The vibrating string and rod free to slide, at both end points, on a vertical guide

Some algebraic properties of the matrices involved in a discrete linear model simulating the transversal motion of a linear vibrating string, or rod, with both end points free to slide on vertical guides, are studied. The discrete linear model, which is conservative, turns out to be, according to the physics of the problem, generally unstable. Nevertheless, it is proved that the linear models are suitable to simulate the motion of the linear string and rod. An extension of the model to the nonlinear string is also considered and some numerical examples are given. Also, in this case, the numerical results seem to be in accordance with the physics of the problem.