What is the crack modeling?
The effect of transverse breathing crack on the dynamics of the rotor-bearing system during shut-down has been considered in the present work. Papadopoulos and Dimarogonas derived the flexibility matrices of the cracked element for flexural vibrations. These flexible matrices are utilized in the present work as proposed in the in the FEM analysis of Sekhar and Prabhu. The flexibility matrix of an uncracked beam/shaft element by ignoring shearing action is given by where EI is the flexural rigidity which represents the bending stiffness and l is the element length.
The phenomenon of opening and closing of the crack during shaft rotation is called breathing action of the crack. Papadopoulos and Dimarogonas23 have illustrated the breathing action of the crack. The crack opens and closes depending on the rotor deflection. If the static deflection is much higher than the rotor vibration which is the case for the large class of machines, then the breathing action of the crack takes place. Considering this case, the crack is closed when = 0 and it is fully open when =.
A crack on the beam element (shown in Fig.1) introduces considerable local flexibility due to strain energy concentration in the vicinity of the crack tip under load. The local flexibility due to the additional strain energy can be represented by a local flexibility matrix C which will be C and C for a fully open crack and half-open, half-closed conditions respectively due to breathing phenomenon of the transverse crack :
Where F0 = and Poisson’s ratio, . The dimensionless compliance coefficients are computed from the derivations discussed in reference21,23. The total flexibility matrix for the cracked section can be obtained by adding the additional flexibility matrix due to crack to the flexibility matrix of the uncracked element, which is given as
[C] = [C0] + [Cc]
From the equilibrium condition
( )T = [T] ( )T,
where the transformation is
The stiffness matrix of the cracked element is written as
[Kc] = [T][C]-1[T]T .
Stiffness matrix of the cracked element, [Kc], replaces the stiffness matrix of the uncracked element while assembling the stiffness matrix of the shaft, [K] in the system equation of motion.