Three displacement functions are introduced to decompose three displacement components so that the three-dimensional equations of motion of a transversely isotropic body are uncoupled. Expanding these functions in terms of orthogonal series, the equations of free vibration problem of a transversely isotropic cylindrical shell with ends simply supported are further simplified to be readily dealt with. As contrast to previous works, modified Bessel function solution with complex arguments is directly adopted for the case of complex eigenvalues. Moreover, for solid cylinders, (modified) Bessel functions of the first kind, even with pure imaginary arguments, are used because of their peculiar properties. Two numerical examples are given to check the correctness of the present method by comparing the results with those of others. Because no assumption is introduced in the paper, the method developed is completely three-dimensionally exact.
Free Vibrations of Transversely Isotropic Cylinders and Cylindrical Shells
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Chen, W. Q., Ying, J., and Yang, Q. D. (November 1, 1998). "Free Vibrations of Transversely Isotropic Cylinders and Cylindrical Shells." ASME. J. Pressure Vessel Technol. November 1998; 120(4): 321–324. https://doi.org/10.1115/1.2842338
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