# Experimental Assessments on the Evaluation of Wire Rope Characteristics as Helical Symmetrical Multi-body Ensembles

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

#### 2.2. Tested Wire Rope Sample Characteristics

_{u}of the cable, and, respectively, the nominal rope length-related mass m.

^{2}= 9.05 mm

^{2}. Respectively, with a rope mass factor w = 0.306, the nominal rope length-related mass, evaluated with widely accepted empirical expression m = 0.01 wd

^{2}, acquires 0.11016 kg/m, which accurately approximates the initially provided value from the producer.

_{wr}of a wire rope was evaluated using Hruska’s approach [8]:

_{steel}denotes the steel Young’s modulus, m is the total number of the rope strands, n denotes the total number of wire layers in each strand, including the king wire that is considered the first layer, and S

_{i}is the cross-sectional area of wire within layer i of a strand j. The angles α

_{i}and β

_{j}respectively denote the lay angle of layer i in strand j and the lay angle of the strand j of the wire rope. Analyzing the denominator of the right-hand-side term in Equation (1), it can obviously be seen that it represents the total net steel area of the normal cross-section of the wire rope. The equation uses the hypotheses of a king wire having α

_{i}= 0 and the elliptical shapes of wires and strands within the normal cross-section of the rope.

_{s}denotes the diameter of the strand circular distribution, D is the wire rope diameter, L is the cable sample length, h

_{s}denotes the strand step, and β is the lay angle of the strand. The evaluation of the lay angle of the wires within the ropes, α

_{i}, can be obviously obtained based on the same procedure.

_{i}and angles α

_{i}and β

_{i}have constant values, Equation (1) acquires a simple expression in respect to these angles. For the inspected rope type, the strand diameter is 2 mm, the strands step is 46 mm, and the wires step (within the strand) is 20 mm. Hereby, H = 0.8152 and the axial stiffness E

_{wr}= 1.712 × 10

^{5}MPa (assuming E

_{steel}= 2.1 × 10

^{5}MPa), which is a value that is framed by the available literature data [23]. In addition, supposing the usual elongation of ε ≅ 1.7%, the stress yield σ ≅ 2910 MPa (also in the range of available values for this kind of small diameter FC steel wire ropes).

#### 2.3. Post-Processing Techniques

_{k}denotes the amplitude, T

_{s}denotes the sampling period, α

_{k}denotes the damping factor, ω

_{k}denotes the angular velocity, ψ

_{k}denotes the initial phase, and j

^{2}= −1. This discrete-time expression can be concisely formulated in the form of the following:

**Y**is considered, with the pencil parameter p, in the following form [25,28]:

**Y**helps to obtain the matrices

**Y**

_{1}and

**Y**

_{2}. Thus,

**Y**

_{1}results from

**Y**without the last column and, respectively,

**Y**

_{2}results from

**Y**, but without the first column [25,26,27,28]:

_{k}, where (k = 1...p), are the generalized eigenvalues of the matrix pencil $\left({Y}_{2}-\lambda {Y}_{1}\right)$. Taking into account that matrices

**Y**

_{1}and

**Y**

_{2}are ill-conditioned, the QZ algorithm is not stable enough to provide adequate results. In this case, it is more accurate to use the following expression:

**Y**

_{1}

^{+}denotes the Moore–Penrose pseudo-inverse matrix of

**Y**

_{1}. In such conditions, the damping b

_{k}and the frequency f

_{k}= 2πω

_{k}results from the eigenvalues of z

_{k}[25,28].

_{k}and the phase ψ

_{k}[25,28] may be found:

## 3. Results

_{cut}= 400 Hz) was provided by the force transducer. For both cases, multiple tests were performed. Two situations for each case were adopted for graphical presentation.

## 4. Discussion

^{5}Nm

^{−1}. The experimental results provided rather different values for the cable rigidity because of dynamic effects (in fact, we discussed the dynamic rigidity). Thus, the values of this parameter acquired the following values: 9.8 × 10

^{5}Nm

^{−1}for the case of m = 8 kg and 24 × 10

^{5}Nm

^{−1}for the case of m = 19 kg.

^{−1}, which presents a 2.1% and 4.7% maximum deviation related to the computed values.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The laboratory experimental setup: (

**a**) General view with a latticed tower, wire rope sample, static charging device, and compute-based acquisition system; (

**b**) detailed view of the upper side of the tower, with the hung rope device and residual motion monitoring transducers; (

**c**) detailed view of the wire rope charging device with an additional mass applied and transducers for monitoring the forces and oscillations, respectively.

**Figure 2.**The evaluation of the wire rope geometrical parameters: (

**a**) The model of the strands wire rope, used for evaluation of the parameters involved in the lay angle computation of the strand and the rope, respectively [23] (p. 25); (

**b**) detailed images of the wire rope used for the experiments, showing longitudinal and sectional views.

**Figure 3.**The raw signals of acceleration in terms of both the timed evolution and the spectral magnitude: (

**a**) First situation within the case of m = 8 kg; (

**b**) second situation within the case of m = 8 kg; (

**c**) first situation within the case of m = 19 kg; (

**d**) second situation within case of m = 19 kg.

**Figure 4.**The raw signals from the force transducer in terms of the timed evolution and spectral magnitude: (

**a**) First situation within the case of m = 8 kg; (

**b**) second situation within the case of m = 8 kg; (

**c**) first situation within the case of m = 19 kg; (

**d**) second situation within the case of m = 19 kg.

**Figure 5.**Joint time-frequency analysis of the absolute motion of the loading mass in terms of the acceleration signal: (

**a**) First situation within the case of m = 8 kg; (

**b**) second situation within the case of m = 8 kg; (

**c**) first situation within the case of m = 19 kg; (

**d**) second situation within the case of m = 19 kg.

**Figure 6.**Timed evolution and related spectral magnitude of the absolute motion in terms of acceleration, recorded at the loading mass: (

**a**) First situation within the case of m = 8 kg; (

**b**) second situation within the case of m = 8 kg; (

**c**) first situation within the case of m = 19 kg; (

**d**) second situation within the case of m = 19 kg. Note: Red circles on the right-side diagrams denote the maximum peaks satisfying the imposed conditions (see text for details).

**Figure 7.**Transfer function of the tested ensemble, comparatively presented with the input and output spectra (magnitudes): (

**a**) First situation within the case of m = 8 kg; (

**b**) second situation within the case of m = 8 kg; (

**c**) first situation within the case of m = 19 kg; (

**d**) second situation within the case of m = 19 kg.

**Figure 8.**Dynamic rigidity spectra: (

**a**) First situation within the case of m = 8 kg; (

**b**) second situation within the case of m = 8 kg; (

**c**) first situation within the case of m = 19 kg; (

**d**) second situation within the case of m = 19 kg.

**Figure 9.**Amplitudes, damping factor, and damping ratio provided by the Prony method as functions of the modal frequencies (according to the first 512 terms of signal decomposition): (

**a**) First situation within the case of m = 8 kg; (

**b**) second situation within the case of m = 8 kg; (

**c**) first situation within the case of m = 19 kg; (

**d**) second situation within the case of m = 19 kg. Red dashed lines within the damping ratio diagrams denote an equivalent damping ratio (see text for details).

**Figure 10.**Double-spectra evolution of the first 512 terms in the exponential function decomposition according to the Prony method with the first test within the case of m = 8 kg: (

**a**) Behavior of each component in respect to the perturbation frequency in the range of interest; (

**b**) overlapped spectral diagrams. The blue continuous thick lines on the graphs denote the response spectra of the linear system, assuming the available terms.

**Figure 11.**Double-spectra evolution of the first 512 terms in the exponential function decomposition according to the Prony method for the second test within the case of m = 8 kg: (

**a**) Behavior of each component in respect to the perturbation frequency in the range of interest; (

**b**) overlapped spectral diagrams. The blue continuous thick lines on graphs denote the response spectra of the linear system, assuming the available terms.

**Figure 12.**Double-spectra evolution of the first 512 terms in the exponential function decomposition according to the Prony method for the first test within the case of m = 19 kg: (

**a**) Behavior of each component in respect to the perturbation frequency in the range of interest; (

**b**) overlapped spectral diagrams. The blue continuous thick lines on graphs denote the response spectra of the linear system, assuming the available terms.

**Figure 13.**Double-spectra evolution of the first 512 terms in the exponential function decomposition according to the Prony method for the second test within the case of m = 19 kg: (

**a**) Behavior of each component in respect to the perturbation frequency in the range of interest; (

**b**) overlapped spectral diagrams. The blue continuous thick lines on graphs denote the response spectra of a cumulative linear system, assuming the available terms.

**Table 1.**Peaks frequencies related to the diagrams in Figure 8.

Case/Situation of Analysis | Frequency (Hz) |
---|---|

m = 8 kg/situation I | 2.499938, 6.749831, 9.99975, 11.24972, 190.9952 |

m = 8 kg/situation II | 9.99975, 11.24972, 188.7453, 219.9945, 301.9925 |

m = 19 kg/situation I | 9.249769, 10.74973, 186.7453 |

m = 19 kg/situation II | 9.249769, 10.74973, 189.4953 |

**Table 2.**Overall damping factors related to the diagrams in Figure 8.

Case/Situation of Analysis | Overall Damping Factor (s^{−1}) |
---|---|

m = 8 kg/situation I | 24.5810 |

m = 8 kg/situation II | 23.8685 |

m = 19 kg/situation I | 12.7252 |

m = 19 kg/situation II | 14.2641 |

**Table 3.**Equivalent damping factors and ratios related to the diagrams in Figure 8.

Case/Situation of Analysis | Equivalent SDoF Damping Ratio (-) | Equivalent SDoF Damping Factor (s ^{−1}) |
---|---|---|

m = 8 kg/sit. I | 0.0312, 0.0312, 0.0312, 0.0312, 0.0192 | 0.4908, 1.3233, 1.9589, 2.2047, 23.0028 |

m = 8 kg/sit. II | 0.0311, 0.0312, 0.0186, 0.0174, 0.0193 | 1.9567, 2.2024, 22.0632, 24.1166, 36.6487 |

m = 19 kg/sit. I | 0.0312, 0.0312, 0.0072 | 1.8117, 2.1062, 8.3916 |

m = 19 kg/sit. II | 0.0312, 0.0312, 0.0077 | 1.8152, 2.1102, 9.1820 |

**Table 4.**Equivalent damping coefficients related to the data in Table 3.

Case/Situation of Analysis | Equivalent SDoF Damping Coefficient (Nsm ^{−1}) |
---|---|

m = 8 kg/situation I | 368.048 |

m = 8 kg/situation II | 353.008 |

m = 19 kg/situation I | 318.896 |

m = 19 kg/situation II | 348.916 |

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**MDPI and ACS Style**

Musca, G.D.; Debeleac, C.; Vlase, S.
Experimental Assessments on the Evaluation of Wire Rope Characteristics as Helical Symmetrical Multi-body Ensembles. *Symmetry* **2020**, *12*, 1231.
https://doi.org/10.3390/sym12081231

**AMA Style**

Musca GD, Debeleac C, Vlase S.
Experimental Assessments on the Evaluation of Wire Rope Characteristics as Helical Symmetrical Multi-body Ensembles. *Symmetry*. 2020; 12(8):1231.
https://doi.org/10.3390/sym12081231

**Chicago/Turabian Style**

Musca (Anghelache), Gina Diana, Carmen Debeleac, and Sorin Vlase.
2020. "Experimental Assessments on the Evaluation of Wire Rope Characteristics as Helical Symmetrical Multi-body Ensembles" *Symmetry* 12, no. 8: 1231.
https://doi.org/10.3390/sym12081231