8 A.C. Batihan and E. Cigeroglu 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-12 10-10 10-8 10-6 10-4 Frequency, w, [rad/s] Frequency, w, [rad/s] Fourth Harmonic, aj4 [m] a 14 , L c =0.2 a 14 , L c =0.4 a 24 , L c =0.2 a 24 , L c =0.4 a 34 , L c =0.2 a 34 , L c =0.4 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-14 10-12 10-10 10-8 10-6 10-4 Fourth Harmonic, aj4 [m] a 14 , a=0.2 a 14 , a=0.5 a 24 , a=0.2 a 24 , a=0.5 a 34 , a=0.2 a 34 , a=0.5 a b Fig. 1.6 Fourth harmonic of each modal coefficient vs. frequency. (a) For ˛ D0:5 and different crack location. (b) For Lc D0:2 m and different crack ratio 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-14 10-12 10-10 10-8 10-6 10-4 Frequency, w, [rad/s] Frequency, w, [rad/s] Third Harmonic, aj3 [m] a 13 , L c =0.2 a 13 , L c =0.4 a 23 , L c =0.2 a 23 , L c =0.4 a 33 , L c =0.2 a 33 , L c =0.4 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-16 10-14 10-12 10-10 10-8 10-6 10-4 Third Harmonic, aj3 [m] a 13 , a=0.2 a 13 , a=0.5 a 23 , a=0.2 a 23 , a=0.5 a 33 , a=0.2 a 33 , a=0.5 a b Fig. 1.7 Third harmonic of each modal coefficient vs. frequency. (a) For ˛ D0:5 and different crack location. (b) For Lc D0:2 m and different crack ratio 1.5 Conclusion In this study, beam with breathing edge crack is modelled by using Euler-Bernoulli beam theory and nonlinear piecewise linear stiffness. Multiple trial functions are used to represent the response in Galerkin’s method where a piecewise linear stiffness matrix based on the slope difference at the crack location is introduced. Harmonic Balance Method with multiple harmonics is used to convert nonlinear ordinary differential equation into a set of nonlinear algebraic equations. It is observed from the results that effect of crack parameters on the natural (resonance) frequency of the cracked beam is insignificant. However, it is observed that harmonics of the response are affected from the crack parameters significantly; hence, this information can be used for the crack detection. Both crack depth and crack location affects the amplitudes of the harmonics of each modal coefficient. As crack ratio increases the amplitudes of the harmonics also increase, however the order of magnitudes of the harmonics are not affected by the crack ratio. The crack location affects amplitudes of the harmonics as well as the order of magnitudes of the harmonics. Depending on the crack location, a harmonic of a different modal coefficient becomes dominant. This fact explains the necessity of using multiple trial functions.
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