NOMENCLATURE
CHAPTER 3 Theoretical Background & Modelling Methodology
3.8 Optimization of welding process parameters
where σy is yield strength of mild steel (2.5×108 N/m2), E is Young’s modulus (2.1×1011 N/m2).
The restraint force is calculated as shown in Equation 3.88.
F = AEε* (3.88)
where A is the cross sectional area of the bar and ε* is the inherent strain. The corresponding moment, M can be calculated as follows:
M = ∑ (F1 l1 + F2 l2) (3.89)
where F1 and F2 are the forces at different nodes and l1,l2 are their corresponding distances from the centre of rotation.
those in which some optimization is required, the method does not point out to the best settings of parameters to carry out the experiments.
3.8.3 Taguchi method: Taguchi’s philosophy is an efficient tool for the design of high quality manufacturing system. Dr. Genichi Taguchi, a Japanese quality management consultant, has developed a method based on orthogonal array experiments, which provides much-reduced variance for the experiment with optimum setting of process control parameters. Thus, the integration of design of experiments (DOE) with parametric optimization of process to obtain desired results is achieved in the Taguchi method. Orthogonal array (OA) provides a set of well-balanced (minimum experimental runs) experiments and Taguchi’s signal-to-noise ratios (S/N). S/N ratios are logarithmic functions of desired output serve as objective functions for optimization. This technique helps in data analysis and prediction of optimum results.
The S/N ratio takes both the mean and the variability into account. It is the ratio of the mean (signal) to the standard deviation (noise). The ratio depends on the quality characteristics of the product/process to be optimized [187]. The standard S/N ratios are (i) smaller the better (SB), (ii) larger the better (LB) and (iii) nominal is best (NB). The optimal setting is the parameter combination, which has the highest S/N ratio. The three signal-to-noise ratios can be defined as follows.
(i) Smaller-the-better:
2
10 1
10log 1
n i yi
n
(3.90)
It can be measured by taking the mean of sum of squares of measured data.
(ii) Larger-the-better:
10 2
1
1 1
10 log
n
i i
n y
(3.91)
It can be measured by taking the mean of sum of squares of reciprocal of measured data. It is generally calculated by taking the reciprocals of measured data and then taking the S/N ratio as in the smaller-the-better case.
(iii) Nominal-the-best:
2
10 2
1
10 log 1
n
ni
(3.92)
It can be measured by taking the square of mean divided by variance. This case arises when a specified value is most desired i.e. neither a smaller nor a larger value is desirable.
3.8.4 Analysis of variance (ANOVA)
ANOVA is a statistical technique, which can infer some important conclusions based on analysis of the experimental data. The method is very useful to reveal the level of significance of influence of factor(s) or interaction of factors on a particular response. It separates the total variability of the response (sum of squared deviations about the grand mean) into contributions rendered by each of the parameter/factor and the error. Thus
T F e
SS SS SS (3.93)
where
2 1
( )
p
T j m
j
SS
(3.94)
and
SST= Total sum of squared deviations about the mean
j = Mean response for jth experiment
m= Grand mean of the response
P = Number of experiments in the orthogonal array SSF = Sum of squared deviations due to each factor SSe= Sum of squared deviations due to error
In ANOVA table, the mean square deviation is defined as:
)
( )
SS Sum of squared deviat MS DF Degree of fre
ion
edom (3.95)
Fisher’s F ratio (Variance ratio) is defined as:
MS for the
Mean for a te erro
rm
r m
F ter (3.96)
Depending on F-value, P-value (probability of significance) is calculated. If the P-value for a term appears less than 0.05 (95% confidence level) then it can be concluded that, the effect of the factor(s)/ interaction of factors is significant on the selected response [185].
3.8.5 Grey relational analysis
In grey relational analysis, experimental data i.e. measured features of quality characteristics of the product are first normalized ranging from zero to one. This process is known as grey relational generation. Next, based on normalized experimental data, grey relational coefficient is calculated to represent the correlation between the desired and actual experimental data. Then overall grey relational grade is determined by averaging the grey relational coefficient corresponding to selected responses. The overall performance characteristic of the multiple response process depends on the calculated grey relational grade. This approach converts a multiple- response- process optimization problem into a single response optimization situation, with the overall grey relational grade as the objective function. The optimal parametric combination is then evaluated by maximizing the overall grey relational grade. In grey relational generation, the normalized data corresponding to Lower-the-Better (LB) criterion can be expressed as:
max ( ) ( ) ( ) max ( ) min ( )
i i
i
i i
y k y k
x k y k y k
(3.97)
For Higher-the-Better (HB) criterion, the normalized data can be expressed as:
( ) min ( ) ( ) max ( ) min ( )
i i
i
i i
y k y k
x k y k y k
(3.98)
where x ki( )is the is the value after the Grey relational generation, min y ki( ) is the smallest value of y ki( )for the kth response. An ideal sequence is x k and k0( ) ( 1, 2, 3, 4, 5...). The grey relational coefficient ( )i k can be calculated as
min max max
( ) ( )
i
oi
k k
(3.99)
where oi ||x k0( )x ki( ) ||= difference of the absolute value x ko( )and x ki( ); is the distinguishing coefficient 0 1. min jmin i kmin||x k0( )x kj( ) || = the smallest value of oi ; and max jmax i kmax||x k0( )x kj( ) || = largest value of oi. After averaging the grey relational coefficients, the grey relational grade ican be computed as:
1
1 ( )
n
i i
k
n k
(3.100)
where n = number of process responses. The higher value of grey relational grade corresponds to intense relational degree between the reference sequencex ko( )and the given sequence ( )x ki . The reference sequence x ko( )represents the best process sequence. Therefore, higher grey relational grade means that the corresponding parameter combination is closer to the optimal.
However, Equation 3.98 assumes that all response features are equally important. But, in practical case, it may not be so. Therefore, different weightages were assigned to different response features according to their relative priority. In that case, the equation for calculating overall grey relational grade (with different weightages for different responses) is modified as shown below:
1
1
( )
n k i k
i n
k k
w k
w
(3.101)
where iis the overall grey relational grade for ith experiment. i( )k is the grey relational coefficient of kth response in ith experiment and wk is the weightage assigned to the kth response.
3.8.5 Confirmatory experiment
After evaluating the optimal parameter settings, the next step is to predict and verify the enhancement of quality characteristics using the optimal parametric combination. The estimated Grey relational grade
^ using the optimal level of the design parameters can be calculated as:
^ _
1
( )
o i
m m
i
(3.102)
where mis the total mean Grey relational grade,
_
iis the mean Grey relational grade at the optimal level, and “o” is the number of the main design parameters that affect the quality characteristics. That means that the predicted or estimated Grey relational grade (optimal) is equal to the mean Grey relational grade plus the summation of the difference between overall mean Grey relational grade and mean Grey relational grade for each of the factors at optimal level.
In Grey-based Taguchi method, the only performance feature is the overall Grey relational grade; and the aim should be to search a parameter setting that can achieve highest overall Grey relational grade. The Grey relational grade is the representative of all individual performance characteristics. In the present study, objective functions was selected in relation to parameters of bead geometry and mechanical properties, and all the responses were given equal weightage.