Vibration Analysis of a Rigid Circular Disk Embedded in a Transversely Isotropic Solid

Abstract

The analytical treatment of normal, rocking, and torsional forced time-harmonic vibrations of a rigid circular disk in a transversely isotropic full-space is revisited. A complete discussion on frequently used contact assumptions of adhesive and smooth models is given, and the effects of different contact models on the results of each vibration mode are discussed. With the aid of appropriate dynamic Green’s functions, the in-plane mode of vibration of the disk is treated analytically for the first time, and the results are expressed in terms of the solution of a Fredholm integral equation. For all four vibration modes, the relations for the contact stress, the resultant force acting on the disk, and the dimensionless compliance factor are given. The available closed-form results in the literature corresponding to the static loading are exactly recovered as the limiting cases of the current study. Furthermore, the results are verified with the special case of an isotropic full-space. By virtue of the contour integration technique as well as the residue theorem, a robust numerical integration technique is proposed to overcome the difficulties in numerical evaluation of the semiinfinite integrals appearing in the obtained Fredholm integral equations of all four modes of vibration. Some plots are provided to present the lateral, vertical, and rocking compliance factors for different transversely isotropic materials. The effects of material anisotropy on the results are also highlighted.