TAGUCHI METHOD USED FOR THEORETICAL RESEARCH

Eugen MERTICARU

Technical University “Gh. Asachi” of Iasi, Romania

Taguchi method is a method generally used for experimental and industrial studies. This method combines the engineering techniques with statistical techniques in order to improve the quality costs through optimization of the design of the product and of the manufacturing process. The paper presents how Taguchi method can be used for theoretical studies in order to plan the study, to minimize the calculus volume and to put to good use the theoretical results. As an example, a concrete study is presented.

Key words: dynamic model, cam mechanism, oscillating follower

1. INTRODUCTION

Generally, the Taguchi method is used for more than 20 years in the developed countries for experimental and industrial studies in order to increase the quality of products with reduced costs.

Usually, when an instability of the characteristics of a product is found, the causes of this instability are wanted in order to reduce them or even to eliminate them. The ways to reduce or to eliminate these causes may be often very expensive. The strategy of Taguchi method is diametrally opposed to this action: instead of removing the disturbing factors, this method tries to minimize the impact of them on the system, that is, a combination of system parameters (controlled factors) is looked for so that the system becomes insensible to the disturbing factors. The seeking of the good combination of system parameters (controlled factors) in order to optimize the system so that it becomes insensible to disturbing factors, is experimentally perfomed with reduced costs.

The present work shows how the Taguchi method may be used for theoretical studies in order to see the trend of a system when its parameters are changed. The advantages of this method, when is used for theoretical studies, are the following: easy planning of the study; the calculus volume is reduced; the mathematical model includes, also, the interactions between the studied controlled factors and allows the establishing of their effects, too; classification of the significance of the effects of factors and their interactions is possible; testing of the significance of the mathematical model is relatively simple, using the SNEDECOR test; graphical representation of the average effects of the factors and their interactions is simple and suggestive.

2. PLANNING OF THE STUDY

As an example, the influence of some controlled factors on the jump of the follower at a cam mechanism with oscillating follower was performed. The cam mechanism is presented in figure 1, and it consists of: 1- cam, 2- follower, 3- follower roller, 4,5- kinematic elements. This mechanism is used in the structure of a textile machine.

The technological force Fu, acting on element 4 in point N (Fig. 1) is considered to have the components: Fux = Fu and Fuy = 0, and it is always opposed as sense to the relative translation motion of element 4 along element 5.

For calculus, an electric motor has been considered, with power Pn=1,1 (kW) and rotation speed nm=920 (rot/min).

Taguchi method used for theoretical research 319

ϕ ϕψ ψ0

2 DD

θ2 ϕD

ω1B A

C

E F

GH

N

ϕ4 = ϕ

5

x

y

1

2

3

45

6

6

6

Fig. 1 – Mechanism scheme

The influence of four factors on jump of the follower was studied. Thus, the calculus was performed for different values of: - the technological force Fu, acting on element 4; - backlash j in kinematic pair cam-follower; - cam rotation speed n1; - the moment of inertia Jt of the follower and of the kinematic elements linked to the follower (elements 2, 3, 4 and 5).

Also, two laws of motion were used: the law of motion with sine acceleration and the law of motion with cosine acceleration.

The studied factors were denoted accordingly with table 1. Two levels of values was established for factors, accordingly with table 2.

Table 1 Factor notation A B C D The factor Fu j n1 Jt

Table 2 The factor A (Fu) B (j) C (n1) D (Jt) level 1 A1: Fu= 0 (N) B1: j=0.016 (mm) C1: n1=180 (rot/min) D1: Jt=1⋅Jt

level 2 A2: Fu=65 (N) B2: j=0.128 (mm) C2: n1=300 (rot/min) D2: Jt=4⋅Jt

The method requires to determine the coefficients of a mathematical model such as:

Y Med A B C D A B A C A D B C B D C D= + + + + + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ (12)

where: Y – the answer of the system; Med – the average of the factors effects; A,B,C,D – the factors effects; AB, AC, AD, BC, BD, CD – the effects of the interactions between factors.

The mathematical model, under general matrix form, describing the effects of the factors and of the interactions between factors has the following expression:

Eugen MERTICARU 320

[ ] [ ],1 ,1 ,1 ,

,1 ,2 ,1 , 1

, ,1 , ,

...... ... ... ...

...

i j i j kn nt

i i i k i i ji i j

i j i k j i k j ki j

If f If fY Med Ef Ef Ef Af Af Af

If f If f= =≠<

⎛ ⎞⎡ ⎤⎛ ⎞ ⎜ ⎟⎡ ⎤ ⎢ ⎥ ⎡ ⎤= + ⋅ + ⋅ ⋅⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎜ ⎟⎣ ⎦⎝ ⎠ ⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

∑ ∑ (13)

where: Y - the answer of the system; Med – the average of the factors effects; Efi,1 – the mean effect of factor i at level 1;

[ ]iAf - the matrix of factor i, which has the form: 10⎡ ⎤⎢ ⎥⎣ ⎦

for factor having the value at level 1 and 01⎡ ⎤⎢ ⎥⎣ ⎦

for

factor having the value at level 2; [ ]tiAf - the transposed of matrix of factor i; k – the number of levels of

the considered factor; n – number of considered factors; ,1 ,i j hIf f - the effect of the interaction between the factor i at level 1 and the factor j at level h.

To calculate the average of the factors effects, the following formula was used:

1

1 s imN

iis im

M e d YN =

= ⋅ ∑ (14)

where: Nsim – total number of theoretical simulation performed on the system answer Y; Yi – the theoretical values of the parameter Y at simulation i.

To calculate the mean effects of factors:

,1

,

iN

ij hh

i ji

YEf Med

N== −∑

(15)

where: Efi,j – the mean effect of factor i at level j; Ni – number of simulations for which the factor i is at level j; Yij,h – the answer of the system at simulation of order h for which the factor i is at level j.

For interactions:

,1 , 1, ,1 ,i j k i jk i j kIf f M Med Ef Ef= − − − (16)

where: Ifi,1fj,k – interaction between factor i at level 1 and factor j at level k; Mi1,jk – the average of the answers Y when the factor i is at level 1 and factor j is at level k; Efi,1 and Efj,k – the mean effects of factor i at level 1 and respectively of factor j at level k.

The calculus followed a complete factorial plan. The complete factorial plan is presented in table 3. Once the effects of factors and of the interactions were calculated, we have to establish if those effects

are significant, that is they are associated with the corresponding factors, or if they are the effects of the uncontrolled factors.

In order to test the significance of the mathematical model, the SNEDECOR test was used, [10], that consists in the comparison between the variance of the factor or interaction and the residual variance of the model, based on the Fisher criterion.

The residual variance of the model is the variance that is not explained by the controlled factors. It is the variance of the deviations between the mathematical model and the answers obtained by simulation, and it is calculated with the relation:

2

Rrez

rV

N= ∑

(17)

where: r – the residuums, calculated as difference between the answer obtained by simulation Y and the answer calculated with the above presented mathematical model, Y~.

r = Y-Y~ (18)

Taguchi method used for theoretical research 321

Table 3 The factors and their levels No.

A B C D Simulation result

1 1 1 1 1 R1 2 1 1 1 2 R2 3 1 1 2 1 R3 4 1 1 2 2 R4 5 1 2 1 1 R5 6 1 2 1 2 R6 7 1 2 2 1 R7 8 1 2 2 2 R8 9 2 1 1 1 R9

10 2 1 1 2 R10 11 2 1 2 1 R11 12 2 1 2 2 R12 13 2 2 1 1 R13 14 2 2 1 2 R14 15 2 2 2 1 R15 16 2 2 2 2 R16

Nrez – number of degree of freedom for residuums:

Nrez = Nsim – Ngl (19)where Ngl – number of degrees of freedom for the considered model, Ngl = 11.

The variance for factor i was calculated with the relation:

( ) ( )2

1sim

i ii i

NV Eniv niv

= ⋅⋅ − ∑ f

(20)

and the variance for the interaction between factors i and j was calculated with relation:

( ) ( ) ( )2, 1 1

simi j i j

i i j j

NV Iniv niv niv niv

= ⋅⋅ − ⋅ ⋅ − ∑ f f

(21)

Testing the significance for the factor i or for the interaction between factors i and j consists in comparing the calculated value of the Fisher criterion, given by relation:

maxi

R

VFV

= , respectively ,max

i j

R

VF

V=

(22)

with the value of the Fisher criterion chosen from a table, FT , for a risk of 5% (for a coefficient of reliability of 95%) and for the numbers of degrees of freedom:

1 1i iv ngl niv= = − , respectively ( ) ( )1 , 1 1i j i jv N niv niv= = − ⋅ − , 2 rezv N= (23)

which, for our models means: and 1 1v = 2 5v = , thus from the table, the value FT(0,95;1;5)=6,61 was selected.

3. DYNAMIC MODEL

This model is proposed in order to study the influence of the elasticity of the contact cam-follower on the jump of the follower. The model takes into account only the elasticity of the contact cam-follower, the other elements of the mechanism being considered as rigid.

If we consider the elasticity of the contact cam-follower, between cam and follower occurs a relative motion. In order to study the relative motion cam-follower a dynamic model with one degree of mobility was used, as presented in figure 2 [5], [9].

Eugen MERTICARU 322

M

0

j

0x>x <0 x=

Kc Kc

exterior flank interior flank

KcM

cK

x<0

KcM

cK

x>j

Fig. 2- Dynamic model with one degree of mobility with elasticity of contact

We consider that the relative motion cam-follower takes place along the common normal direction at the contact point.

In figure 2 the notations represent: M is the reduced mass of the mechanism (kg); j is the backlash in higher pair cam-follower (mm); x is the relative displacement cam-follower along the common normal direction at the contact point (mm); Kc is the rigidity of contact cam-follower (N/mm).

We consider the axis x along the common normal direction at the contact point. The origin of x axis is chosen at contact point of follower with the interior flank of cam. The positive sense of x axis is chosen toward exterior of cam and the negative sense of x axis is chosen toward interior of cam.

The differential equations of relative motion cam-follower, for the model in figure 2, are: - for contact with interior flank (x≤0):

31n

cM x K x R⋅ + ⋅ =ii

(1)

- for follower jump (0<x<j):

31M x R⋅ =ii

(2)

- for contact with exterior flank (x≥j):

( ) 31n

cM x K x j R⋅ + ⋅ − =ii

(3)

In relations (1), (2) and (3) the notations represent: R31 – the static reaction of cam on follower

(neglecting friction); 2

2

d xxdt

=ii

where t = time (s); 109

n = exponent corresponding to elastic force of contact.

The elastic force of contact was expressed by the following equation (linear contact cam-follower roller) [3], [5], [11], [12]:

nelc cF K x= ⋅ (4)

where: Felc is the elastic force of contact (N); Kc is the rigidity of contact cam-follower (N/mm); x is the contact deformation (mm).

The rigidity of the contact cam-follower was calculated with the relation [3], [5], [11], [12]:

( )8

4 98,05 10c wK l= ⋅ ⋅ (5)

Taguchi method used for theoretical research 323

where lw is the length of the contact between cam and follower roller (mm). In relations (1), (2) and (3):

1 2

1 2

M MMM M

⋅=

+

(6)

where: M1 is the reduced mass of cam (kg); M2 is the reduced mass of follower (kg).

1 21

4 cama

b

JMD⋅

= and 2 22

4 tachet

b

JMD⋅

= (7)

1cama r rmJ J J= + (8)

2

2 4 5 32

ctachet r r r

vJ J J J m⎛ ⎞

= + + + ⋅⎜ ⎟ω⎝ ⎠

(9)

In relations (7)-(9): Db1 is the diameter of circle with center in A, tangent to common normal “n-n”; Db2 is the diameter of circle with center in D, tangent to common normal “n-n”; Jr1 is the moment of inertia of cam reduced to cam; Jrm is the moment of inertia of motor reduced to cam; Jr2 is the moment of inertia of follower reduced to follower; Jr4 is the moment of inertia of element 4 reduced to follower; Jr5 is the moment

of inertia of element 5 reduced to follower; 2

32

cvm⎛ ⎞⋅ ⎜ ⎟ω⎝ ⎠

is the moment of inertia of follower roller reduced to

follower.

( )1 02 cos cosbD l d= ⋅ ⋅ α − ⋅ ψ + ψ + α (10)

2 2 cosbD l= ⋅ ⋅ α (11)

where: l is the length DC of follower; d is the distance between cam axis and follower axis AD; α is the pressure angle of the mechanism.

4. RESULTS AND CONCLUSIONS

The motion equations (1), (2), (3) were numerically integrated using computer calculation. The calculus was performed only for the rising phase of the follower. The results were put to good use as described above. As an example, in figures 3 and 4 there is presented the variation of relative displacement cam-follower x versus time, for the two laws of motion used (the law of motion with sine acceleration and the law of motion with cosine acceleration) and for the case when the factors in table 1 are the values as follows: A1B2C1D1 (see tables 2 and 3).

In figures 3 and 4 it can be observed that the relative displacement x becomes greater than zero approximately at time time=0.025 s. The point when follower jump occurs is marked with a small circle. Also, it can be seen that just one follower jump and no impact occur during the rising phase of the follower (x does not reach the value of backlash j=0.128 mm, therefore, no impact occurs).

Also, it can be observed that, at the beginning of rising phase, the follower “deepens” in cam profile (x<0) and then the jump follows.

The calculus results yielded that the reaction force R31 becomes equal to zero or it changes its sign approximately when time is time=0.025 s. Thus, the follower jump occurs at that moment.

The significance of the effects of the factors and interactions between factors, on the relative displacement x, for the law of motion with sine acceleration, is shown in table 4.

Eugen MERTICARU 324

Table 4 The factor

(interaction) The variance of the factor (interaction)

Calculated Fisher criterion

Fmax

Fisher criterion from table

FT

Significance of factor

(interaction) A(Fu) 7.48225E-05 16627.22222 S B(j) 0 0 N

C(n1) 5.0625E-06 1125 S D(Jt) 1.80625E-05 4013.888889 S AB 2.40741E-35 5.34981E-27 N AC 6.0025E-06 1333.888889 S AD 2.35225E-05 5227.222222 S BC 6.01853E-36 1.33745E-27 N BD 0 0 N CD 4.69225E-05 10427.22222

6.61

S Residual variance VR 4.5E-09

Table 5 The factor

(interaction) The variance of the factor (interaction)

Calculated Fisher criterion

Fmax

Fisher criterion from table

FT

Significance of factor

(interaction) A(Fu) 5.11297E-05 138.6276185 S B(j) 2.1025E-10 0.00057005 N

C(n1) 9.79064E-06 26.54532623 S D(Jt) 2.26576E-05 61.43146128 S AB 2.56E-10 0.000694092 N AC 7.03082E-07 1.906264124 N AD 2.82072E-06 7.64780766 S BC 2.7225E-10 0.00073815 N BD 4.16025E-09 0.011279669 N CD 2.17858E-06 5.906764494

6.61

N Residual variance VR 3.68827E-07

In table 4, the factors with significant effect where denoted in the last column of table 4 with “S”. It can be observed that, for the law of motion with sine acceleration, the factors with significant effect

are: A, C and D, that is, the force Fu, the cam speed rotation n1 and, respectively, the moment of inertia of the follower Jt. The interactions between factors with significant effect are: AC, AD and CD.

The significance of the effects of the factors and interactions between factors, on the relative displacement x, for the law of motion with cosine acceleration, is shown in table 5.

In table 5, the factors with significant effect where denoted in the last column of table 5 with “S”. It can be observed that, for the law of motion with cosine acceleration, the factors with significant

effect are: A, C and D, that is, the force Fu, the cam speed rotation n1 and, respectively, the moment of inertia of the follower Jt. The interaction between factors with significant effect is AD.

The following conclusions may be drawn: Taguchi method was used for the theoretical study of the jump phenomenon to a cam mechanism with

oscillating follower. For the theoretic study a dynamic model with elasticity of the contact cam-follower was used. The influence of the following factors on the jump of the follower was studied: the technological force Fu, the backlash in the bilateral higher pair cam-follower j, the cam rotation speed n1, the moment of inertia of the follower. Two laws of motion were used: the law of motion with sine acceleration and the law of motion with cosine acceleration. The calculus was performed only for the rising phase of the follower.

In order to avoid the jump of the follower, it is necessary to increase Fu and to decrease the speed of the cam and the mass of the follower. For the law of motion with sine acceleration, the jump occurs at lower cam speed than for the law of motion with cosine acceleration. This is because the law of motion with sine acceleration has absolute values of the acceleration bigger than the law of motion with cosine acceleration.

Taguchi method used for theoretical research 325

Taguchi method can be successfully used for theoretical studies in order to reduce the volume of calculation, to plan the study and to put to good use the theoretical results.

Fig. 3 – Relative displacement x versus time when factors are at

levels A1B2C1D1 and the law of motion is with sine acceleration

Fig. 4 – Relative displacement x versus time when factors are at levels A1B2C1D1 and the law of motion is with cosine

acceleration

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