## 1 Introduction  
Crack damages are common faults in large structures due to the long time of corrosion, creep and high cyclic stresses. As the crack damage is hard to detect in the early stage, it has become severe threats to the operation safety of equipment and attracts the attentions of many researchers. In order to develop effective techniques for detecting the crack damages, a large number of investigations have been published to analyze the nonlinear dynamics of cracked rotors and to gain insights into the dynamic response characteristics [1; 2; 3]. A reliable diagnosis method for cracks requires comprehensive insights into the dynamic behaviors of cracked rotor-bearing systems, and an accurate dynamic model is needed. Generally, journal bearings possess the property of anisotropy, which makes the dynamics of rotor-bearing systems much more complex [4]. Besides, the speed-dependent characteristic of bearings [5], which denotes that the stiffness and damping coefficients vary with the rotating speed, is also a common phenomenon in rotor-bearing systems. Current research is mostly concentrated on simply supported cracked rotors or supported by isotropic bearings and the anisotropic rotor-bearing systems without cracks. The speed-dependent characteristic of journal bearings is also seldom considered in previous studies. With respect to the studies on cracked rotors, the linear fracture mechanics and the strain energy release rate approach were often used to calculate the variation of local stiffness due to cracks [6]. Recently, many crack models, which could provide more reasonable descriptions for the breathing behaviors, were proposed [7; 8]. [PERSON] et al. [9] investigated the time-variant flexibility of cracked shaft due to bending and torsion loads with a 3D finite element model and a nonlinear contact procedure. Lately, they studied the local flexibility of cracked shafts under harmonic type of loadings [10]. [PERSON] and [PERSON] [11] developed a model for cracked rotating shafts using the rigid finite element method, in which the crack was modeled as two sections connected by several spring-damping elements of variable stiffness. [PERSON] et al. [12] addressed the breathing mechanism of a transverse crack and evaluated the local stiffness loss using the cohesive zone model. [PERSON] et al. [13] used non-symmetric bending principles to develop a new crack breathing model. [PERSON] et al. [14; 15] investigated the effects of interval uncertain parameters on the dynamic behaviors of a rotor system with crack in the shaft. Although these investigations provide effective ways for simulating the dynamic behaviors of cracked rotors, the bearings are mostly ignored or simplified as isotropic ones. As the periodically time-variant characteristic caused by the anisotropic bearings can lead to the parametric excitation phenomenon [16; 17], it should be appropriately dealt with in order to study the dynamic responses of realistic cracked rotor-bearing systems. Regarding the anisotropic rotor-bearing systems, a large number of investigations have also been published in the past few decades. In order to quantitatively simulate the vibration of such systems, several accurate models have been developed in recent years. [PERSON] et al. [18] addressed a 3D finite element method for the stability analysis of asymmetric rotating rotors basing on the modal theory. [PERSON] et al. [19] suggested a 3D finite element procedure for studying the frequency characteristics of anisotropic rotor-bearing systems. [PERSON] et al. [20] presented a 3D finite element-based model order reduction method for parametric resonance and whirling analysis of anisotropic rotor-bearing systems. [PERSON] et al. [21] studied the stochastic boundaries of unstable regions of an uncertain asymmetric rotors using a 3D finite element rotor model. [PERSON] et al. [22] investigated the Sommerfeld effect at forward and backward critical speeds in a rigid rotor shaft system with anisotropic supports.