Output frequency response function-based analysis for nonlinear ...

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of system (22) at o ¼ 8.1 rad/s corresponding to different values of c2 can be obtained through FFT of the time domain output response. Then using Eq. (23), it can be derived from Eq. (11) that FðjoÞ T ¼ j1ðjoÞ; kj2ðjoÞ; ...; k ‘ j ‘ðjoÞ T ¼ðc T cÞ 1 c T YYðjoÞ. Therefore, the OFRF of system (22) with respect to nonlinear parameter c2 in the case of c1 ¼ c3 ¼ c4 ¼ 0 is obtained as Yðjo; c2Þ ¼ð2:060893505718041e þ 002 2:402014548824790e þ 002iÞ þ k 1 ð 5:14248529981906 þ 5:35676372314361iÞc2 þ k 2 ð0:08589533966805 0:08827649204263iÞc 2 2 þ k 3 ð 8:068953639113292e 004 þ 8:248154776018186e 004iÞc 3 2 þ k 4 ð4:598423724418538e 006 4:686570228695798e 006iÞc 4 2 þ k 5 ð 1:679591261850433e 008 þ 1:708497491564935e 008iÞc 5 2 þ k 6 ð4:056287337706451e 011 4:120496550333245e 011iÞc 6 2 þ k 7 ð 6:544911009113156e 014 þ 6:641760366680977e 014iÞc 7 2 þ k 8 ð6:976300614229155e 017 7:073928662624432e 017iÞc 8 2 þ k 9 ð 4:713366512185836e 020 þ 4:776287453573993e 020iÞc 9 2 þ k 10 ð1:827866445826756e 023 1:851299290299388e 023iÞc 10 2 þ k 11 ð 3:098310700824303e 027 þ 3:136656793561425e 027iÞc 11 2 . Based on this function, Eq. (16) can be further computed as Yðjo; cÞ 2 ¼ p0 þ cp1 þ c 2 p2 þ þc 2‘ p2‘ þ ¼ð1:001695593467675e þ 005Þþk 1 ð 4:693027791051078e þ 003Þc2 þ k 2 ð1:329525858242289e þ 002Þc 2 2 þ k 3 ð 2:55801250200731Þc 3 2 þ k 4 0:03645314106899c 4 2 þ k 5 ð 3:968756773045435e 004Þc 5 2 þ k 6 0:01517275811829c 6 2 þ . ARTICLE IN PRESS X.J. Jing et al. / Mechanical Systems and Signal Processing 22 (2008) 102–120 113 Note that this is an alternating series and it holds that p i 4 p iþ1 and p i ! 0. Hence the series may keep decreasing when c is going larger and within its radius of convergence. By following the similar method demonstrated above, the OFRFs of system (22) with respect to nonlinear parameters c 1, c 2, c 3 and c 4 of different cases can all be obtained, for instance Y(jo;c1), Y(jo;c3), and Y(jo;c4) (The other nonlinear parameters are zero if not appearing in the function). The results are shown in Figs. 2–4. Fig. 2 shows that the variation of the magnitude of the OFRFs with respect to each nonlinear parameter. It can be seen that the larger the nonlinear terms, the larger the effect they have on the system output frequency response; there is a good matching between the theoretical computation results and the simulation results to which they have been fitted, and there is also a good match between the theoretical computation results and the simulation tests (for parameter c 3) which are independent of the fitted simulation results. From both Figs. 2 and 3 it can also be seen that the system output frequency response is much more sensitive to the variation of the nonlinear parameters when they are small. Once the value of a nonlinear parameter is sufficiently large, then the sensitivity will tend to be zero. From comparisons between these four nonlinear terms, it can be concluded that the system output frequency response is more sensitive to the variation of the nonlinear parameter c4 when the values are small; however, when the values of each nonlinear parameters are sufficiently large, the system output spectrum is more sensitive to the nonlinear parameter c2. From Fig. 4 it can be seen that the convergence of the OFRFs are all very fast. This can also be understood that the energy disperses quickly with the nonlinear order going larger. It is noted that the ratio functions of c 2 and c 3 go up much faster than that of c 1, especially c 2. This implies that the radius of
114 Magnitude Magnitude of the sensitivity functions (10e-5) 350 300 250 200 150 100 50 ARTICLE IN PRESS X.J. Jing et al. / Mechanical Systems and Signal Processing 22 (2008) 102–120 Output frequency response fuctions c1 Simulation data c4 Simulation data c2 Simulation data c3 Simulation data Simulation tests 0 1 2 3 4 5 6 7 8 9 x 107 0 Nonlinear parameters c1,c2,c3 and c4 Fig. 2. Output frequency response functions with respect to c 1 to c 4, respectively. 10 5 0 -5 -10 -15 Sensitivity of OFRF to nonlinear parameters 1 2 3 4 5 6 7 8 9 10 11 x 106 Nonlinear parameters c1, c2 and c3 Fig. 3. Sensitivity function of the OFRFs with respect to c1 to c3, respectively. convergence of the output spectrum corresponding to c 2 should be larger. Simulation tests verify that the system is still stable when c 2 ¼ 10 17 where the magnitude of the output spectrum is 0.0216, while the system tends to be unstable when c 1 tends to be larger than 10 8 . c1 c2 c3
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: 112 For clarity in discussion, let
Page 15 and 16: 116 ARTICLE IN PRESS X.J. Jing et a
Page 17 and 18: 118 fundamental results, techniques
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Output frequency response function-based analysis for nonlinear ...

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