V. I. Erofeyev, A. V. Serov, P. A. Smirnov

Torsional and bending waves play an important role in the formation of vibration fields in engineering structures. We propose mathematical model of torsional vibrations of rod by taking the geometric nonlinearity into account. We seek the solution in the terms of harmonic wave with amplitude and phase slow changing in time and space. We are passing the shortened equation of quasi-harmonic wave envelope using the method of averaging of "fast" variables. The envelope evolution is described by nonlinear Shrodinger equation in a coordinates moving with the group velocity. The modulation instability of quasi-harmonic waves is possible under certain conditions. Such possibility can be determined with the help of Lighthill criterion. Solutions for equations are found and analyzed for rod lying on a nonlinear elastic foundation. Dispersion dependences and self-modulation of bending waves are studied.