## 1 Introduction  
In recent years, owing to the large demand for civilian buildings, ocean engineering, military structures, and membrane structures [1; 2], many large-span structures have been widely designed for public uses [3; 4; 5]. It is well known that those structures are generally light and flexible so they are sensitive to dynamic loads, such as wind loads, seismic loads, etc. [6]. There are several examples [7; 8; 9] in which fluid pressure induced by the vibration of light and flexible structures has a significant effect on the mechanical system. However, the prediction of the fluid pressure induced by the vibration is very difficult; and there are very few pertinent investigations in the open literature. Due to the lack of understanding in this field, it is necessary to embark on studies, including experimental, theoretical, and numerical simulation studies [10; 11; 12].  
Based on the potential flow theory [13; 14] and the hypothesis that the vibration mode of a structure can be a sinusoidal function shape, [PERSON] [15] deduced a significant result that the added mass of a planar membrane in a rectangle is 0.68 divided by the mass ratio, \(m/\rho l\). Actually, his conclusion only fulfilled the 1-D condition. [PERSON], [PERSON] [16] proposed a new method to investigate the behavior of an elastic shell submerged in an unbounded fluid. In his paper, the fluid was assumed to be an elastic medium, and some analytical solutions written as a Fourier series were proposed. Several publications exist that investigated VIV (vortex-induced vibration) problems. [PERSON] et al. [17] studied flow-induced vibration of a cylindrical shell conveying a compressible fluid. [PERSON] [18] addressed a new approach to estimate the pressure of a circular shell beneath a free surface.  
[PERSON] and [PERSON] [14] carried out an experimental investigation of wave-induced flow around a cylinder in different conditions with several Keulegan-Carpenter ratios and Reynolds numbers. In [PERSON]'s work [19], numerical analysis of several wind towers located on the same building was conducted following the verification of a benchmark model against the wind tunnel data. Some new findings were produced via experiments, but those findings did not include discussion via theoretical analysis.  
Unfortunately, in the papers presented above, the incompressible fluid hypothesis was mostly taken into account [20; 21], which can simplify the mathematical difficulty in the analytical solution, but also changes the physical rules of the FSI problems [22; 23; 24; 25]. In addition, very few studies were devoted to vibrations of a cylindrical shell.  
Based on the reasons above, this paper is the first to consider a cylindrical structure vibrating in the radius direction in both compressible and incompressible fluid [26; 27; 28]. Additionally, some analytical solutions were obtained. In addition, a new mathematical model about the damping ratio of the cylindrical shell vibrating in the fluid was derived and validated. This study enables a novel analytical prediction that can be practically utilized in the design and analysis of cylindrical shell structures interacting with wind loads, earthquakes, blast loading, and so on. This new mathematical technique can be extended to the other research fields. The rest of this study is organized as follows: The governing equations are presented in Section 2. In Section 3, the numerical verifications are investigated and discussed, and finally a summary and conclusions are presented in Section 4.