DYNAMIC AND ENERGETIC ANALYSES OF A STRING/SLIDER NON-LINEAR COUPLING SYSTEM BY VARIABLE GRID FINITE DIFFERENCE

In this paper, a string/slider non-linear coupling system with time-dependent boundary condition is considered. One partial differential equation (PDE), describing the transverse small-amplitude vibration of the string, non-linearly coupled with one ordinary differential equation (ODE), describing the horizontal displacement of the slider, are derived by Hamiltonʹs principle. This is a moving boundary problem since the unknown position of the slider has to be determined as a part of the solutions. A transformation of the variable that converts the original non-stationary boundary conditions to a set of fixed boundary conditions is proposed to avoid the increased complication and loss of accuracy associated with unequal space intervals near the moving boundary. The finite difference method with variable grid is employed to show the numerical results of the coupling effect between the string and slider. Finally, some periodic motions of the moving boundary are assigned to show the divergence of the string vibrations.