# Application of Dynamic Time Warping (DTW) in Comparing MRT Signals of Steel Ropes
## 1 Introduction
\[\mathrm{g\;N}=b_{0}+\left(b_{1}+b_{3}\cdot\lg\frac{D}{d}\right)\cdot\left(\lg \frac{S}{d^{2}}-0.4\cdot\lg\frac{R_{o}}{1770}\right)+b_{2}\cdot\lg\frac{D}{d}+ \lg f_{d}+\lg f_{L}+\lg f_{E} \tag{1}\]  
\[\lg N_{u}=\frac{\lg N}{K_{je}(\textit{user of rope})}\;\;\;\;\;K_{je}>1.0 \tag{2}\]  
where \(b_{\mathrm{i}}\)--constants for the wire ropes of the important rope constructions; \(f_{d}\), \(f_{L}\), and \(f_{\mathrm{c}}\)--the endurance factors; \(d\) [mm]--nominal rope diameter; \(D\) [mm]--sheave diameter, \(S\) [N]--rope tensile force; \(R_{0}\) [N/mm\({}^{2}\)]--nominal tensile strength; \(1\) [mm]--bending length. Figure 2: Examples of structural design solutions for steel wire ropes: (**a**) point contact between wires with stress concentration (left) and linear contact with stress distribution (right) [13]; (**b**) standard strand (left) and compacted strand (right) [14]; (**c**) Bridon Dyform® 6 and Bridon Tiger® Full Lock Guide ropes by Bekaert [15]; (**d**) Casar Stratoplast M [10]; (**e**) AreclorMittal wire rope solutions [16]. During operation, the wires and strands of the rope are subjected to simultaneous bending, torsion, and tension, which is described by a MIMO (Multi-Input Multi-Output) model. Since rope degradation depends on unknown user-specific variables and material scatter, the service life must be treated as a random variable. Thus, the failure risk is defined by Equation (3), in line with ISO/IEC Guide 73:2002. \[Risk=Probability\ of\ event\ \text{-Consequence\ of\ event} \tag{3}\]  
The safe operation of wire ropes exposed to  
* Corrosion;
* Premature rust and oxidation;
* Mechanical wear;
* Internal and external wire breaks;
* Fatigue in bends;
* Tension breaks;
* Core failure fatigue;
* Shear breaks;
* Heat damage;  
requires continuous technical monitoring. This is mandated by EU legislation [18] and national regulations [19]. Over the past 150 years, the sudden failure of steel wire ropes has led to numerous critical incidents in mine hoists, elevators, cableways, offshore cranes, and suspension bridges. Notable examples include the Robinson Deep Mine disaster [20], the Stresa-Mottarone cable car collapse [21], and the Morbi bridge failure [22; 23]. Common causes include undetected corrosion, fatigue degradation, and diagnostic errors linked to limitations of the current inspection techniques or human factors. In contrast to military systems--where daily inspections and redundancy protocols are enforced--most civilian users rely on periodic MRT or visual checks. However, visual inspections assess only -20% of the rope's cross-section, and MRT interpretation can be ambiguous due to signal noise, overlap, and indirect measurement [22; 23]. To improve diagnostics, methods such as acoustic emission (AE) [24], ultrasonics [25], FBG strain sensing [26], weak magnetic excitation [27], and modal analysis [28] have been proposed. Still, practical adoption is limited by equipment cost, complexity, and sensitivity to conditions. Both MRT and MPM share inherent weaknesses: they rely on indirect magnetic symptoms, suffer from defect signal averaging, and depend on operator interpretation. This introduces a trade-off between  
* Type I errors (false positives)--premature rope discard and increased carbon footprint (up to 2300 kg CO\({}_{2}\) per ton of steel),
* Type II errors (false negatives)--missed critical damage with safety risks. Moreover, advanced MRT algorithms typically focus on localized anomalies like broken wires or pitting corrosion [29]. They rarely address the deeper mechanical changes preceding visible failure such as magnetic permeability shifts, strand rearrangement, or changes in rope geometry due to elongation or compaction. These effects are particularly problematic in polymer-filled and compacted ropes, where progressive elongation during service life alters the relative position of rope segments between inspections. As a result, signal synchronization becomes increasingly unreliable over time, directly impacting discard criteria and diagnostic consistency [16].