Vibrations of non-homogeneous media and the prediction of mechanical resonance in human tibia

An analysis of longitudinal and transverse vibration in thin rods and Euler-Bernoulli beams is presented in which consitutive and geometric properties are allowed to vary with spatial coordinates. The specific form of cross-sectional areas and material properties that are adopted are used to model the mechanical resonances of the human tibia. Three specific models are presented. The transeverse vibrations in a beam with linear cross-sectional taper are addressed. This model is considered with both free-free and torsion spring-mass boundary conditions. A second model for transverse vibration in a beam with three piecewise linearly tapered segments is also analyzed. This model is considered with free-free boundary conditions. A model for longitudinal vibration in a beam with general cross-sectional areas and Young's modulus with free-free boundary conditions is also presented. For each of these beams, parameters are chosen to model data from the human tibia. The low-order frequencies of vibrations are computed and a comparison of other results in the literature is made.