## 2 Model Design  
### Modified Noncircular Rolling Bearing Quasi-Dynamic Model  
Quasi-dynamic analysis is an accurate bearing analysis method, because it comprehensively considers the microscopic contact and the rheological characteristics of the lubricant. Quasi-dynamic analysis can accurately calculate the dynamic characteristics of a rigid rolling bearing, but is not suited for a flexible bearing, of which the raceways would be noncircular following structural deformation; thus, a modified quasi-dynamic model for noncircular rolling bearings is needed. Figure 1 shows a sketch of the force of roller bearing. Cutting the roller into slices, the \(k\)-th slice of the \(j\)-th roller experiences contact forces \(Q_{1 jk}\) and \(Q_{2 jk}\), and drags forces \(T_{1 jk}\) and \(T_{2 jk}\) from the inner and outer raceways through oil films, normal force \(F_{cj}\) and tangential force \(f_{cj}\) from the cage, and oil-gas mixture frictional force \(F_{gj}\), as well as its own inertial force, \(F_{zj}\), and inertia moment \(M_{xj}\) due to rolling. According to the Hertz theory, forces \(Q_{1 jk}\) and \(Q_{2 jk}\) between the roller slice and the raceways can be obtained from contact approaches \(\delta_{1 jk}\) and \(\delta_{2 jk}\) of the slice and raceways, respectively. When the ring deforms, the raceway becomes a nonideal cylindrical surface. The deviations of the \(k\)-th slice of the \(j\)-th roller at the local surface from the ideal cylindrical surface can be expressed as \(\delta_{1 jk}\) and \(\delta_{2 jk}\). These two deviations can be superimposed with approaches \(\delta_{1 jk}\) and \(\delta_{2 jk}\), based on the small deformation assumption. By further considering oil film thickness \(h_{1 jk}\) and \(h_{2 jk}\) at the contact area, the actual contact approaches, \(\delta^{\prime}_{1 jk}\) and \(\delta^{\prime}_{2 jk}\), between the roller slice and the noncircular raceway can be obtained. Thus, contact loads \(Q^{\prime}_{1 jk}\) and \(Q^{\prime}_{2 jk}\) of the rolling object and the noncircular raceway can be obtained, as expressed in Equation (1). \(C_{k}\) is the correction of the roller radius, considering the Roller Profile Modification and chamfer [16]. In addition, the roller balance equation can be established as Equation (2). The lubricant parameters in the model are calculated using the Hertz contact theory, the Dowson-Higginson formula [17], and the five-parameter rheological model [18]. The five-parameter rheological model has been experimentally verified and contains the viscosity-pressure property and the viscosity-temperature property of the oil. \[ \begin{array}{l}\delta^{\prime}_{1 jk}=\delta_{1 jk}+h_{1 jk}+\delta_{1 jkh}+C_ {k}\\ Q^{\prime}_{1 jk}=\left(\frac{s^{\prime}_{1 jk}\cdot^{0.8}_{1 jk}}{3.81}\right)^{ 1/0.9}\cdot\left(\frac{\pi\cdot E_{1}}{2(1-v_{1}^{2})}\right)\\ \delta^{\prime}_{2 jk}=\delta_{2 jk}+h_{2 jk}+\delta_{2 jkh}+C_{k}\\ Q^{\prime}_{2 jk}=\left(\frac{s^{\prime}_{2 jk}\cdot^{0.8}_{1 jk}}{3.81}\right)^{ 0.9}\cdot\left(\frac{\pi\cdot E_{2}}{2(1-v_{2}^{2})}\right)\\ C_{k}=\left\{\begin{array}{l}0;\frac{L-L}{2}\leq(k-0.5)\cdot\frac{L}{2}\leq \frac{L+L}{2}\\ (R^{2}-\frac{L}{4})^{0.5}-(R^{2}-x_{k}^{2})^{0.5};\,0<k<\frac{L-L_{l}}{2}\cup \frac{L+L_{l}}{2}<k<l_{e}\\ \\ \frac{1}{n}\sum\limits_{k=1}^{n}\left(T_{1 jk}-T_{2 jk}+P_{1 jk}-P_{2 jk}\right) +F_{cj}+F_{gj}+F_{yj}=0\\ \\ \frac{1}{n}\sum\limits_{k=1}^{n}\left(-Q^{\prime}_{1 jk}+Q^{\prime}_{2 jk} \right)-f_{cj}+F_{zj}=0\\ \\ \frac{D_{0}}{2}(\frac{1}{n}\sum\limits_{k=1}^{n}\left(T_{1 jk}+T_{2 jk}\right) -f_{cj})+M_{xj}=0\\ \\ \frac{1}{n}\sum\limits_{k=1}^{n}\left(-Q^{\prime}_{1 jk}x_{k}+Q^{\prime}_{2 jk} x_{k}\right)+M_{yj}=0\\ \\ \frac{1}{n}\sum\limits_{k=1}^{n}\left(-P_{1 jk}x_{k}+P_{2 jk}x_{k}\right)+M_{zj}= 0\\ \end{array}\right.\end{array}  \tag{2}\]  
Figure 1: Sketch of the force of roller bearing: (**a**) Model of the roller–raceway interaction; (**b**) forces and moments acting on a roller.