FEA has been performed on ANSYS 19.2 for all printed designs, the nonlinear properties of PA 12 material were used for the simulation. It is assumed that the parts manufactured by MJF are nearly isotropic [40][38]. Frictional contacts have been applied between waves as well as plate to wave with friction coefficient of 0.20 [42]. The simulation setup was identical with experimental setup as boundary condition were same by fixing the bottom plate and applying displacement from the top plate.
Figure 15 illustrates the comparison of experimental and simulation results for the loading curve as well as unloading curve for each printed sample. Overall, the comparison showed close agreement between experimental and simulation results. The experimental and simulation results of linear region of each spring are same but there was small deviation in nonlinear region which can be due to buckling, the contact area for the graded dimension or the residual powder remain inside the waves during compression testing. Design A3 has the highest deviation in nonlinear region because of the difference of cross-sectional areas of the waves as middle wave having circular while top and bottom waves have rectangular cross-sectional areas, due to which the middle wave tends to slide or move out which resulted the deviation in experimental and FEA results.
Stress distribution of each design was shown in Fig. 16 which highlights that stress is evenly distributed in each designed wave spring.
The compression pattern of simulation as well as for experiment was almost identical as Fig. 17 shows comparison of simulation and experimental pattern for A4 design.
5.1. Further parametrization
The experimental and simulation results have close agreement, also noted that these morphological changes contribute significantly in the variation of mechanical properties so need to be investigated more, hence the following supplementary designs for FGWS were designed by making some more parametric variations. Design A6 is almost same as design A2 as the only difference is variation in thickness of each wave which is three times more in design A6. Design A7 is related to design A1, having round waves instead of flat as it was in design A1. Design A8 and design A9 are the modified shapes of design A4 as they have the combination of flat and round waves i.e., in design A8 has top and bottom round waves of same diameter but thicker than the middle flat wave while A9 having top and bottom wave of same diameter but thinner than the middle flat wave. Design A10, design A11 are variant forms of design A3 as in all the variation of thickness of waves is from the middle wave i.e., the middle wave is thicker than the top and the bottom wave. A10 has all flat waves while A11 has all round waves. Design A12 and design A13 are related to design A5 i.e., design A12 has variable thickness from inside to outside and provided the curvature between these within each wave while design A13 has variable thickness of top face and bottom face provided between the curvature which are opposite in both faces within each wave. All the designs i.e., design A1 till design A13 have the same designed mass and simulated with the same criteria for the ease in comparison among all designs. The description along with the variations for the supplementary designs are summarized in Table 5.
Table 5
Design parameters along with variations for the supplementary designs
Design #
|
Name
|
Diameter (mm)
|
Width of strip (mm)
|
Thickness of strip (mm)
|
Diameter of strip (mm)
|
Height (mm)
|
Mass (g)
|
A6
|
Variable thickness (variation 1.5 mm)
|
35
|
3.75
|
3.50–0.5
|
-
|
17.49
|
14.04
|
A7
|
Round (variation 1mm)
|
35
|
-
|
-
|
3.2–1.2
|
18.9
|
13.12
|
A8
|
Flat wave between constant (maximum) round waves
|
35
|
4.5
|
1
|
3
|
18.9
|
13.67
|
A9
|
Flat wave between constant (minimum) round waves
|
35
|
5.5
|
2
|
1.5
|
17.34
|
13.20
|
A10
|
Flat waves with thickness variation from the middle wave.
|
35
|
4.0
|
3 − 2
|
-
|
18.7
|
14.5
|
A11
|
Round waves with variation from middle
|
35
|
-
|
-
|
3.20–2.70
|
18.8
|
13.78
|
A12
|
Semicircular cross-sectional waves (same curve)
|
35
|
5
|
3–2
1.5–1
|
-
|
18.6
|
14.8
|
A13
|
Semicircular cross sectional waves wave spring (opposite curve)
|
35
|
5
|
3–2
1.5–1
|
-
|
18.8
|
14.6
|
The complete assemble of all supplementary designs along with the variation are shown in Fig. 18.
The parametric changes such as more variations in thickness and replacing flat waves by round waves is to study the compression trend by making the top wave soft and precedent waves much stiffer to get the different distinct stages in the same curve of load and compression. Also, the round waves always improve the energy absorption of structure, so the variation of diameter is considered for the effect on energy absorption as well as buckling phenomena. Researchers have designed by the combination of different lattice structures to manufacture hybrid structures which has improved the mechanical properties of the structure [16], the same approach is used to design and manufacture the springs by the combination of round and flat waves to study the effect of stacking behavior of these waves on mechanical properties such as stiffness and energy absorption. Morphology of flat surfaces were changed to semi flat surfaces as in design A12 and design A13 to enhance the effect of fillet by designing the curved surfaces for providing the smooth transformation of force as well smooth contact point between the surfaces.
The FEA of the above designed springs have been performed and discussed in two groups on the basis of their behavior depicted in load vs compression curves i.e., 1st group comprises of design A6 to design A9 while second group is of design A10 to design A13. FEA results for the 1st group is shown in Fig. 19.
In design A6, the behavior of the springs is consisted of four sub-stages. In the 1st stage, the load is nearly zero till the fully compression of 1st wave, then load is transferred to second wave which resulted the load-bearing capacity linearly increased till the fully compressed second wave. After that the load is transferred to third wave which results sudden increase in the load-bearing capacity as the stiffness of the waves is increased as shown in Fig. 19 (a). The same phenomena can be seen in design A7 which consists of round waves instead of rectangular waves. The only difference between the behavior of design A6 and A7 is the distinct stages of compression in load versus compression graph as in design A6, four different stages while in design A7 is three, also design A6 has more load-bearing capacity than A7.
The design A8 and A9 consists of FGWS, in which central wave is of rectangular cross sectional while top and bottom are round cross-sectional waves. In these designs, initially, spring shows linear behavior and then its behavior changes to nonlinear. It is also noted that, in design A8, the top and bottom waves are stiffer than the middle wave, so during the linear stage, the top and bottom waves compressed simultaneously, then the middle wave compressed while design A9 is vice versa of design A8 as in design A9 the top and bottom waves are softer than the middle one, so the spring start compressing form the top as 1st wave is fully compressed and then it transfers loads to lower subsequent waves.
The non-behavior has different potential advantages in different applications in which the load needs to be reduced gradually or dynamically at different stages [43]. Previously, the same authors investigated the stiffness and energy absorption of variable dimension helical springs by changing the means diameter of the spring, pitch and wire diameter and found that the spring performance improved as compared to simple helical springs [44]. The studies have been made for this nonlinear behavior of springs for different application such in robot mechanisms [45], electrostatic actuators [46] and also nonlinear springs are modelled by using the rubber material [47]. FEA results for the stress distribution in designs for group 1 shown in Fig. 20.
In continuation to the FEA results of the 2nd group of supplementary designed wave springs, A10 design initially shows linear and then nonlinear behavior once all waves were fully compressed, showing high stiffness while A11 design does not have high stiffness, lower load-bearing capacity but energy loss is more in this particular designs as can be seen by the width of hysteresis loop of this design. In both design A11 and A12 the variable dimensions are from middle wave i.e.; the maximum thickness or maximum diameter of the strip is for the middle due to which compression took place from top and bottom simultaneously.
A12 design shows the smooth transfer of force and have load-bearing capacity less than design A10 and greater than design A11. Semi-flat wave cross sectional area increases the stiffness of the spring, which can be seen in design A13, results high load- bearing capacity of all the supplementary designs. The load-bearing curve has smooth bend which shows the stability of the spring as well smooth transmission of load among the waves. The FEA results are shown in Fig. 21.
FEA results reveals that the stress distribution in the second group of supplementary designs is also uniform along all the surfaces of waves as shown in Fig. 22.