# Design and Analysis of Novel Low-Cost Linear Vibration Motor for an Electronic Cigarette
## 4 Design and Optimization
### Spring Shape Designs  
According to the linear stress analysis, the Lorentz force of the prototype vibration motor is 0.059 N, with a flux density of 0.273 T and input voltage of 4.35 V, which is applied on the top plate via the direction of the vibration. The von Mises stress distribution on the prototype spring is shown in Figure 16. Six areas have been selected to view the von Mises stress distribution on a single spring rib. The von Mises stress of each area is the average value of the six areas. The von Mises stresses on six areas are shown in Table 2, which shows that the distribution is not a declining distribution but a U-shaped distribution that first drops and then rises. The maximum value is 25.50 MPa at A0, and minimum value is 10.62 MPa at A3. The yield stress of SUS 301 EH is 1080 MPa, and FoS of the prototype spring is about 42.35 (>>1), which means the system can work safely under this condition. However, in order to significantly reduce the resonance frequency of the prototype spring, four parts on a spring rib have been selected as design parameters, as shown in Figure 17.  
The design parameters Round #1, Round #2, Width #1, and Width #2 are 0.28, 1.05, 0.30, and 0.40 mm respectively, which are within the manufacturing limitations. Round #1 is the first parameter related to A0 and A1. The bigger the round radius, the higher the maximum von Mises stress is, and the lower the resonance frequency is. From Figure 18 and Table 3, it is clear that the von Mises stress at A1 of type #2 has an 11.89% increase compared with the prototype spring. At the same time, due to the area reduction on A1 and increased displacement, the von Mises stresses in the other areas of type #2 have not significantly changed. Round #2 is the second parameter related to A0 and A1. When the round radius was increased from 0.65 to 1.05 mm, there was no significant increase in the von Mises stress at A0 and A1 because the location of round #2 is a little bit far away from A0 and A1. However, the resonance frequency still slightly decreased. This means that the type #4 spring with a smooth shape of round #2 can be easier to vibrate than the prototype spring. The von Mises stress distribution for different round #2 parameters is shown in Figure 19. The analysis results are compared in Table 4.  
Figure 16: von Mises stress distribution on prototype spring shape. \begin{table}
\begin{tabular}{c c} \hline \hline
**Areas** & **von Mises Stress (MPa)** \\ \hline A0 & 25.50 \\ A1 & 22.03 \\ A2 & 17.31 \\ A3 & 10.62 \\ A4 & 16.14 \\ A5 & 20.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: von Mises stress at selected areas on prototype spring shape. Figure 19: von Mises stress distribution on spring with different round #2 parameters. Figure 17: Four design parameters on prototype spring. Figure 18: von Mises stress distribution on spring with different round #1 parameters. Width #1 is the third parameter related to A4. When width #1 decreases from 0.40 to 0.30 mm, there is a significant increase in the von Mises stress at A4 due to the area reduction. Meanwhile, the maximum von Mises stress has a 1.29% increase at A0 according to the bigger displacement at the top plate with a reduction in resonance frequency (\(-2.49\%\)). In Figure 20, the von Mises stresses in the other areas do not have much difference when decreasing width #1. The analysis results are compared in Table 5.  
Width #2 is the fourth parameter related to A5. The von Mises stress at A5 has a significant increase of about 21.40% when width #2 decreases from 0.50 to 0.40 mm. The maximum von Mises stress of type #8 is 25.88 MPa, which has a 1.49% increase at A0 due to the bigger displacement at the top plate. The resonance frequency has a 3.32% reduction compared to the prototype spring. The analysis results are shown in Figure 21 and Table 6.  
From the analysis results of the four parameters, round #1, width #1, and width #2 have a bigger influence on the resonance frequency by about 2.72%, 2.49%, and 3.32% decrease, respectively.