7 Nonlinear System Identification for Joints Including Modal Interactions 89 7.3.2.5 Time Integration The system can now be expressed as a set of ODEs. These can easily be numerically integrated using ode45inMATLAB. The initial conditions were taken as the displacement and velocity after the fifth peak of each hammer test. 7.3.2.6 Numerical Computation of Instantaneous Natural Frequency and Damping It is desirable to be able to compare the modal maps derived from the Hilbert transform, to those from the model identified by RFS. It is of course possible to simply apply the same Hilbert transform-based analysis from Sect. 7.3.1.1 to the time response of the RFS model; however, this process is slow and sometimes requires manual tuning of the fitting parameters to obtain clean results. It would be preferable to be able to compute model map directly from the RFS model. The approach used here is based on the Harmonic Balance method and Chapter 4 of [12] which discusses how describing functions can be used to analyse transient signals. It is first assumed that the equations of motion have an underlying linear system, with a relatively weak non-linear term: m1Rq1 Cc1Pq1 Ck1q1 Cf1.q1; q2; Pq1; Pq2/ D0 (7.26) m2Rq2 Cc2Pq2 Ck2q2 Cf2.q1; q2; Pq1; Pq2/ D0 (7.27) where all of the non-linear terms are collected into the functions f1./ and f2./. To apply this method to RFS, it was therefore necessary to separate the linear coefficients from those associated with the non-linear terms. It is then assumed that the response is harmonic with exponential decay and has components at both !d;1 and !d;2. This can be expressed in a multidimensional time domain (or hypertime) as follows [13]: q1.t1; t2/ DA1e s1t1 CB 1e s2t2 (7.28) q2.t1; t2/ DB2e s1t1 CA 2e s2t2 (7.29) where sj D j Ci!d;j. Note that the amplitudes of the dominant components Aj are to be specified, whilst the crosscomponents Bj are unknown and must be solved for; these may turn out to be complex, which indicates a phase offset. If it is assumed that the damping and natural frequency vary slowly in time, known as the quasi-static assumption [12], then the following differentiation rule applies: @ @tj esjtj s je sjtj (7.30) Once these expressions are substituted into the equations of motion Eqs. 7.26 and 7.27, the coefficients of esjtj can be equated to yield the following four equations: .m1s 2 1 Cc1s1 Ck1/A1 CF1 D0 (7.31) .m1s 2 2 Cc1s2 Ck1/B1 CG1 D0 (7.32) .m2s 2 1 Cc2s1 Ck2/B2 CG2 D0 (7.33) .m2s 2 2 Cc2s2 Ck2/A2 CF2 D0 (7.34) where Fj andGj are the amplitudes from the non-linear terms, which are unknown. These equations are complex, so both the real and imaginary parts must sum to zero. Again invoking the quasi-static assumption, it can be assumed that the response is approximately locally periodic with frequencies !d;j. This allows us to evaluate the effect of the non-linearity over one cycle in the time domain, which is equivalent to computing the describing function of the non-linear terms. If there is some energy dissipation, then Fj; Gj will turn out to be complex. This process is shown in Eq. 7.35. Aj; Bj IFFT !fj.q1; q2; Pq1; Pq2/ FFT !Fj; Gj: (7.35)
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