# Vibration Responses of a Coaxial Dual-Rotor System with Supporting Misalignment
## 1 Introduction  
Misalignment is one of the most common difficulties in the operation of rotating machinery. Misaligned ball bearings in a rigid rotor system have a strong influence on the bearing stiffness coefficients. Angular misalignment of bearings can introduce extra axial force, while parallel misalignment can result in additional moment loads [1]. The parallel misalignment fault of a flexible coupling magnifies the displacement response at 2\(\times\) amplitude, whereas the angular misalignment response is captured at 2\(\times\) and 4\(\times\) amplitudes [2]. Parallel and angular misalignments can both induce second-harmonic generation, but the influence of angular misalignment is more significant than that of parallel misalignment [3]. Cardan shaft misalignment causes an unbalanced force, which is the main load of bolt fatigue fracture of a high-speed train motor [4]. The misalignment of aerostatic journal bearings increases the dynamic stiffness coefficients of a rotor system [5]. Misalignment of a fluid film bearing of finite length increases the value of the critical bearing stability limit [6]. It is obvious that misalignment has a significant influence on the dynamic characteristics of a rotor system. There are many studies on the influence of misalignment on different types of rotor systems. [PERSON] et al. [7] investigated dynamic characteristics of a misaligned rigid rotor system with flexible squirrel cage supports. [PERSON] et al. [8] studied the impact of a gear coupling parallel misalignment on a gear-coupling multi-span rotor system by combining the signal processing method and the finite elementtheory. [PERSON] [9] presented a method that applied current signal detection and analysis techniques to diagnose an induction motor fault resulting from installation misalignment of a rotor shaft. [PERSON] et al. [10] investigated the heat transfer characteristics of a misalignment under purge flow at second stage vane endwall through heat/mass transfer experiments using the naphthalene sublimation method and CFD simulations. [PERSON] et al. [11; 12] investigated the dynamic responses of dynamic spatially misaligned rotors inside a diesel engine multiple unit system. [PERSON] et al. [13] described the incremental inductance method to detect static misalignment of machines. [PERSON] et al. [14; 15] proposed a novel identification algorithm to simultaneously estimate the unbalance, misalignment, and active magnetic bearing stiffness parameters. [PERSON] et al. [16] studied the vibration response characteristics of a generator rotor under misalignment and stator short-circuit coupling fault. The above studies have achieved significant progress in revealing vibration characteristics of misaligned series-connected rotor systems. However, the dynamics of a dual rotor with misalignment are different from that of series-connected rotor systems which are rotors in an electric motor rotor system [17] and a planetary gearbox rotor system [18]. The different rotors of a series-connected rotor system are connected by a coupling. Each rotor of a series-connected rotor system has the same rotational speed and direction of rotation. On the contrary, a coaxial dual-rotor structure employs an inter-shaft bearing to connect an inner and an outer rotor. These two rotors of a coaxial dual-rotor system can co-rotate or counter-rotate at different speeds. In most cases, the inner rotor is often supported by more than two bearings, making it a redundantly supported system. The special structure of a dual-rotor system coupled with a harsh working temperature environment makes the coaxial dual-rotor system of an aeroengine more prone to misalign. [PERSON] et al. [19] presented a theoretical model of the nonlinear response characteristics of an aeroengine dual-rotor-bearing system with flexible coupling misalignment faults in the inner rotor. [PERSON] et al.