## 2 Theoretical Derivation of the Two Dimension Tire Model
### Flexible Ring on an Elastic Continuous Sidewall Ring Tire Model  
The schematic diagram of the proposed two dimension tire model for the heavy load radial tire with a large flat ratio is illustrated in Figure 3. Meanwhile, the three-dimensional coordinate system, named as _OXYZ_, is shown, in which _XOY_ shows the planar circumferential vibration characteristic of the heavy load radial tire, and _YOZ_ and _XOZ_ illustrate the cross-sectional and out-of-plane vibration characteristic of the heavy load radial tire.  
The main focus of the paper is the circumferential vibration of the heavy load radial tire, so the theoretical modeling and experimental modal analysis of the heavy load radial tire with a large flat ratio is limited within _XOY_.  
With the coupling feature between the radial and tangential deformation of the flexible belt ring and the continuous sidewall ring, the two dimension tire model with a flexible ring on an elastic continuous foundation is developed from the flexible beam on an elastic continuous sidewall beam tire model.  
Some important assumptions are highlighted as:  
1. The cross-sections of the flexible belt ring and the continuous sidewall ring keep plane feature and is perpendicular to the middle section during the tire deformation;
2. The temperature and the inflation pressure keep stable during the tire deformation; and
3. Considering the decoupling feature between the out of-plane vibration and the planar circumferential vibration, the out of-plane vibration of the tire is ignored.  
The flexible ring on continuous sidewall ring tire model consists of four parts:  
1. The radial and tangential stiffness of the sectional sidewall spring model;
2. The continuous sidewall ring (marked in blue in Figure 3);
3. The flexible belt ring (marked in green in Figure 3); and
4. The wheel with the concentrating mass.  
Figure 3: Schematic diagram of the flexible ring on an elastic continuous foundation tire model.  
The radial stiffness of the tire sidewall is modeled with the radial spring \(k_{s71}\) (connecting the flexible belt ring and the continuous sidewall ring) and \(k_{s72}\) (connecting the continuous sidewall ring and the wheel). Additionally, the tangential stiffness of the tire sidewall is featured with the tangential spring \(k_{s91}\) (connecting the flexible belt ring and continuous sidewall ring) and \(k_{s92}\) (connecting the continuous sidewall ring and the tire wheel).  
A polar coordinate system with the origin point located at the ring center is adapted to the theoretical model. Two polar coordinate systems, named as \(XOY\) and \(X_{1}O_{1}Y_{1}\), are defined and respectively refer to the polar coordinate system with a non-rotating (fixed) and a rotating coordinate origin frame in Figure 4. \(P\),\(P_{s}\) are the arbitrary points of, respectively, the flexible belt and the continuous sidewall. \(u_{\theta}\), \(u_{r}\) respectively refer to the tangential and radial deformation of \(P\) at rotating coordinate \(X_{1}O_{1}Y_{1}\), while \(u_{s\theta}\), \(u_{sr}\) respectively refer to the tangential and radial deformation of \(P_{s}\) at rotating coordinate \(X_{1}O_{1}Y_{1}\).  
The interval angles between \(P\)-axis and \(X\)-axis, \(P\)-axis and \(X_{1}\)-axis are respectively \(\varphi\) and \(\theta\). \(\Omega\) is the angular velocity of the rotating ring with respect to the fixed center \(O\)/\(O_{1}\)\":  
\[\varphi=\theta+\Omega t \tag{2}\]  
The transformation between the coordinate \((x,y)\) of the rotating coordinate frame and coordinate \((x^{*},y^{*})\) of the non-rotating (fixed) coordinate frame is referred as:  
\[\left\{\begin{array}{l}x=x^{*}\cos(\Omega t)-y^{*}\sin(\Omega t)\\ y=x^{*}\sin(\Omega t)+y^{*}\cos(\Omega t)\end{array}\right. \tag{3}\]