A discrete conservative model for the linear vibrating string and rod

AbstractIn this paper, we shall suggest and study a conservative discrete model for the linear vibrating string and rod fixed at the end points. We shall prove that the difference systems involved in our models may be seen as second-order unconditionally stable finite difference schemes of the classical equations of the linear vibrating string and vibrating rod. If the forces acting on the string (or rod) are conservative the total energy of the discrete solutions of our models is conserved and we can prove that we have stability for every choice of the time step Δt. We have considered both hinged and clamped rod; the constrains are naturally included into the model and the conservation of energy is still proved by giving a suitable definition of potential energy. Some numercial examples are presented

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

AbstractThe 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

A regularization method for discrete Fourier polynomials.

AbstractIn this paper we present a regularization method for discrete Fourier polynomials in one and two variables. The easy relation between the discrete Fourier coefficients and those of the regularized polynomial is proved. A good automatic choice of the regularization parameter σ is proposed and some numerical results are presented

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

AbstractSome 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