## 2 Ply Angle Prediction  
The 684N model, a diaphragm type used in the past with the K-TGV high-speed train, consisted of a rubber composite material reinforced with six layers of nylon fiber and inner and outer rubber. The ply angle was measured at seven points from the bead part of the upper plate (A) to the bead part of the lower plate (G), as shown in Figure 1[5]. Airbags have a total of three pairs of equally angled piles, and the ply angles are opposite in the neighboring piles. The ply angle starts at point A and gradually increases until it passes the outermost points C and D, and the ply angle gradually decreases as it goes to the inner points E, F, and G.  
The author presented a calculation equation for the ply angle in an airbag using the radius and initial angle in the reference [6] in 2001 as Equation (1). In this paper, an equation for calculating the angle was additionally proposed and verified. Since the airbag has an axisymmetric shape, a change in the radial position means a change in length in the circumferential direction. When the airbag length in the circumferential direction is increased by being pressurized or inflated, the thickness of the airbag reduces and the length in the latitudinal direction increases. Figure 1: Measured cord angles of the 684N diaphragm-type air spring at the designated points from the reference [5]. As shown in Figure 2, in a rectangle with the length in the same circumferential direction as the base and the length in the same longitudinal direction as the height, as the radius increases, the base increases. Regarding the change in thickness and height, in the first case, only the thickness changes and the height remains the same, and in the second case, there is no change in the thickness and the surface area of the airbag is the same before and after the deformation. Finally, the third case is a case in which the height and thickness are changed in proportion to the Poisson's ratio due to the change in length in the circumferential direction. In each case, when the initial angle is \(\theta\) and the initial radial position is \(r\), the angle at the final position \(R\) can be expressed as \(\alpha\), and can be expressed by Equations (1)-(3). \[\alpha=tan^{-1}\bigg{(}\bigg{(}\frac{R}{r}\bigg{)}tan\theta\bigg{)} \tag{1}\]  
\[\alpha=tan^{-1}\bigg{(}\bigg{(}\frac{R}{r}\bigg{)}^{2}tan\theta\bigg{)} \tag{2}\]  
\[\alpha=tan^{-1}\bigg{(}\bigg{(}\frac{R}{r}+\
u\bigg{(}\frac{R}{r}-1\bigg{)} \bigg{)}tan\theta\bigg{)} \tag{3}\]  
Initial Shape  
Figure 3 is a graph showing the change in ply angle at each position when changing from the radial position 137 to 340 mm after setting the ply angle of 11 degrees at the initial radius of 267 mm for the airbag model in Figure 4. Using the exponent of \(\bigg{(}\frac{R}{r}\bigg{)}\) in Equations (1) and (2), Equation (1) and (2) are denoted as Angle_1 st and Angle_2 nd in Figure 3, respectively. Equation (3) is expressed as Angle_poisson. Equation (3) with the Poisson's ratio set to 0.3 and 0.5 are located between Equations (1) and (2). As the Poisson's ratio increases, Equation (3) moves from Equation (1) to Equation (2). Considering the change in thickness, Equations (1) and (2) represent the minimum and maximum ranges of the angle change of the airbag surface. Figure 2: Conceptual diagram of angle prediction equations. The angle distribution according to the longitudinal position from the upper plate to the lower plate of the airbag in Figure 1 is shown in Figure 5 for each ply by radial position and ply angles based on the shape of the 684N air spring. Figure 5 shows the angle calculated by setting the upper bead's radial position of 282 mm and the initial angle of 36 degrees in the angle calculation equations. As a result of comparing the three angle calculation formulas for each ply, all three proposed equations are in good agreement with the measured data.