# Application of Dynamic Time Warping (DTW) in Comparing MRT Signals of Steel Ropes
## 2 Materials and Methods
### MRT Methods
#### 2.1.1 Numerical Model of Search Coils  
Search coils LF1 and LF2--main components of the MRT measurement path--are the generic name of a widespread class of sensors consisting of loops of conducting wire exposed to a magnetic field **B** and generating an output voltage \(V_{C}\) via [PERSON]'s induction law [33; 34]. For a coil, shown in Figure 4a, made with \(N_{T}\) closed turns spanning an area \(A\) with normal unit vector **n** and boundary \(\partial A\), the output voltage \(V_{c}\) is given by the total rate of change of the linked magnetic flux \(\Phi\) as in Equation (4). The negative sign (from [PERSON]'s law) means that the induced electromotive force \(emf\) tends to generate a current which, by the right-hand rule, gives rise to a field opposing the variation in the flux. A typical  
Figure 3: Measuring track for MRT tests: (**a**) GP-type measuring head with a measuring wheel equipped with an incremental encoder [6]; (**b**) MD121 flaw defectograph [32]. distribution of the magnetic field in the vicinity of a round-twisted rope is illustrated in Figure 4b. The distribution of the magnetic field contains information about  
* The structural features of the rope--each outer strand generates a clear cyclic change in the magnetic field. Internal and medullary convolutions are also a source of weak cyclic magnetic anomalies. * The technical condition of the rope. \[V_{c}=-\frac{d\Phi}{dt}=-L_{c}\frac{di}{dt}=-\frac{d}{dt}\iint_{A}\textbf{Bn}dA =\iint_{A}\frac{\partial\textbf{B}}{dt}ndA-\oint_{\partial A}\textbf{v}x\textbf{ B}dl\] (4) where
* \(\rm V_{c}\) is the induced voltage across the coil,
* \(\Phi\) is the magnetic flux through the surface \(A\),
* \(L_{c}\) denotes the inductance of the coil, and \(di\) is the electric current flowing through it,
* \(\bf B\) is the magnetic flux density,
* \(\rm n\) is the unit normal vector to the surface,
* \(\rm v\) is the velocity of the conductor. Figure 4: The following were imaged: (**a**) a search coil [35]; (**b**) distribution of the magnetic field in the vicinity of the round-twisted rope inside the MRT measuring head—the external strands of the rope are a clear source of cyclostationary harmonic magnetic anomalies revealing the spike of rope twisting [36]; (**c**) mapping local defects and corrosion in the LF signal [37]. The output voltage can be derived from Equation (4) for three canonical cases as follows:  
\[-V_{c,i}=\frac{\partial\Phi}{\partial t}=\left\{ \begin{array}{l}A_{c}\frac{dB}{ dt}\\ A_{c}B\omega\\ A_{c}\
abla Bv\end{array} \right. \tag{5}\]  
where  
* \(A_{c}\) is the effective cross-sectional area of the coil,
* \(\omega\) is the angular velocity of rotation,
* \(\
abla B\) represents the spatial gradient of the magnetic field. The first case relates to a fixed coil in a time-varying magnetic field. The second case involves a coil rotating at angular speed \(\omega\) in a constant magnetic field. The third case concerns a coil translating at speed \(v\) in a constant field with a gradient. The dominant component of the induced voltage signal in coils LF1 and LF2 is the first case (Equation (5)), generated mainly by local wire damage (cracks, abrasion, spot corrosion). The second and third cases result from the design features of the measuring head and the lack of peripheral symmetry of the cylindrical coils and the longitudinal symmetry of the distribution of the magnetic field near the rope. The second case reproduces information about the torsional vibrations of the rope and the asymmetry of the arrangement of the rope strands around the circumference. The third case reflects the information about the longitudinal vibrations of the rope, the asymmetry of the strand pitch, and the influence of changes in the speed of the rope scanning on the position of the magnetic plane of the neutral measuring head. The signal from a coreless coil is described by Equation (6).