Abstract
In this article, the dynamical boundary value problem for a vibrating string attached by a point mass at one end is studied. The model we studied is : a one dimensional uniform string with constant density is fixed at one end, with the other end attached by a point mass, and this point mass is attached to the ground with a spring. In this system, there are kinetic energy composed resulted from vibrating and potential composed of gravity, stored energy of string, and potential of spring. Then we apply usual formulation in classical mechanics to form the corresponding equation of motion. We begin with investigating the existence and stability of equilibrium. When the explicit solution is not clear, we use integral curves to verify the existence of steady state. And stability analysis is done via the local minimum of energy, which is verified by the Sturm-Liouville problem corresponding second variation of energy function. Then we discuss the extended model with linear viscous term, apply Galerkin method to guarantee the existence of weak solution.. Finally, we discretize the system with respect to space parameter, and obtain an ODE system, then discuss the existence of steady states and stability. After calculations and discussions, we find that the two main functions influence the system: the stored energy of string and spring potential, when they are both convex, the steady state and the system are stable, however, the boundary conditions could destabilize the motion of the string with high stiffness. On the contrary, the instability due to the string can not be stabilized by the boundary potential.2. Equations of Motion 63. Steady States 94. Stability Analysis 135. Simple Viscous Case 166. Discretized Model 307. Final Remarks 39Appendix A. Some Examples for Steady States 40Appendix B. An Example for Stability Analysis 54Appendix C. A Basis for Galerkin Method 59Appendix D. Some Examples for Stability Analysis of Discretized Model 61