#### Introduction

A force sensor can be obtained from a well-matched combination of a known stiffness and a displacement or strain sensor. Typically commercially available force sensors have been designed for high stiffness (> 1e6 N/m) to measure large forces (>> 10 N) which makes them unsuitable for applications requiring mN or even sub-mN resoluti0n. For such an application JPE designed a custom force sensor with a resolution of 0.1 mN and a range of ±3 N.

#### Excellent performance

- Large range vs large resolution: dynamic range = 6e4 using laser interferometry displacement sensor
- 0.1 mN resolution due to low stiffness
- Low sensor drift / capable of DC force measurement
- Possibility to implement 2nd sensor (Eddy Current) for absolute force sensing.

#### Concept

Rigid mass $m$ is connected to the fixed world via 2 parallel leaf springs. 2 Parallel leaf springs are well-known for a play-free linear guidance that comprise the following stiffness:

$C_x=2C_{x\ }=2\frac{12EI}{L^3}$

A contactless displacement sensor measures the displacement of $m$ in relation to the fixed world as a consequence of the applied force. The sensor used is a laser interferometer with nm resolution. With a stiffness of $1e5\ N/m$, the resolution becomes $0.1\ mN$. In the application in which the force sensor was used, $1e5\ N/m$ was the lowest acceptable value (in relation to the dynamics of the application) but if a lower stiffness of the leaf springs is acceptable even better resolutions can be reached.

#### Stiffness calibration

Manufacturing tolerances influence the stiffness. E.g. the stiffness changes with the leaf spring thickness to the 3rd power. Therefore the sensor is calibrated using precision weights; first the moving mass $m$ is determined with the use of a precision weight ($m_2$) and the resulting 2 eigen frequencies ($\omega_1$ and $\omega_2$): $m=\frac{m_2}{\left(\frac{\omega_1^2}{\omega_2^2}-1\right)}$. Hereafter the force cell is placed in multiple angular configurations (with and without extra weights) to determine the stiffness curve ($F$ vs. $x$) with $F=mg\cdot \sin(\alpha).$

#### Linearity

The derived stiffness-linearity of the 2 parallel leaf springs is within 2%.

#### Parasitic displacements

When the 2 parallel leaf springs move in $x$, the sag ($z$) in height is: $z=\frac{3}{5}x^2/L$. In this particular case the maximum sag is $1.8 \mu m$ at end-of-stroke (which is negligible).

#### Overview

#### Integrated sensor

#### Features

- Monolithic part
- Matching materials for sensor and monolith for temperature stability
- Vacuum compatibility
- Force input at L/2 of the leaf springs for pure linear guidance (no parasitic normal forces)
- Symmetry over the width eliminating parasitic forces and moments
- Mechanical end-stops preventing the leaf springs from exceeding the yield stress ($\sigma_{0.2}$)
- Volume claim: 50 mm x 50 mm x 50 mm