# Measuring Viscosity Using the Hysteresis of the Non-Linear Response of a Self-Excited Cantilever

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{1}and R

_{2.}Each stage has its own capacitance (C

_{1}= 2.37 × 10

^{−10}F and C

_{2}= 5.14 × 10

^{−9}F) and works effectively in a different range of frequencies (typically associated with the oscillation of the cantilever in air or liquid, respectively). The transfer function of the PS shown in Figure 1b is given by

_{1}and R

_{2}(used in this work). As can be observed, the approximation used in Equation (3) introduces negligible errors.

## 3. Results

#### 3.1. Experimental Measurements

_{0}= 139.4 kHz and Q = 240, by sweeping the frequency of a classical external excitation scheme and fitting a Lorentzian curve to the measured frequency response.

_{1}of the PS and then sweeping the value of the potentiometer R

_{2}up and down, while recording the oscillation frequencies of the closed-loop. Four different solutions of water and glycerol were used: (1) pure water, (2) water + 5% glycerol (v/v), (3) water + 10% glycerol (v/v), and (4) water + 15% glycerol (v/v). These corresponded to medium viscosities of, respectively, 1.005 × 10

^{−3}, 1.239 × 10

^{−3}, 1.384 × 10

^{−3}, and 1.650 × 10

^{−3}Pa s at 20 °C [30]. The density of the water–glycerol solutions does not change significantly with the concentration of glycerol (only $~2.5\%$ in this range [30]) and, therefore, the solution density was assumed to be constant and equal to the density of water (998 kg/m

^{3}at 20 °C [30]) throughout this work.

_{1}(R

_{1}= 6.11 kΩ) and polarity (p = −1 with P = 1) were chosen so that sudden jumps in oscillation frequency were observed when sweeping R

_{2}[24,27].

_{2}up and down while the cantilever was immersed in the four viscous solutions. It was observed that when R

_{2}was swept up (Figure 2a), the position of the sudden jump from low to high frequencies changed with the viscosity of the medium. Higher viscosities required a higher value of R

_{2}(bigger phase-shift imposed by the PS on the feedback loop) to jump.

_{2}down (Figure 2b), this dependence was less evident, and the sudden jump from high to low frequencies occurred for similar values of R

_{2}. Furthermore, it was observed that the position of the jumps from low to high frequencies (increasing R

_{2}) and from high to low frequencies (decreasing R

_{2}) did not match. This defined an hysteresis region delimited by two bifurcations, as also found in [27].

#### 3.2. Modeling of the System Behavior

#### 3.2.1. Equation of Motion

_{1}, R

_{2,}and on the oscillation frequency of the closed-loop ${\omega}_{osc}$).

_{1}= 1.0553, a

_{2}= 3.7997, b

_{1}= 3.8018, and b

_{2}= 2.7364.

_{1}and R

_{2}and can be modeled using Equation (3). In this case, it was assumed that the delay ${\tau}_{\mathrm{PS}}$ is the proportionality constant between the phase-shift that it introduces in the feedback loop (given by Equation (3)) and its oscillation frequency ${\omega}_{osc}$, as

#### 3.2.2. Solving for the Oscillation Frequency of the Loop, ω_{osc}

_{2}only (P, R

_{1,}and ${\omega}_{R}$ are fixed in the model).

_{1}and R

_{2}are fixed, resulting in a constant ${\tau}_{\mathrm{PS}}\left({\omega}_{R}\right)$. Finally, the system of Equation (10) is integrated for a chosen time interval (4 ms), with ${x}_{1}\left(t\right)=0$ and ${x}_{2}\left(t\right)=0.1$ when $\left(-{\tau}_{loop}-{\tau}_{\mathrm{PS}}\left({\omega}_{R}\right)\right)\le t\le 0$, as the past history of the function. The solution from the solver is interpolated with the Matlab function deval to obtain evenly time-spaced results (time step of 10

^{−8}s). Finally, the oscillation frequency of the system is obtained by detecting the maximum of the Power Spectral Density (PSD), calculated from the Fast Fourier Transform (FFT) of the time deflection signal ${x}_{2}\left(t\right)$ after transients are removed.

_{2}swept up (red line and symbols of Figure 2a).

_{2}in the region away from the jump (R

_{2}= 5 kΩ and R

_{2}= 6 kΩ, see Figure 2a). Increasing R

_{2}delays the time displacement signal (top row), but the phase space (${x}_{1}\left(t\right)$ vs. ${x}_{2}\left(t\right)$) and oscillation frequency remain essentially unaltered (middle and bottom rows, respectively). Figure 3b shows the case around the jump region (R

_{2}= 0.9 kΩ, R

_{2}= 1.0 kΩ and R

_{2}= 1.1 kΩ, see Figure 2a). Here, it can be observed that the time displacements in this region are no longer described by pure sinusoids, but that the motion already contains components at different frequencies (top row, blue and orange curves), due to the eminency of the sudden jump between different oscillation frequencies (see also [24]). By increasing R

_{2}, a sudden change in the time displacement signal occurs (yellow curve, R

_{2}= 1.1 kΩ), which corresponds to the jump to a higher frequency of oscillation. This jump can also be seen in the sudden change of the shape of the phase space (middle row) and by the normalized PSD curves (bottom row).

#### 3.3. Simulation Results

#### 3.3.1. Dependence of the Oscillation Frequency with Viscosity and Potentiometer R_{2}

_{2}up and down. The results are shown in Figure 4. In these simulations, the viscosity of the medium was varied between η = 0.2 × 10

^{−3}Pa s and η = 2.0 × 10

^{−3}Pa s, in steps of η = 0.04 × 10

^{−3}Pa s, while R

_{2}was increased or decreased between 0 and 10 kΩ, in variable steps (narrower in the jump region), for constant R

_{1}= 6.11 kΩ and polarity p = −1 (P = 1).

_{2}. The simulation proceeded row by row, with constant viscosity, while R

_{2}was swept up. The added mass and damping coefficients were initialized with the frequency of the previous calculated point, then the value of R

_{2}was incremented, and the system of Equation (10) was solved for the new oscillation frequency. This method proceeded until the potentiometer R

_{2}was fully swept, as indicated by the horizontal green arrows in the top row of Figure 4b. After the complete sweeping of R

_{2}, the viscosity of the system was increased. In this case, the added mass and damping coefficients were initialized with the oscillation frequency and R

_{2}of the first point of the previous viscosity row. Then, the value of viscosity was incremented and the system of Equation (10) was solved to determine the oscillation frequency of the first point of the new viscosity row, as indicated by the red arrows in Figure 4b. The simulation protocol was the same for the case of decreasing R

_{2}, as indicated by the colored arrows in Figure 4b, bottom row.

_{2}required to jump from low to high (top row) and from high to low frequencies (bottom row) with the viscosity of the medium. Nevertheless, the position of the sudden frequency jump was less sensitive to the viscosity when R

_{2}was swept down (bottom row). This frequency dependence is color-mapped in Figure 4b, with the jump from low to high frequencies (increasing R

_{2}, top row) delimited by the line between the blue and yellow areas, and the jump from high to low frequencies (decreasing R

_{2}, bottom row) delimited by the line between the yellow and red areas.

_{2}causes a higher amplitude of deflection since the oscillation frequency of the loop gets closer to the natural frequency of the cantilever. Higher amplitude deflection is also observed in low-viscosity mediums, which can be explained by the reduced damping induced by the cantilever–fluid interaction. Note that the magnitude of the values of amplitude of oscillation are arbitrary and depend on the chosen values of B and G (see Equation (10)) used in the simulations.

#### 3.3.2. Sensing Modalities

_{2}) and the right panel of Figure 5 shows the jump from high to low frequencies (decreasing R

_{2}). The red and green circles indicate the values of potentiometer R

_{2}for which the jumps were experimentally registered for each solution (different viscosities), as shown in Figure 2a,b, respectively. The thick, dashed, black lines delimit the jump region while the thin, white lines represent the jump region of the opposite panel.

_{2}up and down, while the self-excited cantilever is immersed in a solution of constant density and viscosity. By measuring the values of R

_{2}required for the first jump, from low to high frequency (sweeping up), and for the second jump, from high to low frequency (sweeping down), one can then determine the viscosity. Indeed, the difference between these values, or the width of the hysteresis, is univocally connected to the viscosity of the medium, as shown by the purple double arrows in both panels. The width of the hysteresis increases (non-linearly) with the viscosity of the medium. The second working modality is termed threshold mode. In this case, the sensor should be self-oscillating in a solution whose viscosity changes with time. It is this change in the viscosity of the medium that triggers the jump between oscillation frequencies.

_{2}swept up), jumping from low to high frequencies. This is shown by the decreasing red arrow on the left panel. Conversely, if the viscosity of the medium increases, the system follows the behavior indicated in the right panel (R

_{2}swept down), jumping from high to low frequencies. This is also indicated by the respective red arrow on the right panel.

## 4. Discussion

_{2}only and to clearly separate between the jumps in frequency when R

_{2}was swept up or down, defining the hysteresis region.

_{2}and the oscillation frequency ${\omega}_{osc}$ are simultaneously updated. When sweeping R

_{2}up with constant ${\omega}_{osc}$, then ${\tau}_{\mathrm{PS}}$ increases. If ${\tau}_{\mathrm{PS}}$ gets bigger than the threshold value required to jump from low to high frequencies, the jump occurs. However, on the other hand, if the jump occurs, the oscillation frequency ${\omega}_{osc}$ suddenly increases (with constant R

_{2}) and, therefore, ${\tau}_{\mathrm{PS}}$ suddenly decreases. In this case, ${\tau}_{\mathrm{PS}}$ may become again smaller than the threshold value required to jump from low to high frequency, and the system will go back to the low-branch solution (the reasoning is equivalent but opposite when R

_{2}is swept down). In summary, when simultaneously updating R

_{2}and ${\omega}_{osc}$ in ${\tau}_{\mathrm{PS}}\left({\omega}_{osc}\right)$ of Equation (8), the solution of Equation (10) jumps back and forth between the two solution branches. The hysteresis region is then measured as the R

_{2}interval that causes the system to alternate between the two branches. Above a certain value of R

_{2}, the system remains definitely in the upper solution branch, since R

_{2}gets sufficiently big (high ${\tau}_{\mathrm{PS}}$) to guarantee that the decrease in ${\tau}_{\mathrm{PS}}$ that occurs when the system jumps to the high-solution branch does not get lower than the threshold value. This mechanism allows us to conjecture that the system shows two fold bifurcations, leading to the sudden jumps and defining the hysteresis region.

_{2}or decreasing the viscosity $\eta $ is physically equivalent, since the jumps from low to high frequencies follow the same path. A physical interpretation of this phenomenon is based on the following observation: An increase on either ${\omega}_{osc}$ or $\eta $ is responsible for an increase of the ratio between Equations (5) and (6), ${c}_{\mathrm{A}}\left({\omega}_{osc}\right)/{m}_{\mathrm{A}}\left({\omega}_{osc}\right)$. So, assuming that the system is oscillating in the low-frequency branch and that viscosity increases, then $\eta \uparrow \stackrel{}{\Rightarrow}{c}_{\mathrm{A}}\left({\omega}_{osc}\right)/{m}_{\mathrm{A}}\left({\omega}_{osc}\right)\uparrow $. On the other hand, if R

_{2}decreases, then the delay imposed by the PS in the loop also decreases. This forces the cantilever to compensate, by increasing its phase and, consequently, the oscillation frequency of the loop and the ratio ${c}_{\mathrm{A}}\left({\omega}_{osc}\right)/{m}_{\mathrm{A}}\left({\omega}_{osc}\right)$, or, schematically: ${R}_{2}\downarrow \stackrel{}{\Rightarrow}{\tau}_{\mathrm{PS}}\downarrow \stackrel{}{\Rightarrow}{\varphi}_{\mathrm{CT}}\uparrow \stackrel{}{\Rightarrow}{\omega}_{osc}\uparrow \stackrel{}{\Rightarrow}{c}_{\mathrm{A}}\left({\omega}_{osc}\right)/{m}_{\mathrm{A}}\left({\omega}_{osc}\right)\uparrow $.

_{2}or increasing the viscosity $\eta $ (the conclusions are opposite when the system is oscillating in the high-solution branch and viscosity decrease or R

_{2}increase, as also seen in Figure 5).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic of the experimental setup. The deflection of the cantilever is detected by a four-quadrant detector, naturally delayed, amplified, saturated, controllably delayed with an adjustable Phase-Shifter (PS), and, finally, fed back to the exciting piezo. (

**b**) Schematic of the two-stage PS. The second stage was used to control the imposed shift in the feedback loop with the potentiometer R

_{2}. (

**c**) Phase-shift introduced by the PS shown in (

**b**) as function of the oscillation frequency of the closed-loop given by the transfer function (Equation (2), solid lines) and the sigmoid approximation (Equation (3), dashed lines), for different polarities of the piezo.

**Figure 2.**Experimental oscillation frequencies of the self-excited microcantilever as function of the phase-shift introduced by the PS, with fixed R

_{1}and polarity (p = −1). (

**a**) Potentiometer R

_{2}swept up; (

**b**) potentiometer R

_{2}swept down. Insets detail the jump regions and G

_{5}, G

_{10,}and G

_{15}indicate the concentration of the glycerol solutions.

**Figure 3.**Solving system of Equation (10) for different values of increasing R

_{2}, for the case of the cantilever oscillating in water. (

**a**) Top, middle, and bottom rows show the time deflection, phase space, and normalized Power Spectral Density (PSD) of the signals far from the jump region for different values of R

_{2}, respectively. (

**b**) Time deflection, phase space, and normalized Power Spectral Density (PSD) of the signals close to the jump region, for different values of R

_{2}.

**Figure 4.**Dependence of the oscillation frequencies on the viscosity of the medium and value of R

_{2}when the potentiometer was swept up (upper row) or swept down (lower row). (

**a**) Jump region when sweeping R

_{2}, for different viscosities. (

**b**) Color map of the oscillation frequencies and (

**c**) color map of the amplitude of oscillation, for the full range of potentiometer and viscosities used.

**Figure 5.**Left and right panels show insets of the oscillation frequency map when R

_{2}was swept up and down, respectively (Figure 4b, top and bottom rows), with thick, dashed, black lines delimiting the jumps. The red and green circles represent the experimental data measured and presented in Figure 2a,b, for the four glycerol solutions. The thin, white, dashed lines represent the jump delimitation of the opposite panel, for an easier visualization of the two distinct sensing modalities proposed.

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**MDPI and ACS Style**

Mouro, J.; Paoletti, P.; Basso, M.; Tiribilli, B. Measuring Viscosity Using the Hysteresis of the Non-Linear Response of a Self-Excited Cantilever. *Sensors* **2021**, *21*, 5592.
https://doi.org/10.3390/s21165592

**AMA Style**

Mouro J, Paoletti P, Basso M, Tiribilli B. Measuring Viscosity Using the Hysteresis of the Non-Linear Response of a Self-Excited Cantilever. *Sensors*. 2021; 21(16):5592.
https://doi.org/10.3390/s21165592

**Chicago/Turabian Style**

Mouro, João, Paolo Paoletti, Michele Basso, and Bruno Tiribilli. 2021. "Measuring Viscosity Using the Hysteresis of the Non-Linear Response of a Self-Excited Cantilever" *Sensors* 21, no. 16: 5592.
https://doi.org/10.3390/s21165592