E. Özkaya

Group — theoretic approach to axially accelerating beam problem

SummaryTransverse vibrations of a beam moving with time dependent axial velocity have been investigated. Analytical solutions of the problem are found using the systematic approach of Lie group theory. Group classification with respect to the arbitrary velocity function has been performed using a newly developed technique of equivalence transformations. From the symmetries of the partial differential equation, the way of deriving exact solutions for the case of arbitrary velocity is shown. Special cases of interest such as constant velocity, harmonically varying velocity and exponentially decaying velocity are investigated in detail. Finally, for a simply supported beam, approximate solutions are presented for the exponentially decaying and harmonically varying cases.

Three-to-one internal resonances in a general cubic non-linear continuous system

A general continuous system with an arbitrary cubic non-linearity is considered. The non-linearity is expressed in terms of an arbitrary cubic operator. Three-to-one internal resonance case is considered. A general approximate solution is presented for the system. Amplitude and phase modulation equations are derived. Steady state solutions and their stability are discussed in the general sense. The sufficiency condition for such resonances to occur is derived. Finally the algorithm is applied to a beam resting on a non-linear elastic foundation.