6 Nonlinear Vibrations of a Beam with a Breathing Edge Crack 85 Fig. 6.1 Beam with a breathing edge crack Fig. 6.2 (a) SDOF representation of a beam with a breathing edge crack, (b) Nonlinear force due to piecewise linear stiffness Since the crack beam can be represented by two linear systems, two different trial functions corresponding to these two states are required. For the period of vibration where the crack is closed, the first eigenfunction of the undamaged beam is used as a trial function; whereas, the first eigenfunction of a beam with an open crack is used in the case of open crack. The details and derivation of these eigenfunctions can be found in [13], where the response of the left and right sides of the beam with respect to the crack are treated separately. Therefore, the first eigenfunction of a beam with an open crack is composed of two functions named as L(x) for the beam segment on the left side of the crack and R(x) for the beam segment on the right side of the crack. Utilizing the first eigenfunction of a beam with an open edge crack and applying Galerkin’s Method, ko and kn can be obtained as follows, ko DEI 0 @ Lc Z 0 L.x/ d4 L.x/ dx4 dxC L Z Lc R.x/ d4 R.x/ dx4 dx1 A ; (6.8) kn Dkeq ko: (6.9) Equation of motion of the nonlinear system given in Fig. 6.2 can be given as follows meq Ra.t/ Cceq Pa.t/ Ckoa.t/ Cfn.t/ DF.t/ Lf ; (6.10) where fn(t) is the nonlinear force which depends on the slope difference. In this study, slope difference at the crack location of a beam with an open edge crack is used in order to identify the contact and separation of the piecewise linear stiffness.
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