Numerical Analysis on Natural Vibration of Cylindrical shell with Different Cross-section

. Cylindrical shell in engineering has different cross-section, i.e., circular section, oval section, rectangular section, and so on. In this work, numerical analysis is conducted on natural vibration of cylindrical shell with different cross-section, and the difference is studied between modal frequency and modal shape from different section shell. Numerical results show that, with the same length, width /thickness and boundary condition, the first 10 order modal frequency from circular section shell is nearly 60% bigger than that from the rectangular section shell. This indicates that the load carry ability of circular cylindrical shell is much better than rectangular cylindrical shell, which will help to support the design of cylindrical shell in engineering.


Introduction
Cylindrical shell has wildly applications in engineering, such pipeline, water tank and pressure hull of submarine. But the section of cylindrical shell may be different For the vibration of circular cylindrical shell, much work has been done. Arnold and Warburton [1] applied the energy method to establish the governing equation of natural vibrato circular cylindrical shell. Leissa [2] summarized recent works on free vibration of cylindrical shell, and he also gave the suitable range for different methods. It's a great work. Zhang [3] uses the wave propagation approach to analyze the free vibration of circular cylindrical shell with finite shell length. For vibration of non-circular cylindrical shell ,available research work is a bit fewer. Klosner and Pohle [4][5] studied the natural vibration of oval cylindrical shell with both ends simply-supported. Sewall [6] conducted series experiments to study the vibration of oval cylindrical shell, and experimental data was plotted in figures and tables for researchers. Baudak [7] applied spline matching method to investigate the free vibration of non-circular cylindrical shell, and modal frequency was calculated. Suzuki and Leissa [8][9] constructed a new Lagrange function, and use it to study the vibration of non-circular cylindrical shell with thickness varying along circumferential direction. In this work, numerical analysis is conducted on natural vibration of cylindrical shell with different cross-section, and the difference is studied between modal frequency and modal shape from different section shell.

Research object
There are two research objects in this work, one is circular cylindrical shell, the other is rectangular cylindrical shell. The shell length is 8000mm, thick ness is 10mm. Radius of circular section is 2600mm, the length and width of rectangular section are also 2600mm with fillets of 800mm.

Boundary condition
In numerical simulation, the boundary condition of both circular and rectangular cylindrical shell is clamped for both shell ends.

Materials
In numerical simulation, material of shell model is 022Cr17Ni12Mo2 steel, parameters of material is shown in Table.1. The density is 7850 kg/m 3 , Elastic module is 200GPa, the Poisson's ratio is 0.3, the Yield strength is 177MPa.

Model Mesh
The mesh work for shell model is conducted in ANSYS software, the element size is 50mm.

Comparative analysis of modal parameters
Comparative analysis is conducted for modal parameters of cylindrical shell with different sections, results are thown in Table 2. The difference of modal frequency is defines as

100%
cir rec Where, cir f and rec f are respectively modal frequency from circular and rectangular cylindrical shell.
It's noted that, compared to rectangular cylindrical shell, model frequencies from circular cylindrical shell is much bigger. The increase is more than 60%, and it varies with change of mode order. Furthermore, increase of the seventh mode order is the biggest for the first ten modes, and it is nearly 127.6%. This phenomenon could be explained, , and circular section has good load carry ability in circumferential direction, while the rectangular section has curvature mutation and flexural stiffness will decrease, and modal frequency will finally be reduced. Therefore, circular section could be applied to replace rectangular section cylindrical shell to enhance structural load carrying ability or increase structural model frequency.

Conclusion
In this work, numerical analysis is conducted on natural vibration of cylindrical shell with different cross-section, and the difference is studied between modal frequency and modal shape from different section shell. Numerical results show that, with the same length, width /thickness and boundary condition, the first 10 order modal frequency from circular section shell is nearly 60% bigger than that from the rectangular section shell. This indicates that the load carry ability of circular cylindrical shell is much better than rectangular cylindrical shell. Besides, modal shape from circular and rectangular shell in the same order is also obvious different. Conclusions in this work will help to support the design of cylindrical shell in engineering.