Until now, the problems of longitu\-di\-nal - transverse vibrations of objects with moving boundaries were solved mainly in a linear setting, the energy exchange through the moving boun\-dary and the interaction between longitudinal and transverse vibrations were not taken into account. The paper presents a new nonlinear mathematical model of transverse vi\-bra\-tions of an unbounded rope with a moving boundary, in which geometric nonlinearity is taken into account. The obtained mathematical model allows one to describe high-intensity vi\-bra\-tions of a rope with a moving boundary.