#
Evaluating Contact-Less Sensing and Fault Diagnosis Characteristics in Vibrating Thin Cantilever Beams with a MetGlas^{®} 2826MB Ribbon

^{*}

## Abstract

**:**

^{®}2826MB ribbons onto the surface of thin cantilever polymer beams are examined and statistically evaluated in this study. Excitation of the beam’s free end generates magnetic flux from the vibrating ribbon (fixed near the clamp side), which, via a coil suspended above the ribbon surface, is recorded as voltage with an oscilloscope. Cost-efficient design and operation are key objectives of this setup since only conventional equipment (coil, oscilloscope) is used, whereas filtering, amplification and similar circuits are absent. A statistical framework for extending past findings on the relationship between spectral changes in voltage and fault occurrence is introduced. Currently, different levels of beam excitation (within a frequency range) are shown to result in statistically different voltage spectral changes (frequency shifts). The principle is also valid for loads (faults) of different magnitudes and/or locations on the beam for a given excitation. Testing with either various beam excitation frequencies or different loads (magnitude/locations) at a given excitation demonstrates that voltage spectral changes are statistically mapped onto excitation levels or occurrences of distinct faults (loads). Thus, conventional beams may cost-efficiently acquire contact-less sensing and fault diagnosis capabilities using limited hardware/equipment.

## 1. Introduction

_{2}O [9] or H

_{2}O

_{2}[10], which bind to suitable surface coating, will change its mass distribution and consequently its vibration characteristics (resonant frequencies). Hence, shifted resonant frequencies indicate a significant concentration of substances on it and, accordingly, the environment. This is the operational principle of magnetoelastic (magnetostrictive) sensors, which allow for monitoring dangerous substances in hostile environments without requiring human presence on the field since the strip vibration signal can obviously be remotely recorded and assessed.

^{®}2826MB) is attached at the clamped end (in contrast to [19,20,21]), with a low-cost reception coil suspended above the ribbon surface. The raw voltage induced in a contact-less manner is recorded using a conventional oscilloscope, without circuits for preliminary conditioning/filtering or amplification. Thus, the objective of obtaining sensing and diagnostic capabilities out of a low complexity (in terms of hardware and operation) setup, by investing in the optimization of the algorithmic framework used, is possible: It is shown that although the beam is only excited at the free end, the sensing of the beam excitation level and diagnosis of different structural changes (magnitude/position on the beam) are both achievable. By virtue of the current results, conventional long, flexible beams equipped with magnetoelastic elements may be used: (i) either for deducing the level of excitation (due to external forces, for instance) suffered by the beam (or any structure connected to it) or (ii) for detecting and localizing faults (loads) of different magnitude affecting the beam for a given level of excitation.

## 2. Materials and Methods

^{TM}K2004E01), a 25 mm long ribbon of Metglas

^{®}2826MB magnetoelastic material and a low-cost Vishay IWAS reception coil (normally used for wireless charging). The beam is 3D-printed in FDM (fused deposition modeling) mode with a PET-G filament and is used in a cantilever arrangement with one end clamped, as presented in Figure 1. The opposite (free) end is fixed to the exciter rod, thereby receiving the vibration of the user-defined profile. For this purpose, an external waveform generator (SIGLENT SDG 5122) is connected to the exciter. The magnetoelastic ribbon is fixed on the beam surface near the clamp with glue, whereas the reception coil is fixed 5 mm above the ribbon, thus bearing no contact with it. The distance of 5 mm was selected from sensitivity tests, as described in [24]. Magnetic flux created by the vibrating (along with the beam) ribbon induces voltage in the reception coil circuit, which is recorded with a digital oscilloscope. Based on the analysis of the recorded voltage’s spectral characteristics, sensing and fault diagnosis results may be obtained. Especially in terms of fault diagnosis, this approach based on using only one signal is representative of real-life applications (bridges, flexible structures and so on), because the excitation signal is often unavailable (or hard to measure) with only the structure’s response signal being available.

#### 2.1. Methodology for the Evaluation of the Sensing Characteristics of the Setup

_{0}: Frequency values in both (or all) groups follow a similar distribution.

H

_{1}: Frequency values in both (or all) groups follow different distributions.

_{0}referred to as the null hypothesis and H

_{1}as the alternative hypothesis. Note that there is no available information on whether the data in the previously mentioned groups follow a normal distribution; hence, non-parametric statistical tests must be used to choose between the null and alternative hypotheses in (1) at a given risk level α (usually equal to 0.05). The latter is the probability of rejecting H

_{0}even though it is true. Such non-parametric statistical tests include the Kolmogorov–Smirnov two-sample test and the Kruskal–Wallis test [25,26]. As its name suggests, the Kolmogorov–Smirnov two-sample test is designed to address a hypothesis problem such as that in (1), when comparisons between only two groups are considered. The null hypothesis is accepted (or rejected) based on the distance between the empirical distributions of data for each group estimated using the associated data. On the other hand, the Kruskal–Wallis test may be used for two or more groups of data [25,26,27] and provides an answer to the question of whether data in the groups under consideration follow similar statistical distributions. If these distributions have similar shapes, then the Kruskal–Wallis test accepts (or rejects) the null hypothesis based on whether the medians of all groups are sufficiently (in some statistical sense) close [25]. Furthermore, the Kruskal–Wallis test may be used with groups containing 5 or 6 data values, as shown in cases presented in [25,26], respectively. These characteristics motivated the choice of the Kruskal–Wallis test to address the hypothesis testing problem (1), which will be presented later (Section 3.1). The Kruskal–Wallis test is coded in most software packages like SPSS

^{®}or MATLAB

^{®}, with the relevant routines using data provided to instantly compute the probability value (referred to as the p-value), which evaluates the evidence against the null hypothesis. A lower p-value indicates more important evidence against accepting the null hypothesis. As will be explained in Section 3.1 and Section 3.2, the p-value may offer valuable information for quantifying the uncertainty (risk) involved when deciding on whether two or more frequency groups feature significant similarities: in other words, whether these groups potentially overlap in part or not.

#### 2.2. Methodology for Evaluating Fault Diagnosis Characteristics of the Setup

- The voltage signal considered was filtered and subsampled (details are given in Section 3.2 in [24]);
- Discrete-time stochastic AutoRegressive (AR) time-series representations were identified on the signal resulting from step 1 (thus modeling its spectral characteristics), and the discrete-time AR poles corresponding to specific bands of the dominant frequencies were computed and plotted on the z-plane;
- Using the AR poles from step 2, the corresponding continuous-time poles were computed and plotted on the s-plane, thus enabling the calculation of the natural frequencies ω
_{n}and damping ratios ζ for the considered bands of dominant frequencies.

_{n}and damping ratios ζ for the considered bands of dominant frequencies is not needed. This fact is particularly promising because in [24], no conclusive (or even indicative) evidence of z-plane pole areas being able to be mapped onto fault cases (test scenarios) was found.

_{0}: Pole locations inside the considered groups follow a similar distribution.

H

_{1}: Pole locations inside the considered groups follow different distributions.

_{0}is the null hypothesis potentially corresponding to (neighboring) s- or z-plane areas with significant overlapping and H

_{1}is the alternative hypothesis designating substantially separable areas at a given risk level. As with the frequency data used to evaluate the setup sensing characteristics, there is no available knowledge of pole locations following a normal distribution. Then, non-parametric statistical tests should be used to decide between the null and alternative hypotheses in (2) at a given risk level α (usually equal to 0.05 or 5%), which is the probability of the examined pole areas not being considered as overlapping (or, in other terms, that H

_{0}is rejected) even though they are. The Kruskal–Wallis test may again be used to solve the hypothesis testing problem (2), as will be presented later (Section 3.2).

## 3. Results and Discussion

#### 3.1. Results of the Statistical Evaluation of Sensing Characteristics

_{0}(groups with a similar distribution of data) is rejected in favor of H

_{1}(groups with a different distribution of data) at a risk level equal to α = 0.05, as explained in Section 2.1. In such cases, the two groups considered are mutually distinguishable at the indicated risk level, meaning that the previously mentioned mapping is valid, again, at the indicated risk level. On the other hand, the shaded cells correspond to cases where H

_{0}(groups with a similar distribution of data) is accepted for the pair of groups under consideration, at a risk level equal to α = 0.05. Then, the two considered excitation frequencies would result in significantly overlapping groups of frequency peaks in the band of interest, meaning that no exclusive one-to-one mapping is possible. An examination of Table 3, Table 4, Table 5 and Table 6 basically validates the conclusions drawn by visual inspection of (the corresponding) Figure 3, Figure 4, Figure 5 and Figure 6. The frequency band around 1400 Hz provides statistically non-overlapping groups at a risk level of 0.05 (Table 4) for beam excitation frequencies up to 115 Hz. Then, the excitation levels of 115 Hz and 130 Hz create overlapping groups (at a risk level of 0.05), since the p-value computed with the Kruskal–Wallis test is 0.055 (see the intersection between the ninth line and the tenth column), or just larger than 0.05, which leads to accepting the null hypothesis. At the same time, p-values just larger than the risk level indicate a statistical tendency of being close to rejecting H

_{0}. In other words, even though the 1400 Hz band does not actually allow for distinguishing between the excitation levels of 115 and 130 Hz, it would be relevant to look at other frequency bands for the null hypothesis being rejected when comparing the groups associated with the levels of 115 and 130 Hz. The band at 316 KHz offers this possibility since the p-value (related to comparing groups created by the excitation levels of 115 Hz and 130 Hz—Table 3) computed with the Kruskal–Wallis test is 0.0077 (see the intersection between the ninth line and the tenth column). However, in the band around 316 KHz, all groups created by the excitation levels above 130 Hz are statistically similar, as indicated by a p-value of the Kruskal–Wallis test statistic equal to 0.2738 > 0.05 when comparing all three groups corresponding to the excitation levels of 130, 145 and 160 Hz. Then, distinguishing between the excitation levels of 115 and 130 Hz in the 316 KHz band may only be achieved in conjunction with p-values from the band around 2100 Hz in Table 6. The latter indicates that the group resulting from the beam excited at 130 Hz cannot be mistaken for that associated with the excitation of 145 Hz at the considered risk level of 0.05. Thus, frequency peak groups resulting from excitation levels up to 145 Hz may be distinguished in a one-to-one comparison using the proposed setup and methodology. Figure 3, Figure 4, Figure 5 and Figure 6 and Table 3, Table 4, Table 5 and Table 6 suggest that a meaningful solution for distinguishing groups associated with excitation levels up to 145 Hz from a group associated with the excitation levels of 160 Hz is not available in all four bands considered.

#### 3.2. Results of the Statistical Evaluation of Fault Diagnosis Characteristics

_{0}is systematically rejected for any comparison of N-1C against the A, B or C-1C fault cases, at α = 0.05, since in all such cases, the p-value is lower than α. In other words, the (imaginary parts of the) poles from the N-1C configuration do not follow a similar distribution as those from the A, B or C-1C configurations at α = 0.05, and faults may be systematically detected. Again, in Table 7, H

_{0}is systematically rejected for any comparison of N-1C against the A, B or C-1C fault cases at α = 0.05. Hence, all –1C fault cases have different impacts on the pole (imaginary) locations, meaning that all –1C faults may be identified. Note, however, that the p-value for comparing the A-1C and B-1C configurations is notably higher (see the intersection between the third line and the fourth column), although smaller than α = 0.05. This is related to the overlap between the two groups, as seen in Figure 7b, which was commented upon earlier on. The same conclusions in terms of detection and identification may be drawn for the –BN configurations with a careful examination of Table 8. Again, comparing the B-BN and C-BN configurations results in a higher-than-usual p-value (although again smaller than α = 0.05), which is related to the slight overlap of groups designating the B-BN and C-BN fault cases in Figure 8b. Lastly, Table 9 allows for addressing the issue of distinguishing between the fault configurations –1C (small fault magnitude) and –BN (large fault magnitude), as defined in Table 1. Again, the shaded cells correspond to significantly overlapping groups of poles. In general, the p-values are always smaller than α = 0.05, meaning that H

_{0}is systematically rejected at the risk level α = 0.05 for all comparisons between the –1C and –BN configurations. Then, no –BN fault may be mistaken for a –1C fault at the designated risk level. Obviously, these results are valid for cases of a single fault (load) occurrence at a time.

_{0}systematically rejected for any comparison of N-1C against the A, B or C-1C fault cases, at α = 0.05, as shown in Table 10.

_{0}systematically rejected for any comparison of N-BN against the A, B or C-BN fault cases, at α = 0.05. Again, one may distinguish between the s –1C (small fault) and –BN (large fault) configurations in Table 1 using rotation angles of the discrete-time poles in the z-plane to form Table 12. As in the continuous-time case, the p-values are always smaller than α = 0.05, meaning that H

_{0}is systematically rejected at the risk level α = 0.05 for all comparisons between the –1C and –BN configurations. Then, no –BN fault may be mistaken for a –1C fault at the designated risk level, even using discrete-time AR poles. As in the continuous-time case, these results are valid for cases of a single fault (load) occurrence at a time.

_{0}at the designated risk level. This, in turn, means that the results presented both for sensing and for fault diagnosis purposes are rather conservative. In terms of evaluating sensing characteristics, for instance, if more data per group were available, then comparisons between certain groups at the band around 1400 Hz (which indicated overlapping groups due to the p-values being marginally higher than α = 0.05) could yield results toward rejecting H

_{0}, thus enabling a distinction between the groups considered.

## 4. Conclusions

^{®}2826MB ribbon attached to its surface was statistically evaluated in terms of contact-less sensing and fault diagnosis characteristics. The vibration of the beam’s free end creates the emission of magnetic flux by the Metglas

^{®}ribbon (fixed on the opposite end), which induces a voltage in a coil suspended over the film. This voltage is, hence, obtained in a contact-less manner and is recorded with an oscilloscope. The voltage signal analysis showed that shifting of the dominant frequencies may result either from changes in the excitation frequency provided to the beam or from faults (loads) of various magnitudes and positions on the beam when the latter vibrates at a given frequency. A statistical framework based on the formulation of statistical hypothesis problems was introduced to evaluate such frequency-shifting characteristics, which led to two main results. First, a mapping between the vibration frequency level of the beam and the resulting frequency shifts observed in the recorded voltage was statistically established. Hence, sensing properties were obtained because the vibration level of the beam may be deduced by monitoring frequency shifting patterns in the voltage signal, with the uncertainty in the process quantified. Second, s-plane or z-plane areas containing poles corresponding to shifted frequencies of the voltage signal (modeled as per Section 2.2) were statistically linked to faults (loads) of specific magnitude and position affecting the beam. Hence, fault diagnosis properties were obtained because the occurrence, magnitude and position of faults (loads) on the beam may be deduced by checking the s-plane or z-plane pole locations, with the uncertainty in the process quantified. Future work will involve validating the setup’s sensing and fault diagnosis performance for low excitation frequency and/or high beam deflection and/or cases of multiple fault occurrence. It would be equally useful to evaluate the impact of integrating multiple sensing sets (ribbon and coils) on the beam.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Experimental setup indicating a load (coin) at position A, other load positions B and C (middle and left) and the arrangement of the reception coil and magnetoelastic film.

Fault Case (Test Scenario) | Load Used | Load Mass (g) | Load Position (From Free End) |
---|---|---|---|

N-1C | No load | 0 | n/a |

A-1C | EUR 1 cent | 2.3 | A (35 mm) |

B-1C | EUR 1 cent | 2.3 | B (185 mm) |

C-1C | EUR 1 cent | 2.3 | C (360 mm) |

N-BN | No load | 0 | n/a |

A-BN | Bolt + nut | 6 | A (35 mm) |

B-BN | Bolt + nut | 6 | B (185 mm) |

C-BN | Bolt + nut | 6 | C (360 mm) |

Excitation Frequency (Hz) | Test Series | Number of Experiments |
---|---|---|

0 | s0 | 6 |

10 | s1 | 6 |

40 | s2 | 6 |

55 | s3 | 6 |

70 | s4 | 6 |

85 | s5 | 6 |

100 | s6 | 6 |

115 | s7 | 6 |

130 | s8 | 6 |

145 | s9 | 6 |

160 | s10 | 6 |

**Table 3.**p-values for the comparison of two groups with the Kruskal–Wallis test in the band of 316 KHz.

Excitation | 0 Hz | 10 Hz | 40 Hz | 55 Hz | 70 Hz | 85 Hz | 100 Hz | 115 Hz | 130 Hz | 145 Hz | 160 Hz |
---|---|---|---|---|---|---|---|---|---|---|---|

0 Hz | N/A | 0.0032 | 0.0032 | 0.0038 | 0.0036 | 0.0036 | 0.0032 | 0.0032 | 0.0036 | 0.0036 | 0.0032 |

10 Hz | 0.0032 | N/A | 0.0027 | 0.0032 | 0.0031 | 0.0031 | 0.0028 | 0.0027 | 0.0031 | 0.0031 | 0.0027 |

40 Hz | 0.0032 | 0.0027 | N/A | 0.0032 | 0.0031 | 0.0031 | 0.0028 | 0.0027 | 0.0031 | 0.0031 | 0.0027 |

55 Hz | 0.0038 | 0.0032 | 0.0032 | N/A | 0.0037 | 0.0037 | 0.0033 | 0.0032 | 0.0037 | 0.0037 | 0.0032 |

70 Hz | 0.0036 | 0.0031 | 0.0031 | 0.0037 | N/A | 0.0086 | 0.0032 | 0.0031 | 0.0036 | 0.0036 | 0.0031 |

85 Hz | 0.0036 | 0.0031 | 0.0031 | 0.0037 | 0.0086 | N/A | 0.3880 | 0.4844 | 0.0086 | 0.0086 | 0.0031 |

100 Hz | 0.0032 | 0.0028 | 0.0028 | 0.0033 | 0.0032 | 0.3880 | N/A | 0.8474 | 0.0203 | 0.0203 | 0.0045 |

115 Hz | 0.0032 | 0.0027 | 0.0027 | 0.0032 | 0.0031 | 0.4844 | 0.8474 | N/A | 0.0077 | 0.0077 | 0.0027 |

130 Hz | 0.0036 | 0.0031 | 0.0031 | 0.0037 | 0.0036 | 0.0086 | 0.0203 | 0.0077 | N/A | 1.0000 | 0.1620 |

145 Hz | 0.0036 | 0.0031 | 0.0031 | 0.0037 | 0.0036 | 0.0086 | 0.0203 | 0.0077 | 1.0000 | N/A | 0.1620 |

160 Hz | 0.0032 | 0.0027 | 0.0027 | 0.0032 | 0.0031 | 0.0031 | 0.0045 | 0.0027 | 0.1620 | 0.1620 | N/A |

**Table 4.**p-values for the comparison of two groups with the Kruskal–Wallis test in the band of 1400 Hz.

Excitation | 0 Hz | 10 Hz | 40 Hz | 55 Hz | 70 Hz | 85 Hz | 100 Hz | 115 Hz | 130 Hz | 145 Hz | 160 Hz |
---|---|---|---|---|---|---|---|---|---|---|---|

0 Hz | N/A | 0.0038 | 0.0036 | 0.0037 | 0.0036 | 0.0032 | 0.0032 | 0.0020 | 0.0033 | 0.0027 | 0.0032 |

10 Hz | 0.0038 | N/A | 0.0037 | 0.0038 | 0.0037 | 0.0032 | 0.0032 | 0.0021 | 0.0034 | 0.0028 | 0.0032 |

40 Hz | 0.0036 | 0.0037 | N/A | 0.0036 | 0.0036 | 0.0031 | 0.0031 | 0.0020 | 0.0033 | 0.0026 | 0.0031 |

55 Hz | 0.0037 | 0.0038 | 0.0036 | N/A | 0.0036 | 0.0032 | 0.0032 | 0.0020 | 0.0033 | 0.0027 | 0.0032 |

70 Hz | 0.0036 | 0.0037 | 0.0036 | 0.0036 | N/A | 0.0031 | 0.0031 | 0.0020 | 0.0033 | 0.0026 | 0.0031 |

85 Hz | 0.0032 | 0.0032 | 0.0031 | 0.0032 | 0.0031 | N/A | 0.0027 | 0.0017 | 0.0029 | 0.0023 | 0.0027 |

100 Hz | 0.0032 | 0.0032 | 0.0031 | 0.0032 | 0.0031 | 0.0027 | N/A | 0.0190 | 0.0105 | 0.0037 | 0.0144 |

115 Hz | 0.0020 | 0.0021 | 0.0020 | 0.0020 | 0.0020 | 0.0017 | 0.0190 | N/A | 0.0555 | 0.0051 | 0.1380 |

130 Hz | 0.0033 | 0.0034 | 0.0033 | 0.0033 | 0.0033 | 0.0029 | 0.0105 | 0.0555 | N/A | 0.2410 | 0.5751 |

145 Hz | 0.0027 | 0.0028 | 0.0026 | 0.0027 | 0.0026 | 0.0023 | 0.0037 | 0.0051 | 0.2410 | N/A | 0.0926 |

160 Hz | 0.0032 | 0.0032 | 0.0031 | 0.0032 | 0.0031 | 0.0027 | 0.0144 | 0.1380 | 0.5751 | 0.0926 | N/A |

**Table 5.**p-values for the comparison of two groups with the Kruskal–Wallis test in the band of 1800 Hz.

Excitation | 0 Hz | 10 Hz | 40 Hz | 55 Hz | 70 Hz | 85 Hz | 100 Hz | 115 Hz | 130 Hz | 145 Hz | 160 Hz |
---|---|---|---|---|---|---|---|---|---|---|---|

0 Hz | N/A | 0.0037 | 0.0036 | 0.0038 | 0.0036 | 0.0036 | 0.0035 | 0.0032 | 0.0020 | 0.0033 | 0.0027 |

10 Hz | 0.0037 | N/A | 0.0036 | 0.0038 | 0.0036 | 0.0036 | 0.0035 | 0.0032 | 0.0020 | 0.0033 | 0.0027 |

40 Hz | 0.0036 | 0.0036 | N/A | 0.0037 | 0.0036 | 0.0036 | 0.0035 | 0.0031 | 0.0020 | 0.0033 | 0.0026 |

55 Hz | 0.0038 | 0.0038 | 0.0037 | N/A | 0.0037 | 0.0037 | 0.0036 | 0.0032 | 0.0021 | 0.0034 | 0.0028 |

70 Hz | 0.0036 | 0.0036 | 0.0036 | 0.0037 | N/A | 0.0036 | 0.0035 | 0.0031 | 0.0020 | 0.0033 | 0.0026 |

85 Hz | 0.0036 | 0.0036 | 0.0036 | 0.0037 | 0.0036 | N/A | 0.0084 | 0.0031 | 0.0020 | 0.0033 | 0.0026 |

100 Hz | 0.0035 | 0.0035 | 0.0035 | 0.0036 | 0.0035 | 0.0084 | N/A | 0.0570 | 0.0068 | 0.0062 | 0.0074 |

115 Hz | 0.0032 | 0.0032 | 0.0031 | 0.0032 | 0.0031 | 0.0031 | 0.0570 | N/A | 0.1380 | 0.0303 | 0.0917 |

130 Hz | 0.0020 | 0.0020 | 0.0020 | 0.0021 | 0.0020 | 0.0020 | 0.0068 | 0.1380 | N/A | 0.0555 | 0.3173 |

145 Hz | 0.0033 | 0.0033 | 0.0033 | 0.0034 | 0.0033 | 0.0033 | 0.0062 | 0.0303 | 0.0555 | N/A | 0.2410 |

160 Hz | 0.0027 | 0.0027 | 0.0026 | 0.0028 | 0.0026 | 0.0026 | 0.0074 | 0.0917 | 0.3173 | 0.2410 | N/A |

**Table 6.**p-values for the comparison of two groups with the Kruskal–Wallis test in the band of 2100 Hz.

Excitation | 0 Hz | 10 Hz | 40 Hz | 55 Hz | 70 Hz | 85 Hz | 100 Hz | 115 Hz | 130 Hz | 145 Hz | 160 Hz |
---|---|---|---|---|---|---|---|---|---|---|---|

0 Hz | N/A | 0.0037 | 0.0036 | 0.0038 | 0.0036 | 0.0035 | 0.0037 | 0.0033 | 0.0020 | 0.0032 | 0.0020 |

10 Hz | 0.0037 | N/A | 0.0036 | 0.0038 | 0.0036 | 0.0035 | 0.0037 | 0.0033 | 0.0020 | 0.0032 | 0.0020 |

40 Hz | 0.0036 | 0.0036 | N/A | 0.0038 | 0.0036 | 0.0035 | 0.0036 | 0.0033 | 0.0020 | 0.0031 | 0.0020 |

55 Hz | 0.0038 | 0.0038 | 0.0038 | N/A | 0.0038 | 0.0036 | 0.0038 | 0.0035 | 0.0021 | 0.0033 | 0.0021 |

70 Hz | 0.0036 | 0.0036 | 0.0036 | 0.0038 | N/A | 0.0035 | 0.0036 | 0.0033 | 0.0020 | 0.0031 | 0.0020 |

85 Hz | 0.0035 | 0.0035 | 0.0035 | 0.0036 | 0.0035 | N/A | 0.0068 | 0.0032 | 0.0019 | 0.0030 | 0.0019 |

100 Hz | 0.0037 | 0.0037 | 0.0036 | 0.0038 | 0.0036 | 0.0068 | N/A | 0.0750 | 0.0071 | 0.0051 | 0.0071 |

115 Hz | 0.0033 | 0.0033 | 0.0033 | 0.0035 | 0.0033 | 0.0032 | 0.0750 | N/A | 0.0555 | 0.0105 | 0.0555 |

130 Hz | 0.0020 | 0.0020 | 0.0020 | 0.0021 | 0.0020 | 0.0019 | 0.0071 | 0.0555 | N/A | 0.0190 | 1 |

145 Hz | 0.0032 | 0.0032 | 0.0031 | 0.0033 | 0.0031 | 0.0030 | 0.0051 | 0.0105 | 0.0190 | N/A | 0.0190 |

160 Hz | 0.0020 | 0.0020 | 0.0020 | 0.0021 | 0.0020 | 0.0019 | 0.0071 | 0.0555 | 1 | 0.0190 | N/A |

**Table 7.**p-values for the comparison of two groups in the s-plane with the Kruskal–Wallis test for the fault detection and localization of –1C cases.

Fault Case | N-1C | A-1C | B-1C | C-1C |
---|---|---|---|---|

N-1C | N/A | 0.0039 | 0.0039 | 0.0039 |

A-1C | 0.0039 | N/A | 0.0104 | 0.0039 |

B-1C | 0.0039 | 0.0104 | N/A | 0.0039 |

C-1C | 0.0039 | 0.0039 | 0.0039 | N/A |

**Table 8.**p-values for the comparison of two groups in the s-plane with the Kruskal–Wallis test for the fault detection and localization of –BN cases.

Fault Case | N-BN | A-BN | B-BN | C-BN |
---|---|---|---|---|

N-BN | N/A | 0.0039 | 0.0039 | 0.0039 |

A-BN | 0.0039 | N/A | 0.0039 | 0.0039 |

B-BN | 0.0039 | 0.0039 | N/A | 0.0374 |

C-BN | 0.0039 | 0.0039 | 0.0374 | N/A |

**Table 9.**p-values for the comparison of two groups in the s-plane with the Kruskal–Wallis test for fault localization and magnitude estimation between the –1C (small fault) and –BN (large fault) cases.

Fault Case | A-1C | B-1C | C-1C | A-BN | B-BN | C-BN |
---|---|---|---|---|---|---|

A-1C | N/A | 0.0104 | 0.0039 | 0.0039 | 0.0039 | 0.0039 |

B-1C | 0.0104 | N/A | 0.0039 | 0.0039 | 0.0039 | 0.0039 |

C-1C | 0.0036 | 0.0036 | N/A | 0.0038 | 0.0036 | 0.0035 |

A-BN | 0.0038 | 0.0038 | 0.0038 | N/A | 0.0038 | 0.0036 |

B-BN | 0.0036 | 0.0036 | 0.0036 | 0.0038 | N/A | 0.0035 |

C-BN | 0.0035 | 0.0035 | 0.0035 | 0.0036 | 0.0035 | N/A |

**Table 10.**p-values for the comparison of two groups in the z-plane with the Kruskal–Wallis test for the fault detection and localization of –1C cases.

Fault Case | N-1C | A-1C | B-1C | C-1C |
---|---|---|---|---|

N-1C | N/A | 0.0163 | 0.0039 | 0.0039 |

A-1C | 0.0163 | N/A | 0.0065 | 0.0039 |

B-1C | 0.0039 | 0.0065 | N/A | 0.0374 |

C-1C | 0.0039 | 0.0039 | 0.0374 | N/A |

**Table 11.**p-values for the comparison of two groups in the z-plane with the Kruskal–Wallis test for the fault detection and localization of –BN cases.

Fault Case | N-ΒΝ | A-ΒΝ | B-ΒΝ | C-ΒΝ |
---|---|---|---|---|

N-BN | N/A | 0.0039 | 0.0039 | 0.0039 |

A-BN | 0.0039 | N/A | 0.0039 | 0.0039 |

B-BN | 0.0039 | 0.0039 | N/A | 0.0374 |

C-BN | 0.0039 | 0.0039 | 0.0374 | N/A |

**Table 12.**p-values for the comparison of two groups in the z-plane with the Kruskal–Wallis test for fault localization and magnitude estimation between the –1C (small load) and –BN (big load) cases.

Fault Case | A-1C | B-1C | C-1C | A-BN | B-BN | C-BN |
---|---|---|---|---|---|---|

A-1C | N/A | 0.0065 | 0.0039 | 0.0039 | 0.0039 | 0.0039 |

B-1C | 0.0065 | N/A | 0.0374 | 0.0039 | 0.0039 | 0.0039 |

C-1C | 0.0039 | 0.0374 | N/A | 0.0039 | 0.0039 | 0.0039 |

A-BN | 0.0039 | 0.0039 | 0.0039 | N/A | 0.0039 | 0.0039 |

B-BN | 0.0039 | 0.0039 | 0.0039 | 0.0039 | N/A | 0.0374 |

C-BN | 0.0039 | 0.0039 | 0.0039 | 0.0039 | 0.0374 | N/A |

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## Share and Cite

**MDPI and ACS Style**

Sultana, R.-G.; Davrados, A.; Dimogianopoulos, D.
Evaluating Contact-Less Sensing and Fault Diagnosis Characteristics in Vibrating Thin Cantilever Beams with a MetGlas^{®} 2826MB Ribbon. *Vibration* **2024**, *7*, 36-52.
https://doi.org/10.3390/vibration7010002

**AMA Style**

Sultana R-G, Davrados A, Dimogianopoulos D.
Evaluating Contact-Less Sensing and Fault Diagnosis Characteristics in Vibrating Thin Cantilever Beams with a MetGlas^{®} 2826MB Ribbon. *Vibration*. 2024; 7(1):36-52.
https://doi.org/10.3390/vibration7010002

**Chicago/Turabian Style**

Sultana, Robert-Gabriel, Achilleas Davrados, and Dimitrios Dimogianopoulos.
2024. "Evaluating Contact-Less Sensing and Fault Diagnosis Characteristics in Vibrating Thin Cantilever Beams with a MetGlas^{®} 2826MB Ribbon" *Vibration* 7, no. 1: 36-52.
https://doi.org/10.3390/vibration7010002