Darpe et al. (2003a) analyzed the coupling of the lateral and longitudinal vibrations due to the presence of transverse surface crack in a rotor for the non-rotating and rotating conditions. The steady state unbalance response of a cracked rotor with a single centrally situated crack subjected to periodic axial impulses was investigated experimentally. Darpe et al. (2003b) studied a simple Jeffcott rotor with two transverse surface cracks and the effect of the interaction of two cracks on the breathing behavior and on the unbalance response of the rotor. They noticed the significant changes in the dynamic response of the rotor when the angular orientation of one crack relative to the other was varied. Darpe et al. (2004) studied the coupling between longitudinal, lateral and torsional vibrations together for a rotating cracked shaft with a response-dependent non-linear breathing crack model. Crack signatures were obtained by using external excitations and the excitation in one mode led to an interaction between all the modes, as the couplings were accounted. The co-existence of frequencies of other modes in the frequency spectra of a particular mode and the presence of sum and difference of frequencies around the excitation frequencies and its harmonics were used as the indicators for crack diagnosis. Darpeet al. (2006) formulated equations of motion of the rotor with a transverse surface crack with a bow, and analyzed the steady state and the transient response of the rotor. They assessed the effect of the residual bow on the stiffness characteristic of the rotating cracked shaft and observed that the usual level of bow might not significantly influence the stiffness variation and the nonlinear nature of crack response was not significantly altered.

Chondros (2005) developed a variational formulation for the torsional vibration of a cylindrical shaft with a circumferential crack. The Hu-Washizu-Barr (1955, 1966) variational formulation was used to develop the differential equation and boundary conditions of the cracked rod. He reported independent evaluations of crack identification methods in rotating shafts and compared with methods using the continuous crack flexibility theory.

Di and Law (2006) provided different types of damage models of a frame element, whose elemental matrices were decomposed into their eigen value and eigen vector matrices. These eigen-parameters were included in a flexibility-based and sensitivity-based model-updating algorithm for the condition assessment of a plane frame structure. Lei et al. (2007) proposed a finite element model for vibration analysis of a crankshaft with a slant crack in the crankpin and investigated the influence of the crack depth on the transient response of a cracked crankshaft.

Lee and Chung (2000) presented a nondestructive evaluation procedure for identifying the crack (i.e. the location and the size of the crack), in a one-dimensional beam-type structure using the natural frequency data. Lowest four natural frequencies were obtained of the cracked structure by FEM and the approximate crack location was obtained by Armon's Rank-ordering method (1994). The actual crack location was identified by Gudmundson's equation (1982) using the determined crack size and aforementioned natural frequencies. Morassi (2001) dealt with the identification of a single crack in a vibrating bar based on the knowledge of the damage-induced shifts in a pair of natural frequencies. The crack was simulated by an equivalent linear spring, which was connected between two segments of the bar. The analysis was based on an explicit expression of the frequency sensitivity to damage and enables non-uniform bars under general boundary conditions to be considered. Owolabi et al. (2003) developed a method to detect the presence of a crack in beams and determine its location and size, based on experimental modal testing methods. Changes in natural frequencies and frequency response function amplitudes as a function of crack depths and its locations were used in the crack detection methodology. Most of the literatures discussed on crack identification methods are based on free vibrations.

Most of the model based crack detection and diagnostics are based on the procedure that the experimental measurements from prototype structures are compared with predicted measurements from a corresponding finite element model. One way to compare the data is to reduce the number of degrees of freedom in the analytical model. This is generally achieved by implementing the condensation scheme. Guyan (1965) introduced a static reduction method, based on the assumption that inertia terms are negligible at low frequency of excitations, in order to reduce mass and stiffness matrices by eliminating DOFs corresponding to slave nodes (e.g. where no force is applied). Hence, any frequency response functions generated using the reduced equation of motion is exact only at zero frequency. As the excitation frequency increases, the inertia terms neglected become more significant. Dharmaraju et al. (2004, 2005) and Tiwari and Dharmaraju (2006) developed algorithms for identifying the crack flexibility coefficients and subsequently estimation of the equivalent crack depth based on the forced response information. They outlined the condensation scheme for eliminating the rotational degree of freedoms at crack element nodes based on the physical consideration of the problem, which was otherwise difficult to eliminate. However, the main practical limitation of the algorithm was that the location of the crack must be known a priori. Also the algorithm used the Euler-Bernoulli beam theory in the beam model without considering the damping in the system. Recently, Karthikeyan et al. (2007a and b) developed an algorithm for the crack localization and the sizing in a cracked beam based on the free and forced response measurements without applying condensation schemes, which was the main practical limitation in application of the work. Moreover, the method was also based on the mode shape measurement, which is relatively difficult to measure. The hybrid reduction scheme was extended by Karthikeyan et al. (2008), by including the damping, to the crack localization and sizing algorithm based on purely the forced response measurements, which is a more controlled and accurate way of measurements. The main contribution of the paper was the elimination of measurement of rotational dofs completely, which otherwise difficult to measure accurately. The application of the regularization technique helped in estimating both the crack size and its location, iteratively, without which the algorithm might lead to unbounded estimates of crack parameters. The converged value of the crack depth ratio and corresponding crack location, up to a desired accuracy, is considered as the final size and the location of the actual crack in the cracked beam. For illustrations, beams with the simply support and cantilever end conditions have been considered. The convergence of the algorithm has been found to be very fast and the robustness of the algorithm was tested by adding the measurement error in the resonant frequency measurement and the measurement noise in force