High-order accurate equations describing vibrations of thin bars

A method for deriving one-dimensional wave propagation equations in thin inhomogeneous anisotropic bars based on the mathematical homogenization theory for periodic media is used to obtain equations governing the longitudinal and transverse vibrations of a homogeneous circular bar. The equations are derived up to $O(\varepsilon^8)$ terms and take into account variable body forces and surface loads. Here, $\varepsilon$ is the ratio of the bar's typical thickness to the typical wavelength.