6 Nonlinear Vibrations of a Beam with a Breathing Edge Crack 87 ka0 Cfn0 D0 c! 2 64 1 0 : : : 0 p 3 75 8 ˆ< ˆ: ac1 : : : acp 9 >= >; C 2 64 k m!2 0 : : : 0 k m!2p2 3 75 8 ˆ< ˆ: as1 : : : asp 9 >= >; C 8 ˆ< ˆ: fns1 : : : fnsp 9 >= >; 8 ˆ< ˆ: Fs1 : : : Fsp 9 >= >; D0 2 64 k m!2 0 : : : 0 k m!2p2 3 75 8 ˆ< ˆ: ac1 : : : acp 9 >= >; Cc! 2 64 1 0 : : : 0 p 3 75 8 ˆ< ˆ: as1 : : : asp 9 >= >; C 8 ˆ< ˆ: fnc1 : : : fncp 9 >= >; 8 ˆ< ˆ: Fc1 : : : Fcp 9 >= >; D0: (6.18) 6.4 Results and Discussion In order to study the effect of parameters of breathing edge crack, case studies are carried out on a clamped-clamped beam by using five harmonics. For the case studies, the following beam properties are used: L D 1m, I D 2:667 10 8 m4, ED206GPa, D7;850kg/m3, AD8 10 4 m2, D0:3, c eq D159:24Ns/m, Lf D0:9m, F.t/ D100sin.!t/N. In Fig. 6.3, fundamental resonance frequencies for different crack ratios and crack locations are plotted. It is observed that as crack ratio increases, the change in fundamental resonance frequency is more significant. Studying Fig. 6.3b, it is observed that for the case of Lc D0:2, which is very close to the location of the sign change in the slope of the beam, the effect of crack ratio on the fundamental resonance frequency is insignificant. Therefore, in crack detection problems, depending on the crack location, fundamental resonance frequency change may not give accurate results; hence, alternative features should be considered for crack detection. Due to breathing effect of the crack, in some part of a cycle the crack remains open and in the rest of the cycle the crack remains closed. For this reason nonlinear force is almost zero for some period and nonzero in the rest of the cycle as shown in Fig. 6.4, where, the variation of the nonlinear force at the fundamental resonance frequency is plotted for one cycle at crack location of Lc D0:2m. Even though very little change in fundamental resonance frequency is observed for Lc D0:2m depicted from Fig. 6.3, the nonlinear force is affected significantly as the crack ratio changes. Fig. 6.3 Variation of fundamental resonance frequency with respect to (a) crack ratio, (b) crack location
RkJQdWJsaXNoZXIy MTMzNzEzMQ==