20 Obtaining Nonlinear Frequency Responses from Broadband Testing 223 QD Q0 Qc1 Qs1 : : : QcN QsN 2 R.2NC1/l X D X0 Xc1 Xs1 : : : XcN XsN 2 R.2NC1/n ED E0 Ec1 Es1 : : : EcN EsN 2 R.2NC1/.sCm/ (20.15) Using (20.15), the variables are rewritten in compact form as follow x.t/ D.T. / ˝In/X q.t/ D.T. / ˝Il/Q e.t/ D.T. / ˝IsCm/E (20.16) where T. / is a vector gathering the trigonometric functions as T. / D 1 p2 cos.k1 / sin.k1 / : : : cos.kN / sin.kN / 2R.2NC1/ (20.17) The time derivative of x.t/ can be written using a linear operator as dx dt D! d d D! dT. / d ˝ In X D!Œ.T. /r/ ˝In X (20.18) with rD 2 6 6 6 4 0 r1 : : : rN 3 7 7 7 5 withrj D 0 kj kj 0 (20.19) Substituting Eqs. (20.16), (20.18) into (20.3) and applying Galerkin procedure gives !.r˝In/X D.I.2NC1/ ˝Ac/XC.I.2NC1/ ˝Bc/E QD.I.2NC1/ ˝C/XC.I.2NC1/ ˝D/E (20.20) Rearranging, the following residue equation is obtained h.Q;!/ Q G.!/E.Q/ D0 (20.21) with G.!/ D.I.2NC1/ ˝C/ƒ 1.I .2NC1/ ˝Bc/ C.I.2NC1/ ˝D/ ƒD!.r˝In/ .I.2NC1/ ˝Ac/ (20.22) The Fourier coefficients of the nonlinear terms are computed using alternating-time-frequency method (AFT) [4], that takes advantage of the fast Fourier transform to compute E Q FFT 1 !q.t/ !e.p.t/; q.t/; Pq.t// FFT !E (20.23) 20.3.2 Continuation of Periodic Solutions In order to track a branch of periodic solutions, a predictor-corrector method based on pseudo-arclength parametrization is used. Denoting JQ and J! the Jacobian matrices with respect to Qand !, respectively, the tangent vector t.i/ at a point .Q.i 1/;!.i 1// along the branch reads
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