# Elasto-Kinematics and Instantaneous Invariants of Compliant Mechanisms Based on Flexure Hinges

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Comprehensive Analytical Model of Flexure Hinge Kinematics

#### 2.1. Analytical Solution

## 3. Analytical Deduction of Instantaneous Invariants for Compliant Mechanism

## 4. Numerical Examples and Experimental Evidence

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of the Skewness of the Curvature Tensor

## Appendix B. Other Relations Regarding the Equilibrium

## Appendix C. A Useful Trick to Avoid the Singularities of Some Integrals

## Appendix D. Fully Analytical Solution of Polodes (Fixed and Mobile) and Inflection Circle for a Flexure Hinge Loaded by a Concentrated Moment

## References

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**Figure 3.**Flexure hinge loaded by forces and moment applied to a generic point of the rigid body connected to it.

**Figure 4.**Generic configuration of two rigid bodies (fixed and mobile) connected by a flexure hinge.

**Figure 8.**Flexure-hinge loaded by two loading paths with the same final loads but different rate-trends.

**Table 1.**Experimental and predicted position of rotation centre and final end angle for four lengths of flexible joint.

Flexible Joint Lengths [mm] | Experimental Centre of Rotation [mm] | Predicted Centre of Rotation [mm] | Experimental Final Angle [deg] | Predicted Final Angle [deg] |
---|---|---|---|---|

360 | (87.3, −238.6) | (83.7, −242.9) | 41.6 | 42.7 |

270 | (85.2, −169.4) | (85.3, −168.2) | 33.0 | 32.6 |

180 | (70.7, −100.1) | (71.4, −97.4) | 22.2 | 21.5 |

90 | (40.3, −47.1) | (41.5, −38.9) | 12.4 | 10.2 |

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**MDPI and ACS Style**

Iandiorio, C.; Salvini, P.
Elasto-Kinematics and Instantaneous Invariants of Compliant Mechanisms Based on Flexure Hinges. *Micromachines* **2023**, *14*, 783.
https://doi.org/10.3390/mi14040783

**AMA Style**

Iandiorio C, Salvini P.
Elasto-Kinematics and Instantaneous Invariants of Compliant Mechanisms Based on Flexure Hinges. *Micromachines*. 2023; 14(4):783.
https://doi.org/10.3390/mi14040783

**Chicago/Turabian Style**

Iandiorio, Christian, and Pietro Salvini.
2023. "Elasto-Kinematics and Instantaneous Invariants of Compliant Mechanisms Based on Flexure Hinges" *Micromachines* 14, no. 4: 783.
https://doi.org/10.3390/mi14040783