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A computational technique is proposed for the Galerkin discretization of axially moving strings with geometric nonlinearity. The Galerkin discretization is based on the eigenfunctions of stationary strings. The discretized equations are simplified by regrouping nonlinear terms to reduce the computation work. The scheme can be easily implemented in the practical programming. Numerical results show the effectiveness of the technique. The results also highlight the feature of Galerkin's discretization of gyroscopic continua that the term number in Galerkin's discretization should be even. The technique is generalized from elastic strings to viscoelastic strings.

A numerical method is proposed to simulate the transverse vibrations of a viscoelastic moving string constituted by an integral law. In the numerical computation, the Galerkin method based on the Hermite functions is applied to discretize the state variables, and the Runge-Kutta method is applied to solve the resulting differential-integral equation system. A linear iterative process is designed to compute the integral terms at each time step, which makes the numerical method more efficient and accurate. As examples, nonlinear parametric vibrations of an axially moving viscoelastic string are analyzed.

A numerical algorithm is proposed for computing nonlinear vibration of axially accelerating viscoelastic strings. Based on independent functions, the variational principle is used to discretize the governing equation into a set of differential/algebraic equations. Numerical examples are presented.

A modified finite difference method is presented to simulate transverse vibrations of an axially moving string. By discretizing the governing equation and the stress-strain relation at different frictional knots, two linear sparse finite difference equations are obtained, which can be computed alternatively. The numerical method makes the non-linear model easier to deal with and of small truncation errors. It also shows stable for small initial values, so it can be used in analyzing the non-linear vibration of viscoelastic moving string efficiently. A practical way of testing the precise of the numerical results is given by using a conservative quantity. Numerical examples are presented and dynamical analysis is given by using the numerical results.