A superposition method of reflected wave for moving string vibration with nonclassical boundary

An analytical wave solution for the vibrational response and energy of an axially translating string in any propagation cycle

“…In this paper, the vibrational response and energy of the traveling string system in any propagation cycle shown in Fig. 1 The governing equation of motion can be obtained [5,7,24] ( )…”

“…The earlier investigations have been reviewed by Hong and Pham [4]. Recent developments of the traveling wave solution (or d'Alembert's solution) of an axially traveling string, which is the focus of this paper, can be achieved by Gaiko and Van Horssen [5] and Chen et al [6][7][8][9].…”

“…And this investigation exerts a profound effect on our studies. Chen et al [6][7][8][9]17] extended the d'Alembert's solution to a finite traveling string with various boundary conditions. Due to multiple reflections, the reflected wave superposition method was proposed for a traveling string with two boundaries.…”

A wave solution for energy dissipation and exchange at nonclassical boundaries of a traveling string

“…They obtained the response to the initial conditions using the method of D'Alembert. Chen et al [9][10][11] investigated a reflected wave superposition method, which was proposed for a finite axially traveling string with classical and nonclassical boundaries at the two ends, and analyzed the total mechanical energy as well as its time rate of change. Huang et al [12] investigated the dynamic responses of high-dimensional nonlinear models of a fixed-fixed translating beam, and analyzed continuous energy exchange and the beating waveform phenomenon of high-dimensional models.…”