Nonlinear effects related to vibrations of long elastic waveguides: formulation of nonlinear equations

A number of effects arising during vibrations of long elastic waveguides can not be explained in the context of the linear theory. One of these effects is redistribution of energy between the longitudinal and transverse vibration modes. Nonlinear equations describing such effects have been studied in momentless approximation with application to vibrations of strings and they account for tension of the string axis as a result of transverse vibrations. The authors have studied a possibility of generalization of these equations for the case of elastic waveguides taking into account internal moments and lateral forces arising during bending of the waveguide