# Design and Analysis of Novel Low-Cost Linear Vibration Motor for an Electronic Cigarette
## 3 Analysis Methods
### Mechanical Analysis  
The static analysis of the mechanical parts is intended to calculate the effects of the constant loads on the structure while ignoring the effects of the inertia and shock that are commonly found when the applied load changes abruptly. The dynamic analysis of the mechanical parts is used to determine the vibrational characteristics. There are two analyses in this sub-section. One is linear stress analysis to check the von Mises stress distribution on the spring. The other is modal analysis to determine the mode frequencies and mode shape from the structures of the spring. The boundary conditions of the mechanical analysis are shown in Figure 5. To observe the boundary conditions easily, the upper case and lower case have been made transparent, and the red areas are the fixed parts for mechanical analyses.  
In the static mechanical analysis [8], the governing equation is as follows:  
\[[K_{m}][x]=\{F_{Lorentz}\}, \tag{5}\]  
where \([K_{m}]\), \([x]\), and \(\{F_{Lorentz}\}\) are the stiffness matrix of the vibration motor, displacement vector, and Lorentz force vector, respectively. The Lorentz force is applied on the top plate to obtain the static analysis results. For an isotropic material, the stresses and strains can be represented by column matrices as:  
\[\{\sigma\}=[D]\{\epsilon\}, \tag{6}\]  
\[[D]=\frac{E}{(1+\
u)(1-2\
u)}\left[\begin{array}{cccccc}1-\
u&\
u&\
u&0&0&0 \\ \
u&1-\
u&\
u&0&0&0\\ \
u&\
u&1-\
u&0&0&0\\ 0&0&0&\frac{1-2\
u}{2}&0&0\\ 0&0&0&0&\frac{1-2\
u}{2}\\ 0&0&0&0&0&\frac{1-2\
u}{2}\\ \end{array}\right], \tag{7}\]  
where \([D]\), \(E\), and \(\
u\) denote a constitutive matrix, Young's modulus, and [PERSON]'s ratio, respectively.  
Mathematically, the von Mises stress is a scalar value of stress that can be computed from the Cauchy stress tensor components in a general three-dimensional (3D) domain. A material starts yielding when the von Mises stress reaches a value known as the yield strength, \(\sigma_{y}\). The von Mises stress is used to predict the yielding of a material under multiaxial loading conditions using results from a simple uniaxial tensile test.  
Figure 5: The boundary conditions of the linear vibration motor.  
The Cauchy stress tensor, \(\sigma_{ij}\), consists of nine components that completely define the state of stress at a point inside a material in the deformed state:  
\[\left[\sigma_{ij}\right]=\left[\begin{array}{ccc}\sigma_{11}&\sigma_{12}& \sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\end{array}\right], \tag{8}\]  
\[\sigma_{v}^{2}=\frac{1}{2}[(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{22}-\sigma_ {33})^{2}+(\sigma_{33}-\sigma_{11})^{2}+6(\sigma_{23}^{2}+\sigma_{31}^{2}+ \sigma_{12}^{2})^{2}], \tag{9}\]  
where \(\sigma_{v}\) is the von Mises stress, and subscripts 1, 2, and 3 denote the x-direction, y-direction, and z-direction, respectively. In Figure 6, \(\sigma_{11}\), \(\sigma_{22}\), and \(\sigma_{33}\) represent the normal stresses, while \(\sigma_{12}\), \(\sigma_{23}\), and \(\sigma_{31}\) are the shear stresses.  
The factor of safety (FoS) expresses how much stronger a system is than it needs to be for an intended load [9]. It is defined as the ratio of the yield stress to the working stress for ductile materials. The working stress is selected as the maximum von Mises stress in this study.  
\[Factor\ of\ Safety=\frac{Yield\ stress\ \sigma_{y}}{Maximum\ von\ Mises\ stress}. \tag{10}\]