8 Resonant Analysis of Systems Equipped with Nonlinear Displacement-Dependent (NDD) Dampers 73 In order to solve Eq. (8.26) and for omitting the secular terms one let AD 1 2 a.T1/ e i .T1/ (8.27) Substituting Eq. (8.27) and its conjugate and derivatives into Eq. (8.26) and separating the imaginary and real parts leads to da dT1 D a 1 4 ˇ1a3 1 8 ˇ2a5 5 64 ˇ3a7 7 128 ˇ4a9 Cksin. C T 1/ (8.28a) d dT1 D kcos. C T1/ a (8.28b) To eliminate the explicit time dependence of the right-hand sides of (8.28a) and (8.28b) one let D T1 (8.29a) Or d dT1 D d dT1 (8.29b) Hence (8.28a) and (8.28b) can be rewritten as follows da dT1 D a 1 4 ˇ1a3 1 8 ˇ2a5 5 64 ˇ3a7 7 128 ˇ4a9 Cksin. / (8.30) d dT1 D C kcos. / a (8.31) Periodic solution of the externally excited system correspond to the stationary solutions of Eqs. (8.30) and (8.31), where both aand become constant, that is da dT1 D0 (8.32) d dT1 D0 (8.33) Suppose ã and Q refer to the stationary solution of a and , thus Substituting (8.30) and (8.31), respectively in (8.32) and (8.33), results in QaC1 4 ˇ1Qa3 C1 8 ˇ2Qa5 C 5 64 ˇ3Qa7 C 7 128 ˇ4Qa9 ksin Q D0 (8.34a) C kcos Q Qa D0 (8.34b)
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