# Stiffness Considerations for a MEMS-Based Weighing Cell

^{1}

^{®}, Technische Universität Ilmenau, Max-Planck-Ring 12, 98693 Ilmenau, Germany

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Stiffness of the MEMS-Based Weighing Cell

#### 2.1.1. Analytical Stiffness Calculation

_{2}of the transmission lever (Figure 3) is sufficient as the single system variable. These assumptions lead to a conservative system, except for the force applied to the shuttle ($F$):

#### 2.1.2. Numerical Stiffness Calculation

^{3}and in the rigid parts around 3400 µm

^{3}. Parameter studies showed that these meshing combinations yield a convincing compromise between computational time and model accuracy. As model material, anisotropic silicon (density ρ = 2330 kg/m

^{3}, stress tensor: σ

_{11}= σ

_{22}= σ

_{33}= 1.66 × 10

^{5}MPa, τ

_{21}= τ

_{31}= τ

_{32}= 64,000 MPa) was used directly from the ANSYS material library. The parts connected to the frame are secured with fixed supports and the force is introduced on the upper part of the shuttle, the probing area as indicated in Figure 4a. The deflection s is calculated at the point of the force application. The system stiffness was derived assuming a force of ${F}_{y}=-200$ µN in the direction of motion and the out-of-plane stiffness assuming a force of ${F}_{z}=-50$µN in out-of-plane direction, according to Figure 4 and Equation (13):

#### 2.2. Microfabrication of MEMS-Based Weighing Cells

#### 2.3. Experimental Determination of the System Stiffness

## 3. Results and Discussion

#### 3.1. Theoretical Stiffnesses Evaluation

#### 3.2. Analysis of the Influences of Fabrication Tolerances on the System Stiffness

#### 3.2.1. Measured Influencing Geometry Parameters

_{z}= 0.14 µm) of the roughness of the lower 50 µm etch depth (R

_{z}= 0.57 µm). Since the etch rate decreases in depth with longer etching and ends at 200 cycles for the 100 µm thick component layer, the etch riffles got smaller the deeper the structures are etched.

#### 3.2.2. Sidewall Angle Influence on the Stiffness

#### 3.3. Experimental Investigation of the System Stiffness

#### 3.4. Discussion of the Theoretical and Experimental Results

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic illustration of the working principle of the MEMS-based weighing balance including the mechanical parts (black), the electrical parts and the electrical periphery (red). (1) upper lever, (2) lower lever, (3) shuttle, (4) coupling element, (5) transmission lever, V

_{s}voltage of the sensor signal, V

_{A}Voltage to actuate the shuttle in balance; (

**b**) Photo-stacked overview image of the microfabricated MEMS-based weighing cell.

**Figure 2.**Schematic illustration of a single flexure hinge with indicated elastic deformation as used for the calculation and geometric parameters: maximum height of the hinge H, minimal hinge size h, width (out-of-plane thickness) of the hinge w, length of the hinge l, bending angle φ, force F and momentum M.

**Figure 3.**Simplified schematic of the MEMS-based weighing cell that was used for the analytical approach. Black: unloaded state, dashed grey: loaded/deflected state.

**Figure 4.**(

**a**) Illustration of sections of the meshed FEM model (software ANSYS); (

**b**) weighing cell in an unloaded and a loaded state with the force F

_{y}; (

**c**) FEM simulation of an unloaded hinge G and the principal stress distribution if the hinge is loaded with a force F

_{y}.

**Figure 5.**Schematic illustration of the workflow for the fabrication for a MEMS-based weighing cell based on a silicon-on-insulator substrate.

**Figure 6.**Illustration of the direct force measurement device used for measuring the stiffness of the MEMS-based weighing cell: (1) slit aperture; (2) beam balance; (3) loading button; (4) MEMS-based weighing cell chip; (5) joint; (6) permanent magnet; (7) coil; (8) deflection mirror; (9) interferometer.

**Figure 7.**Force-displacement measurement device used for the stiffness measurement of the MEMS based weighing cell (

**a**) overall with a magnification to the MEMS holder and the loading button; (

**b**) side view from the camera used for positioning (

**c**) front view with the MEMS.

**Figure 8.**(

**a**) scanning electron microscopy (SEM) images showing a flexure hinge, (

**b**) exemplary device layer sidewall image as recorded by laser-scanning microcopy (LSM), and (

**c**) an according topography profile, (

**d**) SEM image of the cross section of a cutted flexure hinge with a designed height of h = 7 µm, (

**e**) SEM image of the cross section with a designed height of h = 5 µm.

**Figure 9.**Dependence of the etching quality on the single flexure hinge stiffness, (

**a**) in direction of motion and (

**b**) in out-of-plane direction.

**Figure 10.**(

**a**) Recorded compensation current signal from one cycle with 10 s integration time; (

**b**) corresponding measured displacement signal MEMS of one calibration cycle with 10 s integration time; (

**c**) Force-displacement characteristics per cycle; (

**d**) long-term measurement of MEMS stiffness of the system W 3-1 C1.

**Figure 11.**Eccentric test, (

**left**): geometrical illustration of probing area, (

**right**): stiffness measurement results in direction of motion of system W 3-1 C1.

**Figure 12.**Stiffness measured in the direction of motion compared to the calculated analytical values. (

**Left**): values of the stiffness results, (

**right**): percentage representation of the deviation from the measured stiffness (green) to the analytical stiffness with ideal geometry (dark blue) and to the calculations using the adjusted geometry (light blue).

System | Hinge Size in µm | Lever Size in µm | Dir. | Stiff. Single Hinge in Nµm | Analytical Solution in N/m | Numerical Solution in N/m | Deviation in% | Stiffness Ratio |
---|---|---|---|---|---|---|---|---|

W 3-1 C1/C2 | h = 9 l = 100 | l_{AD} = 2000 l _{FG} = 1400 | −y | 13.18 | 26.6 | 27.6 | 3.8 | 5.1 |

−z | - | 139.7 | - | |||||

W 3-2 | h = 7 l = 200 | l_{AD} = 2000l _{FG} = 1400 | −y | 3.27 | 6.95 | 7.2 | 3.6 | 10.8 |

−z | - | 78 | - | |||||

W 3-3 | h = 15 l = 100 | l_{AD} = 2000l _{FG} = 1400 | −y | 55.46 | 112 | 111.95 | 0.04 | 1.8 |

−z | - | 200.4 | - | |||||

W 2-1 | h = 9 l = 100 | l_{AD} = 1700 l _{FG} = 1000 | −y | 13.18 | 44.6 | 45.5 | 2.0 | 4.2 |

−z | - | 192.3 | - |

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**MDPI and ACS Style**

Wedrich, K.; Cherkasova, V.; Platl, V.; Fröhlich, T.; Strehle, S.
Stiffness Considerations for a MEMS-Based Weighing Cell. *Sensors* **2023**, *23*, 3342.
https://doi.org/10.3390/s23063342

**AMA Style**

Wedrich K, Cherkasova V, Platl V, Fröhlich T, Strehle S.
Stiffness Considerations for a MEMS-Based Weighing Cell. *Sensors*. 2023; 23(6):3342.
https://doi.org/10.3390/s23063342

**Chicago/Turabian Style**

Wedrich, Karin, Valeriya Cherkasova, Vivien Platl, Thomas Fröhlich, and Steffen Strehle.
2023. "Stiffness Considerations for a MEMS-Based Weighing Cell" *Sensors* 23, no. 6: 3342.
https://doi.org/10.3390/s23063342