## 3 Modeling of the Metallic Braided Tube  
### Mathematical Model of the Metallic Braided Tube  
The structure of the braided corrugated hose is shown in Figure 1; the metallic braided hose covers the outside of the bellows to protect the internal structure and there is a contact between the metallic braided hose and the internal bellows. The metallic braided hose is made up of several strands of metal wires woven in a certain order and angle along the axis of the bellows. Depending on the desired mechanical properties of the braided corrugated hoses, the geometry, size, the weaving angle, and the number of wires per strand vary greatly, which makes it more difficult to generate geometric models. Therefore, a metallic braided modeling method based on the modified helix line is proposed. Figure 2 shows the metallic braided tube's basic component. Each strand is constituted by wires, and the wires in each strand have the same motion trajectory. By analyzing the structure of the metallic braided tube, this crossed spiral structure can be equated to the trajectory formed by any point on the outer circumference of the metallic braided corrugated hose rotating around the central axis and moving along the central line at the same time, so the shape of any wire in the braided tube is a cylindrical spiral. Therefore, the wire's motion trajectory can be described in Figure 3, in which \(r\) is the radius of the \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Specification** & **C** & **Si** & **Mn** & **P** & **S** & **Cr** & **Ni** \\ \hline
304 stainless steel & \(\leq\)0.08 & \(\leq\)1 & \(\leq\)2 & \(\leq\)0.045 & \(\leq\)0.03 & \(\leq\)18 - 20 & \(\leq\)8 - 10.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The chemical composition of 304 stainless steel. Figure 1: Schematic diagram of the braided corrugated hose. braided tube, \(p\) is the pitch, \(\alpha\) is the braiding angle, and \(\theta\) is the angle of rotation. The equation of motion [10] can be expressed as:  
\[\left\{ \begin{array}{l}x=rcos(\theta)\\ y=rsin(\theta)\\ z=\frac{\theta}{2\pi}\cdot p\end{array} \right. \tag{1}\]  
For the braided tube, the braiding angle depends on the value of pitch \(p\). When \(p\) is large, the weaving speed is fast, so the braiding angle is small; when \(p\) is small, the weaving speed is slow, so the weaving angle is large. The relationship between braiding angle \(\alpha\) and pitch \(p\) can be expressed as:  
\[\tan(\alpha)=\frac{2\pi r}{p} \tag{2}\]  
Formula (1) is the wire trajectory in one direction, while the metallic braided tube is interlaced in two directions. Therefore, by transforming the sign in the y-direction of Formula (1), the other direction of the wire trajectory can be obtained. For the places where the clockwise and counterclockwise wires intersect, the previous scholar [11] used the constraint equation to constrain the degree where they coincide, but this method did not quite fit compared with the realistic situation. According to some research [12], a sinusoidal motion can be added to the helix line. Indeed, the radius of the helix line is not constant, and it will become bigger or smaller like a sine function. To avoid the clockwise and counterclockwise wires intersecting at the same point, we set different direction's wires to have different radii, as Figure 4 shows; therefore, the helix lines in different directions will not interference when intersecting, and the improved radius Formula of helix line can be expressed as:  
\[r=r_{0}+h\cdot\sin(N\cdot\theta) \tag{3}\]  
where \(r_{0}\) is the nominal radius, \(h\) is the maximum distance to the nominal radius, and \(N\) is the number of jumps accomplished by the fibers in one pitch length. The complete geometric equation of modified helix line can be expressed as:  
\[\left\{ \begin{array}{l}x=(r_{0}+h\cdot\sin(N\cdot\theta))\cdot\cos(\theta) \\ y=(r_{0}+h\cdot\sin(N\cdot\theta))\cdot\sin(\theta)\\ z=\frac{\theta}{2\pi}\cdot p\end{array} \right.