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The use of the Taguchi method and principal component analysis for the sensitivity analysis of a dual-purpose six-bar mechanism F C Chen1∗ , Y F Tzeng2 , W R Chen1 , and M H Hsu1 1 Department of Mechanical Engineering, Kun Shan University, Tainan, Republic of China 2 Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science and Technology, Yen-Chao, Kao-Hsiung, Republic of China The manuscript was received on 16 March 2008 and was accepted after revision for publication on 21 July 2008. DOI: 10.1243/09544062JMES1106

Abstract: In this paper, the Taguchi method and the principal component analysis were applied to a dual-purpose six-bar mechanism for investigating the influence of manufacturing tolerance and joint clearance on the quality of the mechanism. Experiments were carried out based on the orthogonal array from the Taguchi method, which calculated the S/N ratios of the positional and angular errors of a dual-purpose six-bar mechanism. Using the principal component analysis, the S/N ratios were transformed into a multiple performance index to further understand the effect of the control factors on the quality of the six-bar mechanism. Using the analysis of response table and response diagram, the key dimensions of the mechanism could be identified and their tolerance optimized, i.e. decreasing the tolerance of important dimensions and increasing the rest, with the objective of simultaneously improving the quality of the mechanism and reducing the cost. Keywords: Taguchi method, pincipal component analysis, sensitivity analysis, dual-purpose sixbar mechanism, quality design

1

INTRODUCTION

Hsien 710, Republic of China. email: [email protected]

link moves in a series of specified motions guided by the input link, the mechanism is known as a motion generation mechanism. Complex mechanisms are common in engineering applications. These mechanisms are generally a combination of two of the above three types of mechanisms that satisfy specific engineering requirements. One of such mechanisms is a dual-purpose six-bar linkage [1], as shown in Fig. 1. The function relationship between the input and output links is y = x 2 , and the resultant locus of point P of the mechanism is a straight line. This mechanism possesses both the characteristics of a function generator and a path generator and is therefore classed as a dual-purpose six-bar mechanism [1]. The accuracy of linkages passing through prescribed positions is of paramount importance in the design of mechanisms. Even with the most precise design, there will be inevitable constraints in the prototype, such as manufacturing tolerance, joint clearances, and the deflection and thermal deformation of links, all

JMES1106 © IMechE 2009

Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science

The main function of mechanisms is to deliver and transform motion, an example of which is the transformation of continuous rotation of an input link into an intermittent reciprocating motion or the changing of the input–output speed ratio. Linkage mechanisms can be classified into three types: function generation, path generation, and motion generation mechanisms. When the relationship between the output link and the input link satisfies specific functions, the mechanism is classified as a function generation mechanism. If the position of a point on the output link relative to the input link traces a specific path, the mechanism is called a path generation mechanism. If the output ∗ Corresponding

author: Department of Mechanical Engineering,

Kun Shan University, 949 Da Wan Road, Yung-Kang City, Tainan

734

F C Chen, Y F Tzeng, W R Chen, and M H Hsu

of which affect the accuracy of the mechanism. These factors ultimately affect the performance of the designed mechanism. Inadequate design of tolerance on link lengths and joint clearances may result in mechanical error of appreciable magnitude, as was documented in the analysis of linkages in the past few decades. In earlier studies, Garnett and Hall [2] investigated the effects of tolerance and clearance in linkage design. A few researchers [3–6] then introduced variations to linkage synthesis and presented a mechanism synthesis method accounting for manufacturing tolerances and cost by function generation problems. Dhande and Chakraborty [3] proposed a stochastic model, also known as the equivalent linkage model, for analysing the mechanical error of a four-bar function generator by considering the net effect of tolerance and clearance on the length of the equivalent link. Using the equivalent linkage model, Mallik and Dhande [7] performed the analysis and synthesis of four-bar path generators. Lee and Gilmore [8] proposed an effective link length model to generalize the equivalent linkage concept and carried out a sensitivity study on infinitesimal motion. Faik and Erdman [9] introduced a non-dimensional sensitivity coefficient for individual links of a mechanism rather than for the entire system. The work was later extended to the sensitivity synthesis of fourbar linkages with three and four prescribed precision positions [10]. Ting and Long [11] presented a general theory to determine the sensitivity of tolerances

to the performance quality of mechanisms and also a technique to identify robust designs. They demonstrated the effect of tolerance specification on performance quality and showed that performance quality can be improved significantly by tightening key tolerance while loosening less major ones. Zhu and Ting [12] used the theory of performance sensitivity distribution to study the sensitivity of a system to variations. They defined a tolerance box of a mechanism as the contracting circumscribing box of the design sensitivity ellipsoid. Caro et al. [13] proposed an efficient tolerance synthesis method that computes the optimal tolerance box of a selected robust manipulator by finding the largest tolerance box of a mechanism. Expectedly, tightening tolerance would significantly increase the cost of manufacture, whereas relaxing tolerance would result in assembly problems that would potentially compromise the overall performance of the system. The key is thus specifying a set of optimal dimensional tolerances that minimizes the cost of manufacturing while still ensures that the system fulfils its designed function. In 1980, Taguchi proposed the concepts of designing parameters and tolerances. The philosophy of the Taguchi method [14–16] is to achieve a robust engineering design through optimizing design parameters against sensitivity to parameter variations. This is in contrast to the traditional method of controlling the source of the variation at all costs. Despite the successful application of the Taguchi method to improving manufacturing processes and to the development of products, its application on the synthesis of mechanism is only reported by Kota and Chiou [17] in 1993. The design parameters were link dimensions, and these were assigned as the control factors with their corresponding tolerances as the noise factors. Using the orthogonal array of the Taguchi method, the optimum dimension of a path generator of a mechanism was synthesized. Most published Taguchi applications to date are related to the optimization of a single performance characteristic. Also of much interest is the handling of the more demanding multiple performance characteristics (MPCs) [18, 19]. When optimizing a process or product with MPCs, the objective is to determine the best design parameters, which will simultaneously optimize all the quality characteristics of interest to the designer. The more frequently used approach is to assign a weighting factor for each response. The difficulty, however, lies in determining the weighting value itself for each response. The primary ‘weighting’ method is using engineering judgement together with past experiences to optimize MPCs [20]. The consequent results often include some uncertainties in the decision-making process. Using fuzzy logic [21], the MPCs can be easily dealt with by setting up a reasoning procedure for each performance characteristic and transforming them into a single value termed as



Fig. 1

Dual-purpose six-bar mechanism

Taguchi method and principal component analysis for a six-bar mechanism

the multiple performance index (MPI). However, as the number of quality characteristics increases, the forming of the fuzzy rule base becomes increasingly complex. The principal component analysis was proposed by Pearson and developed into a computational method by Hotelling [22]. In research studies, it is common to encounter conditions by which variable parameters could be inter-related [23]. The current pressing need is to reduce the number of variables and to make them independent or, at the least, having linear interrelationships, i.e. make them the so-called potential variables. Clearly, using fewer potential variables or components to effectively represent the complex inter-relationship between parameters in a structure is extremely cost-effective. In this case, the principal component analysis can be used to achieve such an objective. In this paper, the Taguchi orthogonal array is used to conduct experiments for calculating the S/N ratios of the positional and angular errors of a dual-purpose six-bar mechanism. The principal component analysis is then used to determine the main contributor of the S/N ratios to the quality characteristics. The S/N ratios are then transformed into an MPI for analysing the effect of the controlled parameters on the quality of the six-bar mechanism. On understanding the effect of each parameter, decisions are then made to either reduce the tolerances to increase quality or to increase the tolerances to reduce cost.

factors, whereas those that are difficult or expensive to control are assigned as noise factors. 2.1

Ideal function, input signal, and output response

The dual-purpose six-bar mechanism studied herein is shown in Fig. 3. The mechanism is an amalgam of a function and a path generator. During its motion, the output link, in this case output 1, will pass through precise angular positions and the coupler (C), output 2, will pass through a series of points. Its quality characteristic is ‘the smaller the errors are, the better the performance is’ and this applies to the angle of the output link and the position of C. When the angle of the input link is known, the position of all links in the six-bar linkage can be derived using the vector loop method as follows θ2 = 2 tan−1

−F1 ±

E12 + F12 − G12 G1 − E 1

r1 sin θ1 + r2 sin θ2 + r4 sin θ4 θ3 = tan r1 cos θ1 + r2 cos θ2 + r4 cos θ4 2 2 2 −F ± E + F − G 2 2 2 2 θ8 = 2 tan−1 G2 − E 2 ⎛ ⎞ −r1 sin θ1 − r5 sin θ5 ⎜ −r8 sin θ8 − r9 sin θ9 ⎟ ⎟ θ7 = tan−1 ⎜ ⎝ −r cos θ − r cos θ ⎠


(1)

1

5

(2) (3)

(4)

5

−r8 cos θ8 − r9 cos θ9

2 TAGUCHI METHOD

Fig. 2 Taguchi method

−1

1

All man-made machines or set-ups are classified as engineering systems according to the Taguchi method. As shown in Fig. 2, an engineering system generally consists of four sections: signal factor, control factor, noise factor, and output response. Signal factor is the input from the user to the system for the specified output response. If the system’s output response changes with the input signal, the system is considered to possess dynamic characteristics according to the Taguchi method. Parameters that are easy or inexpensive to control are usually chosen as the control

735

where θi is the angular position of the vector r i and E1 = 2r2 (r1 cos θ1 + r4 cos θ4 )

(5a)

F1 = 2r2 (r1 sin θ1 + r4 sin θ4 )

(5b)

G1 =

(5c)

r12

+

r22

−

r32

+

r42

+ 2r1 r4 cos(θ1 − θ4 )

E2 = 2r8 (r1 cos θ1 + r5 cos θ5 + r9 cos θ9 )

(5d)

Fig. 3 Vector loop diagram of the dual-purpose six-bar mechanism Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science

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F2 = 2r8 (r1 sin θ1 + r5 sin θ5 + r9 sin θ9 )

(5e)

G2 = r12 + r52 − r72 + r82 + r92 + 2r1 r5 cos(θ1 − θ5 ) + 2r1 r9 cos(θ1 − θ9 ) + 2r5 r9 cos(θ5 − θ9 ) θ2 = θ5 + α

(5f ) (5g)

At this point, the coordinates of the coupler point C can be expressed as (xc , yc ) = (x1 + r1 cos θ1 + r5 cos θ5 , y1 + r1 sin θ1 + r5 sin θ5 )

(6)

For the six-bar mechanism, the angular displacement of the output link of the function generator ψs (=θ8 ) deviates from the ideal value ψi with the difference εs = ψs − ψi known as the structural error. The combined positional error of the links due to manufacturing tolerance and clearances at the joints is known as mechanical error and is defined as εm = ψm − ψs . Therefore, the error of output 1 between the ideal and actual mechanisms (i.e. the ideal function model for the quality characteristic prediction) is given as y1 = ε = εs + εm = (ψs − ψi ) + (ψm − ψs ) = ψm − ψi (7) As the structural error is much less than the mechanical error [24, 25], equation (7) can be simplified as y1 = ε ∼ = εm , whose characteristic is described as ‘the-smaller-the-better’, with an ideal value of 0. For a dual-purpose six-bar mechanism, the positional error of the path-generating mechanism also has the characteristic of ‘the-smaller-the-better’ and therefore has an ideal value of 0. Therefore, the ideal function model can be expressed as y2 = |Ex |

(8)

y3 = |Ey |

(9)

where Ex and Ey are deviations between the actual and the ideal positions in the X and Y directions, respectively. The input signal φ comes from the angular position of the input link. For every 5◦ made by the input link, a total of five angles (M1 , M2 , . . . , M5 ) [1] are recorded as the input signal in order to compute the position of the coupler and the angle of the output link of the six-bar mechanism. The difference between the actual position and the ideal position is then taken as the output response. This difference has the characteristics of ‘the-smaller-the-better’ and therefore has an ideal value of 0. The expression of S/N ratio can be Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science

expressed as S/N = η = 10 log

1/n

1

n j=1

yij2

,

i = 1, 2, 3

(10)

where n = 5 is the number of the output responses. In this paper, the Taguchi method is first applied to determine three S/N ratios of the X and Y positions of the coupler and the angle of the output link. The principal component analysis is then used to calculate the main contributor to the S/N ratios of the quality characteristics. The S/N ratios are subsequently transformed to an MPI according to the contribution level to further analyse the effects of the control factors on the quality of the dual-purpose six-bar mechanism. 2.2

Control factors and levels

Control factors are parameters in a mechanism that can be easily or economically controlled. By means of the vector loop method, the position of each link of the dual-purpose six-bar mechanism can be determined. The associated design parameters are thus the tolerances of the links: r1 , r2 , r3 , r4 , r5 , r7 , r8 , r9 , as well as angles θ4 , θ9 , and α. The nominal values of the parameters of this mechanism are listed in Table 1 [1]. The net change in the equivalent link’s length due to manufacturing tolerance is taken as the equivalent link length tolerance, σ = 0.0025 mm, and the angular tolerance of σ = 0.01. In two-level tolerance design experiments, the factor levels are established as [16] Level 1 = Nominal − σ

(11a)

Level 2 = Nominal + σ

(11b)

2.3

Noise factors and levels

Noise factors are unique operating parameters that are not easy or immensely costly to control. Consequently, the selection and setting of these factors are vital in the development of a widely applicable and

Table 1

Control factors and nominal values

Factor

Design parameter

Nominal value

A B C D E F G H I J K

r1 r2 r3 r4 r5 r7 r8 r9 θ4 θ9 α

4.291 93 2.378 11 0.863 56 6.195 45 1.629 85 1.933 68 2.833 94 3.308 39 40.326◦ –41.755◦ 125.232◦



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robust product or manufacturing technique. For a sixbar mechanism, it is recognized that there are various factors that are difficult to control. These include manufacturing errors, the extension and compression of links, and lateral bending of links. Another factor is the operating temperature that will have an effect on the thermal expansion or compression of the link length. As the aim of this paper is to determine the sensitivity of the dimensions of the links, these noise factors will not be considered herein.

orthogonal array as an example, the initially required full factorial 211 = 2048 sets of experiments can be significantly reduced to 12 sets with similar results. Moreover, interaction among factors would be evenly distributed to each column, another feature of this method, thereby ensuring that the effect of interaction is minimized.

2.4

The objective of principal component analysis [22] is to use the least number of independent variables to describe a large number of raw data by deriving weighting values from the available information. As it is not a subjective decision, the weighting value is a good method of attributing each variable the required level of importance and at the same time treating the general information objectively. In simple terms, the principal component analysis has the capability of finding variations within gathered information. Suppose a product has two quality characteristics and that the axis system in two-dimensional has a point (X1 , X2 ) = (1, 1). Using coordinate transformation, the new axis system Y1 and Y2 can be obtained as shown in Fig. 4. From Fig. 4, it can be seen that the point (X1 , X2 ) = (1, 1) when located in the new axis system has a relative relationship with Y1 and Y2 as almost zero. Moreover, on the two-dimensional plane view, its changes are mainly at Y1 with very little variation at Y2 .

Experimental layout

Experiments were conducted in accordance to the L12 orthogonal array with the afore-defined input signal, and control factors assigned into the array are shown in Table 2. It also shows the complete assignment of the L12 orthogonal array with the inner columns assigned for the control factors and the outer columns for the values of the input signal. The Taguchi method uses an orthogonal array to plan for experiments and for analysing results. In the L12 orthogonal array, A1 and A2 are compared by using the average of the results from experiments 1–6 and that from experiments 7–12. It is noted that in column A in the A1 experiments, both B 1 and B 2 occurred three times. The same is observed for the A2 experiments. This is termed ‘balanced’ or ‘orthogonality’. The main effect of B (i.e. the average difference between B 1 and B 2 ) is separable from the comparison between A1 and A2 . As C 1 and C 2 are replicated the same number of times within A1 and A2 , C is also said to be in balance with A. Similarly, D, E, F, G, H, I, J, and K are all in balance with A. Likewise, column B balances with A, C, D, E, F, G, H, I, J, and K and column C balances with A, B, D, E, F, G, H, I, J, K and so on. In short, the average effects of all factors are balanced with one another. Using orthogonal array can substantially reduce the time and cost of developing a new product or technique, thereby increasing the competitiveness of the product in the open market. Taking the L12 (211 )

Table 2

3

PRINCIPAL COMPONENT ANALYSIS

Fig. 4

Experimental layout M1

L12

A

B

C

D

E

F

G

H

I

J

K

1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 2 2 2 2 2 2

1 1 1 2 2 2 1 1 1 2 2 2

1 1 2 1 2 2 2 2 1 2 1 1

1 1 2 2 1 2 2 1 2 2 1 1

1 1 2 2 2 1 1 2 2 1 1 2

1 2 1 1 2 2 1 2 2 1 2 1

1 2 1 2 1 2 2 2 1 1 1 2

1 2 1 2 2 1 2 1 2 2 1 1

1 2 2 1 1 2 1 1 2 2 1 2

1 2 2 1 2 1 2 1 1 1 2 2

1 2 2 2 1 1 1 2 1 2 2 1


Coordinate system

y11

y21

M2 y31

y12

y22

M5 y32

...

y15

y25

y35


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When the axes X1 and X2 are rotated counterclockwise by an angle θ , the original point relative to the new axis system is given as Y1 = (cos θ )X1 + (sin θ )X2

(12a)

Y2 = (− sin θ )X1 + (cos θ )X2

(12b)

√ If θ = 45◦ , the above equations give (Y1 , Y2 ) = ( 2, 0), and as shown in Fig. 4, the objective of transforming the Y1 axis into the largest variable has been achieved. Before carrying out the principal component analysis, the units of the quality characteristics are checked. If they were different, standardization has to be carried out, ensuring that the average value is 0 and that the standard deviation is 1. If there were m number of experiments, the control factors A, B, . . . , K represent the design parameters in the experiments, xi ( j) represents the jth measured value of the quality characteristics of the ith experiment, and p represents the number of the quality characteristics. Therefore, the quality characteristic matrix X can be expressed as follows ⎡ ⎤ x1 (1) x1 (2) · · · x1 (p) ⎢ x2 (1) x2 (2) · · · x2 (p) ⎥ ⎢ ⎥ X=⎢ . (13) .. .. ⎥ ⎣ .. . ··· . ⎦ xm (1) xm (2) · · · xm (p) where xi ( j)(i = 1, 2, . . . , m; j = 1, 2, . . . , p) The principal component analysis is then applied to determine the principal components of the independent factors Yp ’s, which then replace the original values. The method establishes a first-order linear expression of Y1 , Y2 , . . . , Yp and can be expressed as: Y1 = a11 x1 (1) + a21 x1 (2) + a31 x1 (3) + · · · + ap1 x1 (p) Y2 = a12 x1 (1) + a22 x1 (2) + a32 x1 (3) + · · · + ap2 x1 (p) ........................................................... ........................................................... Yp = a1p x1 (1) + a2p x1 (2) + a3p x1 (3) + · · · + app x1 (p) (14) where Y1 , Y2 , . . . , Yp are mutually independent, [a11 , a21 , . . . , ap1 ] is the first eigenvector of the correlation matrix [a1p , a2p , . . . , app ] is the pth eigenvector and so on. Generally, if there were p number of variables in the raw data, there should also be p principal components after transformation. The main difference between the principal components and the original variables is that the former is mutually independent, derived by stripping the inter-dependence properties of the latter through linear transformation. In most of the cases in which principal component analysis is used, the first few components contain a large part of the total variance, and the original p-dimensional Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science

data set can then be approximated by a q-dimensional (q < p) data set without substantial loss of information by discarding the p − q highest order principal components. 4

RESULTS AND DISCUSSION

4.1

Principal component analysis

By standardizing the signal and control factors in the L12 orthogonal experimental set-up, the dual-purpose six-bar mechanism is analysed. The position errors of the coupler in the X and Y directions, η1 and η2 , and the error in the angle of the output link, η3 , are analysed for the S/N ratios. Table 3 shows the computer-simulated experimental results. Prior to the principal component analysis, the original data are standardized to an average value of 0 and a standard deviation of 1. The standardization equation is shown as follows xˆ i ( j) =

¯ j) xi ( j) − x( σ

(15)

where σ =

m i=1

m ¯ j) = x(

¯ j)]2 [xi ( j) − x( m−1

(16)

xi ( j) m

i=1

(17)

The standardized S/N ratios are shown in Table 4. The values are then input into a computational software to calculate the correlation matrix. This process is to determine whether the principal component analysis is applicable to this set of variables. If the variables do not possess any correlation, then the returned correlation value is 0 and the analysis ceases. From Table 5, it can be seen that the correlation of the positional accuracy of the coupler in the X and Y directions is as high as 0.954, suggesting that if the accuracy in the X direction is improved, the accuracy in the Y direction would improve simultaneously. There is also Table 3

Experimental results

L12

η1

η2

η3

1 2 3 4 5 6 7 8 9 10 11 12

30.596 30.772 32.008 24.680 29.750 25.323 30.850 24.819 24.070 30.404 25.573 23.945

37.950 35.599 36.272 31.601 36.495 30.841 36.035 30.544 30.488 37.043 30.955 30.752

38.600 38.804 39.659 33.007 39.407 33.543 38.844 33.330 33.992 40.406 34.652 33.077



Table 4 L12

Standardized S/N ratios

η1s

1 2 3 4 5 6 7 8 9 10 11 12

η2s

0.8947 0.9497 1.3359 −0.9537 0.6304 −0.7528 0.9741 −0.9103 −1.1443 0.8347 −0.6747 −1.1834

Table 5

Accuracy in X direction Accuracy in Y direction Angle accuracy

Table 8

0.7107 0.7780 1.0597 −1.1325 0.9767 −0.9559 0.7911 −1.0261 −0.8079 1.3059 −0.5904 −1.1095

L12

Second principal component, Y2

Third principal component, Y3

MPI

1 2 3 4 5 6 7 8 9 10 11 12

1.7297 1.3555 1.8690 −1.6063 1.4554 −1.5318 1.4597 −1.7196 −1.7393 1.8677 −1.2537 −1.8863

−0.4741 0.2051 0.3012 −0.2715 −0.1104 0.0868 0.1040 0.0695 0.0552 −0.0469 0.2218 −0.1406

0.1650 0.1073 0.1746 0.1458 −0.2372 0.1373 0.1226 0.0766 −0.2429 −0.3302 −0.0761 −0.0428

1.6765 1.3230 1.8248 −1.5652 1.4112 −1.4870 1.4229 −1.6707 −1.6935 1.8122 −1.2166 −1.8376

Correlation matrix

Accuracy in X direction

Accuracy in Y direction

Angular accuracy

1.000

0.954

0.967

0.954

1.000

0.956

0.967

0.956

1.000

a significant correlation between the positional accuracy in the X direction and the angle of the output link, with a corresponding correlation value of 0.967. Using the correlation matrix, the eigenvalues and eigenvectors can be computed, representing the magnitudes and directions of the principal components, respectively. These are shown in Tables 6 and 7. The eigenvectors are multiplied by the standardized values, as shown in the following equation ⎡ ⎤ ⎡ ⎤⎡ ⎤ Y1 0.5780 0.5757 0.5784 η1s ⎣Y2 ⎦ = ⎣0.4568 −0.8155 0.3553 ⎦ ⎣η2s ⎦ 0.6762 0.0588 −0.7343 Y3 η3s

(18)

The principal components of the quality characteristics (i.e. Y1 , Y2 , Y3 ) can be determined as shown in

Table 8. Finally, in order to analyse multi-quality characteristics, the three principal components are further assessed according to the weighting of their contribution and combined as an MPI. The explanatory power of the ith component to the overall variation can be defined in the following equation as λi Explanatory power of ith component = p i=1

1 2 3

Eigenvalue

Proportion explained (%)

Proportion accumulated (%)

2.9180 0.0491 0.0329

97.27 1.64 1.09

97.27 98.91 100.00

Table 7

Eigenvectors Principal component

Input factor

1

Accuracy in X direction Accuracy in Y direction Angle accuracy

0.5780 0.5757 0.5784


2 0.4568 −0.8155 0.3553

3 0.6762 0.0588 −0.7343

λi (19)

where λi is the ith eigenvalue of the correlation matrix. The explanatory power of each principal component, i.e. the weighting, is shown in Table 6. By analysing the contribution level of each principal component according to its weighting, the MPI can be expressed as ⎡ ⎤ Y1 [MPI] = [0.9727 0.0164 0.0109] ⎣Y2 ⎦ (20) Y3 The MPI obtained by synthesizing the three principal components is shown in Table 8. 4.2

Table 6 Weighting of principal component Principal component

Principal components and MPI

First principal component, Y1

η3s

1.3922 0.6194 0.8407 −0.6948 0.9140 −0.9446 0.7627 −1.0422 −1.0606 1.0941 −0.9071 −0.9738

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Effects of control factor

Table 9 shows the response of each control factor to the MPI, and Fig. 5 shows the response diagram. Table 9 reveals that the absolute values of the first and second levels of every control factor are the same. This is because in an L12 orthogonal array, every control factor has two levels. By assessing the ‘max.–min.’ range, the effect of each control factor on the MPI can be evaluated. It is clear from Table 9 and Fig. 5 that with the exception of I and K , all control factors have generally significant effect on the MPI and are therefore classified as key dimensions. The decisions made are that to improve the quality of the mechanism, the tolerance must be tightened, whereas to reduce manufacturing costs, the tolerance of non-key dimensions, such as control factors I (θ4 ) and K (α) can be relaxed. Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science

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Table 9

Response table Factors

Levels

A

1 2 Max.–min.

Ranking

B

0.5305 −0.5305 1.0611 5

C

0.4805 −0.4805 0.9610 7

D

−0.5522 0.5522 1.1045 4

E

0.4525 −0.4525 0.9049 8

Fig. 5

4.3

0.5885 −0.5885 1.1770 2

G

0.5556 −0.5556 1.1112 3

0.6358 −0.6358 1.2715 1

H −0.4518 0.4518 0.9035 9

I

J

0.0097 −0.0097 0.0194 11

−0.4880 0.4880 0.9759 6

K −0.0846 0.0846 0.1692 10

Response diagram

Analysis of variance

The analysis of variance is fundamentally similar to the analysis of the max.–min. range in the variation response table. The main difference is that the former can separate the total variability of the MPIs, which are measured by taking the sum of the squared deviations from the mean of the MPIs and dividing them into contributions by each of the control factors and the experimental error. The max.–min. range method, in contrast, displays the effect of the entire range, including those caused by experimental errors. From the result of the analysis of variance shown in Table 10, the variations, V , caused by each control factor on the MPI, as well as the effect of the control factors on the quality characteristic variation, can be observed. The main control factors that can effectively reduce the variations and contribute to the MPI are identified as G (15.97 per cent), E (13.68 per cent), F (12.20 per cent), C (12.05 per cent), A (11.12 per cent), J (9.41 per cent), B (9.12 per cent), D (8.09 per cent), and H (8.06 per cent). Table 10

F

Analysis of variance

Source

df

S

V

%

F -test

Confidence (%)

A B C D E F G H I J K Total

1 1 1 1 1 1 1 1 1 1 1 11

3.378 2.770 3.660 2.457 4.156 3.704 4.850 2.449 0.001 2.857 0.086 30.368

3.378 2.770 3.660 2.457 4.156 3.704 4.850 2.449 0.001 2.857 0.086 2.761

11.12 9.12 12.05 8.09 13.68 12.20 15.97 8.06 0.00 9.41 0.28 100.0

84.4 69.3 91.5 61.4 103.9 92.6 121.3 61.2

98.84 98.59 98.92 98.41 99.05 98.94 99.19 98.41

71.4

98.63


The F -test can also be used to identify the control factor having significant effect on the performance characteristics. For comparison, the results from significance testing of the factors are also shown in Table 10. Generally, a change in the control factor has a significant effect on the performance characteristics when the F -test value is large. The F -test is simply a ratio of sample variances. When this ratio becomes sufficiently large, then the two sample variances are accepted as being unequal at a certain confidence level.

5

CONCLUSIONS

The Taguchi orthogonal array experimental method has been applied to compute the S/N ratios of the positional and angular errors of a dual-purpose sixbar mechanism. The principal component method was then applied to analyse the main contributors to the S/N ratios of the quality characteristics, from which weighting values were calculated and transformed to an MPI. This index was used to study the effect of each control factor on the quality characteristics of the six-bar mechanism to determine whether it was required to tighten tolerance to improve the quality of the mechanism. From the analysis of the response and table diagram, it was shown that all link dimensions, except θ4 and α, have significant effects on the MPI and were therefore classed as key dimensions. The decisions made were that the tolerance of these key dimensions could be tightened to improve the quality of the mechanism, or the tolerance of the two non-key dimensions, θ4 and α, could be relaxed to reduce manufacturing costs. The results from this paper have shown that by using the Taguchi JMES1106 © IMechE 2009


and the principal component methods to analyse a dual-purpose six-bar mechanism, the effect of each control factor on the MPI could be systematically investigated. ACKNOWLEDGEMENTS The authors are grateful to the National Science Council of the Republic of China for supporting this research under Grant No. NSC 95-2221-E-168-010.

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