#
Gas-Flow Sensor Based on Self-Oscillating and Self-Sensing Cantilever^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup and Behavior Description

_{2}) are set. The dynamic excitation of the cantilever takes place via a DDS (direct digital synthesis) generator and the resulting cantilever oscillation is detected frequency sensitive with a look-in amplifier. Because of the resonance frequency shift due to the gas-flow, a PLL (phase locked loop) was used to ensure that the cantilever permanently oscillates at the resonance point. Resonance amplitude and resonance frequency are detected. Parallel to this, the static deflection of the cantilever is measured.

_{0}depends on the velocity 𝑣

_{0}according to

_{𝐷}depends on the geometry of the body and the flow conditions, quantified by the Reynolds number

_{𝑔𝑎𝑠}—the gas density, 𝜂

_{𝑔𝑎𝑠}—the dynamic viscosity of the gas and w—the cantilever width. For the output signal of the sensor (voltage of the Wheatstone bridge) follows

_{0,gas}—the resonance frequency in gas) follows

_{𝑔𝑎𝑠}- the density of the gas and 𝜂

_{𝑔𝑎𝑠}the dynamic viscosity of the gas. The 1st term describes the influence of movement of the cantilever on the damping, it increases with decreasing viscous layer thickness δ, which gets smaller with rising frequency ω

_{0.gas}, and the second term is the so-called Stokes’ term. This means that the damping of the oscillating cantilever is determined by Equation (12) for small flow velocities and satisfies the dependence of (10) for larger velocities. A simple approach is to add both parts together

## 3. Results

_{2}. All 3 measured variables - the resonance amplitude, the resonance frequency and the static deflection – are changed due to the gas-flow:

- The static deflection increases quadratically with the flow velocity according to equation (5). In the lower flow range, the measurement is inaccurate due to the very small responds and the noise of the static measurement.
- The resonance amplitude drops with 1/v. It has a high sensitivity in the flow range of 1–10 m/s. In this range it decreases by 50%.
- The resonance frequency increases linearly with the flow velocity. The change is small: $\frac{\Delta {f}_{0}/{f}_{0}}{\Delta {\nu}_{0}}\approx 0,1\cdots 0,2\%/\frac{m}{s}$. The size of the resonance frequency and its change as a function of the gas-flow is gas type dependent.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Self-actuated cantilever with integrated read-out; (

**b**) gas-flow sensor enclosure with cantilever and preamplifier; (

**c**) diagram of PLL based sensor control with signal separation.

**Figure 2.**(

**a**) Measured dependency of resonance amplitude, resonance frequency and (

**b**) bending of the cantilever sensor on the gas-flow through a tube with 1 mm diameter (dotted line—fitted curve, dashed line—simplified behavior, symbols - measured values, blue—air, red—CO

_{2}).

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## Share and Cite

**MDPI and ACS Style**

Zöllner, J.-P.; Durstewitz, S.; Stauffenberg, J.; Ivanov, T.; Holz, M.; Ehrhardt, W.; Riegel, W.-U.; Rangelow, I.W.
Gas-Flow Sensor Based on Self-Oscillating and Self-Sensing Cantilever. *Proceedings* **2018**, *2*, 846.
https://doi.org/10.3390/proceedings2130846

**AMA Style**

Zöllner J-P, Durstewitz S, Stauffenberg J, Ivanov T, Holz M, Ehrhardt W, Riegel W-U, Rangelow IW.
Gas-Flow Sensor Based on Self-Oscillating and Self-Sensing Cantilever. *Proceedings*. 2018; 2(13):846.
https://doi.org/10.3390/proceedings2130846

**Chicago/Turabian Style**

Zöllner, Jens-Peter, Steve Durstewitz, Jaqueline Stauffenberg, Tzvetan Ivanov, Mathias Holz, Waleed Ehrhardt, W.-Ulrich Riegel, and Ivo W. Rangelow.
2018. "Gas-Flow Sensor Based on Self-Oscillating and Self-Sensing Cantilever" *Proceedings* 2, no. 13: 846.
https://doi.org/10.3390/proceedings2130846