Analysis of a nonlinear model of the vibrating rod (Q1364601)

The authors introduce a nonlinear model of the vibrating rod by substitution of the rod, of mass \(M\) and length \(\ell\), by a system of \(n\) longitudinal springs linked by \(n+1\) hinges of mass \(M/n+1\). The model is studied as a Hamiltonian system, depending on the parameter \(h=\ell/n\). The motion equations are obtained for the transversal deplacement \(\eta_i(t)\) of the hinges. These equations describe the motion provided some consistency conditions are satisfied, i.e. that the stress (tension, bending moment, shear stress) is not such as to destroy the system. Conditions of this kind my be taken into account by transforming the equations into inequalities. The proof is given of global existence and uniqueness theorem, for all \(h>0\), for the solution satisfying Cauchy initial data. The proof is extended to the case \(h\to 0\), and to the case of a generic external force.