Abstract
A three-dimensional elasticity-based continuum model is developed for describing the free vibrational characteristics of an important class of isotropic, homogeneous, and completely free structural bodies (i.e., finite cylinders, solid spheres, and rectangular parallelepipeds) containing an arbitrarily located simple inhomogeneity in form of a spherical or cylindrical defect. The solution method uses Ritz minimization procedure with triplicate series of orthogonal Chebyshev polynomials as the trial functions to approximate the displacement components in the associated elastic domains, and eventually arrive at the governing eigenvalue equations. An extensive review of the literature spanning over the past three decades is also given herein regarding the free vibration analysis of elastic structures using Ritz approach. Accuracy of the implemented approach is established through proper convergence studies, while the validity of results is demonstrated with the aid of a commercial FEM software, and whenever possible, by comparison with other published data. Numerical results are provided and discussed for the first few clusters of eigen-frequencies corresponding to various mode categories in a wide range of cavity eccentricities. Also, the corresponding 3D mode shapes are graphically illustrated for selected eccentricities. The numerical results disclose the vital influence of inner cavity eccentricity on the vibrational characteristics of the voided elastic structures. In particular, the activation of degenerate frequency splitting and incidence of internal/external mode crossings are confirmed and discussed. Most of the results reported herein are believed to be new to the existing literature and may serve as benchmark data for future developments in computational techniques.
Highlights
The rapid advancement in contemporary industries towards high precision in engineering applications calls for more accurate predictions of the dynamic behavior of mechanical systems
Consider the group of linear, macroscopically homogeneous, isotropic, and traction-free canonically shaped elastic bodies depicted in Fig. 1, as described below: Fig. 1a, Fig. 1b, Fig. 1c, Fig. 1d,and Fig. 1e
At these crossovers, corresponding to specific eccentricities, the previously split eigen-frequencies become degenerate. This implies that beyond the crossover eccentricity, the stiffness of structure in one vibration mode will interchange place with another
Summary
The rapid advancement in contemporary industries towards high precision in engineering applications calls for more accurate predictions of the dynamic behavior of mechanical systems. It is noteworthy that due to the excellent properties of Chebyshev polynomials in numerical operations, the adopted method is well known to predict more frequencies and modes with higher convergence rate, better numerical stability, and improved accuracy in comparison with other types of admissible functions such as simple algebraic polynomials, in the 3-D vibration analysis of an elastic bodies where numerical instability may occur with a great number of terms of admissible functions [38,41,53] This fact was underlined by Zhou et al [40] where they demonstrated that by using Chebyshev polynomials instead of simple polynomials as the admissible functions, the immunity against ill-conditioned behavior in computing eigenfrequencies of completely free solid and annular thick circular plates can be greatly enhanced. The computed complete spectrum of eigen-frequencies and mode shapes can reveal the physical characteristics of the problem and serve as the benchmark for assessment of other numerical or asymptotic solutions
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
