Dynamic analysis of an axially moving viscoelastic string by the Galerkin method using translating string eigenfunctions

Transverse motion of an axially moving viscoelastic string is analyzed via the Galerkin method, with the emphasis of the effect of Galerkin truncation bases on bifurcation and chaos. The transverse motions of the string are governed by a nonlinear partial-differential equation. Based on stationary or translating string eigenfunctions, the Galerkin method is applied to reduce the equation into a set of ordinary differential equations. By use of the Poincare maps, the dynamical behaviors are identified based on the numerical solutions calculated via the fourth-order variable-step Runge–Kutta–Merson method. The acceptability of two kinds of string eigenfunctions and the plausibility of the low order Galerkin truncation in computing longtime nonlinear dynamics are discussed. Numerical simulations revealed that periodic, quasi-periodic and chaotic motions occur in the transverse vibrations of the axially moving viscoelastic string.