**Acknowledgments**

The current solution method is also compared with the finite element model (ANSYS) for shells under free-free boundary condition. In the FEM model, the shell surface is divided in‐ to 8000 elements with 8080 nodes. The calculated natural frequencies are compared in Ta‐

In most techniques, such as the wave approach, the beam functions for the analogous boun‐ dary conditions are often used to determine the axial modal wavenumbers. While such an approach is exact for a simply supported shell, and perhaps acceptable for slender thin shells, it may become problematic for shorter shells due to the increased coupling of the ra‐ dial and two in-plane displacements. To illustrate this point, we consider relatively shorter and thicker shell (*l*=8*R* and *R* =39*h*). The calculated natural frequencies are compared in Ta‐ ble 9 for a clamped-clamped shell. It is seen that while the current and FEM results are in good agreement, the frequencies obtained from the wave approach (based on the use of

> **m = 1 m = 2 FEM Ref. [32] present FEM Ref. [32] Present**

 3229.8 4845.5 3230.3 5146.0 8075.8 5139.8 1882.8 2350.2 1880.9 3850.7 4775.6 3848.9 899.59 985.48 898.18 2017.8 2303.4 2014.1 896.97 919.01 896.56 1390.9 1479.2 1388.9 1501.9 1517.45 1501.6 1676.4 1714.0 1676.0 2386.1 2402.05 2386.0 2472.5 2501.8 2472.6

**Table 9.** Comparison of the natural frequencies for a circular cylindrical shell with clamped-clamped boundary

rection, which represents a significant advancement over many existing techniques.

The exact solution method can be readily applied to shells with elastic boundary supports. Since the above examples are considered adequate in illustrating the reliability and accuracy of the current method, we will not elaborate further by presenting the results for elastically restrained shells. Instead, we will simply point out that the solution method based on Eqs. (27) is also valid for non-uniform or varying boundary restraint along the circumferential di‐

An improved Fourier series solution method is described for vibration analysis of cylindri‐ cal shells with general elastic supports. This method can be easily and universally applied to a wide variety of boundary conditions including all the 136 classical homogeneous boun‐ dary conditions. The displacement functions are invariantly expressed as series expansions in terms of the complete set of trigonometric functions, which can mathematically ensure

conditions, *L*=0.502 m, *R*=0.0635 m, *h*=0.00163 m, σ=0.28, E=2.1E+11 N/m3, ρ=7800 kg/m3.

bles 8. An excellent agreement is observed between these two solution methods.

230 Advances in Vibration Engineering and Structural Dynamics

beam functions) are significantly higher, especially for the lower order modes.

**n**

**4. Conclusion**

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (No. 50979018).
