86 A.C. Batihan and E. Cigeroglu Positive slope difference implies a closed crack; thus, a nonzero nonlinear force; whereas, a negative slope difference implies an open crack; hence, zero nonlinear force. Therefore, fn(t) can be written as fn.t/ D 8 < : kna.t/ if a.t/ d R dx d L dx ˇ ˇ ˇxDLc >0 0 if a.t/ d R dx d L dx ˇ ˇ ˇxDLc <0 : (6.11) 6.3 Application of Harmonic Balance Method Letting D!t, a(t) can be written as follows utilizing multi harmonics a. / Da0 CX p acpcos.p / CX p aspsin.p /; (6.12) where a0, acp and asp are the bias term, coefficient of sine and cosine components, respectively. Similarly, nonlinear forcing fn can be written as fn . / Dfn0 CX p fncpcos.p / CX p fnspsin.p /; (6.13) where fn0 D 1 2 2 Z 0 fn . /d ; (6.14) fncp DX p 0 @ 1 2 Z 0 fn . /cos.p /d 1 A; (6.15) fnsp DX p 0 @ 1 2 Z 0 fn . /sin.p /d 1 A: (6.16) Substituting Eqs. 6.12 and 6.13 into Eq. 6.10 and collecting sine and cosine terms lead to the following relation for the pth harmonic keq meq!2p2 asp ceq!pacp Cfnsp Fsp sin.p / C keq meq!2p2 acp Cceq!pasp Cfncp Fcp cos.p / Ckeqa0 Cfn0 D0; (6.17) where Fcp and Fsp are the cosine and sine components of the pth harmonic of the external forcing. From Eq. 6.17, the following set of nonlinear equations are obtained
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