The perturbation method in problems of the dynamics of inhomogeneous elastic rods

Abstract

The regular perturbation method (the small-parameter method) is developed in order to investigate the dynamics of weakly inhomogeneous rods with arbitrary distributed loads and boundary conditions of various types leading to self-conjugate boundary-value problems. The approach rests on the introduction of a perturbed argument, namely, the Euler variable, and a suitable representation of the eigenfunctions. It enables one to carry out uniform constructions of the basis and the eigenvalues, as well as the frequencies with any required accuracy in terms of the small parameter using quadratures of known functions. To illustrate the effectiveness, an example involving inhomogeneous rods with hinged left-hand ends and free right-hand ends and with box-shaped and circular cross-sections whose dimensions depend linearly on the coordinate are investigated and computed.