## 2 Analysis Method
### Numerical Analysis Method  
The updated Lagrangian formulation is assumed to have the current configuration \(V(x,t)\) as the reference configuration. The initial configuration is the space occupied by an object at time \(t=0\) s, while the current configuration is at a specific time, as seen in Figure 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Item** & **Material** & **Density (kg/m\({}^{3}\))** & **Young’s Modulus (GPa)** & **Poisson’s Ratio** \\ \hline Armature & Permalloy & 8300 & 127 & 0.3 \\ Bobbin & Vectra E130i & 1610 & 15 & 0.43 \\ CDP & Aluminum & 2750 & 71 & 0.29 \\ Gasket & Rubber & 1300 & 0.004 & 0.5 \\ PMs & N50H & 7500 & 155 & 0.3 \\ SDP & Soniphor T10 um & 1200 & 0.13 & 0.36 \\ Voice coil & Copper & 8960 & 110 & 0.34 \\ The others & SUS 304 & 7850 & 192 & 0.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The material properties of the BA receiver and gasket. Figure 3: X-ray photos of the BA receiver (**a**) before and (**b**) after the drop impact test. Hence, the governing equations in the updated Lagrangian form are the conservation of mass, momentum, and energy, as follows, respectively:  
\[\rho J=\rho_{0}, \tag{1}\]  
\[\frac{\partial\sigma}{\partial x}+\rho b=\rho\ddot{u}(x,t), \tag{2}\]  
\[\rho\frac{\partial\dot{u}^{int}}{\partial t}=D(x,t)\sigma, \tag{3}\]  
where \(\rho\) is the material density of the current configuration, \(J\) is the Jacobian matrix, \(\rho_{0}\) is the material density of the initial configuration (\(t=0\) s), \(b\) is the acting force per mass, \(x\) is the position, \(u(x,t)\) is the displacement, \(\dot{u}(x,t)\) is the velocity, \(\ddot{u}(x,t)\) is the acceleration, and \(\omega^{int}\) is the internal energy change. The rate of deformation \(D(x,t)\) is given by  
\[D(x,t)=\frac{\partial\dot{u}(x,t)}{\partial x}. \tag{4}\]  
For a linear elastic material:  
\[\sigma(x,t)=E\int_{0}^{t}D(x,t)dt=Eln(F(x,t)), \tag{5}\]  
where \(F(x,t)\) is the deformation gradient. The initial conditions are as follows:  
\[\sigma(x,0)=\sigma_{0}(x), \tag{6}\]  
\[u(x,0)=0, \tag{7}\]  
\[\dot{u}(x,0)=\dot{u}_{0}(x). \tag{8}\]  
The essential boundary conditions are  
Figure 3:0  
where \(\overline{\dot{u}}\) is the velocity vector, and \(\Gamma_{v}\) is the specified velocity boundary. The traction boundary conditions are  
Figure 3:1  
Figure 3:2 Initial and current configurations. where \(n\) is the unit-length direction vector in the current configuration, \(t_{x}\) is the stress vector, and \(\Gamma_{t}\) is the specified surface force boundary. Compared to the implicit dynamic analysis [23], the explicit dynamic analysis has some advantages regarding solving the complex impact-contact problem, for which the contact time is truly short, such as a low analysis time with high numerical accuracy. The explicit dynamic analysis used a central difference rule to integrate the equations of motion explicitly through time, using the kinematic conditions at one increment to calculate the kinematic conditions at the next increment. At the beginning of the increment, the program solved for the dynamic equilibrium, which states that the product of the nodal mass matrix \(M\) and the nodal accelerations \(\ddot{u}\) is equal to the net nodal forces, i.e., the difference between the external applied forces \(F\) and the internal element forces \(I\); the superscript \(i\) refers to the increment number:  
Figure 3:3  
The accelerations at the beginning of the current increment (time \(t\)) were calculated by  
Figure 3:4  
As the explicit procedure always used a lumped mass matrix, solving for the accelerations was trivial. There were no simultaneous equations to solve, and the acceleration of any node was determined completely by its mass and the net force acting on it, which made the nodal calculations very inexpensive.