Conservative linear models for the vibrating string and rod free to slide, at one end point, on a vertical guide

The motion of a linear vibrating string, or rod, fixed in one point and free to slide on a vertical guide in the other one, is studied. Only transversal vibrations are considered. The motion is simulated by a discrete conservative model. In order to prove that the total energy is conserved, a matrix form of the potential and kinetic energy is used. The same technique may be used to simplify the proof of energy conservation theorem given by the authors in a previous paper. The numerical method related to the model turns out to be unconditionally stable. Some examples of the motion simulated by the model are given.