On the normal modes of free vibration of inhomogeneous and anisotropic elastic objects

Abstract

The Hamilton's principle approach to the calculation of vibrational modes of elastic objects with free boundaries is exploited to compute the resonance frequencies of a variety of anisotropic elastic objects, including spheres, hemispheres, spheroids, ellipsoids, cylinders, eggs, shells, bells, sandwiches, parallelepipeds, cones, pyramids, prisms, tetrahedra, octahedra, and potatoes. The paramount feature of this calculation, which distinguishes it from previous ones, is the choice of products of powers of the Cartesian coordinates as a basis for expansion of the displacement in a truncated complete set, enabling one to analytically evaluate the required matrix elements for these systems. Because these basis functions are products of powers of x, y, and z, this scheme is called the xyz algorithm. The xyz algorithm allows a general anisotropic elastic tensor with any position dependence and any shape with arbitrary density variation. A number of plots of resonance spectra of families of elastic objects are displayed as functions of relevant parameters, and, to illustrate the versatility of the method, the measured resonant frequencies of a precision machined but irregularly shaped sample of aluminum (called a potato) are compared with its computed normal modes. Applications to materials science and to seismology are mentioned.

Lattice thermal conductance in nanowires at low temperatures: Breakdown and recovery of quantization

“…The vibrational frequencies in such wires can be studied with a resonant ultrasound spectroscopy method by Visscher et al, for example. 16 Finally, we remark on the fact that the wires composed of softer materials are more appropriate to observe the quantization of thermal conductance. According to the discussion given above, the temperature range where the quantization of thermal conductance ͑or plateau͒ is expected to appear is ⌬T = T c − T b .…”

Finite-temperature elasticity of fcc Al: Atomistic simulations and ultrasonic measurements

“…C 11 and C 44 considering the material as an isotropic solid ---accurate elastic moduli of the solid can be determined from collected RUS spectra using a multidimensional, iterative fitting approach (RuSpec (Magnaflux Quasar, Albuquerque, NM) that minimizes the root-mean-square (RMS) error between the measured and calculated resonant frequencies for a sample of known mass and dimensions. The calculated resonant frequencies were determined by minimization of the Lagrangian equation corresponding to the free body vibrations of an elastic solid, using RayleighRitz method 50,53 . Figure 9 shows the measured Young's and shear moduli and Poisson's ratio of the polycrystalline aluminum sample as determined by RUS over the 297-773 K temperature range.…”

Simple model for the vibrations of embedded elastically cubic nanocrystals

“…When the elastic constants are not isotropic, a numerical approach such as the one known as Resonant UltraSound (RUS) 9 can find mode frequencies and displacement fields of free nanoparticles. [10][11][12] What has been missing up until now is a description of the vibrations of an elastically anisotropic sphere embedded in an isotropic matrix.…”