Mathematical formulation
Based on Eq. (3), the effects of the above mentioned process variables on the magnitude of both average wire amplitude (Aa) and kerf width (Wk) has been evaluated by computing the values of the different constants of Eq. (3) and utilizing the relevant data from Table 5. The mathematical models for correlating Aa and Wk in addition with the considered process variables were obtained by Eqs. 4 and 5:
Table 5 Plan for central composite rotatable second-order design: different controlling parameters and results
$$ \begin{array}{l}{\mathrm{A}}_{\mathrm{a}} = 5.98371\ \hbox{--}\ 1.01661\kern0.75em {\mathrm{W}}_{\mathrm{t}}\hbox{--}\ 0.01284\kern0.75em {\mathrm{W}}_{\mathrm{s}}\hbox{--}\ 0.11263\kern0.75em {\mathrm{F}}_{\mathrm{r}}\hbox{--}\ 0.05458\kern0.75em {\mathrm{S}}_{\mathrm{v}}\\ {}\kern1.32em + 0.21820\kern0.75em {{\mathrm{W}}_{\mathrm{t}}}^2+\kern0.37em 0.00342\kern0.75em {{\mathrm{W}}_{\mathrm{s}}}^2+\kern0.61em 0.01601\kern0.75em {{\mathrm{F}}_{\mathrm{r}}}^2+\kern0.49em 0.00037\kern0.37em {{\mathrm{S}}_{\mathrm{v}}}^2\\ {}\kern1.40em \hbox{--} \kern.68em 0.03961\kern0.61em {\mathrm{W}}_{\mathrm{t}}{\mathrm{W}}_{\mathrm{s}}+0.03675\kern0.85em {\mathrm{W}}_{\mathrm{t}}{\mathrm{F}}_{\mathrm{r}}\hbox{--}\ 0.00371\ {\mathrm{W}}_{\mathrm{t}}{\mathrm{S}}_{\mathrm{v}}\\ {}\kern1.40em \hbox{--}\ 0.02046\ {\mathrm{W}}_{\mathrm{s}}{\mathrm{F}}_{\mathrm{r}} + 0.00165\ {\mathrm{W}}_{\mathrm{s}}{\mathrm{S}}_{\mathrm{v}} + 0.00081\ {\mathrm{F}}_{\mathrm{r}}{\mathrm{S}}_{\mathrm{v}}\end{array} $$
(4)
$$ \begin{array}{l}\mathrm{Wk} = \kern0.8em 80.7126\kern0.37em \hbox{--} \kern0.61em 2.0166\kern0.49em {\mathrm{W}}_{\mathrm{t}}\kern0.37em \hbox{--} \kern0.37em 0.2562\kern0.37em {\mathrm{W}}_{\mathrm{s}}\ \hbox{--} \kern0.37em 0.4518\kern0.37em {\mathrm{F}}_{\mathrm{r}}\kern0.5em \hbox{--} \kern0.5em 0.2192\kern0.37em {\mathrm{S}}_{\mathrm{v}}\ \\ {}\kern1.8em +\kern0.61em 0.3631\kern0.61em {{\mathrm{W}}_{\mathrm{t}}}^2\kern0.36em +\kern0.61em 0.0168\kern0.73em {{\mathrm{W}}_{\mathrm{s}}}^2\kern0.36em +\kern0.49em 0.0743\kern0.73em {{\mathrm{F}}_{\mathrm{r}}}^2+\kern0.36em 0.0015\kern0.61em {{\mathrm{S}}_{\mathrm{v}}}^2\ \\ {}\kern2.1em \hbox{--} \kern0.30em 0.1031\kern0.37em {\mathrm{W}}_{\mathrm{t}}{\mathrm{W}}_{\mathrm{s}} + 0.1113\kern0.37em {\mathrm{W}}_{\mathrm{t}}{\mathrm{F}}_{\mathrm{r}}\ \hbox{--}\ 0.0143\ {\mathrm{W}}_{\mathrm{t}}{\mathrm{S}}_{\mathrm{v}}\ \hbox{--}\ 0.0789\ {\mathrm{W}}_{\mathrm{s}}{\mathrm{F}}_{\mathrm{r}}\\ {}\kern1.75em + 0.0064\ {\mathrm{W}}_{\mathrm{s}}{\mathrm{S}}_{\mathrm{v}} + 0.0039\ {\mathrm{F}}_{\mathrm{r}}{\mathrm{S}}_{\mathrm{v}}\end{array} $$
(5)
Checking the accuracy of the model
The adequacy of the above two proposed models have been tested through the analysis of variance (ANOVA). The usual method for testing the adequacy of a model is carried out by computing the F-ratio of the lack of fit to the pure error and comparing it with the standard value. If the F-ratio calculated is less than the standard values, the postulated model is adequate (Nain et al. 2015; DIPP Motion Pro User s Manual). The calculated F-ratios were found to be higher than the tabulated values with a 95% confidence level and hence the models were considered to be adequate. Another way of determining the accuracy of the fitted regression model is to find the coefficient of determination (R2). In all the three cases that the values of determination coefficient (R2) and adjusted determination coefficient (adj. R2) are more than 90% which confirms good significance of the model. The results of the analysis justifying the closeness of fit of the mathematical models have been enumerated, as shown in Tables 6 and 7. The p-values of the models are also found to be less than 0.05, which verifies that the model is acceptable. It is concluded that the evolved models given by Eqs. (4) and (5) are quite adequate and demonstrate the independent, quadratic and interactive effects of the different machining parameters on the average wire amplitude and kerf width criteria values.
Table 6 ANOVA analysis for Wire amplitude (Aa)
Table 7 ANOVA analysis for Kerf width (Wk)
Parametric influence on average wire amplitude
The influence of wire tension, wire running speed, flow rate and servo voltage on average wire amplitude can be shown in Fig. 3. Average wire amplitude decreases with the increase of wire tension and wire running speed. However, it increases with dielectric flow rate. Servo voltage has a weak influence on average wire amplitude. One of the most effecting parameters of wire vibration amplitude in wire EDM process is wire tension. Figure 4 shows wire shape difference under wire tension 0.5 and 4.0 N. Within considerable range, an increase in wire tension significantly increases the cutting speed and accuracy due to the sharp straightness of the wire.
When the wire running speed has a lower value, the amplitude slightly increases. The debris exclusion from the discharge gap is a little difficult at lower wire running speed because there is no high-speed flow of working fluid around the wire. Then, the debris stagnation occurs around the wire, which causes unstable machining and larger amplitude of wire vibration. When the wire running speed is higher, the debris is smoothly excluded.
Dielectric flow rate is the rate at which the dielectric fluid is circulated. Flow rate of the working fluid from jet nozzles is important for efficient machining. One of the forces exerted on the wire is the dielectric flow such that as the flow rate increases around the wire, the movement of the wire speeds up and thus the average wire amplitude increases.
Servo voltage acts as the reference voltage to control the wire advances and retracts. Figure 3 shows that there is little decrease of average wire amplitude with change of servo voltage from 50 to 70 V. After that, the average wire amplitude increases slightly.
The effect of both wire tension and servo voltage on average wire amplitude is shown in the contour graph of Fig. 5. It can be concluded that the average wire amplitude has maximum value higher than 4.0 μm of dark green color when wire tension values ranges between 1.0 N and 0.5 N with servo voltage values ranges between 65 V and 75 V. However, the average wire amplitude has minimum value less than 2.0 μm of light green when wire tension higher than 3.5 N and servo voltage between 65 V and 75 V. Thus, it can be concluded that to minimize average wire amplitude, it is better to make the value of wire tension ranges between 3.5 N and 4.0 N in addition with servo voltage ranges between 65 V and 75 V.
Figure 6 shows the effect of wire running speed and dielectric flow rate on average wire amplitude. It is found that the average wire amplitude has maximum values ranges between 2.5 μm and 3.0 μm of green color at the lower part of the figure when wire running speed values ranges between 3.0 m/min and 10.0 m/min. However, the average wire amplitude has minimum values ranges between 2.0 μm and 2.5 μm of light green at the higher part of the figure when wire running speed values ranges between 4.0 m/min and 15.0 m/min for the same range of dielectric flow rate (0 to 8.0 L/min). Thus, it can be concluded that to minimize average wire amplitude, it is better to make the value of wire running speed ranges between 4.0 m/min and 15.0 m/min for the same range of dielectric flow rate (0 to 8.0 L/min).
Parametric influence on kerf width
Figure 7 shows the relationship between kerf width and the effecting parameters such as wire tension, wire running speed, flow rate and servo voltage. Kerf width decreases directly with the increase of wire tension and wire running speed. However, it increases with the increase of flow rate and servo voltage. When the wire tension increases, the straightness of the wire increases and thus decreases the average wire amplitude and so decreases the resulted kerf width. It can be noticed that the kerf width decreases with the wire tension, which agrees with the variation of wire vibration amplitude with the wire tension shown above. Therefore, the increase of wire amplitude results in the increase of kerf width. In other words, the wire amplitude and the machined kerf width can be decreased by increasing wire tension also in fine wire EDM.
Figure 8 shows the effect of wire tension and servo voltage on kerf width. It can be shown that the maximum kerf width of value higher than 72 μm is located at the region of color dark green when the wire tension has value ranges between 0.7 N and 0.5 N with servo voltage values ranges between 69 V and 71 V. However, the minimum kerf width of value less than 66 μm is located at the region of color dark blue when the wire tension has values ranges between 3.5 N and 4.0 N with servo voltage values ranges between 65 V and 75 V. Thus, it can be concluded that when wire tension values ranges between 3.5 N and 4.0 N in addition with servo voltage ranges between 65 V and 75 V, minimum kerf width values resulted.
The contour graph relating between kerf width with both of wire running speed and dielectric flow rate can be shown in Fig. 9. The maximum kerf width of values ranges between 70 μm and 71 μm that located at the region of color moderate green when the wire running speed has value ranges between 5.0 m/min and 8.0 m/min with dielectric flow rate values ranges between 7.0 L/min and 8.0 L/min. whereas, the minimum kerf width of value less than 66 μm is located at the region of color dark blue when the wire running speed has values ranges between 10.5 m/min and 11.5 m/min with dielectric flow rate values ranges between 1.5 L/min and 2.5 L/min. Thus it can be concluded that to minimize kerf width, it is better to make the value of wire running speed ranges between 10.5 m/min and 11.5 m/min in addition with dielectric flow rate ranges between 1.5 L/min and 2.5 L/min.
Optimality search
For the purpose of achieving stable wire electrical discharge machining, optimal combination of the various effecting process-variables such as the wire tension, wire running speed, flow rate and servo voltage, can be analyzed based on the developed mathematical models. The optimal search was formulated for the various process variable conditions based on minimizing average wire amplitude and kerf width values. The optimal combination of various process variables thus obtained within the bounds of the developed mathematical models and contour graphs. The optimal values resulted have been listed, as shown in Table 8.
Table 8 Optimal values of WEDM parameters