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 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 Box-Hunter planning with input parameters to find the regression equation, creating a basis for developing user interface to show the required graph Tóm tắt - Trong ngành kỹ thuật nói chung Cơ khí nói riêng, thực nghiệm công việc quan trọng nhằm chứng minh lý thuyết; xây dựng mối quan hệ thông số đầu vào - đầu để phân tích xu hướng, diễn biến thơng số, tìm thơng số tối ưu nhằm đáp ứng mục tiêu cụ thể đó… Xử lý số liệu thực nghiệm mang tính định đến q trình thực nghiệm Việc xử lý số liệu tính tốn thủ cơng ứng dụng máy tính Bài báo trình bày việc sử dụng phần mềm Matlab xây dựng thuật toán xử lý số liệu cho quy hoạch thực nghiệm quay bậc Box-Hunter với thơng số đầu vào để tìm phương trình hồi quy, tạo sở xây dựng giao diện với người sử dụng thể 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; BoxHunter; Matlab Problem statement processing algorithms used in Matlab for second order rotatable designs Box-Hunter with inputs and 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 appropriate inputs now becomes a popular issue in Mechanical Engineering today 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 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 (-1,-1,+1) X3 (-1,+1,+1) (0,0,+) (+1,+1,+1) (+1,-1,+1) (0,-,0) (-,0,0) (0,+,0) X2 (+,0,0) (-1,-1,-1) (0,0,-) X1 (+1,-1,-1) (-1,+1,-1) (+1,+1,-1) Figure Box-Hunter diagram, with k = The structure of the uniform rotatable planning of k elements includes: the orthogonal part comprises n1 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:  b x +  k y = b0 + i i i =1 bij xi x j +  bii xi2 (1) i =1 In [1], after solving the systems of matrices equations, we obtain the closed form of the coefficients and the variance as follows: n b0 = a1 y k j − a2 j =1 n  x y ij (2) j i =1 j =1 n bi = a3  x y , ij i =  k j (3) j =1 n biu = a4  x x uj y j , ij u  i; i , u =  k (4) j =1 n k n k bii = a5  xij2 y j + a6  xij2 y j − a7  yi (5) Sb20 = a1.Sts2 (6) Sb2i = a3 Sts2 (7) Sb2iu = a4 Sts2 (8) Sb2ii = ( a5 + a6 ) Sts2 = a7 Sts2 (9) j =1 i =1 j =1 i =1 The regenerated variance is defined by the experiments at the center planning: ( y n0 u Sts2 = − y0 u =1 ) (10) n0 − with the regenerated degree of freedom: fts = n0 – (11) Total squared residuals of objective function values is calculated by the regression equation: n ( S pt =  yi − yi i =1 ) (12) Total squared residuals of objective function values is calculated empirically: n0 ( Stn =  yu0 − yu0 u =1 ) (15) Normalized Fisher coefficient is calculated as: k k i , j =1; i j in the regression equation Residual variance is calculated as: S pt − Stn S pt − Stn Sdu = = f du n − h − n0 − (13) The degree of freedom of the residual variance: fdu = n - h - (n0 - 1) (14) where, h: the quantities of parameter bi has significant Ft = Sdu2 Sts2 (16) 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 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 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 anfa anfa]';x1(20)=0; x2 = [1 -1 -1 1 -1 -1 0 anfa anfa]';x2(20)=0; x3 = [1 1 -1 -1 -1 -1 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 In this example, experimental data is used in [7] Data import: Matrix X, Y; a; the variation of factors; ; coefficient Student tb; Fb Increase ; decrease the variation of factors presented in [2] The results consisting of coefficient b, conclusions of the regression equation are then displayed as: Calculated b and variance Sb: eq (2) (9) Testing the significant of coefficients b: F Removing the insignificant coefficients T Temporary regression equation Retesting the significant coefficients: least square method Calculated generated variance Sts; Residual variance Sdu; Ft: eq (12)  (16) Testing the compatible of temporary regression equation: Ft < Fb Figure 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 F T The regression equation with level  Real variance Figure Human-machine interface End Figure Flowchart of data processing algorithms 3.2.2 Implementation and results We code the data processing programs based on the proposed algorithm in Figure The full program is 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 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 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: Box-Wilson; BoxBehnken; TNT, TNR… REFERENCES [1] Lưu Đức Bình, Quy hoạch thực nghiệm lĩnh vực Cơ khí, Đại học Đà Nẵng, Bài giảng cao học ngành Công nghệ chế tạo máy,2013 [2] Lưu Đức Bình, Nghiên cứu ảnh hưởng thơng số cơng nghệ đến chất lượng q trình gia công tia lửa điện, Luận án Tiến sĩ kỹ thuật, Đại học Đà Nẵng, 2012 [3] PGS.TS Nguyễn Hữu Lộc, Phân tích quy hoạch thực nghiệm, Đại học Quốc gia TP HCM, 2012 [4] PGS.TS Nguyễn Doãn Ý, Quy hoạch xử lý số liệu thực nghiệm, Nhà xuất Xây dựng, Hà Nội, 2006 [5] GS TSKH Nguyễn Phùng Quang, Matlab Simulink dành cho kỹ sư điều khiển tự động, Nhà xuất Khoa học Kỹ thuật, Hà Nội, 2005 [6] Hoàng Vĩnh Sinh, Trần Xn Tùy, Lưu Đức Bình, Tối ưu hố q trình gia cơng cắt dây tia lửa điện với mục tiêu đạt suất gia cơng cao nhất, Tạp chí Khoa học & Công nghệ Trường Đại học Kỹ thuật, số 84 (69-73), 2011 [7] Lưu Đức Bình, Xây dựng giao diện máy tính cho việc lựa chọn thông số công nghệ máy cắt dây tia lửa điện, Tạp chí Khoa học & Cơng nghệ, Đại học Đà Nẵng, số 54, 2012 (The Board of Editors received the paper on 15/02/2014, its review was completed on 13/03/2014) ... 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... 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... Fb Increase ; decrease the variation of factors presented in [2] The results consisting of coefficient b, conclusions of the regression equation are then displayed as: Calculated b and variance