Passive Control of Differential Algebraic Inclusions - General Method and a Simple Example

Abstract

In this chapter, we consider a master system consisting of a nonlinear differential inclusion and an algebraic equation of constraint (resulting in a Differential Algebraic Inclusion (DAI) system). This system is coupled to a nonlinear energy sink (NES) corresponding to a one degree-of-freedom essentially nonlinear differential equation. We examine how a resonance capture can lead to a reduced order dynamical system. To obtain this reduced order model, we describe a multiple time scale analysis governed by the introduction of multi-timescales via a small parameter \(\varepsilon \) that is finite and strictly positive. The mass of the NES is small versus the mass of the master system, and it governs a mass ratio defining the small parameter \(\varepsilon \). The first timescale is the fast scale. Introducing the Manevitch complexification leads to the definition of slow time envelope coordinates. These envelope coordinates either do not directly depend on the fast time scale or do not depend on this fast time scale via introduction of the so-called Slow Invariant Manifold (SIM). The slow time dynamics of the master system components is analyzed through introduction of equilibrium points, corresponding to periodic solutions, or singular points (governing bifurcations around the SIM), corresponding to quasi-periodic behaviors. We present a simple example of semi-implicit Differential Algebraic Equation (DAE), including a friction term coupled to a cubic NES. Analytical developments of a 1:1:1 resonance case permit us to predict passive control of a DAI by a NES.