12 The Importance of Phase-Locking in Nonlinear Modal Interactions 127 12.3.1 The Backbone Curves of the Asymmetric Case From Table 12.1 it can be seen that, for the asymmetric case, all nonlinear parameters, ˛i, are non-zero. Additionally, the ratio between the linear natural frequencies, !n1W!n2, is approximately 1:3 . Therefore, it seems likely that a similar ratio will exist between the response frequencies, and hence the case where r D3is considered. From Eq. (12.19) this leads to 4 !2 n1 ! 2 r1 U1 C3˛1U3 1 C2˛3U1U2 2 C3˛2U2 1U2eCj .3 1 2/ D0; (12.20a) 4 !2 n2 9! 2 r1 U2 C2˛3U2 1U2 C3˛5U3 2 C˛2U3 1e j .3 1 2/ D0: (12.20b) As we are concerned with the phase difference between the modes, we consider the case where both modal amplitudes are non-zero. Therefore, the imaginary components of Eq. (12.20) both lead to sin.3 1 2/ D0, which may be satisfied by 3 1 2 D0; ; : : :, thus enforcing phase-locking between the modes. The real components of Eq. (12.20) may then be written 4 !2 n1 ! 2 r1 C3˛1U2 1 C2˛3U2 2 Cp3˛2U1U2 D0; (12.21a) 4 !2 n2 9! 2 r1 U2 C2˛3U2 1U2 C3˛5U3 2 Cp˛2U3 1 D0; (12.21b) where p D C1 when W 3 1 2 D0; 1 when W 3 1 2 D : (12.22) Here, the backbone curves associated with the solutions to the p DC1 case (i.e. where the linear modes are in-phase) are denotedSC1 , and the solutions to the p D 1 case (i.e. where the linear modes are in anti-phase) are denotedS 1 . The backbone curves SC1 and S 1 , found using Eq. (12.21) are shown in Fig. 12.2, along with the response of the system when subject to forcing in the first linear mode (i.e. the second mode is unforced). A linear, proportional damping model is used for this forcing case, i.e. the damping term in theith linear equation of motion is 2 !ni Pqi,where is the modal damping ratio, which is equal for both modes. The forcing applied to the first linear mode is sinusoidal, at amplitude P1. In the case U1 (×10− 4 ) U2 (×10− 5 ) 0 0.5 1 1.5 2 2.5 3 3.5 0 1 2 3 4 S+ 1 S−1 Forced response Fig. 12.2 The backbone curves and a forced response of the asymmetric beam. This is shown in the projection of the amplitude of the fundamental component of first linear mode, U1, against that of the second linear mode, U2. The backbone curves S C 1 and S 1 are represented by a grey line and a red line respectively, whilst the forced response is represented by a blue line
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