UNCORRECTED PROOF Journal of Sound and Vibration (2002) 000(0), 00–00 doi:10.1006/jsvi.5173, available online at http://www.idealibrary.com on LETTERS TO THE EDITOR ESTIMATING THE COMPLEX TRANSFER FUNCTION OF A NON-LINEAR SYSTEM M. J. Hinich Applied Research Laboratories, The University of Texas at Austin, Austin, TX 78713-8029, U.S.A. E-mail: hinich@mail.la.utexas.edu and E. M. A. M. Mendes y Department of Government, The University of Texas at Austin, Burdine Hall 536D, Austin, TX 78712- 1087, U.S.A. E-mail : emendes@mail:la:utexas:edu (Received 29 January 2002 1. INTRODUCTION In the literature, the identification of non-linear systems using functional series has been approached in many different ways. Since the earlier work by Wiener [1] great attention has been devoted to find algorithms for estimating the coefficients of the so-called Volterra models [2]. These models are a direct result of the work of Frechet [3] on the theory of functionals. Examples of recent development on the field can be found in references [4, 5] and references therein. A method is presented in this paper for estimating the linear and quadratic complex transfer function of a weakly non-linear system. The input to the system that is excited by a signal especially constructed for the purpose. The excitation signal is a sum of sinusoids with the same amplitude and pseudo-randomly jittered phases that are selected for the experiment and recorded. To motivate the non-linear frequency-domain linear plus quadratic model presented in section 2, consider the following simple example. Assume that a non-linear system has only one output and the system response satisfies the homogenous non-linearly perturbed differential equation: y 00 ðtÞþ ly 0 ðtÞþ kyðtÞþ dy 2 ðtÞ¼ 0; ð1Þ where l and d are small positive constants. This equation describes weakly non-linear oscillations of the type found in rotating machinery and in non-linear electric circuits [6, section 5.1]. An approximate solution for the linearized approximation of this equation 3B2 YJSVI : 5173 PROD.TYPE: COM ED: RAVI: PAGN: MVA SCAN: MANGALA pp.1--9 (col.fig.: NIL) y Permanent address: Departamento de Eletricidade, Funda - c * ao de Ensino Superior de S * ao Jo * ao Del Rei, Pra - ca Frei Orlando 170-Centro, S * ao Jo * ao Del Rei, MG-36307.904, Brazil. 0022-460X/02/$35.00 # 2002 Elsevier Science Ltd. All rights reserved.