A study on the establishment of experimental data processing algorithms in mechanical engineering

THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(79) 2014, VOL 1 1 A STUDY ON THE ESTABLISHMENT OF EXPERIMENTAL DATA PROCESSING ALGORITHMS IN MECHANICAL ENGINEERING NGHIÊN CỨU XÂY DỰN[.]

THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(79).2014, VOL 1 1

A STUDY ON THE ESTABLISHMENT OF EXPERIMENTAL DATA

PROCESSING ALGORITHMS IN MECHANICAL ENGINEERING

NGHIÊN CỨU XÂY DỰNG THUẬT TOÁN XỬ LÝ SỐ LIỆU THỰC NGHIỆM

TRONG LĨNH VỰC CƠ KHÍ

Luu Duc Binh

The University of Danang, University of Science and Technology; Email: ldbinh@dut.udn.vn

Abstract - In the engineering sector in general and in particular

Mechanical engineering, experiment is crucial to prove the theory;

build a relationship between the input - output parameters to

analyze trends, developments between the parameters, or to find

out the optimal parameters to meet specific goals Experimental

data processing is decision to the process of experimentation The

data processing can be calculated manually or by computer

applications This paper presents the use of Matlab software to

build data processing algorithms for level 2 Box-Hunter planning

with 3 input parameters to find the regression equation, creating a

basis for developing user interface to show the required graph

Tóm tắt - Trong các ngành kỹ thuật nói chung và Cơ khí nói riêng,

thực nghiệm là công việc hết sức quan trọng nhằm chứng minh lý thuyết; xây dựng mối quan hệ giữa các thông số đầu vào - đầu ra

để phân tích xu hướng, diễn biến giữa các thông số, hoặc tìm ra

bộ thông số tối ưu nhằm đáp ứng mục tiêu cụ thể nào đó… Xử lý

số liệu thực nghiệm mang tính quyết định đến quá trình thực nghiệm Việc xử lý số liệu có thể bằng tính toán thủ công hoặc ứng dụng máy tính Bài báo này trình bày việc sử dụng phần mềm Matlab xây dựng thuật toán xử lý số liệu cho quy hoạch thực nghiệm quay đều bậc 2 Box-Hunter với 3 thông số đầu vào để tìm

ra phương trình hồi quy, tạo cơ sở xây dựng giao diện với người

sử dụng thể hiện các đồ thị cần thiết

Key words - experiments; data processing; planning; Box-Hunter;

Matlab

Từ khóa - thực nghiệm; xử lý số liệu; quy hoạch thực nghiệm;

Box-Hunter; Matlab

1 Problem statement

Technically, among various methods proposed in the

literature to solve problems, resorting to solve the so-called

‘extremal’ problem in order to obtain the optimal

conditions or the optimal values of a MIMO system In

general, according to the number of objects, the systems to

be controlled or optimized are usually complex Therefore,

most of the solutions can be gained experimentally

An active strategy to implement the experiments is

proposed by Ronald Fisher in 1935 to solve the biological

problems Furthermore, G Box, Wilson, Hunter, Cohran

develop an improvement based on the theory of

experimental extreme mathematics around 1950 In

Vietnam, experimental planning is first applied from the

1970s

The most important task of the experimental planning

is the data processing after the experiments This process

can be implemented manually that leads to spend a lot of

time and effort However, there are some cases that cannot

be executed

Computer applications in data processing are the

inevitable trend today However, the use of commercial

software for data processing, such as Minitab, Stata, SPSS,

etc just solves the "flame" of the problem Meaning that, it

is used to input the data and to deduce the results, while

processing algorithms are entirely a "secret technology" of

the software vendors

The initiative in data processing algorithms for all kinds

of different experimental planning will help researchers

understand the experimental nature and the initiative in

data processing as well as allow to establish the interaction

impacts and the influence of high order of factors

For instance, in [6, 7], the authors proposed data

processing algorithms used in Matlab for second order rotatable designs Box-Hunter with 3 inputs and 1 output If the number of inputs is more than 3, it is difficult to geometrically perform the relationship between the inputs and the output, as well as hard to analyze the influence of each input on the output How to choose 3 appropriate inputs now becomes a popular issue in Mechanical Engineering today

2 Preliminaries in Box-Hunter Planning

Box-Hunter planning is a mixed second order rotatable designs This is currently the best planning due to the fact that the rotatable nature makes the accuracy of the regression equation be the same for all the spatial elements, which have the same distance to the center of the planning The combination of the uniform properties and the rotary properties makes the variance be constant in certain areas from the planning center

Figure 1 Box-Hunter diagram, with k = 3

The structure of the uniform rotatable planning of k elements includes: the orthogonal part comprises n1

X1

X2

X3

(-1,+1,-1)

(-1,+1,+1) (-1,-1,+1)

(+1,-1,+1) (+1,+1,+1)

(-1,-1,-1) (0,0,+)

(0,0,-)

(0,+,0) (0,-,0)

(-,0,0)

(+,0,0)

2 Luu Duc Binh experiments constructed fully empirically, 2k experiments

at points (*) and n0 experiments at the center

With k = 3, the corresponding numbers of experiments

are: n1 = 8; 2.k = 6; n0 = 6; the arm  = 1,682 [1, 3, 4]

The regression equation has a form as:

2

k

i i i j ij i j ii i

In [1], after solving the systems of matrices equations,

we obtain the closed form of the coefficients and the

variance as follows:

2

3 1

n

j

=

4

1

n

j

=

0

1

Sb =a S ts (6)

3

i

4

iu

b =a S ts

(8)

ii

b = a +a S ts =a S ts (9) The regenerated variance is defined by the experiments

at the center planning:

( )

0 0

2 1

0 1

n

u u ts

S

n

=

−

=

−



(10) with the regenerated degree of freedom:

Total squared residuals of objective function values is

calculated by the regression equation:

( )2 1

n

i

=

Total squared residuals of objective function values is

calculated empirically:

0 0 1

n

u

=

=  − (13) The degree of freedom of the residual variance:

fdu = n - h - (n0 - 1) (14)

where, h: the quantities of parameter bi has significant

in the regression equation

Residual variance is calculated as:

2

0 1

du

du

S

Normalized Fisher coefficient is calculated as:

2

2

du t ts

S F S

3 Data processing algorithms

3.1 Flowchart algorithm

The regression equation performing the relationship between input-output is established by three following tasks [1]:

- Estimate the coefficient “b”

- Check the meaning of the coefficients b

- Check the compatibility of the regression equation Then, the flowchart algorithm is described as in Figure 2

3.2 Data processing algorithm in Matlab

For the convenience in usage and management, our programs are coded in many files (file.m) In this paper, we briefly present some main steps; the full program is described in [2]

3.2.1 Import of Data

The data import is created by file “Solieu.m” with data such as: Student number, Fisher number; marginal value of coded variables; coefficient a1  a7; experimental matrix X and experimental results Y

% Box-Hunter design with 3 factors

% Data

tb = 3.365; %t(p,f) Student coefficient with P=0.99 and f=m-1=5

anfa = 1.682;

Fbang = 10.2; % fisher with F(0,01;9;5)

lb = [-1.682 -1.682 -1.682]; ub = [1.682 1.682 1.682]; % the variation of factors

k = 3;

a = [0.1663 0.0568 0.0732 0.125 0.0625 0.0069 0.0568]; % a: handbook

% Matrix x1 = [1 1 1 1 1 1 1 1 anfa -anfa]';x1(20)=0;

x2 = [1 1 1 1 1 1 1 1 0 0 anfa -anfa]';x2(20)=0;

x3 = [1 1 1 1 1 1 1 1 0 0 0 0 anfa -anfa]';x3 (20)=0;

%y: data from experiment

y = [25.86 23.12 25.23 24.40 25.76 22.88 25.07

24.08 26.51 21.30 25.14 24.81 25.81 25.26

25.82 25.80 25.73 25.59 25.89 25.55]';

THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(79).2014, VOL 1 3

In this example, experimental data is used in [7]

Figure 2 Flowchart of data processing algorithms

3.2.2 Implementation and results

We code the data processing programs based on the

proposed algorithm in Figure 2 The full program is

presented in [2]

The results consisting of coefficient b, conclusions of the regression equation are then displayed as:

Figure 3 Results

However, for the simple understanding of the results as well as to perform these results in geometrical form, the author has built a human-machine interface as showed in Figure 4

Figure 4 Human-machine interface

The right hand side of the interface shows the data import, the coefficients b and the regression equation In addition, it also shows the optimal input values when the output reaches the extremes

The left hand side of the interface describes the graphs The graphs present the relationship between the two out of three input elements and the output element; the remaining

Calculated b and

variance Sb:

eq (2)  (9)

Calculated generated

variance Sts; Residual

variance Sdu; Ft:

eq (12)  (16)

Testing

the compatible

of temporary regression

equation:

F t < F b

The regression

equation with level 

Real variance

End

Data import:

Matrix X, Y; a; the

variation of factors; ;

coefficient Student t b ; F b

Testing

the significant

of coefficients b:

Retesting the significant coefficients:

least square method

T

F

T

F

Temporary

regression equation

Increase ;

decrease the variation of factors

Removing the insignificant coefficients

4 Luu Duc Binh input value can be selected by the optimal value or a

random value in a defined domination by a simple and

visual operation on the interface Simultaneously, the

interface also represents the contour line of graph in order

to choose the optimal value for those two inputs

4 Conclusion

This paper proposes the data processing algorithms for

Box-Hunter planning with three inputs It is the basis for

the computer application in experimental data processing

In comparison with the manual data processing which

encounters with some problems such as taking a long time,

being confused easily and the error caused by rounding the

results, the proposed algorithm is rapidly, reliable with

high accuracy, visual performance It can be applied in

scientific research with good results

In future work, we will invest in the algorithms for the

other experimental plannings such as: Wilson;

Box-Behnken; TNT, TNR…

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(The Board of Editors received the paper on 15/02/2014, its review was completed on 13/03/2014)