## 5 Conclusions  
We investigated the distribution pattern of the magnetic induction density \(B_{\rm z}\) of a permanent magnet and the relationship between the anchor cable tensile force and magnetic induction density through theoretical model derivation, simulations, and laboratory experiments. A new anchor cable force monitoring technique based on the magnetic intensity distribution of permanent magnets was proposed. Our conclusions are as follows:  
1. By combining the molecular circulation model and the Biot-Savart law, an analytical expression for the spatial magnetic field distribution of a rectangular permanent magnet was derived. This expression was then coupled with the stress-strain characteristics of the steel strand to establish a theoretical monitoring model that relates the anchor cable's tensile force to the magnetic induction density of the permanent magnet. By monitoring the magnetic signal, the magnitude of the anchor cable force can be determined, thereby enabling effective anchor cable force monitoring. 2. Using MATLAB, the distribution pattern of the magnetic induction density \(B_{\rm z}\) and the force-magnetism relationship in the anchor cable force monitoring model were studied. The results show that the z-component of the magnetic induction density \(B_{\rm z}\) is symmetrically distributed about the central point. On the surface of the permanent magnet, the variation in \(B_{\rm z}\) forms a \"downward concave\" curve, and as the lift-off height z increases, \(B_{\rm z}\) transforms into an \"upward convex\" curve, approximately following a quadratic distribution. Notably, the value of \(B_{\rm z}\) decreases as the tensile force increases. 3. The indoor experimental results indicate that the magnetic induction density \(B_{\rm z}\) decreases as the tensile force increases, demonstrating a negative correlation consistent with the simulation results. The quadratic function effectively describes the relationship between the force and the magnetic field. Repeatability analysis of the test data shows that the standard deviation of the magnetic induction density \(B_{\rm z}\) collected under the design loads is less than 1.5%, indicating low data dispersion and good stability of the monitoring technique. These results verify the effectiveness and applicability of the proposed monitoring technique. 4. Given the constraints of the present study, we investigated the magnetic signal distribution characteristics along the \(z\)-axis of the rectangular permanent magnet sensing structure, along with their correlation with the applied load. Further research could address the three-dimensional magnetic signal distribution and its associated relationship with the load. 5. The feasibility and applicability of all of the sensing structures were confirmed through theoretical derivation, model validation, and laboratory testing. However, during the fabrication process of the sensor structure, alignment errors between the Hall probe and the centerline of the permanent magnet, along with the lift-off height of the Hall probe, can impact the measurement accuracy. If the Hall probe is misaligned from the centerline, the measurement results will fail to reflect the actual variation in magnetic field strength, resulting in inaccurate measurements. During the fab \begin{table}
\begin{tabular}{c c c c} \hline
**Types of Sensor** & **Accuracy** & **Reliability** & **Cost** \\ \hline The sensor proposed in this paper & High & High & Low \\ \hline Vibrating-wire sensors & High & High & Moderate \\ \hline Magnetic sensors & Ultrahigh & Moderate & High \\ \hline Optical fiber sensors & Ultrahigh & Moderate & Ultrahigh \\ \hline \end{tabular}
\end{table} 
Table 5: Performance comparison of different sensors. rication of the sensor, improving the machining precision can ensure that the Hall probe consistently measures the magnetic field strength along the centerline of the permanent magnet.