## 3 Theoretical Analysis
### Model Derivation
For example, the error for Block D at a release height of 0.6 m and a slope of 45\({}^{\circ}\) is 9.4%. For Block L under the same conditions, the error is 4.59%. And during the directional shift in the rolling process, the two-dimensional shape that is approximately perpendicular to the rolling axis resembles an ellipse, as shown in Figure 20:4 However, calculations are still performed assuming a circular shape, resulting in errors. In contrast, for spherical blocks, the roundness of any cross-section is one, so even if there is a directional shift during rolling, the two-dimensional cross-section perpendicular to the rolling axis remains consistent with the one used for calculation, resulting in smaller errors between the calculated and theoretical values. 3. For some blocks with small contact areas with the slope surface and a high center of gravity, rocking may occur during the rolling process, consuming some energy. For example, blocks I, J, K, and L exhibit this phenomenon, and it becomes more pronounced as the contact area with the slope surface decreases. The position of the rolling axis oscillates with the change in the center of gravity, causing the entire rolling process to be unstable. However, for blocks rolling on either side, the rotational inertia keeps them in a balanced state, as shown in Figure 20:5 In contrast, blocks J and K, which have the same circularity and volume as blocks G and H, do not exhibit rocking during rolling because they have larger contact areas with the slope surface and a lower center of gravity. As shown in Figure 28, no energy loss occurs due to rocking during rolling, reducing the error between the calculated and experimental values. Figure 20:6 Comparison of rolling speeds for a 60\({}^{\circ}\) slope. For blocks with 0.33 < \(\lambda\) < 1, it is necessary to consider the situation where the rolling axis changes during the rolling process. Therefore, for such blocks, both the high and low values of the calculated speeds are given using Formulas (10) and (11), denoted as the block with index \({}^{\prime}\) and the block with index \({}^{\prime\prime}\), as shown in the figures below. In Figure 29, the experimental speed of Block D closely matches the theoretical high-speed value, and the average error between the experimental values and the theoretical high values is 4.92%. This is because Block D has a significantly different rolling resistance arm, moment of inertia, and minimum circumscribed circle radius for its two rolling cross-sections. When such-shaped blocks roll on a flat slope, a longer rolling distance is required to change the rolling axis, so the entire process mainly involves rolling around the axis  
Figure 20:7 Block rolling without swinging, maintaining the initial rolling axis. Figure 20:8 Deviation of the rolling axis during block rolling. Figure 20:9 Test block sways left and right during rolling. with a lower rolling resistance. The actual speed from a 0.6 m height release is situated between the high and low theoretical values. During the actual rolling process of the model experiment, there was a slight deviation in the rolling axis, as shown in Figure 26, but the deviation was small, and the block did not complete the change of the rolling axis after rolling off the slope. However, the process of changing the rolling axis involves an increase in the rolling resistance arm, which increases the energy consumption during rolling, leading to the rolling speed leaning towards the lower calculated value. For the cases where the experimental speed is less than the minimum theoretical speed in Figures 30 and 31, the reason is the energy loss caused by the slight jumping and rocking of Blocks E and F during rolling. This characteristic is similar to that of disc-shaped blocks. However, due to the similar moment of inertia and rolling resistance arm for rolling around the different axes, the rotational inertia required for rolling around the initial rolling axis cannot be maintained.