Correction Factors of the Approximate Theories for Axisymmetric Modes of Longitudinal Waves in Circular Rods

Abstract

The longitudinal waves guided by rods are widely used in broad engineering fields. Approximate theories are required to improve the understanding of the longitudinal wave propagation in finite rods in particular. The correction factors are commonly used in the vibration analyses of beams and plates, but are seldom adopted to the longitudinal wave propagation in rods in a similar manner. In this paper, the longitudinal and radial displacements in axisymmetric problems of circular rods are expanded in infinite power series of the radial coordinate. By using Hamilton’s principle, an infinite one-dimensional system of equations of motion is established. The high-order components of stress and strain, and their relations are introduced to obtain the infinite one-dimensional system for the axisymmetric wave propagation in elastic rods. A proper truncation of the infinite equations leads to an approximate theory of a specific order. To improve the truncated equations, some high-order components of strain are multiplied by the correction factors. The correction factors for the first- to fourth-order approximations are systematically determined to ensure that the cutoff frequencies are the same as the exact values calculated by the Pochhmammer–Chree equation. The frequency spectra, via the well-known Pochhmammer–Chree equation and the approximate theories of order one to four, are presented for comparison in a region where the longitudinal wave is not attenuating and the wavelength in the axial direction is longer than the diameter of the rod. Compared to the approximate theories without correction factors, the approximate theories with correction factors show some advantages in accuracy when the branches are high.