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

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 .
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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