A NEW CONVERGENCE CRITERIA OF VOLTERRA SERIES FOR HARMONIC INPUTS F. Thouverez Ecole Centrale de Lyon Mechanical Engineering Department UMR .5513 :36 avenue Guy de Collongue Lyon 69131 France Abstract The use of Volterra series is often limited by its con- vergence radius. We will study in this paper a pro- cedure to estimate a ma.jorant of this convergence radius. The criterion which will be developed here is associated to a sinusoidal input and to the first order frequency response of a non-linear system. In order to establish our criterion. the dynamical equa- tions will be re-written by means of the harmonic balance method to estimate the frequency response. The Volterra series obtained with this equation has a convergence radius which can be calculated with the complex variable theory. We will show that the estimation error due to the harmonic balance method does not. change the estimation of the con- vergence domain. To illustrate our results, we will apply our criterion to a Duffing oscillator and to a two degrees of freedom system with a non linear spring. 1 Introduction Actual structures often exhibit a non-linear be- havior. This non-linear behavior is due to com- plex junctions (bolting, solder, ... ), to the mechani- cal characteristics (composite materials, ... ) and ge- ometrical effects (large displacement, large strain). The analytical solution of such problem does not exist in most cases. On the other hand , it is gen- erally possible to express this solution a._c; a power expansion by means of the Volterra series theory. This kind of development , as the Taylor series, is only valid for a given convergence radius. Lot of developments have been already carried out to de- termine a majorant of the convergence radius [4]. 723 In some cases those criteria can be applied only for time bounded excitation [7) [2) and predict only the existence of a convergence domain. More specific criteria has been also developed [3] [10] which arc easy to use but cannot be applied to any kind of non linear problems. For particular systems as closed- loop systems, we ca.n estimate a convergence radius by means of the complex variables theory [ 1] [5]. This approach will be applied to the harmonic bal- ance equation to find a new kind of convergenn' criterion. We will show that this technic can be applied to multi dimensional systems. 2 Volterra series A large number of nonlinear dynamic systPms can be represented by means of a. Volterra series. In- deed, the solution of a nonlinear system defined by the following differential equation: .C(u) + pnl (u, it, ... )= F(t) (1) where u(t) F(t) .[ pnl output input linear differentia.! opera.tor analytical function of ·u, i1., ... can be expressed as a Volterra series [4]: •XJ u(t) L W;(t) i=l lV, (t) z IT ( ) ( 2) h;(r 1 , ... , r,) F t- T; • R• k=l where h; ( r 1 , ... , ri) is the Volterra kernel of order i.