## 2 Dynamic Model of a Helical-Geared Rotor-Bearing System
### Disk and Helical Gear Pair  
In this paper, the gear pair is assumed to be two rigid disks. The kinetic energy including the axial motion of each disk can be expressed as follows:  
\[T_{d}=\tfrac{1}{2}m_{d}[{(\dot{V}_{d})}^{2}+{(\dot{W}_{d})}^{2}+{(\dot{U}_{d})}^{ 2}]+\tfrac{1}{2}l_{dD}[{(\dot{B_{s}})}^{2}+{(\dot{\Gamma_{s}})}^{2}]+\tfrac{1}{ 2}l_{dD}(\Omega+\dot{a}_{d}){(\dot{B}_{d}\Gamma_{d}-\dot{\Gamma_{s}}B_{d})}+ \tfrac{1}{2}l_{dP}(\Omega+\dot{a}_{d})^{2} \tag{10}\]  
where \(m_{d}\), \(l_{dD}\), and \(l_{dP}\) are the mass, transverse mass moment of inertia, and polar mass moment of inertia of the disk, respectively. The equation of motion of each disk can be derived as:  
\[[M_{d}]\big{[}\ddot{q}_{d}\big{]}+\Omega[G_{d}]\big{[}\dot{q}_{d}\big{]}+[K_{d }][q_{d}\big{]}=\{F_{d}\} \tag{11}\]  
where  
\[\{q_{d}\}=\Big{[} \begin{array}{cccccc}V_{d}&W_{d}&U_{d}&B_{d}&\Gamma_{d}& \alpha_{d}\end{array} \Big{]}^{T} \tag{12}\]  
and \([M_{d}]\) and \([G_{d}]\) are the respective mass and gyroscopic matrices of one gear, and \(\{F_{d}\}\) is the force vector due to disk eccentricity. Figure 3 describes a helical gear dynamic model with 12 generalized coordinates. \(O_{1}\) and \(O_{2}\) are the gear centers when the rotating shafts have no deformation; \(d_{0}\) is the initial gear pair center distance; \(\beta_{0}\) is the initial helix angle of the gear pair. In most previous studies [1; 2; 3; 1; 1; 13; 15; 16], these values are regarded as constant when they are time-independent. Generally, the gear centers will change with time due to the vibration of the rotating shafts. The gear centers can move to \(C_{1}\) and \(C_{2}\). \(G_{1}\) and \(G_{2}\) are the mass centers of the pinion and gear, respectively. Due to the vibrations of the rotating shafts, the centers of the gear and pinion would shift, and displacements (\(V_{d}\), \(W_{d}\), \(U_{d}\), \(B_{d}\), \(\Gamma_{d}\), and \(\alpha_{d}\)) would change with time. This makes the center distance, helix angle, transverse pressure angle, relative position angle, contact ratio, and the relative displacement of the gear pair vary with time. In other words, due to the time-varying motions of pinion and gear, the gear pair center distance and the helix angle are time-dependent; therefore, the gear pair center distance is changed from \(d_{0}\) to \(d\) and expressed as  
\[d(t)=\sqrt{\left(V_{d2}(t)-V_{d1}(t)+d_{0}\right)^{2}+\left(W_{d2}(t)-W_{d1}(t )\right)^{2}+\left(U_{d2}(t)-U_{d1}(t)\right)^{2}} \tag{13}\]  
Similarly, the helix angle is changed from \(\beta_{0}\) to \(\beta(t)\), which is represented as  
\[\beta(t)=\tan^{-1}\frac{\pi d(t)}{p_{z}},\;\beta(t)\bigg{\{} \begin{array}{ ll}>0&\text{if the gear has left hand teeth}\\ <0&\text{if the gear has right hand teeth}\end{array}  \tag{14}\]  
where \(p_{z}\) is the lead of the helical gear pair. \(\Omega_{1}\) and \(\Omega_{2}\) are the rotating speeds of the pinion and the gear, respectively. The transverse geometrical eccentricities are denoted by \(e_{1}\) and \(e_{2}\). \(\eta_{1}\) and \(\eta_{2}\) are the transverse eccentricity angles of the transverse plane. The gear mesh force along the pressure line can be expressed as:  
\[F_{h}(t)=c_{m}\dot{\delta}(t)+k_{m}(t)\delta(t) \tag{15}\]  
The damping coefficient \(c_{m}\) is assumed to be zero. The gear mesh stiffness is accounted for by transverse tooth bending stiffness, transverse tooth shear stiffness, transverse tooth radial compressive stiffness, transverse gear foundation stiffness, Hertzian contact stiffness, axial tooth bending stiffness, axial tooth torsional stiffness, and axial gear foundation stiffness. The gear mesh stiffness is assumed as a periodic function of the mesh period. In the mesh situation, there are three teeth contacts and two teeth contacts, and the gear mesh stiffness will be different. The details of the gear mesh stiffness areshown in [PERSON] et al. [19]. The relative displacement of the gear mesh in a direction perpendicular to contact surfaces is represented by \(\delta(t)\) and defined as follows  
\[