9 Performance Comparison Between a Nonlinear Energy Sink and a Linear Tuned Vibration Absorber for Broadband Control 89 2id1A!C2i 1A!Ci B! 3KBjBj 2 C G 2 ei t1 D0 (9.22) We are interested in the behavior of the system on the stable branches of the SIM. Substituting Eq. (9.18) into (9.22) gives 2id1 !B i B 3K ! BjBj2 .2i 1! i .1 !//BC 1 2i 1 ! 3KBjBj2 D0 (9.23) ExpressingBin polar form and splitting into real and imaginary parts yields d1b D f1.b; / g.b/ ; d1 D f2.b; / g.b/ (9.24) where Dt1 ˇand f1.b; / D36K2 1b 5 24K!2 1b 3 C3GK!bsin C2!2 2 2 1 C ! 2 C2!2 1 b G!2 .!sin C cos / f2.b; / D 1 b 54K2 .!C2 /b5 C6K!.4 1 C!C8 /b 3 C9GK!b2 cos (9.25) 2!2 2!C2 2 C2!2 bCG!2 . sin !cos / g.b/ D 4 27K2b4 12Kb2!2 C 2!2 C!4 According to [16], Eq. (9.24) admits two types of fixed points. The first type is referred as ordinary fixed points and is computed by solving for f1 Df2 D0 and g ¤0. The types of fixed points are referred as folded singularities and are found for f1 Df2 Dg D0. The ordinary fixed points are obtained by solving f1 Df2 D0 for cos , sin and using trigonometric identity. A third order polynomial in Z Db2 is then obtained. The folded singularities are generated by setting f 1 Dg D0 or equivalently f2 Dg D0, giving ij D arctan 3Kb2 i !2 ! ˙arccos 2 6 4 2bi 2K 1b2 i 9Kb2 i 6!2 C2 2!2 1 C !4 C2!4 1 G!q3Kb2 i .3Kb2 i 2!2/ C 2!2 C!4 3 7 5 (9.26) From Eq. (9.26), a condition on the forcing amplitude is obtained as follows Gifs 2bi 2K 1b2 i 9Kb2 i 6!2 C2 2!2 1 C !4 C2!4 1 !q3Kb2 i .3Kb2 i 2!2/ C 2!2 C!4 (9.27) here, the subscript fs stands for folded singularities. 9.4.3 Detached Resonance Curve An important feature that can affect the performance of the NES is the possible presence of detached resonance curves (DRC). This can be analyzed by locating the boundary of the saddle-node bifurcation in Eq. (9.24). Introducing perturbations around the fixed points and linearizing with respect to the perturbation, the so-called variational equation is obtained. By imposing the roots of the characteristic polynomial to be zero, an equation for Z is obtained as
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