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Appendix C. Proof of Lemma 1 The lemma is summarized by the following observation. For clarity, let I ¼ 3. cn ¼ |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} c c ... c n sðjÞn j¼i 1 for i ¼ 1; 2; 3; n ¼ 1½c1 c2 c3Š 1 1 1; n ¼ 2½c1 c2 c3Š ½c1 c2 c3Š 3 2 1; ¼½c1 2 c1 c2 c1 c3 c2 2 c2 c3 c2 3Š n ¼ 3½c2 1 c1 c2 c1 c3 c2 2 c2 c3 c2 3Š ½c1 c2 c3Š 6 3 1; ¼½c3 1 c21 c2 c2 1 c3 c1 c2 2 c1 c2 c3 c1 c2 3 c3 2 c22 c3 c2 c2 3 c3 3Š n ¼ 4½c3 1 c21 c2 c2 1 c3 c1 c2 2 c1 c2 c3 c1 c2 3 c3 2 c22 c3 c2 c2 3 c3 3Š ½c1 c2 c3Š 10 4 1; ¼½c4 1 c3 1 c2 c3 1 c3 c2 1 c22 c21 c2 c3 c2 1 c3 2 c1 c3 2c1 c2 2 c3 c1 c2 c2 3 c1 c3 3 c4 2 c3 2 c3 c2 2 c23 c2 c3 3 c43 Š n ¼ 5½c4 1 c3 1 c2 c3 1 c3 c2 1 c22 c21 c2 c3 c2 1 c23 c1 c3 2 c1 c2 2 c3 c1 c2 c2 3 c1 c3 3 c4 2 c3 2 c3 c2 2 c23 c2 c3 3 c43 Š ½c1 c2 c3Š 15 5 1: ¼½c5 1 c41 c2 c4 1 c3 c3 1 c22 c3 1 c2 c3 c3 1 c23 c21 c3 2 c21 c22 c3 c2 1 c2 c2 3 c21 c3 3 c1 c4 2 c1 c3 2 c3 c1 c2 2 c23 c1 c2 c3 3 c1 c4 3 c5 2 c42 c3 c3 2 c23 c22 c3 3 c2 c4 3 c5 3Š sðiÞ n ¼ PI To complete the proof, the complete mathematical induction can be adopted. An outline for this proof is given here. Note that c n ¼ c n 1 c1; ...; c n 1 ½sð1Þn sðiÞn þ 1 : sð1ÞnŠ ci; ...; c n 1 ½sð1ÞnŠ cI includes all the non-repetitive terms of form c k1 1 ck2 2 c kI I with k1 þ k2 þ þkI ¼ n and 0pk1; k2; ...; kIpn. These terms can be separated into I parts, the ith part of which, i.e., cn 1½sð1Þn sðiÞn þ 1 : sð1ÞnŠ ci, includes all the non-repetitive terms of degree n which are obtained by the parameter ci timing the components of degree n– 1inc n 1 from sð1Þn sðiÞn þ 1tosð1Þn. Assume that the lemma holds for step n. Then for the step n+1, the ith part of the components in c n+1 must be cn½sð1Þnþ1 ðsðiÞn þ þsðIÞnÞþ1 : sð1Þnþ1Š ci which is cn½sð1Þnþ1 sðiÞnþ1 þ 1 : sð1Þnþ1Š ci. This completes the proof of Lemma 1. & References ARTICLE IN PRESS X.J. Jing et al. / Mechanical Systems and Signal Processing 22 (2008) 102–120 119 [1] J.H. Taylor, Describing Functions, An Article in the Electrical Engineering Encyclopedia, Wiley, New York, 1999. [2] D. Graham, D. McRuer, Analysis of Nonlinear Control Systems, Wiley, New York, London, 1961. [3] S. Engelberg, Limitations of the describing function for limit cycle prediction, IEEE Transactions on Automatic Control 47 (11) (2002) 1887–1890. [4] S.R. Sanders, On limit cycles and the describing function method in periodically switched circuits, IEEE Transactions on Circuits and Systems–I: fundamental theory and applications 40 (9) (1993) 564–572. [5] H. Elizalde, M. Imregun, An explicit frequency response function formulation for multi-degree-of-freedom non-linear systems, Mechanical Systems and Signal Processing 20 (2006) 1867–1882. [6] P.W.J.M. Nuij, O.H. Bosgra, M. Steinbuch, Higher-order sinusoidal input describing functions for the analysis of non-linear systems with harmonic responses, Mechanical Systems and Signal Processing 20 (2006) 1883–1904. [7] M. Solomou, C. Evans, D. Rees, N. Chiras, Frequency domain analysis of nonlinear systems driven by multiharmonic signals, in: Proceedings of the 19th IEEE Conference on Instrumentation and Measurement Technology, vol. 1, 2002, pp. 799–804. [8] J.C. Peyton Jones, Automatic computation of polyharmonic balance equations for non-linear differential systems, International Journal of Control 76 (4) (2003) 355–365. [9] V. Volterra, Theory of Functionals and of Integral and Integrodifferential Equations, Dover, New York, 1959. [10] W.J. Rugh, Nonlinear System Theory: the Volterra/Wiener Approach, Johns Hopkins University Press, Baltimore, MD, 1981. [11] I.W. Sandberg, Expansions for nonlinear systems, Bell System Technical Journal 61 (2) (1982) 159–199. [12] I.W. Sandberg, Volterra expansions for time-varying nonlinear systems, Bell System Technical Journal 61 (2) (1982) 201–225. [13] I.W. Sandberg, On Volterra expansions for time-varying nonlinear systems, IEEE Transactions on Circuits and Systems CAS-30 (1983).
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Page 1 and 2: Abstract Mechanical Systems and Sig
Page 3 and 4: 104 experimental data, which is giv
Page 5 and 6: 106 Alternatively, CE(Hn( )) can al
Page 7 and 8: 108 proposed method above as demons
Page 9 and 10: 110 except that Cp,q ¼ c is non-ze
Page 11 and 12: 112 For clarity in discussion, let
Page 13 and 14: 114 Magnitude Magnitude of the sens
Page 15 and 16: 116 ARTICLE IN PRESS X.J. Jing et a
Page 17: 118 fundamental results, techniques

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