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

Abstract

The finite length model of a traveling string can be used to study the lateral vibrations in many engineering devices. The vibrational energy exchange mechanism and its characteristics are very complex, due to the axial movement and the different boundary conditions. A finite length translating tensioned string model with mixed boundary conditions is considered here in order to study the exchange of vibrational energyduring the reflection process. The boundary conditions are respectively at one end a spring-dashpot and the other a fixed boundary, together forming one kind of mixed boundary conditions. An analytical solution and energy expressions for the propagating wave are presented using a reflected wave superposition method. Firstly, a complete cycle of boundary reflections in the string 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 with the vibrational response 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 comparison shows that the reflected wave superposition method solution is achievable for higher translational speeds, even the critical speed, which is not attainable from most numerical methods. The subsequent energy analytical expressions for a traveling string with these mixed boundary conditions are obtained in terms of the superposition of the traveling waves and their reflections. The properties of vibration energy exchange as a function of the translational velocity, the type of boundary and level of damping are discussed. Numerical simulation results proved that the viscous damper results in energy dissipation at the boundary, and the choice of the magnitude and direction of the translational string velocity can affect the energy of the traveling wave.