# Design and Performance Research of a Wearable Airbag for the Human Body
## 3 Vehicle-Pedestrian Collision Finite Element Model
### Validation of the Airbag Inflator Finite Element Model  
This paper uses the Uniform Pressure Method [22,23] to simulate the inflation of an airbag. The pressure inside the airbag is assumed to be uniformly distributed during the inflation process and the gas filling the airbag is simulated by defining the mass and temperature of the gas. It is worth noting that the gas charged to the airbag in the uniform pressure method is an ideal gas with a constant specific heat capacity. Simulating the filling process using the uniform pressure method has the advantages of easy model control and fast calculation. Equation (1) shows the formula to calculate the airbag volume [24]. In terms of divergence theorem, the volume integration can be transformed to surface  
Figure 3: ACH 27 hybrid inflator.  
integration. The surface integration at each time step can be obtained by integrating the surface area of the elements.  
\[V=\int\int\int d_{x}d_{y}d_{z}=\oint xn_{x}d\Gamma\approx\sum_{i=1}^{N}\overline{x _{i}}n_{ix}A_{i} \tag{1}\]  
where \(d_{x}d_{y}d_{z}\) is the volume of the element in the integration, \(\Gamma\) is the surface area of the element in the integration, \(V\) is the volume of the airbag, and \(\overline{x_{i}}\) is average value of element coordinates on the \(x\)-axis. \(n_{ix}\) is the cosine of the angle between the element normal direction and the \(x\)-axis, and \(A_{i}\) is the surface area of the \(i\)th element. The uniform pressure method simulates the unfolding process of an airbag in which the volume of the airbag gradually increases under the reaction induced by pressure, temperature, and mass. The airbag inflation process is adiabatic, the specific heat capacity of the gas is constant, the pressure and temperature inside the airbag are uniform, and the pressure and volume of the uniform pressure method airbag model satisfy the following equations:  
\[pV=mRT \tag{2}\]  
\[p=(\gamma-1)\rho E \tag{3}\]  
\[\gamma=\frac{C_{p}}{C_{v}} \tag{4}\]  
where \(p\) is the gas pressure in the airbag, \(m\) is the gas mass in the airbag, \(R\) is the molar gas constant, \(T\) is the air temperature inside the airbag, and \(\gamma\) is the ratio of constant volume specific heat to constant pressure specific heat of the gas, \(\rho\) is the gas density, and \(E\) is the specific internal energy of the gas.  
In this study, a tank test is conducted to validate the effectiveness of the uniform pressure method utilized in the finite element simulation and to obtain the input curves that characterize the injected gas utilized in the finite element model. These two curves include the temperature variation curve and the gas mass curve. These two curves are assigned to the keyword /PROP/INJECT1 in Radioss. In the tank test, the gas inflator is placed in the closed container, and then detonated. The total mass of the gas is 12 g, the initial temperature is a constant value, 295 K, and the volume of the closed vessel is 28.3 L. The molar mass of the gas is 0.031 kg/mol, and the specific heat capacity at constant pressure is 520 J/(kg\(\cdot\)K). The temperature history curve, in Figure 4a, and the internal pressure history curve, in Figure 5b, are recorded by the temperature sensor and pressure sensor, respectively. The time history curve of gas mass, in Figure 4b, is derived via MTA analysis [25; 26] in terms of the above recorded temperature history curve and internal pressure history curve. The FEA model of the tank test is shown in Figure 5a and it contain 62,100 shell elements. Rigid material is assigned to the shell elements. As shown in Figure 5b, the internal pressure history curves obtained by simulation and testing have excellent agreement.