## 5 Conclusions  
This paper is concerned with the viscoelastic damping effect on the acoustic radiation field. A numerical model was developed to deal with the vibroacoustic responses of a typical ring-stiffened conical shell by using the combination of FEM and BEM. Some conclusions are drawn as follows:  
1. The water medium can reduce the natural frequency and change the modal shapes. Meanwhile, the effect of suppressing vibration responses at a high frequency is better than that at a low frequency;
2. The damping effect at a high frequency is better than that at a low frequency. The increase in damping thickness would effectively reduce the amplitude of radiated noise, but does not change the distribution of the acoustic field;
3. There is a strong correlation between the damping position and the vibration mode of the structure. The damping on the web may achieve the best vibration damping effect, while the damping on the ribbed slab may result in the minimum radiation range.  
It is noted that although a series of numerical research is provided in this paper, a detailed validation study would improve confidence in the results. Thus, experimental analysis of vibration and sound radiation in the ring-stiffened conical shell are needed in the future for further validation.  
Figure 14: Acoustic radiation field in the yoz plane (1500 Hz): (**a**) no damping; (**b**) damping on the plate; (**c**) damping on the web; (**d**) damping on the ribbed slab.  
**Author Contributions:** Conceptualization, H.G. and Z.C.; methodology, Z.C.; investigation, Q.G.; software, Z.C.; validation, Z.C.; formal analysis, W.Z.; data curation, W.Z.; writing--original draft preparation, Z.C.; writing--review and editing, W.Z.; supervision, H.G.; project administration, Q.G.; funding acquisition, Q.G. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by financial support from the financial support from the National Natural Science Foundation of China, grant number U2006229; the Key Research and Development program of Shandong Province, grant number 2019 JZZY010125; and the Key Research and Development program of Shandong Province, grant number 2020 CXGC010701.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** All data, models, or code generated or used during the study are available from the corresponding author by request.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Nomenclature**  
\(\
abla\) Laplace operator, \(\
abla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y ^{2}}+\frac{\partial^{2}}{\partial z^{2}}\)  
\(\Omega\)-  
Flow field computational domain  
\(p(x,y,z)\) Sound pressure  
\(k\) Number of waves per second (m\({}^{-1}\)), \(k=\omega/c=2\pi f\)/\(c\)  
\(\omega\) Angular frequency (rad/s), \(\omega=2\pi f\)  
\(f\) Frequency (Hz)  
\(c\) Sound velocity (m/s)  
\(\rho_{0}\) Density of medium  
\(q(x,y,z)\) Normal velocity  
\(j\) Imaginary unit, \(j=\sqrt{-1}\)  
\(S\) Boundary of \(\Omega\)  
\(V\) Volume  
\(\widetilde{p}\) Weight function  
[K] Stiffness matrix  
[M] Mass matrix  
[C] Damping matrix  
\(\{p_{i}\}\) Sound pressure matrix  
[Q\({}_{i}\)] Acoustic excitation  
\(f_{n}\) Natural frequency  
\(\eta\) Loss factor