Explicit Novel Nonlinear Regression Method to Find Radial Natural Frequencies of Hemispherical Resonator
Tóm tắt
Hemispherical shells are commonly used in percussion instruments such as Hang, Bell, and Singing Bowls as well as other engineering applications. The dynamic properties of a hemispherical shell are important, especially for music which demands close control of the natural frequencies. Theoretically, the dynamic behavior of hemispherical shell is studied by solving very complex differential equations and mostly results are available for thin shells. Finite element analysis (FEA) is thus favored but it requires costly software and special expertise. FEA is also mesh-sensitive and mesh-independent analysis is required. Considering these facts, in this paper, an explicit nonlinear equation is formulated, which is a function of the geometry and material parameters. In this paper, the proposed explicit nonlinear equation is formulated using nonlinear regression with 4000 simulated data points generated using software. It obeys dimensional homogeneity and is validated using numerical and experimental methods. Validation of this equation using numerical data shows RMS error mostly less than 6% and 5% for geometry and material parameters, respectively. Frequency values of two manufactured hemispherical shells are extracted using audio analysis as well as experimental and operational modal analysis. The validation of these experimentally obtained frequencies with the proposed explicit equation gives a maximum absolute error of less than 5%, except in a few higher modes it is up to 7%. The proposed nonlinear explicit equation predicts the first four radial mode frequencies of a hemispherical shell just by entering a few physically significant parameters. It can be used to design shells demanding close control of the frequency values. The model can also be used in optimization.
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