3.1. Three Forces
In the FSLW process, the interaction between the spindle and the specimen during the forming process of the weld may be divided into four phases: the plunging stage (I), dwelling stage (II), welding stage (III), and pulling-out stage (IV) [
22]. The spindle experiences different forces in different phases. For example, the time dependence of the three forces on the spindle under the FSLW welding parameter of 1500 rpm and 70 mm/min are shown in
Figure 3. From
Figure 3, it can be seen that the red, blue and yellow line represent the
Fx,
Fy and
Fz, respectively.
During the plunging stage (Stage I), the tool started to rotate and was pressed down until it plunged into the parent material to the specified depth. In this stage, the tool continued to be pressed into the yet-to-be-softened metal material, and the axial force (
Fz) increased sharply with time [
28], while the lateral force (
Fx) and the feed force (
Fy) increased slightly. Forcellese et al. [
22] claimed that the axial force quickly risen owing to the strength of the deforming material to the pin penetration that prevailed on the softening due to the heat generated by the stirring action of the rotating pin.
After reaching the predetermined downward force, the tool rotated in place in the dwelling stage (Stage II). The friction between the rotating tool and the base material continuously generated heat that flowed into the base material and caused it to reach a thermoplastic state. This ensured an effective joint in the initial weld along the weld advancing direction. As the base material continued to soften and the dwell time of the spindle continued to lengthen,
Fz showed a significant decreasing trend, as shown in
Figure 3. Gao et. al. [
29] pronounced that the axial force decreased sharply in the dwelling stage because the heat generation by shoulder–workpiece friction produced more plasticized material. Moreover,
Fx and
Fy also decreased slightly.
In the welding stage (Stage III), the tool was moving from the region of high material softening to the region of low material softening. All three forces on the spindle increased substantially [
23], and their increases were in the order of
Fz >
Fy >
Fx. After this short period of transition, the welding entered a more stable state, where the stability of
Fx was the best, and its change with time could be ignored.
Fy was in the range of 1400 ± 300 N, and
Fz fluctuated in the range of 5000 ± 1000 N.
During the pulling-out stage (Stage IV), the tool stopped advancing while maintaining a constant rotational speed and pulled away from the weld surface, leaving a keyhole on the surface of the plate. During this stage, the contact area between the tool and the plate decreased continuously, and the three forces on the spindle continued to decay until the tool was completely pulling out the surface of the plate and all three forces decreased to zero.
Based on the above analysis of the entire welding process, it was found that in Stage I, II, and IV, the spindle acted on the plate for a brief period and the three forces fluctuated over a wide range. These two characteristics greatly increased the difficulty analyzing of the relationship between the rotation speed, the welding speed, and the three forces on the spindle, and they also lowered the accuracy of the analysis significantly. For this reason, the time dependence of the axial force in Stage III was used to analyze the variation trends of the welding process parameters as a function of the axial force.
An orthogonal test was designed for the welding process parameters using ω = 1000, 1500, and 2000 rpm and
ν = 30, 50 and 70 mm/min.
Figure 4 shows the time dependence of the three forces for the FSLW welding of aluminum and magnesium under nine sets of welding process parameters.
The results in
Figure 4 show that after the stirring spindle entered the stable feeding phase, the axial force first increased and then decreased. As the welding process parameters changed, the curves for the three forces also changed accordingly. With the stirring spindle rotating at ω = 1000, 1500, and 2000 rpm, the stability of the three forces decreased, and the range of fluctuations increased as the welding speed increased, such as the curve fluctuation in
Figure 4a being smaller than that in
Figure 4b,c. At
v = 30, 50, and 70 mm/min, the three forces fluctuated more frequently as the rotation speed increased. For
Fx, the fluctuation was larger in
Figure 4a–c than that in
Figure 4 d–i. The increase in the three-way force of the spindle was more frequent. At the same time, due to the increase in the rotation speed, the heat input to the weld joint increased, the plasticization of the metal increased [
30,
31], and the range of fluctuations of the three forces became even smaller during the welding process.
In the study of the relationship between the rotational speed, the welding speed, and the three forces on the spindle, the three forces were fitted to a normal distribution. The fitting equation for the normal distribution is shown in Equation (1), the original data were determined by calculating the action time of axis force in force-time curve by MATLAB, and the fitting results are shown in
Figure 5. The three forces basically followed a normal distribution, therefore the average values (μ) of the fitted curves were used to represent the three forces in the analysis of the relationship between the three forces, the rotational speed and the welding speed:
where μ and σ represent the mean value and standard deviation, respectively.
To investigate the effect of the rotation and welding speeds on the three forces on the spindle, a response surface mathematical model with the rotational and welding speeds as independent variables and
Fx,
Fy and
Fz as dependent variables was used, according to the center combination method in the Design Expert 12.0 software. The reliability of the model was verified based on the accuracy of the model [
32,
33]. To facilitate the design of the model, the independent variables were assigned encoded parameter values. The actual and encoded values are shown in
Table 3. Two factors, three-level central composite design matrix were selected to minimizing experimental conditions. The upper and lower limits of each parameter are coded as 1 and −1, respectively. The coded values for the intermediate (0) level were the rest of parameters. A total of nine experiments were required in this central composite design experimental design. The independent variable parameter matrix and the experimental results of the response values are shown in
Table 4.
For this test, a mathematical model was established using a second-order fitting equation, as shown in Equation (2):
where
xi and
xj denote the independent variables,
b0 is a constant term,
bij represents the interaction coefficients of the two input variables,
bii is the second-order effect of parameters
xi,
is a random error, and
Yi is the value of the response function. In this experiment, the two factors and three response equations can be expressed as [
32]:
Calculations using the Design Expert 12.0 software yielded the following response equations of the three forces:
Using the ANOVA module of the Design Expert 12.0 software, the accuracy of the response surface model was tested. In the analysis of variance process, the values of
P and
F reflect the reliability of the model. The greater the values of
F, the smaller the values of
P were, and the higher the reliability of the model was. Generally, a
P value less than 0.05 indicates that the model is significant, and a
P value greater than 0.1 shows that the model is poor and cannot be used reliably. The results in
Table 5 show that the
Fz response surface model and the
v, ω, and
v2 terms were all highly significant. In addition, the closer the coefficient of determination,
R2, between the actually measured value of the response surface function and the predicted value of the model was to 1, the better the fit was between the predicted and experimental values. In the
Fz model, R
2 was as high as 0.98, which indicated a high degree of fitting between the predicted and experimental values.
Figure 6 shows the correlation graphs of experimental and predicted values. From
Figure 6, it was found that the predicted and experimental values of the response surface function were evenly distributed on both sides of the 45° inclined line, indicating that the response surface models of
Fx,
Fy and
Fz had high fitting accuracy and the models were applicable.
Figure 7 shows the response surface plots for the rotational speed, welding speed to the forces
Fx,
Fy and
Fz.
Figure 7a shows that the variation trend of
Fx was clearly different from those of
Fy and
Fz. When ω = 1000 rpm and
v = 30 mm/min,
Fx exhibited the minimum value. As the rotational and welding speed increased,
Fx increased and assumed the maximum value when ω = 2000 rpm and
v = 70 mm/min. On the XY horizontal plane of the response surface plot, the contour line for
Fx was approximately a circular arc. This indicated that the rotational speed and the welding speed interacted very weakly with
Fx and had only an overlaying effect. Furthermore, over the entire variation range of the fitted parameters,
Fx fluctuated in the range of 300–360 N, and the welding process parameters had little effect on the changes of the
Fx values [
34]. Similarly, the
Fy also obtained a minimum at ω = 1000 rpm and
v = 30, and the interaction dropped sharply as it approached 1000 rpm and 30 mm/min. In
Figure 7b, the rotational and welding speeds had an interacting effect on
Fy [
35]. For a given rotational speed, as the welding speed increased,
Fy increased. For a given welding speed, when the rotational speed increased,
Fy first increased and then decreased, with the maximum occurring at approximately ω = 1500 rpm and
v = 70 mm/min. As shown in
Figure 7c,
Fz was approximately symmetric about the horizontal plane of ω = 1500 rpm. At
v = 30 mm/min,
Fz decreased sharply as the rotational speed approached 1000 and 2000 rpm, reaching a maximum at ω = 1500 rpm and
v = 70 mm/min. On the XY horizontal plane, the contour had an elliptical shape, and the interaction of the rotation and welding speeds with
Fz was quite significant. Under this interaction,
Fz changed minimally for ω = 1300–1700 rpm and
v = 40–60 mm/min. Residual plots are shown in
Figure 7d–f, which indicate that the fitting accuracy of all experimental date was up to 97%, and the experimental analysis was dependable.
3.3. Spindle Dynamic Performance Evaluation Model
3.3.1. Consistent Processing of Indices
Consistent processing of the indices can effectively avoid irrational evaluation results, because the various indices had different orders of magnitude. The linear proportional transformation method can effectively retain the size relationships and ratios of the indices before and after the change. Hence, to be as realistic as possible and eliminate the differences between the indices, the linear proportional transformation method was used to transform indices on an extremely large scale.
The two factors, i.e., the forces and vibrations, each have their own standards. In the present parametric study, the three forces possessed extremely large indices: the greater the force, the better the performance was. For vibrations, the indices were small: the smaller they were, the better the performance of the spindle was. For convenience, the reciprocals of the vibration were used and converted into very large indices for the calculation. The linear proportional transformation method was used to standardize the indices by letting:
The evaluation matrix was obtained as follows:
All subsequent calculations of the data were based on the evaluation matrix R.
3.3.2. Establishing Hierarchical Structure Map
Based on the known evaluation indices, it was determined that there were four layers: object layer O, criterion layer C, sub-criterion layer D, and program layer P. An analytic hierarchy process diagram, as shown in
Figure 10, was established. The first layer, object layer O, was for the evaluation of the FSLW spindle dynamic performance. The second layer, criterion layer C, had the two first-level indices—the three forces and vibration—to evaluate the dynamic performance of the FSLW spindle. The third layer, sub-criterion layer D, had second-level indices (six indices including
Fx,
Fy and others) for evaluating the dynamic performance of the FSLW spindle. The fourth layer contained the nine sets of weld process parameters studied herein.
3.3.3. Determining the Weight Vector
According to assumption (ii) above, as shown in
Table 6, a 1–9 scale
aij was used to measure the relative importance of any two criterion layers C
i and C
j to the object layer O.
If the influence of three-dimensional force and vibration on target layer is the same in criterion layer C, then criterion layer C and target layer O are both assigned values of 0.5. In other words, the normalized weight of criterion layer C to target layer O is as follows:
The hierarchical structure included two parts: the criterion layer and the sub-criterion layer. Therefore, according to
Table 6, a pairwise comparison matrix A1 for D1, D2 and D3 with respect to C1 was firstly constructed and then the relative weight vector
w1(3) was calculated. The position weight was set to 0 for factors in the sub-criterion layer that had no effect on C1. Thus, the relative weight vectors
w1(3) of six factors with respect to C1 were constructed. Similarly, the relative weight vectors
w2(3) of each factor with respect to C2 may be constructed using the same method.
Then, a pairwise comparison matrix of the sub-criterion layer with respect to the criterion layers C1 and C2 were constructed: , and , respectively.
Using the sum method, the maximum feature root of each matrix was calculated and the corresponding normalized feature vector:
, .
All layers passed the consistency test.
The weights for factors at positions that had no influence on C1 in the sub-criterion layer were set to 0. Thus, this obtained the relative weight vectors w1(3) of the six factors with respect to C1. Similarly, the relative weight vectors w2(3) of each factor with respect to C2 was obtained using the same method. The relative weight vectors of the sub-criterion layers for the factors of the criterion layer were as follows:
,
.
The matrix formed by the column vectors of the relative weight vector of the sub-criterion layer with respect to the factors of the criterion layer was as follows:
The combined weight of the sub-criterion layer D with respect to the object layer O was .
Based on the combined weight vector, with respect to the dynamic performance of the spindle, a weight of 0.0531 was assigned to Fx, a weight of 0.1303 was assigned to Fy, and a weight of 0.3167 was assigned to Fz. The weight of the spindle vibration was the same with the forces.
The comprehensive weight of the sub criterion layer D to the dynamic performance of the spindle passes the combination consistency test.
Therefore, the third layer passed the combination consistency test. The combined weight of the six factors for their effect on the dynamic performance of the spindle was
3.3.4. Comprehensive Evaluation Using TOPSIS Method
The combined weighting, based on the evaluation matrix R and the parameters on the performance of the spindle, was as follows:
Use Equation (6), a weighted evaluation matrix was constructed as follows:
A positive ideal solution Z* (z1*, z2*,…, z6*) and a negative ideal solution Z− (z1−, z2−,…, z6−) were determined as follows:
,
.
The distances from each parameter to the positive and negative ideal solutions were calculated as follows:
The positive and negative ideal solutions were:
,
.
The relative approach degree B of each parameter to the ideal spindle performance was calculated from , as follows:
.
Based on the relative approach degree B, the spindle performance for each parameter was ranked in order of merit as follows: P1 > P2 > P6 > P5 > P3 > P4 > P7 > P9 > P8.