On the Application of Liapunov’s Direct Method to Linear Lumped-Parameter Elastic Systems

Abstract

A specialized Liapunov functional is presented for stability analysis of linear lumped-parameter elastic systems involving completely general types of forces not explicitly dependent on time. This functional appears to have the simplest possible form capable of providing necessary and sufficient stability conditions for virtually all such systems. Complete analyses are carried out for systems with and without velocity-dependent forces. Simple approaches are given for investigating systems involving “arbitrarily small” and “arbitrarily large” velocity-dependent forces, as well as for determining whether and how the “size” and “structure” of the velocity-dependent forces affect stability. Applications are made to several examples.