No abstract

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.

Appearance of transverse cracks in rotor systems mainly affects their stiffness content. The stability of such systems at steady-state running is usually analyzed by using the Floquet’s theory. Accordingly, the instability zones of rotational speeds are dominated by negative stiffness content in the whirl response in the vicinity of critical rotational speeds. Consequently, an effective stiffness measure is introduced here to analyze the effect of the crack and the unbalance force vector orientation on the intensity of negative potential and stiffness content in the whirl response. The effective stiffness expression is obtained from the direct integration of the equations of motion of the considered cracked rotor system. The proposed effective stiffness measure is applied for steady-state and transient operations using the Jeffcott rotor model with open and breathing crack models. The intensity of negative potential and stiffness content in the numerical and experimental whirl responses is found to be critically depending on the propagation level of the crack and the unbalance force vector orientation. Therefore, this can be proposed as a crack detection tool in cracked rotor systems that either exhibit recurrent passage through the critical rotational speeds or steady-state running.

Aircraft engines, aerospace rotating equipment, gas turbines, compressors, and rotors in several industrial and aerospace applications approach their nominal operational speeds after the passage through at least one of their critical rotational speeds. During the passage through the critical speeds, elevation in vibration amplitudes is usually observed due to the effect of residual unbalance in these real-life applications rotors. In all of the reported literature, the theoretical and numerical simulation results and the related Campbell diagrams suggest that the backward whirl (BW) zone should precede the passage through the critical forward whirl (FW) speed/speeds of such systems. Here, the existence of zones of rotational speeds at which BW orbits are expected to appear will be investigated immediately before and after the passage through the critical FW speed. Accordingly, startup operations of two different configurations of crack-free rotor-disk systems are considered in this numerical and experimental study. It is found out that there exist zone/zones of the shaft rotational speeds at which BW orbits are experimentally captured where these zones are localized immediately after the passage through the critical FW rotational speed during the startup operations. These BW zones are strongly affected by the acceleration of the shaft during the transient startup operations. These findings suggests that the BW should not necessarily precede the critical FW speed as suggested by the related Campbell diagrams.

Rotors have wide applications in several aerospace and industrial heavy-duty systems. In most of these applications, the rotating system reaches its steady state operational speed after the passage through at least one of its critical rotational speeds. In real-life applications, the probable appearance of a residual slight unbalance in the system could cause an elevation in vibration amplitudes at the critical rotational speeds. Accordingly, propagation of cracks in rotating shafts usually influences the level of these vibration amplitudes during start-up and cost-down operations. For such rotating systems, the critical whirl speeds are usually associated with forward and backward whirl responses where it has been always assumed that the backward whirl zone should precede the forward whirl zone. Here, two configurations of cracked rotor-disk systems are considered to study the effect of the angular acceleration and the unbalance force vector orientation with respect to the crack opening direction on the whirl response at the backward whirl zone of rotational speeds. The obtained numerical simulation results are verified through a robust experimental testing for system startup operations. The backward whirl zone is found here to appear immediately after the passage through the critical forward whirl rotational speed. The onset of the backward whirl is also found to be associated with a sharp drop in vibration whirl amplitudes. This backward whirl zone is found to be significantly affected by the unbalance force angle vector orientation and the shaft angular acceleration. More importantly, this zone of backward whirl orbits is not found to be preceding the critical forward whirl zone for the considered cracked shaft-disk configurations.