# Schematic Diagrams Design of Displacement Suppression Mechanism with One Degree-of-Freedom in a Rope-Guided Hoisting System

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- One of the mechanical parts of DSM is capable of swinging within a certain angle.
- (2)
- The mechanism can be locked at the limited position.
- (3)
- It is a 1-DOF planar structure in order to implement easily.

## 2. Topology Design for DSM

_{h}can be derived as

_{h}and n must be positive integers. Thus, n = 4, 6, 8, 10, etc. In this case, let n = 4 or n = 6 to simplify the mechanical structure, that is, four-bar and six-bar linkage mechanisms are chosen for schematic diagram design.

## 3. Connected Relations of Graphs

#### 3.1. Adjacency Relation of Topological Graph

#### 3.2. Adjacency Relation of Mechanism Diagram

#### 3.3. Vertexes Symmetry Identification

^{T}= B, where A and B are two adjacent matrixes of topological graphs, then two topological graphs are isomorphic. This method can be also used for schematic diagram symmetry identification.

^{T}(2, 6), AM (2) = TM (1, 3) × AM (F4) × TM

^{T}(1, 3):

- 1)
- Topological graphs are numbered on the basis of the perimeter loop theory and two vertexes to be identified are both in the canonical perimeter loop.
- 2)
- There is only one branch path outside the canonical perimeter loop.
- 3)
- Topological vertexes in sub-chain are no more than one.

_{0}with its corresponding vertexes numbered in the counterclockwise direction. The new adjacency matrixes are set as AM

_{a}(i) and AM

_{b}(i) during the sequential numbering processes respectively in the counterclockwise and clockwise direction. Thus, the initial matrix AM

_{a}(1) and AM

_{b}(1) satisfy AM

_{a}(1) = AM

_{b}(1) = AM

_{0}, thereby, AM

_{a}(i + 1) and AM

_{b}(i + 1) are derived as Equations (8) and (9), respectively:

_{b}(5) = AM

_{b}(1) by computing each step in Figure 10 with Equations (8) and (9). The corresponding vertex numbers of AM

_{b}(5) are marked in red, which indicates that vertexes 1 and 3 are equivalent adjacency symmetry.

## 4. Computational Identification

- (i)
- Determine label number of mechanical rack: Two vertexes with topological symmetry form a vertex pair. All vertex pairs can be determined by symmetry identification of a topological graph. Among these symmetrical vertex pairs, the vertices between which have topological asymmetry, are chosen as labels of mechanical rack. The vertex labels of alternative rack are put into row matrix a with the length of N
_{a}. - (ii)
- Determine the number of prismatic joints: Since revolute joints are generally used in DSM and more prismatic joints will lead to more different mechanism diagram, one or two prismatic joints are both considered in the process to get mechanism diagrams. N
_{p}is used to denote the number of prismatic joints. - (iii)
- Obtain all possible extended adjacent matrixes: All extended adjacent matrixes of mechanism diagrams with one or two prismatic joints are put into the matrix RecordAM for the following symmetry identifications. Corresponding rack vertex label and the labels of vertexes connected by prismatic joints are stored into matrix LocP. [RecordAM_LocP] =
**RestoreAM_Loca**(AM_{0}, N_{p}, a, n, r), where**RestoreAM_Loca**() is a user-defined MATLAB function. - (iv)
- Symmetry identification of mechanism diagrams: One extended adjacent matrix EAM is taken out from RecordAM to compare with another extended adjacent matrix EAMx one by one. If EAM ≠ EAMx, record EAM in matrix R
_{x}and record the corresponding rack and prismatic joints information from LocP in matrix Lp_{x}; otherwise, abnegate EAM and identify the next one. [R_{x}LP_{x}] =**SymmetryIdentP**(n, r, N_{p}, a, RecordAM, LocP), where**SymmetryIdentP**() is a user-defined MATLAB function. - (v)
- Discard the mechanism diagrams with two consecutive prismatic joints in a six-bar mechanism except those both connected with mechanism rack: Two consecutive prismatic joints mean a mechanical part is connected with other two parts both by prismatic joints, which leads to a more complex structure and more difficult diagram design relative to that in a four-bar mechanism. Since a mechanism rack is immovable, its position can be quickly determined according to required movement. The number of alternative mechanisms is N
_{PLx}=**length**(L_{Px})/N_{p}, where**length**() is a MATLAB function to obtain the matrix length. If the common vertex of two consecutive prismatic joints is not equal to rack label, a corresponding extended adjacent matrix from matrix R_{x}and a matrix of rack and prismatic joints information from Lp_{x}will be recorded in matrixes f_{R}and f_{LP}, respectively. - (vi)
- Abandon six-bar mechanism diagrams with all prismatic joints in four-bar sub-loops and the mechanisms for which any prismatic joint exist in a sub-loop without a mechanism rack: From objective topological graphs, a sub-loop of a six-bar topological graph is four vertexes and will be substituted with four mechanical parts in a conversion process to the mechanism diagram. A six-bar mechanism with all prismatic joints existing in a four-bar sub-loop is almost equivalent to four-bar mechanisms, due to the following movement of the other sub-chain. In addition, if prismatic joints exist in this sub-loop without mechanism rack, there is little possibility to achieve locking in a limited position because the function is achieved by setting prismatic joints and a mechanism rack in the same sub-loop. Therefore, the above conditions can be neglected.

_{G}with dimension of n × n is defined to display the information about two symmetrical vertexes and R

_{G}(p, q) represent the element in the pth row and the qth column of R

_{G}. Thus, R

_{G}(p, q) = 1 indicates that vertexes p and q have topological symmetry.

_{G1}, R

_{G2}, and R

_{G3}calculated from Figure 11 are used to describe the topological symmetry of vertexes in Figure 3, Figure 4 and Figure 5, where

_{1}denotes that vertex pairs with topological symmetry are 1–2, 1–3, 1–4, 2–3, 2–4, 3–4, and vertexes 1, 2, 3, 4 have mutual symmetry. According to this principle, symmetrical vertex pairs and the determined rack positions can be derived as shown in Table 1.

_{i}}

_{1−L}denotes the family string of the ith vertex with L generation matrixes in a topological graph.

## 5. Alternatives for DSM and Results

_{p}= 1 as shown in Figure 14a with single prismatic joint, since there are five minimum sub-loops with prismatic joints while the others have four, there are more mechanical bars to swing, and they need a great driving force. When n = 6, r = 5, and rack label is vertex 4 in Figure 14b, these types of mechanisms have a smaller swing angle, owing to a closed four-bar sub-loop in each of them. In this case, we can choose four-bar mechanisms with single prismatic joints for first priority in the shaft, and then the six-bar mechanism with two prismatic joints if more stable resistant forces are needed.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Topological Graph | Four Bar | Six Bar with r = 5 | Six Bar with r = 6 |
---|---|---|---|

Symmetric vertex pairs | 1~2, 1~3, 1~4, 2~3, 2~4, 3~4 | 2~6, 1~3, 4~5 | 1~4, 2~3, 2~5, 2~6, 3~5, 3~6, 5~6 |

Mechanism rack label | 1 | 1, 2, 4 | 1, 2 |

Topological Graph | Four Bar | Six Bar with r = 5 | Six Bar with r = 6 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Rack | 1 | 1 | 2 | 4 | 1 | 2 | ||||||

N_{p} | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |

N_{R} | 4 | 6 | 7 | 21 | 7 | 21 | 7 | 21 | 7 | 21 | 7 | 21 |

N_{lpx} | 2 | 4 | 5 | 13 | 4 | 12 | 6 | 13 | 4 | 12 | 7 | 21 |

N_{f} | 2 | 4 | 5 | 8 | 4 | 7 | 6 | 7 | 4 | 8 | 7 | 12 |

N_{z} | 2 | 4 | 3 | 6 | 0 | 0 | 3 | 2 | 0 | 5 | 0 | 0 |

Topological Graph | Four Bar | Six Bar with r = 5 | Six Bar with r = 6 | ||||
---|---|---|---|---|---|---|---|

Rack | 1 | 1 | 2 | 4 | 1 | 2 | |

Prismatic joints | N_{p} = 1 | (1,4) | (1,5) | (1,5) | |||

(3,4) | (3,4) | (3,4) | |||||

(4,5) | (4,5) | ||||||

N_{p} = 2 | (1,2).(1,4) | (1,2).(3,4) | (1,5).(3,4) | (1,2).(1,6) | |||

(1,2).(2,3) | (1,2).(4,5) | (3,4).(4,5) | (1,6).(2,3) | ||||

(1,4).(2,3) | (1,5).(1,6) | (1,6).(3,4) | |||||

(2,3).(3,4) | (1,5).(2,3) | (2,3).(5,6) | |||||

(1,5).(3,4) | (3,4).(5,6) | ||||||

(2,3).(4,5) |

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**MDPI and ACS Style**

Yan, L.; Cao, G.; Wang, N.; Peng, W.
Schematic Diagrams Design of Displacement Suppression Mechanism with One Degree-of-Freedom in a Rope-Guided Hoisting System. *Symmetry* **2020**, *12*, 474.
https://doi.org/10.3390/sym12030474

**AMA Style**

Yan L, Cao G, Wang N, Peng W.
Schematic Diagrams Design of Displacement Suppression Mechanism with One Degree-of-Freedom in a Rope-Guided Hoisting System. *Symmetry*. 2020; 12(3):474.
https://doi.org/10.3390/sym12030474

**Chicago/Turabian Style**

Yan, Lu, Guohua Cao, Naige Wang, and Weihong Peng.
2020. "Schematic Diagrams Design of Displacement Suppression Mechanism with One Degree-of-Freedom in a Rope-Guided Hoisting System" *Symmetry* 12, no. 3: 474.
https://doi.org/10.3390/sym12030474