Negative Effective Stiffness Content in Cracked Rotors

The appearance of cracks in rotor systems affects the whirl response in the neighborhood of the critical whirl rotational speeds. The combined effect of the crack depth and the unbalance force vector angle orientation with respect to the crack opening direction on the effective stiffness content of the cracked rotor system in the neighborhood of the critical rotational speed is addressed here. The effective stiffness expression of the cracked system can be obtained from the direct integration of the equations of motion of the cracked rotor system. The cracked rotor equations of motion can be expressed by the Jeffcott rotor or the finite element models. The appearance of cracks in rotor systems converts them into parametrically excited dynamical systems with time-periodic stiffness components. The interaction between the time-periodic stiffness and the external periodic forcing function of the unbalance force significantly alters the effective stiffness content in the system at both transient and steady state operations. For wide range of crack depths and unbalance force vector angles, the effective stiffness has been found to be of negative values. This means that the cracked rotor system tends to have more resistance to deflect towards the center of its whirl orbit and less resistance to deflect away under the unbalance force excitation effect. Consequently, in the negative stiffness content zone of the unbalance force vector angles, the cracked rotor system tends to exhibit a sharp growth in the vibration whirl amplitudes. However, for positive effective stiffness values, the shaft has more resistance to deflect away from its whirl orbit center. Therefore, the cracked rotor system is at higher risk of failure in the negative effective stiffness zone of unbalance force vector angles than the positive effective stiffness zone of these angles.

Application of the Proper Orthogonal Decomposition Method for Cracked Rotors

The application of the proper orthogonal decomposition (POD) method to the vibration response of a cracked rotor system is investigated. The covariance matrices of the horizontal and vertical whirl amplitudes are formulated based on the numerical and experimental whirl response data for the considered cracked rotor system. Accordingly, the POD is directly applied to the obtained covariance matrices where the proper orthogonal values (POVs), and the proper orthogonal modes (POMs) are obtained for various crack depths, unbalance force vector angles, and rotational speeds. It is observed that both POVs and their corresponding POMs are highly sensitive to the appearance of the crack and the unbalance force angle direction in the neighborhoods of the critical rotational speeds. The sensitivity zones of the POVs and POMs to the crack propagation are found to be coinciding with the unstable zones found by the Floquet's theory of the considered cracked system.

A novel amplitude-independent crack identification method for rotating shaft

The shaft crack is one of the most common and serious malfunctions in rotating machines and may lead to catastrophic failure if undetected in time. However, the conventional crack identification methods are amplitude-dependent and thus can be only applied to the crack identification under some specific conditions. In this paper, a novel amplitude-independent crack identification method (AiCIM) is significantly proposed to eliminate the amplitude-dependent property and promote the effectiveness of the crack identification. First and foremost, a fast time-varying vibration phenomenon of the cracked-rotor system is newly found. Through the theoretical analysis, the fast time-varying vibration mechanism of the cracked-rotor system is revealed for the first time. It is indicated that the vibration signal of the cracked-rotor system is modulated by the fast-oscillated instantaneous frequency, which is independent of the amplitude of the vibration signal. AiCIM is then put forward on the basis of the fast time-varying vibration mechanism and matching time–frequency analysis theory. Specially, the amplitude-independent instantaneous frequency of the vibration signal is extracted via the matching time–frequency analysis theory, and the time–frequency representation energy-concentration is enhanced along the instantaneous frequency trajectory. Since instantaneous frequency of the vibration signal carrying the critical fault information is employed to identify the shaft crack, AiCIM is only relevant to the phase of the vibration signal, i.e. amplitude independent. As a result, AiCIM successfully eliminates the dependence on the signal amplitude and is more sensitive to the weak crack. Both the numerical and experimental results demonstrate that AiCIM behaves best to extract the fast-oscillated feature of the fast time-varying vibration induced by the shaft crack in comparison with other time–frequency analysis methods, and AiCIM effectively suppress the effect of noises on the instantaneous frequency estimation because of its amplitude-independent property. Influences of the crack parameters on the nonlinear instantaneous frequency are finally discussed with AiCIM. This study provides a potential way to the online crack identification.