Abstract Summary
In this paper, we studied a vibration of a string. The string is attached to a fixed point on one end while on the other hand, the position of the contact point can vary along a smooth obstacle. By applying a boundary fixing transformation, we transform the problem from a linear problem with moving boundary, to a nonlinear problem with fixed boundary. It is assumed that the vibrations around the stationary position of the string is small. Explicit approximations of the solution are obtained by using a multiple time-scales perturbation method. Depending on the parameters in the problem, it turns out that three different cases for the obstacle boundary condition have to be considered, that is, Dirichlet, or Neumann, or Robin type of boundary conditions. To avoid infinite-dimensional system of ordinary differential equations that occurs in the modal interactions of the string vibrations, characteristic coordinates are used together with a multiple time-scales approach to analyze the string dynamics in terms of traveling waves in opposite directions. A comparison of a direct numerical integration of the PDE problem and and the results obtained by using afromentioned perturbation approach shows an excellent agreement in the results.
