## 2 Materials and Methods
### Theoretical Estimation of Material Flow  
The selected study object is a combine harvester with one axial threshing-separation apparatus, which consists of a transition cone with bolted helical vane (1), a feeding head with two pitch impellers (2), a rotor upper cover with adjustable guide vanes (3), and rotor with rasbars (4) (Figure 1). Material in an axial rotor does not move in a straight line--it spirals due to rotor rotation and the internal guide vane effect. The guide vane with a certain angle controls the axial crop movement speed. The crop movement speed can be accelerated by increasing the guide vane angle and decreasing it to a smaller angle. The guide vane can be adjusted accordingly by the feed rate when it increases, also increasing the guide vane angle to accelerate crops through the threshing-separation unit (Figure 2). The poor design of the guide vanes could lead to poor threshing performance, resulting in crop accumulation, broken grain rate, and rotor blockage [23]. Figure 1: Material flow in the axial threshing–separation unit: 1. Transition cone with bolted helical vane; 2.Feeding head with two pitch impellers; 3. Rotor upper cover with adjustable guide vanes; 4. Rotor with rasbars. Figure 2: Schematic of guide vanes of study object: 1. Rotor threshing rotation; 2. First two fixed-angle guide vanes; 3. Six guide vanes with variable angle; 4. Guide vane linkage; 5. Variable guide vane angle 10–35\({}^{\circ}\); 6. Guide vane movement direction; 7. Material movement. One of the key indicators for characterizing this flow is the material rotation number, the number of helical revolutions the material completes around the rotor axis during its passage. The mathematical modeling approach for predicting the material rotation number and axial velocity based on the nonlinear kinematic theory was developed by [PERSON] and [PERSON] [18; 24]. This method more accurately represents the physical behavior of crop material in axial-flow threshing units compared to classical helical path assumptions [18; 23; 24]. In [PERSON]'s nonlinear kinematic model, the rotation angle \(\theta_{m}(x)\) of the material and its longitudinal position \(x\) along the rotor are related through [24]:  
\[\theta_{m}(x)=\frac{a}{R_{s}}.x^{b} \tag{1}\]  
where: \(a\)--scaling constant (depending on guide vane angle and crop properties), \(b\)--exponent reflecting slippage (usually \(b<1\)), \(R_{s}=R+\frac{\delta}{2}\)--effective radius of the material trajectory, \(x\)--material position along the rotor length, R--radius of the rotor, \(\delta\)--weighted average clearance between the rotor and the concave. The number of material revolutions completed from entry to position \(x\) is [24]:  
\[N(x)=\frac{\theta_{m}(x)}{2\pi}=\frac{a}{2\pi R_{s}}.x^{b} \tag{2}\]  
This function captures the non-uniform, gradually increasing rotational behavior of the material due to slippage and varying axial resistance. To link the combine's adjustable vane angle (\(\gamma\)) to the model, empirical constants \(a\) and \(b\) were assigned based on prior research. Larger vane angles tend to push material more aggressively (lower \(a\)) and reduce slippage (higher \(b\)) [24]. In addition to modeling rotation, [PERSON]'s nonlinear kinematic model provides the actual speed of crop material as it moves through the axial rotor. The material follows a curved path, and its total speed is influenced by both axial movement and rotational swirl [18; 23; 24].