114 N. N. Balaji et al. Fig. 15.2 Results from the seven-harmonic QP simulation, plotting the (a) first and (b ) second natural frequencies as functions of the modal amplitudes. Q1,Q2 domain (which is a planar region) through single lines. The results shown in Fig. 15.2 demonstrate that beyond a certain amplitude, the frequency backbone seems to change direction, indicating strong presence of modal interactions/couplings. Little can be said about such phenomena with the continuation scheme employed presently, since the constraint enforced above for the continuation may end up artificially constraining the system, rendering these features on the backbone plot to have limited physical meaning. One will have to resort to multi-parameter continuation techniques in order to fully explore such features. For the purposes of this study, the following analyses will be restricted to single harmonic simulations. Although this has limited physical significance, it will, at the minimum, enable a better understanding of modal interactions that are not related to internal resonances (which necessarily require a multi-harmonic ansatz). 15.3.1 Two-Component Single Harmonic Balance Results For the system at hand, it must be noted that one may employ the method of multiple scales (MMS, as in [8]) in order to obtain analytical expressions for the nonlinear natural frequency (and damping) characteristics. For the system at hand, the expressions for the natural frequencies are .ω (1) =ω (1) 0 + 3α 8 (Q 2 1 +2Q2 2), and. (15.6a ) ω(2) = ω (2) 0 + 3α 8 (2Q 2 1 + Q 2 2). (15.6b ) Note that these expressions are derived using MMS considering just a single harmonic, thereby enabling direct comparison to the numerical results generated by the quasi-periodic harmonic balance (QP-HB) approach developed above. . represents the book-keeping parameter used in the perturbation analysis. The above expressions correspond to the analysis carried up to an.O( ) accuracy. Figure 15.3 shows a comparison of the resonant frequency backbone surfaces (since these are functions of both the modal amplitudes) computed using the numerical approach (QP-HB) and the analytical approach (MMS). A close match can be observed, especially for small amplitudes. Another aspect can be highlighted by constructing vertical contour sets on the above modal surfaces. These are contour levels drawn for different values of . Q1 an d. Q2 , as shown in Fig. 15.4. These highlight the fact that even when the mode 1 amplitude is fixed (and nonzero), the mode 1 frequency can vary purely due to the changes in the mode 2 amplitude (vice versa for the mode 2 frequency). This fact is also reflected by the analytical dependence in Eq. 15.6. This seems to suggest that in a practical setting, the employment of data from hammer impact tests (which provides a broad spectrum and therefore
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