# An Improved Analytical Model of a Thrust Stand with a Flexure Hinge Structure Considering Stiffness Drift and Rotation Center Offset

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## Abstract

**:**

## 1. Introduction

## 2. Conceptual Illustration

## 3. Conventional Analytical Model of the Thrust Stand

## 4. The Improved Analytical Model of the Thrust Stand

#### 4.1. Hinge Bending Deflection Modeling

#### 4.2. Gravity-Induced Extension Effect

#### 4.3. Rotational Center Offset Effect

#### 4.4. The Improved and Corrected Analytical Model

## 5. Case Study and Discussion

#### 5.1. The Improved and Corrected Analytical Model

- (1)
- Region A is the thinnest region of the hinge, has a length about 1/3 of the entire one, and has the characteristics of a large aspect ratio. The stresses and strains in both bending and tensile deformation are large. It is a key concern in force analysis. Therefore, a controlled structured hexahedral mesh is used instead of an unstructured one for the meshing (see Figure 9b). A three-level hexahedral mesh is established in the thickness direction, and 30 and 100 elements are divided in the axial and width directions, respectively, using “mapped” and “swept” techniques to accomplish the above operations. Accordingly, region A is equivalent to a large number of healthy micro-cantilevers.
- (2)
- Region B contains the part of the hinge root with larger curvature, the hexahedral element is no longer applicable, and the physical field-controlled tetrahedral element is used to build the mesh. Furthermore, in order to avoid a poor-quality mesh in the narrow region of the hinge root, a virtual mesh technique is applied to its root to supplement a circular arc-shaped region (see Figure 9a); this region is only used to distinguish the difference between the meshes, and does not have an actual physical partitioning function (i.e., the machining of the hinge shown in Figure 9a is shaped in one piece).
- (3)
- Region C is the part outside the hinge, which is not the focus of attention, so it is subjected to a coarser free-division tetrahedral mesh.

#### 5.2. Results and Discussion

#### 5.3. Discussion for Thrust Measurement

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Comparison before and after analytical model improvement. (

**a**) The conventional analytical model and pendulum deflection schematic; (

**b**) the new analytical model and pendulum deflection schematic.

**Figure 2.**The designed thrust stand consisting of the compound pendulum (partial). (

**a**) 3D rendering diagram; (

**b**) physical diagram.

**Figure 7.**Schematic diagram of the geometric dimensional changes under axial tension of the flexure hinge.

**Figure 8.**Flexure hinge bending deformation. (

**a**) Axial section diagram; (

**b**) measurement displacement schematic.

**Figure 11.**Comparison of stiffness equation and FEM. (

**a**) Tension line stiffness ${K}_{T\_line}$; (

**b**) offset line stiffness ${K}_{d\_line}$.

**Figure 13.**Bending line stiffness variation of the flexure hinge under gravity-induced extension effect.

**Figure 14.**Comparisons of the stiffness equations and FEM under Case I. (

**a**) Tension line stiffness ${K}_{T\_line}$; (

**b**) offset line stiffness ${K}_{d\_line}$; (

**c**) bending line stiffness ${K}_{p\_line}$; (

**d**) bending line stiffness variation under axial tensile force.

**Figure 15.**Comparisons of the stiffness equations and FEM under Case II. (

**a**) Tension line stiffness ${K}_{T\_line}$, (

**b**) offset line stiffness ${K}_{d\_line}$, (

**c**) bending line stiffness ${K}_{p\_line}$, (

**d**) bending line stiffness variation under axial tensile force.

Parameter | Symbol | Unit |
---|---|---|

Applied thrust | F | N |

Distance from thrust point to torsion center | ${l}_{Th.}$ | m |

Distance from pendulum arm centroid to torsion center | ${l}_{ro.}$ | m |

Distance from counterweight to torsion center | ${l}_{cout.}$ | m |

Distance from measurement point to torsion center | ${l}_{sen.}$ | m |

Thruster mass | ${m}_{Th.}$ | kg |

Pendulum arm mass | ${m}_{ro.}$ | kg |

Counterweight mass | ${m}_{cout.}$ | kg |

The gravitational acceleration | g | ms^{−2} |

Gravity of the whole pendulum | G | N |

Tangential component of the gravity of the whole pendulum | ${G}_{tangent}$ | N |

Deflection angle of the pendulum | $\theta $ | rad |

Horizontal displacement at measuring point | $u$ | m |

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

Length | L | 12 mm |

Width | W | 20 mm |

Height | H | 3 mm |

Minimum thickness | t | 0.1 mm |

Elliptic long axis | a | 6 mm |

Elliptic short axis | b | 1.45 mm |

Young’s modulus | E | 110 Gpa |

Poisson’s ratio | v | 0.3 |

Density | ρ | 8750 kg/m^{3} |

t | b | a | H | L | W | |
---|---|---|---|---|---|---|

[mm] | ||||||

Case I | 0.1 | 1.45 | 1.45~6 | 3 | $2\ast a$ | 20 |

Case II | 0.1 | 1.45 | 6 | 3 | 12 | 10~20 |

$\mathit{t}$ | ${\mathit{m}}_{\mathit{T}\mathit{h}.}$ | ${\mathit{l}}_{\mathit{T}\mathit{h}.}$ | ${\mathit{m}}_{\mathit{r}\mathit{o}.}$ | ${\mathit{l}}_{\mathit{r}\mathit{o}.}$ | ${\mathit{m}}_{\mathit{c}\mathit{o}\mathit{u}\mathit{t}.}$ | ${\mathit{l}}_{\mathit{c}\mathit{o}\mathit{u}\mathit{t}.}$ | |
---|---|---|---|---|---|---|---|

[mm] | [kg] | [m] | [kg] | [m] | [kg] | [m] | |

A1 | 0.1 | 3 | 0.5 | 0.4529 | 0.1377 | 7 | 0.2246 |

A2 | 0.1 | 3 | 0.5 | 0.4500 | 0.1400 | 7 | 0.2200 |

B1 | 0.1 | 2 | 0.5 | 0.4528 | 0.1378 | 4 | 0.2245 |

B2 | 0.1 | 2 | 0.5 | 0.4519 | 0.1385 | 4 | 0.2230 |

C1 | 0.3 | 3 | 0.5 | 0.4656 | 0.1275 | 7 | 0.2450 |

C2 | 0.3 | 3 | 0.5 | 0.4438 | 0.1450 | 7 | 0.2100 |

${\mathit{K}}_{{\mathit{G}}_{\mathit{a}\mathit{x}\mathit{i}\mathit{a}\mathit{l}}}$ | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{{\mathit{G}}_{\mathit{t}\mathit{a}\mathit{n}\mathit{g}\mathit{e}\mathit{n}\mathit{t}}}$ | ${\mathit{F}}_{\mathit{r}\mathit{e}\mathit{v}\mathit{i}\mathit{s}\mathit{e}}$ | ${\mathit{F}}_{\mathit{a}\mathit{b}\mathit{s}\_\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}$ | ${\mathit{F}}_{\mathit{r}\mathit{e}\_\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}$ | |
---|---|---|---|---|---|---|

[N m rad^{−1}] | [N] | |||||

A1 | −4.8976 × 10^{−5} | 0.0996 | −0.0964 | 1.2511 × 10^{−6} | 1.9590 × 10^{−8} | 1.57% |

A2 | −4.8976 × 10^{−5} | 0.0996 | 0.2254 | 1.2998 × 10^{−4} | 1.9590 × 10^{−8} | 0.015% |

B1 | −2.9388 × 10^{−5} | 0.0996 | −0.0894 | 4.0579 × 10^{−6} | 1.1755 × 10^{−8} | 0.29% |

B2 | −2.9388 × 10^{−5} | 0.0996 | 0.0155 | 4.6040 × 10^{−5} | 1.1755 × 10^{−8} | 0.026% |

C1 | −2.9064 × 10^{−4} | 1.5292 | −1.5252 | 1.4831 × 10^{−6} | 1.1626 × 10^{−7} | 7.84% |

C2 | −2.9064 × 10^{−4} | 1.5292 | 0.9246 | 9.8139 × 10^{−4} | 1.1626 × 10^{−7} | 0.022% |

t | u | ${\mathit{F}}_{\mathit{r}\mathit{e}\mathit{v}\mathit{i}\mathit{s}\mathit{e}}$ | ${\mathit{F}}_{\mathit{a}\mathit{b}\mathit{s}\_\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}$ | ${\mathit{F}}_{\mathit{r}\mathit{e}\_\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}$ | |
---|---|---|---|---|---|

[mm] | [um] | [N] | |||

Case 1 | 0.1 | 1 | 1.2974 × 10^{−6} | 2.7602 × 10^{−9} | 0.21% |

Case 2 | 0.1 | 20 | 2.5948 × 10^{−5} | 5.5203 × 10^{−8} | 0.21% |

Case 3 | 0.1 | 100 | 1.2974 × 10^{−4} | 2.7602 × 10^{−7} | 0.21% |

Case 4 | 0.2 | 100 | 3.1343 × 10^{−4} | 4.0241 × 10^{−7} | 0.13% |

Case 5 | 0.3 | 100 | 7.0094 × 10^{−4} | 8.9445 × 10^{−7} | 0.13% |

Case 6 | 0.4 | 100 | 0.0013 | 1.8243 × 10^{−6} | 0.14% |

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## Share and Cite

**MDPI and ACS Style**

Chen, X.; Zhao, L.; Xu, J.; Liu, Z.
An Improved Analytical Model of a Thrust Stand with a Flexure Hinge Structure Considering Stiffness Drift and Rotation Center Offset. *Actuators* **2024**, *13*, 21.
https://doi.org/10.3390/act13010021

**AMA Style**

Chen X, Zhao L, Xu J, Liu Z.
An Improved Analytical Model of a Thrust Stand with a Flexure Hinge Structure Considering Stiffness Drift and Rotation Center Offset. *Actuators*. 2024; 13(1):21.
https://doi.org/10.3390/act13010021

**Chicago/Turabian Style**

Chen, Xingyu, Liye Zhao, Jiawen Xu, and Zhikang Liu.
2024. "An Improved Analytical Model of a Thrust Stand with a Flexure Hinge Structure Considering Stiffness Drift and Rotation Center Offset" *Actuators* 13, no. 1: 21.
https://doi.org/10.3390/act13010021