On the three-dimensional vibrations of a hollow elastic torus of annular cross-section

This paper offers a three-dimensional elasticity-based variational Ritz procedure to examine the natural vibrations of an elastic hollow torus with annular cross-section. The associated energy functional minimized in the Ritz procedure is formulated using toroidal coordinates (r,q, j)({r,\theta , \varphi}) comprised of the usual polar coordinates (r, θ) originating at each circular cross-sectional center and a circumferential coordinate j{\varphi} around the torus originating at the torus center. As an enhancement to conventional use of algebraic–trigonometric polynomials trial series in related solid body vibration studies in the associated literature, the assumed torus displacement, u, v and w in the r, θ and j{\varphi } toroidal directions, respectively, are approximated in this work as a triplicate product of Chebyshev polynomials in r and the periodic trigonometric functions in the θ and j{\varphi} directions along with a set of generalized coefficients. Upon invoking the stationary condition of the Lagrangian energy functional for the elastic torus with respected to these generalized coefficients, the usual characteristic frequency equations of natural vibrations of the elastic torus are derived. Upper bound convergence of the first seven non-dimensional frequency parameters accurate to at least five significant figures is achieved by using only ten terms of the trial torus displacement functions. Non-dimensional frequencies of elastic hollow tori are examined showing the effects of varying torus radius ratio and cross-sectional radius ratio.     

Longitudinal vibrations of elastic rods of variable cross-section (concentrators)

We consider longitudinal vibrations of elastic rods of variable cross-section (concentrators) of conical, exponential, and catenoidal types and obtain analytic expressions for the gain factors of the concentrators in the cases of boundary conditions of the first and second kinds. We numerically study various profiles and compare the gain factors depending on the type of boundary conditions and the eigenvalue number. We note that as the eigenvalue number increases, the curves for the gain factors of both the first and the second kind tend to the limit curves.     

On the steady vibrations of elastic materials with voids

This paper is concerned with the linear elastodynamics of homogeneous and isotropic materials with voids. First, the singular solutions corresponding to concentrated forces in the case of steady vibrations are established. Then, representations of Somigliana type for the displacement field and the change in the volume fraction field are presented. Radiation conditions of Sommerfeld type are derived. The potentials of single layer and double layer are used to reduce the boundary value problems to singular integral equations for which Fredholm's basic theorems are valid. Existence and uniqueness results for exterior problems are established.     

Three-dimensional vibration analysis of prisms with isosceles triangular cross-section

This paper studies the three-dimensional (3-D) free vibration of uniform prisms with isosceles triangular cross-section, based on the exact, linear and small strain elasticity theory. The actual triangular prismatic domain is first mapped onto a basic cubic domain. Then the Ritz method is applied to derive the eigenfrequency equation from the energy functional of the prism. A set of triplicate Chebyshev polynomial series, multiplied by a boundary function chosen to, a priori, satisfy the geometric boundary conditions of the prism is developed as the admissible functions of each displacement component. The convergence and comparison study demonstrates the high accuracy and numerical robustness of the present method. The effect of length-thickness ratio and apex angle on eigenfrequencies of the prisms is studied in detail and the results are compared with those obtained from the classical one-dimensional theory and the 3-D finite element method. Sets of valuable data known for the first time are reported, which can serve as benchmark values in applying various approximate beam and rod theories.     

Note on axial-shear and contour vibrations of prisms

Exact solutions of the equations of the linear theory of elasticity are given for axial-shear modes of vibration of an isotropic, prismatic bar whose normal section is an equilateral triangle or has the equilateral triangle as a module. A family of contour modes is also described for bars with a rhombic section formed of two equilateral triangles and with sections having the rhombus as a module.     

On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle

Existence and uniqueness results are established for weak formulations of initial-boundary value problems which model the dynamic behavior of an Euler-Bernoulli beam that may come into frictional contact with a stationary obstacle. The beam is assumed to be situated horizontally and may move both horizontally and vertically, as a result of applied loads. One end of the beam is clamped, while the other end is free. However, the horizontal motion of the free end is restricted by the presence of a stationary obstacle and when this end contacts the obstacle, the vertical motion of the end is assumed to be affected by friction. The contact and friction at this end is modelled in two different ways. The first involves the classic Signorini unilateral or nonpenetration conditions and Coulomb's law of dry friction; the second uses a normal compliance contact condition and a corresponding generalization of Coulomb's law. In both cases existence and uniqueness are established when the beam is subject to Kelvin-Voigt damping. In the absence of damping, existence of a solution is established for a problem in which the normal contact stress is regularized.The work of the last two authors was supported in part by Oakland University Research Fellowships.     

On the quasistatic effective elastic moduli for elastic waves in three-dimensional phononic crystals

Effective elastic moduli for 3D solid–solid phononic crystals of arbitrary anisotropy and oblique lattice structure are formulated analytically using the plane-wave expansion (PWE) method and the recently proposed monodromy-matrix (MM) method. The latter approach employs Fourier series in two dimensions with direct numerical integration along the third direction. As a result, the MM method converges much quicker to the exact moduli in comparison with the PWE as the number of Fourier coefficients increases. The MM method yields a more explicit formula than previous results, enabling a closed-form upper bound on the effective Christoffel tensor. The MM approach significantly improves the efficiency and accuracy of evaluating effective wave speeds for high-contrast composites and for configurations of closely spaced inclusions, as demonstrated by three-dimensional examples.     

On the application of the constant deflection-contour method in nonlinear vibrations of elastic plates

Summary  This paper concerns the application of the constant deflection-contour method to problems involving nonlinear vibrations. Two specific problems are considered: a clamped circular plate and an annular plate with free inner boundary. For the linear case, the results obtained offer excellent agreement with previous studies, indicating significant potential for the utilization of this method in different nonlinear cases. The analysis may be applied to other types of geometrical structures. Notwithstanding the fact that only a first-term approximation has been made for the deflection function, in conjunction with the Galerkin procedure, excellent agreement has been found. Additional analytical calculations could be made to improve accuracy, indicating that the method could prove particularly useful when employed with a symbolic manipulation package. Received 13 June 2001; accepted for publication 6 November 2001     

Forced vibrations of a boxed shell of square cross-section

We construct the solution of the problem on the steady-state vibrations of a finite boxed shell of square cross-section with symmetry conditions at the shell ends. We present the dispersion curves, find the natural frequencies, and study the stress distribution in the shell. We obtain a simple formula for the approximate analysis of the shell in the case of low-frequency vibrations on the basis of the expansion of the solution in two small parameters and on the Lagrange interpolation formula.     

On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies

Based on rigorous operator theory, a general solution of the three-dimensional elasticity equations of equilibrium for 1D hexagonal quasicrystals is obtained. The solution is expressed in terms of four quasiharmonic functions, which is very simple and useful. The point phonon (phason) force solution of an infinite 1D hexagonal quasicrystal body is derived, all in terms of elementary functions. They can play an important role in numerical simulations such as boundary element method.     

Asymmetric vibrations of an elastic sphere

Historically, the vector Navier equation governing the dynamic response of an elastic, homogeneous, isotropic sphere has been solved using the Helmholtz decomposition of the displacement vector. Further, many of the problems in the literature have been restricted to ones involving axisymmetric geometry. In this presentation, the time-dependent Navier equation is solved using a set of vector spherical harmonics which, previously, has been used primarily in quantum mechanics studies but which seems particularly useful in solving asymmetric problems with nonconservative body forces. Expressions for the displacements, strains, and stresses and a discussion of the vibrations of an elastic sphere are given.Part of the material presented here was developed while the author was on a Developmental Leave at the University of Texas at Austin.     

Low frequency vibrations of elastic bars

The method of expansion of three-dimensional displacements in a double power series of the transverse coordinates is employed to find one-dimensional equations applicable to low frequency vibrations of uniform, elastic, isotropic and anisotropic bars. The axial displacements accompanying torsion are chosen specially for each cross-sectional shape of bar—resulting in the correct, or nearly correct, torsional rigidity. Applications are to bars of elliptic, triangular and rectangular sections, illustrating various independent and coupled extensional, flexural and torsional modes of motion.     

Large amplitude free flexural vibrations of laminated composite skew plates

Here, the large amplitude free flexural vibration behaviors of thin laminated composite skew plates are investigated using finite element approach. The formulation includes the effects of shear deformation, in-plane and rotary inertia. The geometric non-linearity based on von Karman's assumptions is introduced. The non-linear governing equations obtained employing Lagrange's equations of motion are solved using the direct iteration technique. The variation of non-linear frequency ratios with amplitudes is brought out considering different parameters such as skew angle, number of layers, fiber orientation, boundary condition and aspect ratio. The influence of higher vibration modes on the non-linear dynamic behavior of laminated skew plates is also highlighted. The present study reveals the redistribution of vibrating mode shape at certain amplitude of vibration depending on geometric and lamination parameters of the plate. Also, the degree of hardening behavior increases with the skew angle and its rate of change depends on the level of amplitude of vibration.