Strongly nonlinear shear perturbations in discrete and continuous cubic nonlinear systems

Abstract

Systems with constraints, the masses in which move only along guides, can execute strongly nonlinear vibrations. This means that nonlinear phenomena manifest themselves at arbitrary small deviations from equilibrium. The form of vibrations of a single mass is described by elliptical Jacobi functions. The spectrum of these vibrations is found. With an increase in amplitude, the period of vibrations decreases. We deduce equations of strongly nonlinear vibrations of a chain of connected masses. In the continuum limit, we obtain a new nonlinear equation in partial derivatives. We devise transformation of variables leading to linearization of this equation. We implemented a factorization procedure that decreases the order of the equation in partial derivatives from second to first. Exact solutions to the first-order equation describe the slow evolution of the displacement profile in a distributed system. In the absence of preliminary tension of elastic elements in the continued model, traveling waves cannot be achieved; however, time-oscillating solutions like standing waves are possible. We obtain an equation for a field of strongly nonlinear deformations. Its exact solution describes periodic movement in time and space. As well, the period of time oscillations decreases with an increase in amplitude, and the spatial period, in contrast, increases. The product of the vibration frequency multiplied by the spatial period is a constant that depends on the deformation energy. We propose a scheme of the mechanical system producing strongly nonlinear torsional vibrations. We experimentally measured the period of torsional vibrations of a single disc. We show that with an increase in amplitude in the process of vibration attenuation, an increase in the period occurs, which agrees with calculations. We measure the shapes of nonlinear vibrations of a chain of connected discs. A strongly nonlinear behavior of the chain is observed.