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Rayleigh quotient and orthogonality in the linear space of boundary functions, finding accurate upper bounds of natural frequencies of non-uniform beams

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Abstract

Instead of the traditional method of utilizing the extremal values of Rayleigh quotient, in this research we develop an upper bound theory to determine the natural frequencies in a linear space of boundary functions. The boundary function will have the following restrictions: boundary conditions are satisfied, fourth-order polynomial at least with unit leading coefficient. It proves that the maximum value of Rayleigh quotient in terms of kth-order boundary function builds a good upper bound when the \((k - 3)\)th-order natural frequency is approximated. There are four boundary conditions of the beam, so the fourth-order boundary function is unique without having any parameter. On the contrary, the kth-order boundary function has \(k-4\) free parameters, which leave us a chance to optimize them. We employ the orthogonality to derive the higher-order optimal boundary functions from the lower-order optimal boundary functions exactly, which are constructed sequentially by starting from the fourth-order boundary function. We examine the upper bound theory for uniform beams under three types of supports, and the single- and double- tapered beams under cantilevered support. Comparison is made between the first four natural frequencies with the exact/numerical ones, which proves the usefulness of the upper bound theory.

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Acknowledgements

The work of Botong Li is supported by the Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. College of Mechanics and Materials, Hohai University, Nanjing, 210098, Jiangsu, China

    Chein-Shan Liu

  2. Center of Excellence for Ocean Engineering, Center of Excellence for the Oceans, National Taiwan Ocean University, Keelung, 202-24, Taiwan

    Chein-Shan Liu

  3. School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

    Botong Li

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  1. Chein-Shan Liu
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  2. Botong Li
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Correspondence to Botong Li.

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Liu, CS., Li, B. Rayleigh quotient and orthogonality in the linear space of boundary functions, finding accurate upper bounds of natural frequencies of non-uniform beams . Arch Appl Mech 90, 1737–1753 (2020). https://doi.org/10.1007/s00419-020-01693-4

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  • DOI: https://doi.org/10.1007/s00419-020-01693-4

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