The chaotic response of the viscoelastic traveling string: An integral constitutive law

This paper investigates chaotic behaviors of an axially traveling viscoelastic string with geometric nonlinearity. The stress and the strain of the viscoelastic string obey the Boltzmann superposition principle. The Galerkin method is applied to truncate a nonlinear partial-differential-integral equation governing transverse motion into a set of ordinary differential-integral equations. For the string modeled as a standard linear solid, new auxiliary variables are introduced to transform those equations into ordinary differential equations. By use of the Poincare maps, the chaotic behaviors are presented based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented for varying one of the following parameter: the axial traveling speed, the amplitude of tension fluctuation, the viscoelastic exponent and coefficient of the string, while other parameters are fixed.