Analysis of a nonlinear model of the vibrating rod
type: Item
instance of: scholarly article
title: Analysis of a nonlinear model of the vibrating rod
zbMATH Open document ID: 0884.35101
DOI: 10.1007/PL00001418
author: Angela Iannelli
author: Giovanni Prouse
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 1998-04-02T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 74H45
Mathematics Subject Classification ID: 74K10
zbMATH DE Number: 1052941
zbMATH Keywords: longitudinal springs
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: consistency conditions
zbMATH Keywords: generic external force
reviewed by: Silvio Nocilla
MaRDI profile type: Publication
full work available at URL: pl00001418
OpenAlex ID: W4247663992
description: scientific article; zbMATH DE number 1052941