# On Modeling the Bending Stiffness of Thin Semi-Circular Flexure Hinges for Precision Applications

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## Abstract

**:**

## 1. Introduction

## 2. Approaches for Modeling the Bending Stiffness of Flexure Hinges

## 3. Finite Element Modeling of Thin Flexure Hinges

- The domain is divided into three parts: a highly refined central zone (A), a transition zone (B) and a roughly meshed outer zone (C).
- The boundary between the central zone (A) and the transition zone (B) is controlled so that the local normal stress in the x-direction is 10% of its maximum value.
- The transition zone (B) is limited by the transition of the notch contour to the adjacent links.
- In the central zone (A), the element size in the z-direction depends on the minimum element size in the x-y plane and a maximum aspect ratio of the hexahedral elements.
- To reduce the distortion of the elements when adapting to the semi-circular contour, the zones are delimited by cylindrical surfaces.
- The non-matching meshes are connected at their interface by suitable contact elements.
- Elements with quadratic approximation functions are used to avoid shear locking.

## 4. Comparison between Analytical and Finite Element Models

- For the parameter values considered, the results of the 3D-FE model are closer to the 2D-FE plane strain model.
- The minimum notch height h or the $h/R$ ratio in this geometry range has a small influence on the deviation to the 3D-FE model, except for the equation by Tseytlin [16].

## 5. Correction Factors for the Accurate Determination of the Bending Stiffness

## 6. Verification of Results in Application Examples

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

FE | Finite element |

3D | Three-dimensional |

2D | Two-dimensional |

BT | Beam theory |

## Appendix A

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

${x}_{A}$ | Elements along the length of zone A | 10–100 |

${x}_{B}$ | Elements along the length of zone B | 3–30 |

${x}_{C}$ | Elements along the length of zone C | 1–10 |

${y}_{A}$ | Elements along the height of zones A and B | 2–10 |

${y}_{C}$ | Additional elements along the height of zone C | 1–5 |

${s}_{A}$ | Transition of elements along the length of zone A towards the center of the hinge | 0–1 |

${s}_{A}$ | Transition of elements along the length of zone B towards the center of the hinge | 0–1 |

$AR$ | Limiting aspect ratio | 5–10 |

${z}_{A}$ | Elements along the width of zone A | $\frac{b{y}_{A}}{hAR}$ |

${z}_{B}$ | Elements along the width of zone B | $\frac{b{y}_{A}}{\sqrt{10}hAR}$ |

${z}_{C}$ | Elements along the width of zone C | $\frac{b{y}_{A}}{(h+2R)}$ |

Parameter | S${}_{{\mathit{\sigma}}_{\mathit{x}},\mathit{max}}$ | S${}_{\mathit{nodes}}$ |
---|---|---|

Model | $1.000$ | $1.000$ |

${x}_{A}$ | $0.847$ | $0.325$ |

${x}_{B}$ | $0.075$ | $0.081$ |

${x}_{C}$ | $0.000$ | $0.004$ |

${y}_{A}$ | $0.048$ | $0.669$ |

${y}_{C}$ | $0.000$ | $0.003$ |

${s}_{A}$ | $0.249$ | $0.003$ |

${s}_{C}$ | $0.000$ | $0.002$ |

$AR$ | $0.064$ | $0.115$ |

## Appendix B

## References

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**Figure 1.**Simplified representation of a compliant mechanism used in a weighing cell in [1].

**Figure 3.**3D-FE model of the semi-circular flexure hinge: (

**a**) mesh strategy and boundary conditions; (

**b**) deflected state with bending stress $|{\sigma}_{x}|$ in MPa for $\phi ={1}^{\circ}$ (scaled by a factor of 21).

**Figure 5.**Dimensionless out-of-plane normal stress along the dimensionless z-coordinate for different width values b for $h=0.05$ mm.

**Figure 6.**Relative stiffness deviation of the FE models to the analytical equation by Paros and Weisbord [12] for different $b/h$ ratios and: (

**a**) different $h/R$ ratios ($R=3$ mm); (

**b**) $h/R=0.025$.

**Figure 7.**Relative stiffness deviation of the 3D-FE model to the 2D-FE plane stress model considering the correction factor proposed by Zettl et al. [22] for different $h/R$ and $b/h$ ratios.

**Figure 8.**Relative stiffness deviation of the 2D-FE plane stress model to the analytical equation by Paros and Weisbord [12] for different $h/R$ ratios.

**Figure 9.**Relative stiffness deviation of the 3D-FE model to the 2D-FE plane stress model for different $h/R$ and $b/h$ ratios.

**Figure 10.**Relative stiffness deviation of the 3D-FE model to the analytical equation by Paros and Weisbord [12] considering the estimated correction factors for different $h/R$ and $b/h$ ratios.

**Table 1.**Approaches and assumptions for quasi-static modeling of the bending stiffness of notch flexure hinges (C: Circular; CF: Corner-filleted; E: Elliptical; P: Parabolic; PF: Power function; PL: Polynomial).

Approach | Equation/Model by | Contour | Bernoulli’s Hypothesis | Saint-Venant’s Principle | State of Plane Stress |
---|---|---|---|---|---|

Euler-Bernoulli’s beam theory | Paros and Weisbord [12] | C | × | × | × |

Castigliano’s second theorem | Lobontiu [9] | C, CF, E, P | × | × | × |

Principle of virtual work | Li et al. [13] | PF | × | × | × |

Finite Beam-based Matrix Modelling | Zhu et al. [14] | C, E, PF | × | × | × |

Timoshenko’s beam theory | Dirksen and Lammering [15] | C, P | × | × | |

Continuum mechanics | Tseytlin [16] | C, E, P | |||

Nonlinear theory of large deflections of rod-like structures [10] | Linß et al. [17] | C, CF, E, PL | × | × | × |

Henning et al. [18] | C, CF, E, PL | × | × | × | |

Smith et al. [2] | C, E | × | |||

Finite element method | Koster et al. [19] | C | × | ||

Schotborgh et al. [20] | C, CF | × |

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

Total length | L | 15 mm |

Total height | H | 9 mm |

Radius | R | 3 mm |

Minimum notch height | h | 50 $\mathsf{\mu}$m |

Width | b | 10 mm |

Young’s modulus | E | 71 GPa |

Poisson’s coefficient | $\nu $ | 0.33 |

Density | $\rho $ | 2.8 g/cm${}^{3}$ |

Model | ${\mathit{k}}_{\mathit{\phi}}$ (Nmm/rad) |
---|---|

Exact equation by Paros and Weisbord [12] | 16.243 |

Exact equation by Lobontiu [9] | 16.243 |

Approximate equation by Tseytlin [16] | 18.542 |

Design tool by Henning et al. [18] | 16.173 |

Design equation by Linß et al. [17] | 15.887 |

Linear 3D-FE model | 18.029 |

Nonlinear 3D-FE model | 18.032 |

Nr. | h (mm) | R (mm) | b (mm) | E (GPa) | $\mathit{\nu}$ | Material | Application | Ref. |
---|---|---|---|---|---|---|---|---|

1 | 0.04 | 4.98 | 15 | 207 | 0.30 | Steel alloy | Investigation on single hinge | [6,7] |

2 | 0.05 | 8 | 10 | 128 | 0.30 | Beryllium-copper | Watt balance | [31] |

3 | 0.07 | 5 | 25 | 114 | 0.34 | Titanium alloy | Parallel manipulator | [32] |

4 | 0.25 | 1.5 | 20 | 141 | 0.29 | Invar | Positioning stage | [4] |

5 | 0.25 | 5 | 15 | 71 | 0.33 | Aluminum alloy | Amplifier mechanism | [33] |

6 | 0.30 | 5 | 6 | 71 | 0.33 | Aluminum alloy | Investigation on single hinge | [17] |

7 | 0.40 | 15 | 15 | 114 | 0.34 | Titanium alloy | Parallel manipulator | [34] |

8 | 0.50 | 3 | 15 | 71 | 0.33 | Aluminum alloy | Amplifier mechanism | [33] |

9 | 0.60 | 12 | 12 | 207 | 0.30 | Steel alloy | Positioning stage | [35] |

Nr. | $\mathit{h}/\mathit{R}$ | $\mathit{b}/\mathit{h}$ | ${\mathit{f}}_{\mathit{BT}}^{2\mathit{D}}$ | ${\mathit{f}}_{2\mathit{D}}^{3\mathit{D}}$ | ${\mathit{k}}_{\mathit{\phi}}$ (Nmm/rad) | ${\mathit{k}}_{\mathit{\phi}}^{\ast}$ (Nmm/rad) | ${\mathit{k}}_{\mathit{\phi}}^{3\mathit{D}}$ (Nmm/rad) | ${\mathit{e}}_{{\mathit{k}}_{\mathit{\phi}}}$ (%) | ${\mathit{e}}_{{\mathit{k}}_{\mathit{\phi}}^{\ast}}$ (%) |
---|---|---|---|---|---|---|---|---|---|

1 | 0.008 | 375 | 0.997 | 1.097 | 31.53 | 34.48 | 34.45 | −8.486 | 0.090 |

2 | 0.006 | 200 | 0.998 | 1.094 | 17.91 | 19.54 | 19.53 | −8.292 | 0.063 |

3 | 0.014 | 357 | 0.995 | 1.129 | 117.09 | 131.47 | 131.30 | −10.829 | 0.127 |

4 | 0.167 | 80 | 0.936 | 1.088 | 5200.76 | 5292.89 | 5300.39 | −1.880 | −0.141 |

5 | 0.050 | 60 | 0.981 | 1.112 | 1059.51 | 1155.67 | 1155.03 | −8.270 | 0.055 |

6 | 0.060 | 20 | 0.977 | 1.093 | 669.39 | 714.66 | 714.72 | −6.343 | −0.008 |

7 | 0.133 | 37.5 | 0.948 | 1.118 | 7189.32 | 7625.27 | 7629.22 | −5.766 | −0.052 |

8 | 0.167 | 30 | 0.936 | 1.109 | 7856.47 | 8148.34 | 8161.33 | −3.735 | −0.159 |

9 | 0.171 | 20 | 0.934 | 1.083 | 26,778.32 | 27,071.77 | 27,124.71 | −1.277 | −0.195 |

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**MDPI and ACS Style**

Torres Melgarejo, M.A.; Darnieder, M.; Linß, S.; Zentner, L.; Fröhlich, T.; Theska, R. On Modeling the Bending Stiffness of Thin Semi-Circular Flexure Hinges for Precision Applications. *Actuators* **2018**, *7*, 86.
https://doi.org/10.3390/act7040086

**AMA Style**

Torres Melgarejo MA, Darnieder M, Linß S, Zentner L, Fröhlich T, Theska R. On Modeling the Bending Stiffness of Thin Semi-Circular Flexure Hinges for Precision Applications. *Actuators*. 2018; 7(4):86.
https://doi.org/10.3390/act7040086

**Chicago/Turabian Style**

Torres Melgarejo, Mario André, Maximilian Darnieder, Sebastian Linß, Lena Zentner, Thomas Fröhlich, and René Theska. 2018. "On Modeling the Bending Stiffness of Thin Semi-Circular Flexure Hinges for Precision Applications" *Actuators* 7, no. 4: 86.
https://doi.org/10.3390/act7040086