Nonlinear Oscillations of a Chain of Masses in a Liquid

Nonlinear regular and random vibrations of a metamaterial made as a chain of massive elements immersed in a viscous liquid are studied. The nearest neighbors are connected by nonlinear elastic forces. Stokes friction is taken into account. A system of equations for mass oscillations is obtained. In the continuum long-wave approximation, the system is reduced to a simplified partial differential equation with nonlinear and dissipative terms. We find the exact solutions and show that the competition of nonlinearity and absorption can lead to the formation of a steady wave profile shape. For noise waves, the correlation function and intensity spectrum are given. The process of sawtooth wave excitation by distributed external sources is analyzed. The limiting “peak” amplitude is calculated. Waves in tubes of variable cross section filled with metamaterial are considered. Based on the solution of the equation containing coordinate-dependent tube section, an expression is found for the amplitude of the sawtooth wave.