# Dynamic Behavior Analysis of a High-Rise Traction System with Tensioned Pulley Acting on Compensating Rope

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## Abstract

**:**

## 1. Introduction

## 2. Modeling Description

#### 2.1. Description of the System

_{i}, (i = 1, 2, 3, 4) of ropes are both assumed to be uniform;

#### 2.2. Derivation of the Accurate Mathematical Model

_{N}

_{+1}-t

_{N}is time step; and N represents the Nth time step; ${q}_{N}=q\left({t}_{N}\right)$ represents the value of a generalized coordinate at ${t}_{N}$; the vector ${a}_{N}$ of acceleration-like variables is defined by the following recursive relationship:

## 3. Study Case

#### 3.1. Comparison between Traditional Traction System and Tensioned Traction System

#### 3.2. Discussion on Tensioned Pattern

#### 3.2.1. Different Tension Acting on Tensioned Pulley

#### 3.2.2. Different Damping Acting on Tensioned Pulley

#### 3.3. Discussion on Different Running Speed of Traction System

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${l}_{i}\left(t\right)$ | Length of lifting and compensating ropes | ${c}_{p}$ | Damping between tensioned pulley and ground |

${v}_{i}\left(t\right)\text{}$ | Velocity of lifting and compensating ropes | ${F}_{\mathrm{T}}$ | Preload working on tensioned pulley |

${a}_{i}\left(t\right)$ | Acceleration of lifting and compensating ropes | ${A}_{i}({B}_{i})$ | Connecting point between the lifting(compensating) ropes and conveyances |

$L$ | Total lifting height | ${k}_{c}$ | Stiffness between conveyances and guide rails |

${x}_{i}$ | Coordinate measurement of the lifting rope and compensating ropes | ${c}_{c}$ | Damping between conveyances and guide rails |

${C}_{i}$ | Tangent points between the compensating ropes and tensioned pulley | ${f}_{u,l(c),i}(t)$ | Longitudinal interaction forces between the conveyances and lifting(compensating) ropes |

${m}_{p}$(${J}_{p}$) | Mass and moment of inertia of tensioned pulley | ${f}_{y,l(c),i}(t)$ | Transverse interaction forces between the conveyances and lifting(compensating) ropes |

${r}_{p}$ | Radius of the tensioned pulley | ${f}_{p,i}(t)$ | Interaction forces between the tensioned pulley and compensating ropes |

${U}_{{C}_{i}}(t)$ | Longitudinal vibration of conveyances | ${Y}_{{C}_{i}}(t)$ | Transverse vibration of conveyances |

${U}_{i}(x,t)$ | Longitudinal vibration of ropes | ${Y}_{i}(x,t)$ | Transverse vibration of ropes |

${U}_{P}(t)$ | Longitudinal vibration of tensioned pulley | $Theta(t)$ | Rotation vibration of tensioned pulley |

## Appendix A

## Appendix B

## Appendix C

## Appendix D

## Appendix E

## References

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**Figure 1.**Model of bilateral traction system with compensating rope tensioner. (

**a**) schematic diagram of traditional traction system; (

**b**) schematic diagram of tensioned traction system; (

**c**) technical solution of traditional traction system; (

**d**) technical solution of tensioned traction system.

**Figure 2.**Comparison of the longitudinal displacement at the conveyance between tensioned traction system with a terminal tension (blue line) and a traditional traction system (red line): (

**a**) the ascending conveyance; (

**b**) the descending conveyance.

**Figure 3.**Longitudinal natural frequency comparison between tensioned (blue line) and traditional (red line) traction system.

**Figure 4.**Longitudinal displacement of the left lifting rope: (

**a**) tensioned traction system; (

**b**) traditional traction system.

**Figure 5.**Longitudinal displacement of right lifting rope: (

**a**) tensioned traction system; (

**b**) traditional traction system.

**Figure 6.**Comparison of the transverse displacement at the conveyance between a traction system with a terminal tension (blue line) and a traditional traction system (red line): (

**a**) the ascending conveyance; (

**b**) the descending conveyance.

**Figure 7.**Transverse displacement of left lifting rope: (

**a**) tensioned traction system; (

**b**) traditional traction system.

**Figure 8.**Transverse displacement of right lifting rope: (

**a**) tensioned traction system; (

**b**) traditional traction system.

**Figure 9.**The longitudinal response of the bilateral traction system at the ascending conveyance. (

**a**) The red line: terminal tension = $7\times {10}^{4}$ N; (

**b**) the green line: terminal tension = $1\times {10}^{5}$ N; (

**c**) the blue line: terminal tension = $1.5\times {10}^{5}$ N; (

**d**) the yellow line: terminal tension = $2\times {10}^{5}$ N.

**Figure 10.**The longitudinal response of the bilateral traction system at the descending conveyance. (

**a**) The red line: terminal tension = $7\times {10}^{4}$ N; (

**b**) the green line: terminal tension = $1\times {10}^{5}$ N; (

**c**) the blue line: terminal tension = $1.5\times {10}^{5}$ N; (

**d**) the yellow line: terminal tension = $2\times {10}^{5}$ N.

**Figure 11.**The longitudinal response of the bilateral traction system at the tensioned pulley. (

**a**) The red line: terminal tension = $7\times {10}^{4}$ N; (

**b**) the green line: terminal tension = $1\times {10}^{5}$ N; (

**c**) the blue line: terminal tension = $1.5\times {10}^{5}$ N; (

**d**) the yellow line: terminal tension = $2\times {10}^{5}$ N.

**Figure 12.**Time-frequency characteristics of longitudinal response of the bilateral traction system at the ascending conveyance with terminal tension and damping cylinder acting on the tensioned pulley: (

**a**) damping = 0; (

**b**) damping = $5\times {10}^{4}$ N/s; (

**c**) damping = $2\times {10}^{5}$ N/s.

**Figure 13.**Time-frequency characteristics of longitudinal response of the bilateral traction system at the descending conveyance with terminal tension and damping cylinder acting on the tensioned pulley: (

**a**) damping = 0; (

**b**) damping = $5\times {10}^{4}$ N/s; (

**c**) damping = $2\times {10}^{5}$ N/s.

**Figure 14.**Time-frequency characteristics of longitudinal response of the bilateral traction system at the tensioned pulley with terminal tension and damping cylinder acting on the tensioned pulley: (

**a**) damping = 0; (

**b**) damping = $5\times {10}^{4}$ N/s; (

**c**) damping = $2\times {10}^{5}$ N/s.

**Figure 15.**Time-frequency characteristics of rotation of the tensioned pulley with terminal tension and damping cylinder acting on the tensioned pulley: (

**a**) damping = 0; (

**b**) damping = $5\times {10}^{4}$ N/s; (

**c**) damping = $2\times {10}^{5}$ N/s.

**Figure 16.**The variation of largest amplitude of longitudinal vibration of ascending (

**a**) and descending (

**b**) side of tensioned traction system along with time. The black line: damping = 0; the red line: damping = $5\times {10}^{4}$ N/s; the blue line: damping = $2\times {10}^{5}$ N/s.

**Figure 17.**The position statistics of largest amplitude of longitudinal vibration of tensioned traction system. (

**a-1,2,3**): ascending side with damping = 0, $5\times {10}^{4}$ N/s, $2\times {10}^{5}$ N/s between tensioned pulley and ground; (

**b-1,2,3**): descending side with damping = 0, $5\times {10}^{4}$ N/s, $2\times {10}^{5}$ N/s between tensioned pulley and ground.

**Figure 18.**The transverse response of the bilateral traction system at ascending conveyance with terminal tension and damping cylinder acting on the tensioned pulley: (

**a**) damping = 0; (

**b**) damping = $5\times {10}^{4}$ N/s; (

**c**) damping = $2\times {10}^{5}$ N/s.

**Figure 19.**The transverse response of the bilateral traction system at descending conveyance with terminal tension and damping cylinder acting on the tensioned pulley: (

**a**) damping = 0; (

**b**) damping = $5\times {10}^{4}$ N/s; (

**c**) damping = $2\times {10}^{5}$ N/s.

**Figure 20.**Sum of squares and standard deviation of amplitude of tensioned system: (

**a**) sum of squares of amplitude; (

**b**) standard deviation of amplitude.

**Figure 21.**The longitudinal response of the bilateral traction system at the ascending conveyance. The black line: V

_{max}= 12 m/s; the red line: V

_{max}= 13.66 m/s; the green line: V

_{max}= 15 m/s.

**Figure 22.**The longitudinal response of the bilateral traction system at the descending conveyance. The black line: V

_{max}= 12 m/s; the red line: V

_{max}= 13.66 m/s; the green line: V

_{max}= 15 m/s.

**Figure 23.**The longitudinal response of the bilateral traction system at the tensioned pulley. The black line: V

_{max}= 12 m/s; the red line: V

_{max}= 13.66 m/s; the green line: V

_{max}= 15 m/s.

**Figure 24.**The rotation vibration of tensioned pulley. The black line: V

_{max}= 12 m/s; the red line: V

_{max}= 13.66 m/s; the green line: V

_{max}= 15 m/s.

**Figure 25.**Variation of longitudinal natural frequency and excitation frequency comparison under different running speed V

_{max}. The black line: V

_{max}= 12 m/s; the red line: V

_{max}= 13.66 m/s; the green line: V

_{max}= 15 m/s.

Parameter. | Description | Value |
---|---|---|

${\rho}_{1}$(${\rho}_{2}$) | The linear density of lifting(compensating) rope | $9.56\text{}\mathrm{kg}/\mathrm{m}$ |

N | The number of ropes | 4 |

${m}_{{c}_{1}}$(${m}_{{c}_{2}}$) | The mass of bilateral conveyances | $61\text{}\mathrm{t}$ |

$E$ | The elastic modulus of ropes | $1.02\text{}\times \text{}{10}^{11}\text{}\mathrm{pa}$ |

$D\_rope$ | The diameter of ropes | $48\mathrm{mm}$ |

${m}_{p}$ | The mass of tensioned pulley | 4 t |

${r}_{p}$ | The radius of tensioned pulley | 2.25 m |

${k}_{c}$ | The stiffness between conveyances and guide | $2.1\times {10}^{5}\text{}\mathrm{N}/\mathrm{m}$ |

${c}_{c}$ | The damping between conveyances and guide | $1.4\times {10}^{4}\text{}\mathrm{N}/\mathrm{s}$ |

${l}_{initial}$ | The initial lengths of the left lifting ropes | 853 m |

${l}_{end}$ | The final lengths of the left lifting ropes | 30 m |

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**MDPI and ACS Style**

Wang, L.; Cao, G.; Wang, N.; Zhang, Y.
Dynamic Behavior Analysis of a High-Rise Traction System with Tensioned Pulley Acting on Compensating Rope. *Symmetry* **2020**, *12*, 129.
https://doi.org/10.3390/sym12010129

**AMA Style**

Wang L, Cao G, Wang N, Zhang Y.
Dynamic Behavior Analysis of a High-Rise Traction System with Tensioned Pulley Acting on Compensating Rope. *Symmetry*. 2020; 12(1):129.
https://doi.org/10.3390/sym12010129

**Chicago/Turabian Style**

Wang, Lei, Guohua Cao, Naige Wang, and Yunchang Zhang.
2020. "Dynamic Behavior Analysis of a High-Rise Traction System with Tensioned Pulley Acting on Compensating Rope" *Symmetry* 12, no. 1: 129.
https://doi.org/10.3390/sym12010129