An Asymptotic Solution and the Green's Function for the Transverse Vibration of Beams with Variable Properties

An analytical solution procedure for the dynamic response of beams with variable properties is developed by using an asymptotic solution and the Green's function. This asymptotic closed form solution is derived for the transverse vibration of beams under the assumption of slowly varying properties, such as mass, cross-section, tension etc., along the beam length. However, this solution is still found to be very accurate even in the case of large variation, such as step change in cross-section, mass, and tension. Therefore, this derived asymptotic closed form solution and the Green's function can be easily applied to find dynamic responses for various kind of beam vibration problems.