## 3 Results and Discussion
### Effect of Bubbles on Displacement of Elastic Cylinder
#### 3.3.1 Model of Elastic Cylinder Displacement
Since the elastic cylindrical boundary size used in this article is much smaller than the maximum radius of the bubble, when the dimensionless distance is closer, the bubble will wrap the elastic cylinder, and the first wrapped position is the tangent, that is, the intersection of the bubble's vertical direction and the elastic cylindrical boundary. Here, this paper assumes that for the bubble in the expansion period, at the moment when the bubble contacts the elastic cylinder boundary, the intersection position \(\Delta x\) of the vertical line through the bubble generation point and the elastic cylinder boundary already exists inside the bubble and is no longer subject to the thrust of the bubble. Under this condition, it is considered that the time history of the elastic cylinder being subject to the thrust of the bubble is the moment when the bubble expands to a radius of \(r=d\), and there is a gradual movement speed during the expansion process. Based on this, the time when the bubble contacts the elastic cylinder boundary can be obtained as  
\[t=A\frac{d}{R_{d}} \tag{11}\]  
where \(\dot{R}_{d}\) is the bubble velocity when \(r=d\). In addition, the bubble expands fastest at the initial stage, and the bubble radius growth curve tends to be flat as the radius increases. At the moment of contact with the elastic cylinder boundary, the larger the bubble's dimensionless distance, the smaller the bubble growth rate. This algorithm for calculating time for displacement and average velocity has certain problems. The closer the dimensionless distance, the shorter the time, and the farther the dimensionless distance, the longer the time history. Therefore, it is necessary to set the coefficient \(A\) to calculate the time according to the velocity at the moment of contact. Based on the above discussion, the relationship between the dimensionless distance and the time of action of the thrust exerted by the bubble can be obtained as follows:  
\[t= \begin{cases}Ad/\dot{R}_{d}&(\gamma\leqslant 1)\\ BR_{max}\sqrt{\rho_{L}/p_{\infty}-p_{V}}&(\gamma>1)\end{cases}  \tag{12}\]  
Constructing models for both the elastic cylinder and the bubble is essential for analyzing the forces acting on the elastic cylinder. During the bubble's movement, the elastic cylinder will experience its own tension and forces. It is assumed that force is generated during the expansion of the bubble, causing the entire elastic cylinder to deform. The shape of the elastic cylinder is modeled as an arc of a circle. As shown in Figure 15, the dark red region represents the overall deformation of the elastic cylinder, which forms a segment of a circle with a radius of \(r_{1}\). The bold black line represents the initial position of the elastic cylinder. The total length of the elastic cylinder is \(L\). According to the coordinate system shown in the figure, \(-L/2\leq x\leq L/2\), where \(x\) is the position along the cylinder, \(R\) is the bubble radius, \(r_{rod}\) is the radius of the elastic cylinder, \(R_{max}\) is the maximum bubble radius, and \(\rho_{L}\) is the liquid density. Figure 15: Schematic of elastic cylinder under load. (**a**) The expansion process, (**b**) The collapse process. Figure 15a is a schematic of the overall force of the elastic cylinder during the bubble expansion process, and Figure 15b is a schematic of the overall force of the elastic cylinder during the bubble expansion process. To study the displacement of the elastic cylindrical boundary, the elastic boundary curve after being pushed by the bubble is assumed to be an arc, where the maximum displacement in the \(y\) direction of the arc is \(S_{0}\).