On the reflected wave superposition method for a travelling string with mixed boundary supports

Abstract

An analytical vibration response in the time domain for an axially translating and laterally vibrating string with mixed boundary conditions is considered in this paper. The domain of the string is a constant, dependent upon the general initial conditions. The translating tensioned strings possess different types of mixed boundary conditions, such as fixed dashpot, fixed spring-dashpot, fixed mass-spring-dashpot. An analytical solution using a reflected wave superposition method is presented for a finite translating string. Firstly, the cycle of boundary reflection for strings is provided, which is dependent upon the string length. Each cycle is divided into three time intervals according to the travelling speed and direction of the string. Applying D'Alembert's principle and the reflection properties, expressions for the reflected waves under three different non-classical boundary conditions are derived. Then, the vibrational response of the axially translating string is solved for three time intervals by using a reflected wave superposition method. The accuracy and efficiency of the proposed method are confirmed numerically by comparison to simulations produced using a Newmark-β method solution. The energy expressions for a travelling string with a fixed dashpot boundary condition is obtained and the time domain curves for the total energy and the change of energy at the boundaries are given.

Analysis of longitudinal oscillations in a vertically moving cable subject to nonclassical boundary conditions

“…Thus, in order to obtain an order ε accurate approximation of the solution of ū (ξ , s ) , it is necessary and sufficient to construct an order ε accurate approximation of the solution of W (ξ , s ) . From (5) and Appendix B , it follows that W (ξ , s ) has to satisfy: 3) is transformed into a simplified problem (6) . In the following sections, accurate, analytical approximations of the solution W (ξ , s ) of problem (6) are constructed, and by using (4) and ( 5) , accurate approximations of ˜ u of problem (3) can be obtained.…”

“…From (5) and Appendix B , it follows that W (ξ , s ) has to satisfy: 3) is transformed into a simplified problem (6) . In the following sections, accurate, analytical approximations of the solution W (ξ , s ) of problem (6) are constructed, and by using (4) and ( 5) , accurate approximations of ˜ u of problem (3) can be obtained.…”

“…Gaiko and van Horssen in [4] considered transverse vibrations of a traveling cable subject to a moving Dirichlet boundary condition with boundary damping, and in [5] the authors further discussed resonances and vibrations in an elevator cable system due to boundary sway. Chen et al in [6] analyzed vibration responses for an axially translating cable of fixed length for classical mixed boundary conditions, and in [7] these authors studied for a traveling cable the energy dissipation and the energy exchange for fixed boundaries. Recently, researchers started to study longitudinal and transverse vibrations of moving cables or beams with moving nonclassical boundary conditions.…”

Reflection of waves in a waveguide from a boundary with nonlinear stiffness: application to axial and flexural vibrations

“…Some studies concerned initial value problems and free wave propagation and developed solutions based on the d'Alembert's method [10][11][12][13][14]. This paper concerns steady-state, time-harmonic axial and flexural waves in a rod or a beam with a nonlinear boundary stiffness using a wave propagation approach.…”

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.…”