# Design and Experiment of Symmetrical Shape Deployable Arc Profiling Mechanism Based on Composite Multi-Cam Structure

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## Abstract

**:**

## 1. Introduction

## 2. Description of Deployable Arc Profiling Mechanism

#### 2.1. Structure of Mechanism

#### 2.2. Design Principle of Mechanism

## 3. Parameter Design of Deployable Arc Profiling Mechanism

#### 3.1. Pitch Curve of Stationary Cam Groove

_{1}, The No. i (i = 1, 2, . . ., i ≤ n) support linkage is connected to point A (X

_{1A}, Y

_{1A}) on the arc, and the end of this support linkage is point P (X

_{1P}, Y

_{1P}). The projections of points A and P on the X

_{1}-axis are points B and C, respectively. The extension line of support linkage passes through the center of the arc at point O, as shown in Figure 3. The angle between extension line and X

_{1}-axis is α, the radius of the arc is r, the length of arc support linkage is l, the arc length between any two adjacent support linkages is c.

_{1A}+ r)

^{2}+ Y

_{1A}

^{2}= r

^{2}, (X

_{1A}≠ −2r), the coordinate of point A is (X

_{1A}, $\sqrt{{r}^{2}-{(r+{X}_{1A})}^{2}}$). Owing to OA/OB = OP/OC and OA/AB = OP/PC, we can obtain equation:

_{1A}) / r and α = ic / r, so r + X

_{1A}= rcos(ic / r). Take it into Equation (2), we can obtain the pitch curves of point P on stationary cam grooves as:

#### 3.2. Parameter Design of Driving Cam

- Pitch curve design of driving cam grooves, which can drive the support linkages to carry out regular movement.
- Driving motion equation design.
- Motion positioning groove design, which enables driving cam to be driven according to the resulted motion parameters.

#### 3.2.1. Pitch Curve of Driving Cam Grooves

_{2}at the end of No. j (j = 1, 2, . . ., j ≠ n) support linkage shown in Figure 5. In this way, the motion trajectories of remaining support linkages can be obtained by the given straight line and the fixed point. This fixed point O

_{2}is taken as the coordinate origin, and the endpoint E of No. n support linkage is moving on the straight line. F is the projection point of point E on X

_{2}-axis. The motion trajectory of point Q (X

_{2Q}, Y

_{2Q}) that located at the end of No. i support linkage is the pitch curve of driving cam groove.

_{2}-X

_{2}Y

_{2}plane is (X

_{2E}, aX

_{2E}+ b). Vector

**e**is a unit vector along X

_{2}_{2}-axis on O

_{2}-X

_{2}Y

_{2}plane, so the angle between vector

**e**and vector O

_{2}_{2}E is:

_{2}are on the same arc, and point O is the center of the arc, then $\angle E{O}_{2}Q=\angle EOQ/2=(n-i)c/2r$. Since i ≤ n, $\angle E{O}_{2}Q$ is always positive. Therefore, the angle

_{2}Q, since the lengths of OO

_{2}and OQ are equal to r + l, and $\angle {O}_{2}OQ=(i-j)c/r$, we can obtain the length of O

_{2}Q is:

_{2}E can be obtained as:

_{2E}exists, so the Equation (8) needs to be further solved.

_{2}EF. Solve it can get:

#### 3.2.2. Driving Motion Equation

_{2}and rotational motion around point O

_{2}on O

_{1}-X

_{1}Y

_{1}plane, as shown in Figure 5. The trajectory of point O

_{2}and point E on O

_{2}-X

_{2}Y

_{2}plane can be expressed by Equation (3), then, i = j corresponds to point O

_{2}, and i = n corresponds to point E in Equation (3). Thus, the translational trajectory equation of point O

_{2}(base point) on O

_{1}-X

_{1}Y

_{1}plane is:

_{1}-X

_{1}Y

_{1}plane is:

**e**is a unit vector along X

_{1}_{1}-axis direction on O

_{1}-X

_{1}Y

_{1}plane, and

**e**is a unit vector along X

_{2}_{2}-axis direction on O

_{2}-X

_{2}Y

_{2}plane. Vector

**e**is fixed on the symmetrical center line of the mechanism, and vector

_{1}**e**is moved relative to vector

_{2}**e**on the driving cam. φ is the angle between vector

_{1}**e**and vector

_{2}**e**. When

_{1}**e**is parallel to

_{2}**e**and in the same direction, the value of angle φ is zero. When

_{1}**e**moves counterclockwise with respect to

_{2}**e**and through the zero position, the value of φ is a positive number, while when it is clockwise with respect to

_{1}**e**and through the zero position, the value of φ is a negative number. The direction of rotation of φ marked in Figure 7 is the direction of the deployable arc in deploying motion, and φ is a negative value at this time. The rotational motion equation of driving cam relative to stationary cam is a rotation vector of vector

_{1}**e**relative to vector

_{2}**e**. The rotation angle φ of driving cam can be obtained according to the change of rotation vector of vector O

_{1}_{2}E that is relative to vectors

**e**and

_{1}**e**.

_{2}_{1}-X

_{1}Y

_{1}plane, vector O

_{2}E can be derived by the parameter equations of points O

_{2}and E, the angle between vectors

**e**and O

_{1}_{2}E can be calculated. So, the rotation vector O

_{2}E relative to

**e**on O

_{1}_{1}-X

_{1}Y

_{1}plane is:

_{2}E = (X

_{2E}, aX

_{2E}+ b), and rotational change of vector O

_{2}E relative to vector

**e**. The rotation angle φ on driving cam can finally obtained as:

_{2}#### 3.2.3. Motion Positioning Groove

_{2}only constrained one translational degree of freedom (DOF) of driving cam. The other two DOFs are the movement DOF of the base point O

_{2}along the No. j groove and the rotation DOF of the driving cam, which are not constrained. Except for the base point O

_{2}, an additional positioning point G should be added to constrain the rotational DOF. Correspondingly, a motion positioning groove should be designed on stationary cam, whose pitch curve can be calculated by using obtained motion equations of the driving cam.

_{2G}, Y

_{2G}) is on O

_{2}-X

_{2}Y

_{2}plane, the projection point of G on O

_{1}-X

_{1}Y

_{1}plane is G’ (X

_{1G}’, Y

_{1G}’). The trajectory of G’ forms the motion positioning groove.

_{1}-X

_{1}Y

_{1}plane. With the driving cam rotates φ angle according to Equation (14), the projection of point G moves to point G’ on O

_{1}-X

_{1}Y

_{1}plane. According to coordinate transformation of Cartesian coordinate system, the coordinate of point G’ relative to O

_{2}-X

_{2}Y

_{2}plane is:

_{1}-axis on O

_{1}-X

_{1}Y

_{1}plane.

#### 3.3. Pitch Curve of Adjusting Groove

_{1}-axis, and there is no displacement is performed on Y

_{1}-axis. If the endpoint R (X

_{1R}, Y

_{1R}) of any one of arc support linkage is selected as the driving point, the motion component Y

_{1R}of point R on Y

_{1}-axis is equal to the motion component Y

_{1P}of point P on Y

_{1}-axis in Equation (3). In the X

_{1}-axis direction, it is only necessary to subtract the increment r of the arc radius change to keep the position of deploying center point O unchanged. According to the principle of relative motion, the relationship between X

_{1R}and X

_{1P}can be derived, that is, X

_{1R}= X

_{1P}− (−r). At this point, the pitch curve of the adjusting groove can be drawn:

#### 3.4. Driving Parameter of Deployable Arc Profiling Mechanism

_{2}-X

_{2}Y

_{2}plane where the driving cam located is U (X

_{2U}, Y

_{2U}), and the projection point U’ of U on O

_{1}-X

_{1}Y

_{1}plane is U’ (X

_{1U}’, Y

_{1U}’) where the stationary cam located. The driving parameter is d.

_{1}-X

_{1}Y

_{1}plane is:

_{1}-X

_{1}Y

_{1}plane. Then the relationship between driving parameter d and arc radius r can be obtained as (d − X

_{1U}’)

^{2}+ (s − Y

_{1U}’)

^{2}= t

^{2}, solving it can get:

## 4. Analysis and Experiment

#### 4.1. Impact of Parameters a and b

#### 4.2. Prototype Verification

## 5. Conclusions

## 6. Patent

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 14.**Impact of parameters a and b on included angle. (

**a1**) a = −0.5, b = 35, (

**a2**) a = −0.5, b = 0, (

**a3**) a = −0.5, b = −35, (

**b1**) a = −1, b = 35, (

**b2**) a = −1, b = 0, (

**b3**) a = −1, b = −35, (

**c1**) a = −2, b = 35, (

**c2**) a = −2, b = 0, (

**c3**) a = −2, b = −35.

**Figure 15.**Distribution law of maximum and minimum value of included angle. (

**a**) Maximum value of included angle, (

**b**) Minimum value of included angle.

**Figure 17.**Comparison of incision quality between without and with application of deployable arc profiling mechanism.

Parameter | Value |
---|---|

Range of variation of radius r (mm) | 20~50 |

Arc length of segment arc c (mm) | 10.5 |

Length of arc support linkage l (mm) | 70 |

Total number n of arc support linkages in upper part of mechanism | 5 |

Length of push linkage s (mm) | 150 |

Length of transmission linkage t (mm) | 265 |

Given straight line | Y = −X/2 − 35 |

Coordinate of motion positioning point G | (−160, 43) |

Coordinate of connecting point U | (−87, 79) |

Test Arc Radius r (mm) | Corresponding Parameter d (mm) | Measured Arc Radius r (mm) | Radius Deviation (mm) | Accuracy (%) |
---|---|---|---|---|

20 | 213.7 | 19.285 | 0.715 | 96.425 |

25 | 244.2 | 24.300 | 0.700 | 97.200 |

30 | 269.0 | 29.290 | 0.710 | 97.633 |

35 | 288.8 | 34.285 | 0.715 | 97.957 |

40 | 304.8 | 39.295 | 0.705 | 98.238 |

45 | 317.8 | 44.290 | 0.710 | 98.422 |

50 | 328.6 | 49.290 | 0.710 | 98.580 |

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## Share and Cite

**MDPI and ACS Style**

Xu, Z.; Yang, Z.; Duan, J.; Jin, M.; Mo, J.; Zhao, L.; Guo, J.; Yao, H.
Design and Experiment of Symmetrical Shape Deployable Arc Profiling Mechanism Based on Composite Multi-Cam Structure. *Symmetry* **2019**, *11*, 958.
https://doi.org/10.3390/sym11080958

**AMA Style**

Xu Z, Yang Z, Duan J, Jin M, Mo J, Zhao L, Guo J, Yao H.
Design and Experiment of Symmetrical Shape Deployable Arc Profiling Mechanism Based on Composite Multi-Cam Structure. *Symmetry*. 2019; 11(8):958.
https://doi.org/10.3390/sym11080958

**Chicago/Turabian Style**

Xu, Zeyu, Zhou Yang, Jieli Duan, Mohui Jin, Jiasi Mo, Lei Zhao, Jie Guo, and Huanli Yao.
2019. "Design and Experiment of Symmetrical Shape Deployable Arc Profiling Mechanism Based on Composite Multi-Cam Structure" *Symmetry* 11, no. 8: 958.
https://doi.org/10.3390/sym11080958