Simulations of transverse vibrations of an axially moving string: a modified difference approach

A modified finite difference approach to simulate transverse vibrations of an axially moving string is presented. The stress is treated as a new unknown in discretization of the spatial variable. A set of differential-algebraic equations is established based on the discreted governing equation and the constitutive relation. For linear vibrations, a conserved functional is employed to test the algorithm, and the 1, 2, 3, 4-term truncated modal analytical solutions are compared with the numerical solution. For the free nonlinear vibration, a new conserved functional is used to check the algorithm. Effects of the transport speed on the free and forced nonlinear vibrations are numerically investigated.