# Dynamic Pressure Analysis of Hemispherical Shell Vibrating in Unbounded Compressible Fluid
## 5 Conclusions  
A new analytical approach to calculate the pressure of the hemispherical shell vibrating in compressible fluid has been firstly developed. It is rigorously verified by the theoretical approach based on the compressible fluid assumption, which can predict the values of the velocity and pressure very well. It is found that the assumption of the compressibility of fluid plays a main role in the velocity and pressure of the FSI system. In addition, the exact solution and theoretical formulation is obtained by an integrated approach. According to the solutions, it is found that the pressure on the shell surface is proportional to the acceleration of boundary vibration when the fluid is assumed to be incompressible. Meanwhile, under the assumption of compressible fluids, the boundary pressure is proportional to the velocity of the boundary vibration. Considering the velocity of fluids, the amplitude of velocity decreases with the rate of \(1/r\) in the compressible condition, whilst it decreases with the rate of \(1/r^{2}\) in the incompressible condition.  
In addition, the physical implication of the developed formula has been discussed. It is found that the motion of the boundary will propagate to the far field at the speed of sound velocity with the same frequency, but will decrease with distant \(r\). Initially, the motion of the shell compresses the air locally, which will transfer to the far field afterward. The compression of air produces the local acceleration pressure, which is proportional to fluid velocity. The results reveal that the compression force can be exactly expressed by the new formula.  
Furthermore, it can be clarified that incompressible fluid assumption for fluids can induce a large numerical error in various cases. It is also found that under the incompressible fluid assumption, the effect of air pressure is equivalent to the effect of the added mass, which confirms the scientific findings by many previous research studies and common theories that are applied today. While under the compressible fluid consideration, the effect of air pressure is equivalent to the effect of viscous damping, whose ratio is exactly \(\rho_{o}c\). Therefore, our present findings and established formula can be used in the design and application of the resisting wind load of the spherical shell structure. The novel analytical formulation that was formed in this study can be used to predict the vibration of a spherical shell structure interacting with fluids. This new formulation can be applied to the design and analysis of the shell structure in practice.  
Conceptualization, P.L. and S.K.; Methodology, P.L. and B.T.; Validation, P.L., S.K. and B.T.; Formal Analysis, P.L.; Investigation, P.L. and S.K.; Resources, B.T.; Writing-Original Draft Preparation, P.L.; Writing-Review & Editing, S.K. and B.T.; Visualization, P.L.; Supervision, S.K.; Project Administration, B.T.; Funding Acquisition, P.L., S.K. and B.T.  
This work was supported by the national Natural Science Foundation of China (grant numbers 51508238) and by the Jiangsu Postdoctoral Research Plan (grant numbers 1601014B).  
The second author wishes to gratefully acknowledge the Japan Society for Promotion of Science (JSPS) for his JSPS Invitation Research Fellowship (Long-term), Grant No. L15701, at Track Dynamics Laboratory, Railway Technical Research Institute and at Concrete Laboratory, the University of Tokyo, Tokyo, Japan. The JSPS financially supports this work as part of the re-search project, entitled \"Smart and reliable railway infrastructure\". Special thanks to European Commission for H2020-MSCA-RISE Project No. 691135 \"RISEN: Rail Infrastructure Systems Engineering Network\" (www.risen2 rail.eu) [34]. In addition, the collaboration and assistance from EU Cost Action TU1409 (Mathematics for Industry Network) are highly appreciated.  
The authors declare that there are no conflicts of interest regarding the publication of this paper.