Wave Response of the Axially Moving String with Complex Boundary Conditions Based on Characteristic-Line Method and Duhamel’s Integral

This paper extends the application of the characteristic-line method to non-periodic response computations for axially moving strings across arbitrary time spans. It clarifies the reasons for Galerkin’s calculation errors and energy non-conservation in Eulerian descriptions. The dynamic equations are transformed into endpoint reflection ordinary differential equations within the characteristic line domain at first. A rapid recursion semi-analytical method, leveraging Duhamel’s integral, is introduced to tackle complex boundary reflection equations, overcoming limitations in previous studies. This approach reveals the nonconservation of energy in axially moving strings from the perspective of elastic wave propagation and proposes an optimal boundary damping value for vibration attenuation. Numerical simulations using Galerkin and finite difference methods confirm the method’s effectiveness in computing non-periodic responses. The increasing error in Galerkin method with higher axial velocities is attributed to the emergence of local waveforms, as opposed to Galerkin’s use of global trial functions. Energy non-conservation, the energy decreases during elongation and increases during shortening, is explained by changes in waveform length during boundary reflection. Optimal damping effectively dissipates vibrational energy. This study’s rapid recursion method, combining characteristic-line theory and Duhamel’s integral, significantly advances our understanding of dynamic phenomena in axially moving string systems.