Kinematic analysis and slip-roll ratio quantification for conductive roll-ring flexible rings

2.2.1. Analysis of power roll ring speed

Fig. 2(a) illustrates the kinematic analysis and the composite motion principle of the flexible ring based on its geometric configuration. Fig. 2(b) presents the velocity contour plot, revealing a distinct gradient distribution during operation. The velocity peaks at 363 mm/s at contact point A (inner ring interface) and gradually decreases along both paths (A→B→C→D→E and A→K→G→F→E), approaching zero at contact point E (outer ring interface). This distribution results from the superposition of spin and orbital velocities under pure rolling conditions, where the two velocities are equal in magnitude but vary in directional alignment depending on the position.

Furthermore, the velocity vector analysis in Fig. 3 reveals the velocity superposition mechanism at characteristic points: 1. Point A: The spin velocity ($V_2$) and orbital velocity ($V_H$) are in the same direction, resulting in the maximum linear velocity $V_A=V_H+V_2$. 2. Points B and K: The angle between $V_2$ and $V_H$is 45°, yielding resultant velocities $V_B$ and $V_K$ that are lower than $V_A$. 3. Points C and G: With a 90° angle between $V_2$ and $V_H$, the resultant velocities $V_C$ and $V_G$ decrease further. 4. Points D and F: As the angle between $V_2$ and $V_H$ increases to 135°, the resultant velocities $V_D$ and $V_F$ continue to drop. 5. Point E: The spin velocity ($V_2$) and orbital velocity ($V_H$) are in opposite directions, resulting in a near-zero linear velocity $V_E=V_H-V_2\approx$0. The above points summarize the velocity superposition mechanism of the flexible ring.

Fig. 2Kinematic analysis schematic and flexible ring velocity contour plot

a) Kinematic analysis schematic

b) Flexible ring velocity contour plot

Fig. 3Velocity decomposition along flexible rings

The velocity synthesis mechanism was analyzed in conjunction with the velocity vector diagram (Fig. 4(a)). Based on vector addition principles, the theoretical velocity trajectory aligns well with actual conditions. To further validate the model, the results were compared with the operational contour plots of the flexible ring (Fig. 4(b)) to verify kinematic characteristics under pure rolling. The results show excellent agreement with theoretical predictions, demonstrating high reliability.

Fig. 4Flexible ring kinematics

a) Velocity vectors

b) Contours plot

2.2.2. Calculation of the flexible loop's rotational speed

To address the challenge of velocity extraction under coupled rotation and revolution, we adopted Wang Q. et al.’s [11] flexible ring method on an elastic foundation. The instantaneous resultant velocity is precisely acquired via vector summation of $Y$-$Z$ axis velocities and probe monitoring at key locations.

For precise dynamic calculation, the velocity is quantified using the inner ring speed ($V_1$), flexible ring rotation speed ($V_2$), and revolution speed ($V_3$). Contour analysis reveals that the maximum velocity occurs at the inner contact point ($V_{max}=V_2+V_3$), while the minimum occurs at the outer contact point ($V_{max}=V_3-V_2$). By extracting the global maximum and minimum velocities in the $YZ$ plane, $V_2$ and $V_3$ can be decoupled and solved simultaneously.

To validate accuracy, two probes on the flexible ring monitored real-time velocities. As shown in Fig. 7, both points exhibit periodic oscillations with maxima below 370 mm/s, indicating stability. The strong agreement between simulation data and theoretical calculations confirms the method's reliability.

Fig. 5Flexible ring and velocity curve diagram

2.2.3. Calculation of the rolling ratio for flexible rolling rings

The rotation and revolution velocities of the flexible ring are thus obtained. The inner contact velocity ($V_1$) equals the product of the inner ring's rotational speed and radius. The sliding velocity is the difference between $V_1$ and the flexible ring velocity. Accordingly, the slide-to-roll ratio is given by:

As shown in the table, the flexible ring's revolution velocity slightly exceeds its rotation velocity, and their sum is marginally below the inner ring velocity. The very low slide-to-roll ratio indicates negligible macroscopic sliding, confirming near-ideal pure rolling. The slight increase in this ratio under higher loads further demonstrates kinematic stability. These results validate that the proposed velocity vector synthesis method accurately captures the flexible ring’s true motion, providing a reliable basis for dynamic analysis.