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

The finite length translating tensioned string model with mixed boundary conditions is considered in order to study the exchange of vibrational energy during reflection process in this paper. The boundary conditions are respectively at one end a spring-dashpot and the other a fixed boundary, which is one kind of mixed boundary conditions. An analytical solution and energy expressions for the propagating wave are presented, which uses a reflected wave superposition method. Firstly, the cycle of boundary reflection for strings is provided. To simplify the process for obtaining the response, each cycle is divided into three time intervals. Applying D’Alembert’s principle and the reflection properties, expressions for the reflected waves under these mixed boundary conditions are derived. Then, the vibrational response of the axially translating string is solved for three time intervals. 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 these spring-dashpot and fixed boundary conditions are obtained by using the analytical solution and the properties of vibrational energy exchange as a function of the translational velocity, the boundaries and level of damping are discussed