Analytical approximate 3D solution for the longitudinally vibrating cylinder

Summary The paper presents a three-dimensional approximate solution of a longitudinally vibrating cylinder. It is based on the frequency equation for free waves travelling along an infinitely long cylinder which follows from the radial boundary condition on the lateral surface of the cylinder. An equivalent longitudinal traction is calculated by integration of the normal stresses over the end cross section. The full solution of the longitudinally vibrating rod adapted to given axial boundary conditions is derived. The solution is compared to numerical results, and good agreement is obtained.