × ) = 1 (radial force, parallel and
slide), feed force ) = × ) = 1 ,
torque 0 = 1 ⋅ PP , Poisson's ratio ν = ,
Young's modulus ( = × ,
density ρ = × − NJ PP .
The spindle structure of the machine tool is shown in Figure
1.
The 30th Chinese Control and Decision Conference (2018 CCDC)
According to formula (1), the optimum design variable of
this machine tool spindle is
; = >G ' ' ' ' ' // / / / @7
According to the formula (2), the objective function of the
optimal design of the spindle of the model machine is:
I [ = ω
+ ω
π
¦
N D
)D /
)
D
+ +
+
+ @
> + $
N$
N % /
/
(, D
D
)D /
)
N D
+ +
> + $
+
+ @ is the
Where,
(, D
N$
N % /
/
rigidity of the spindle of this type of machine tool.
π
¦
L
'L − G
=
Initial
value
/mm
Optimization
value /mm
L
HYDOXDWH [ = I [ + S [
N D
)D /
)
D
+ +
+
+ @
> + $
(, D
N$
N % /
/
+ ω
Optimized
parameter
is the spindle volume of this type of
× /L
machine tool
/ = / + / , D = / + / + /
According to the formula (4), the fitness function of
constructing the corresponding genetic algorithm is:
= ω
Table 1 The contrast diagram between optimization of data
and original data
' L − G × /L
L =
The maximum evolutionary algebra is selected
MAXGENS = 500. The optimized design of the spindle is
carried out by using the improved genetic algorithm in this
paper, and the optimized results are shown in Table 1.
The maximum evolutionary generation value is chosed
MAXGENS=500. The optimization design of the spindle is
carried out by using the improved genetic algorithm in this
paper, and the optimization result is given by table 1.
π
¦
L
' L − G × /L + FW
=
α
Optimized
parameter
Initial
value
/mm
Optimization
value /mm
d
65
70
L1
62
65
D1
95
90
L2
35
39
D2
100
98
L3
42
44
D3
103
102
L4
105
106
D4
105
103
L5
188
187
D5
110
107
In order to verify the static and dynamic characteristics of
the optimized spindle, it is necessary to establish a finite
element analysis model of the optimized spindle, and
perform corresponding performance analysis, as shown in
Figure 4.
¦ J Mβ ;
M
=
Where , J M [ is a constraint function
4.2 Result Analysis
The number of design variables of the spindle is 11,
considering the quality and efficiency of genetic
optimization. Therefore, the population size POPSIZE =
100 is selected. In order to increase the new crossover
probability in the process of genetic optimization, the
crossover probability is PXOVER = 0.8;To improve the
diversity of groups on the basis of the evolution of groups to
prevent the loss of important information, so select the
mutation probability is PMUTATION = 0.15;
Table 1 is the spindle design parameters optimized before
and after optimization of the comparative data. After
optimization, the stiffness is increased by 6.8% and the
volume is reduced by 4.3% .It can be seen that the
optimization effect is obvious after the optimization design,
which shows the feasibility and superiority of this method.
At the same time, through the example design and the result
analysis, it also shows that the machine tool spindle
structure designed by the optimized design method is more
reasonable, which fully shows the benefit and application
value of the genetic algorithm in the optimization design of
the machine tool spindle components.
REFERENCES
[1]
[2]
[3]
[4]
[5]
I. Chakraborthy, K. Vinay, S.B. Nair, R. Tiwari, Rolling
element bearing design through genetic algorithms,
Engineering Optimisation 35 (6) (2003) 649-659.
Li Hongfang. Chaotic genetic algorithm and structural
optimization design, [D], Tianjin University, 2004
Guan Qiming, Yin Jian, Li Yi, et al. Implementation of
genetic algorithm for optimal design of dynamic
characteristics of machine tool spindle. Journal of Guizhou
University of Technology (NATURAL SCIENCE
EDITION), Vol. second, thirty-third: February, 85-882004
Shen Liang. CAD expert system for helical gear design -genetic algorithm optimization design [D], 2004
Shantanu Gupta, Rajiv Tiwari, Shivashankar B. Nair,
Multi-objective design optimisation of rolling bearings using
genetic algorithms, Mechanism and Machine Theory, 2007
Fig. 4 Cloud chart of displacement
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