An analysis and experimental validation of natural frequencies of elastic ellipsoids with the Rayleigh–Ritz method

Abstract

The Rayleigh–Ritz method is widely employed to analyze free vibrations of elastic solids and structures. For a simple formulation and efficient evaluation of vibrations of elastic ellipsoids, which also include spheres, the Cartesian coordinate system is utilized. It is hoped that the formulation with lengthy expressions can be solved with fewer terms of displacement functions in Chebyshev polynomials for simple evaluations of stiffness and mass matrices of the elastic ellipsoids with the procedure. The vibrations of elastic ellipsoids are calculated with geometric parameters for the validation of the procedure and formulation with known results and the analysis from this study.

Keywords:

• Rayleigh–Ritz method
• vibration
• ellipsoid
• frequency
• Chebyshev
• polynomials
• mode

Acknowledgments

The authors acknowledge TXC-NBU Joint Center of Research receives funding from TXC (Ningbo) Corporation to promote research work for the advances of technology of acoustic wave resonators and applications. The authors are also grateful to Shining3D Company (Yangzhou, Jiangsu Province, China) for the ellipsoid samples with their excellent 3D printing technology.

Authors’ contributions

Conception, revision, and submission of the article were done by Ji Wang (JW) and Jinghui Wu (JHW), while the mathematical formulation, procedure, derivation, calculation, and check were made by RW, LX, TM, and AZ with help from JD and SZ. Reviews and discussions of the study and drafting were also shared and communicated with all authors. All authors have read and agreed to the published this version of the manuscript.

Data availability statement

The data of this study are available from authors and author’s website.

Disclosure statement

The funders and other parties had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China (Grant Nos. 11672142, 11772163, and 12172183) with additional support through the Technology Innovation 2025 Program (Grant 2019B10122) of the Municipality of Ningbo, Zhejiang Province, China.