RHEOLOGICAL PROPERTY ESTIMATION USING NONLINEAR VIBRATION OF A
PARTIALLY FLUID-IMMERSED CANTILEVER BEAM

By

Sarthak Mehulbhai Desai

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Electrical and Computer Engineering—Master of Science

2024

ABSTRACT

Resonant sensors for the measurement of rheological properties like density and viscosity are of-

ten employed in online process-monitoring applications. Micro-acoustic MEMS devices such as

micro-cantilevers and Surface Acoustic Wave devices have been widely used for such measure-

ments. However, due to their scale, these devices measure thin film viscosity and density. Such

measurements are often not comparable to macroscopic measurements obtained through conven-

tional devices. Miniaturized cantilever-based devices provide an interesting alternative as they are

minimally intrusive, like micro-acoustic sensors, yet measure in bulk rheological domain. However,

the interactions between the liquid and the oscillating beam are more complex to model. Such

interactions have been previously modeled using classical linear Euler-Bernoulli beam theory or by

considering an equivalent lumped elements oscillator such as a Duffing or Van der Pol oscillator.

The derived models are subsequently used to relate the liquid’s viscosity and density to measurable

parameters such as resonance frequency 𝑓0 and resonant mode quality factor 𝑄−1. Currently, there

are no exact models in the literature for describing the nonlinear vibration of partially immersed

beams, nor experimental results providing an understanding of how fluid properties affect the

nonlinear vibration characteristics.

This work focuses on first establishing empirical relationships between different experimental

parameters -such as fluid volume, density, viscosity, length of beam, etc.- and a force-excited,

partially immersed beam’s resonant response. Then, it describes an ensemble machine learning

(ML) based approach that models the nonlinear change in the frequency response of the beam

with an increase in excitation amplitude, by measuring the variation in resonance frequency and

quality factor of a selected sensitive mode. These measured quantities are subsequently used as

features on which ensemble ML models are trained to predict the density and viscosity of the tested

fluids. With relatively few (275) training data points, the model can predict viscosity with a 96.65%

(𝑅2, 10-fold cross-validation) score and both density and viscosity with an 89.58% (𝑅2, 10-fold

cross-validation) score. The impact of this work is to provide a proof of concept for a rheological

property sensor that utilizes an ML-based approach for online viscosity and density measurement.

In loving memory of my grandmother, Pushpaben B Desai.

iii

ACKNOWLEDGEMENTS

I wholeheartedly thank the following of my professors, friends, and colleagues:

My thesis advisor, Dr. Sunil K. Chakrapani, for his continuous support and contribution to my

thesis work for the past 1.5 years. My committee members, Dr. Nelson Sepulveda and Dr. Vaibhav

Srivastava, for taking the time to oversee my final defense.

The Physical Ultrasonics, Microscopy, and Acoustics (PUMA) Lab for providing the resources

to conduct my experiments, and my lab mates Yoganandh Madhuranthakam, Zebadiah Miles, and

Subal Sharma for their kind assistance both in and outside the lab.

The instructors of the courses I had the pleasure of being a teaching assistant for, specifically

Dr. Tim Hogan and Matt Meier, for granting me a great learning opportunity. I also thank Brian

Wright for his constant help with lab setup and function.

The staff at the Dept. of Electrical and Computer Engineering. Especially, Ms. Lisa Clark for

guiding me through my transition to a master’s thesis track.

My friends, old and new. In particular, Henry Dsouza, Aakash Gupta, Ankur Kamboj, and

Anirudh Suresh, my friends from Drishti and Robocon SVNIT, and my friends from high school,

for their counsel and motivation, both academically and in routine life. And to my roommates

Brijen, Vishwanath, and Manu, for giving me a home away from home.

Finally, but most importantly, my parents, my brother Amish, and sister-in-law Komal, for their

unwavering support.

iv

TABLE OF CONTENTS

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
1.1 Fluid immersed vibrating beam models . . . . . . . . . . . . . . . . . . . . . .
1.2 Rheology using immersed beams . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 2

BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
2.1 Nonlinear vibration of fluid immersed beam . . . . . . . . . . . . . . . . . . .
2.2 Beam frequency response hypotheses . . . . . . . . . . . . . . . . . . . . . . .

1
1
3

6
6
7

CHAPTER 3

9
9
3.1 Experimental setup .
3.2 Linear and nonlinear parameters . . . . . . . . . . . . . . . . . . . . . . . . . 11

METHODS AND MATERIALS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 4

EMPIRICAL STUDY OF FLUID PROPERTY EFFECT . . . . . . . . . 13
4.1 Proportionality analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Regression analysis .

.

CHAPTER 5

RHEOLOGICAL PROPERTY ESTIMATION THROUGH
MACHINE LEARNING MODELS . . . . . . . . . . . . . . . . . . . . 19
5.1 Training data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Machine Learning methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3 Result and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
. 27
5.4 Future work . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

CHAPTER 6

6.1 Conclusion .

SUMMARY .
.

.

.

.

. 29
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

BIBLIOGRAPHY .

.

.

.

.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

v

CHAPTER 1

INTRODUCTION

1.1 Fluid immersed vibrating beam models

Vibration of beams immersed in fluids have been of interest to multiple fields. Specifically,

there are several applications where the vibration response of a beam has been used to evaluate

fluid properties. These applications scale over several orders of magnitude in length from micro

electro-mechanical systems (MEMES) and nano electro-mechanical systems (NEMS) devices, to

oil rig type of setup. When the vibrating beam is in a vacuum i.e. without any surrounding fluid,

its natural resonant frequencies has been well modeled and understood using analytical methods

for many practical scenarios [1]. However, determining the frequency response of an elastic beam

immersed in a viscous fluid is challenging. A group of work by Sader [2], Green [3], and Van

Eysden [4] have explored analytical solutions with restrictions that the length of the beam must

greatly exceed its nominal width, the amplitude of vibration must be small, and the fluid must be

incompressible in nature. Their proposed analytical model accounts for the loading induced by

the viscous fluid, thus enabling the frequency response to be estimated from a knowledge of the

material and geometric properties of the beam, and the viscosity and density of the fluid. For

a thermal force-driven immersed cantilever, they found that the contribution of viscous effects is

strongly dependent on the dimensions of the beam [2] [3]. Decreasing these dimensions enhances

viscous effects, resulting in increased broadening and shifting of the resonant peak from its value

in a vacuum. In [4], they provide an exact solutions for the three-dimensional flow generated by an

oscillating thin blade in a viscous fluid. Shabani et al. [5] presented a solution for microcantilever

beams, wherein they formulated free vibration frequencies of a cantilever micro-beam submerged

in a bounded frictionless and incompressible fluid cavity. They noted that fluid loading, modeled

as added masses, was greater for lower modes than for higher modes. By varying the fluid density,

it was also shown that the higher modes are of greater importance in denser fluids.

However, most of these approaches are limited to beam vibrations that are considerably smaller

than the cross-section width. This results in a vibrational response that is linear. Further simplifying

1

the necessary modeling effort to capture the frequency response. An analysis of the nonlinear

problem involving finite amplitude vibrations of flexible beams was explored by Aureli et al.

[6]. The structure was modeled using the Euler-Bernoulli beam theory for the case of sharp-edged

beams, and the model was used to analyze and predict the steady-state response of the beam vibrating

under harmonic base excitation. The derived equations of motion included a hydrodynamic function

expressed as the linear combination of the classical Navier–Stokes hydrodynamic function and a

correction term. The correction term captured the effect of hydrodynamic damping resulting from

moderately large oscillations. They also experimentally verified their model. Cagri [7] developed

a nonlinear dynamic model based on the forced Van der Pol oscillator and demonstrated the time-

domain sensitivities of the micro-cantilever to the varying properties of the surrounding fluids and

multi-frequency excitations.

While these articles have explored the case of a beam fully immersed in a fluid, analysis of

a beam that is partially immersed in a viscous fluid is limited in the literature. Abassi et al. [8]

presented an analytical approach to describe the modal behavior of Euler Bernoulli beams partially

immersed in a viscous fluid and validated the solution numerically. However, the solution is only

valid for linear vibration. The nonlinear convective inertial effects in the fluid were neglected, and

the hydrodynamic loading on the beam was a linear function of its displacement. Al-Qaisia et al.

[9] studied the steady-state frequency response of a slender cantilever beam partially immersed in

water and carrying an intermediate mass. The assumptions of the inextensibility condition were

taken as a means to account the inertia nonlinearities, which have a strong influence on the steady

state frequency response curves of the beam system. The nonlinear equation of motion was derived

using the Euler Lagrange method in conjunction with the assumed mode method, and the steady

state responses under the effect of sinusoidal distributed and concentrated loads were obtained.

Uscilowska et al. [10] obtained a closed-form solution of the natural frequency and mode shape

for a partially immersed column with eccentrically located tip mass. The column was modeled as

a uniform Bernoulli–Euler cantilever beam fixed at the bottom with a concentrated mass at the top.

M K. Kwak et al. [11] explored free flexural vibration of a cantilever plate with one end partially

2

submerged in a fluid and the other end fixed.

Presently, there are no articles in the literature that have explored the nonlinear response of a

partially immersed beam. Furthermore, many of the existing studies have partial overlap and have

not been experimentally verified; instead they focus primarily on analytical or numerical models.

Hence, Chapter 3 and 4 of this article presents an experimental exploration of nonlinear vibration

of a partially immersed cantilever beam. It focuses on weak nonlinearities arising primarily from

elastic nonlinearity of the solid beam rather than the geometric nonlinearities. To understand

the effect of the nonlinear response of the beam, a set of hypotheses are first developed, which

are tested using a series of experiments. The experimentally measured properties include linear

resonant frequency and damping ratio, nonlinear frequency shift, and nonlinear damping ratio.

These measured properties were compared for different experimental conditions such as volume

of the fluid, length of immersion, and variation in fluid properties such as viscosity and density.

Subsequently, a regression analysis was carried out to determine the quantitative relationships

between different parameters. Once these relationships were established, measured parameters

were used to generate models for rheological measurement.

1.2 Rheology using immersed beams

In many applications that involve online process and condition monitoring, fluid rheological

properties like viscosity and mass density have high relevance. Traditional laboratory equipments

are often unsuitable due to their space requirements, operating temperature and other physical

constraints. Additionally, sample collection for these devices typically involves manual labor,

which can be both time-consuming and prone to errors. Hence, during the last two decades, there

has been a increased interest in resonant viscosity and mass density sensors. Owing to their reduced

size compared to conventional instruments, their relatively straightforward integrability in a process

line, and potentially low manufacturing costs [12].

Microacoustic sensors, including quartz thickness shear mode (TSM) resonators [13, 14] and

surface acoustic wave (SAW) devices [15], have emerged as effective alternatives to conventional

viscometers [16]. Microcantilevers, often used in atomic force microscopy [17–19], have also

3

proven effective as liquid property sensors, enabling simultaneous measurement of viscosity and

density with sample volumes less than 1 nL [20]. They typically utilize highly sensitive optical

readouts to measure vibration amplitudes. When immersed in liquid, such cantilevers experience

a significant decrease in quality factor due to high dissipation [20], resulting in reduced vibra-

tion amplitudes, and limiting their range to low-viscosity liquids. Other studies have utilized

micromachined cantilevers and doubly clamped beams, driven by Lorentz forces [21–24] or piezo-

electric effects [25, 26], as liquid property sensors, demonstrating their effectiveness for viscosities

measurement up to several Pa·s.

However, these sensors measure viscosity under relatively high shear rates and low vibration

amplitudes, making their results less directly comparable to those from traditional viscometers. For

complex liquids like emulsions, micro-acoustic devices might not adequately capture rheological

effects that are apparent only on a macroscopic scale [27]. Vibrating structures, with their lower

resonance frequencies and higher vibration amplitudes, are generally better suited for a broader

range of viscosity and density of fluids, as well as non-Newtonian and complex liquids [21].

Depending on the particular resonator design, closed-form models, considering structural and fluid

mechanics, may become relatively complex, demanding high modeling effort and computational

power. Several models have been described in recent literature that aim to provide a description

of the interaction of a vibrating, fluid-immersed cantilever and surrounding fluid, for example,

[2, 4, 28]. Most of these models assume the cantilever to be sharp-edged, long, thin, and completely

immersed in fluid. Though some [29] also use partially immersed beams. The models are derived

from Euler - Bernoulli beam theory [28, 30, 31], or by approximating the cantilever as an oscillating

sphere immersed in a liquid[25, 29], or by considering an equivalent lumped elements oscillator

immersed into a liquid [32].

Concerning excitation and readout, in prior listed studies, recording the frequency responses

of piezoelectric or piezoresistive devices [33–35], is a very common technique. In many cases,

resonance frequency 𝑓𝑟 and quality factor 𝑄 (also known as damping ratio) are first evaluated,

which are then related to the liquid’s viscosity and mass density by an appropriate model [20, 32].

4

In [32], a generalized reduced order model is presented, relating resonance frequency 𝑓𝑟 and quality

factor 𝑄 to mass density and viscosity, for a single excitation amplitude. In this generalized models,

any material or geometric nonlinearity parameters are not explicitly considered but are contained

in a single factor in the model.

Both material (elastic) and geometric nonlinearity of a beam are known to effect its frequency

response, resulting in frequency shift and amplitude dependent damping [36]. The nonlinear

frequency shift and damping have also been shown to be dependent on the rheological properties

of the fluid in which the beam is immersed [37]. Hence, utilizing changes in resonance frequency

and quality factor due to immersed fluid dependent nonlinear resonance response of a beam is

worth exploring. Though, a physics-based analytical model would be fairly complex to develop,

and such a model has not been explored in literature for a partially fluid immersed cantilever.

As this work primarily focuses on estimation of fluid rheological properties: dynamic viscosity

and mass density, a machine learning based modeling approach is considered here. In Chapter 5,

we present a vibrating cantilever sensor setup for the measurement of mass density and dynamic

viscosity of fluids that utilizes the effects of material nonlinearity of a partially fluid-immersed

cantilever on its frequency response with a change in excitation amplitude. From the frequency

response, the resonant mode, under 1 KHz, most sensitive to fluid rheological properties, is selected.

For the selected resonant mode, the resonance frequency 𝑓0 and quality factor 𝑄−1 is calculated,

the variation in 𝑓0 and 𝑄−1 with increase in actuation amplitude is measured. The frequency

response in a 100 Hz window around the resonant mode is parameterized by fitting it to sum of

log-normal distribution functions. The coefficients of this parametrized function, as well as features

derived from tracking changes 𝑓0 and 𝑄−1 with respect to amplitude, are used as features to train

ensemble machine learning models to predict mass density and dynamic viscosity of 11 different

test liquids. The measurement is done in rheological domain and hence is comparable to the results

of conventional laboratory instruments.

5

CHAPTER 2

BACKGROUND

2.1 Nonlinear vibration of fluid immersed beam

The effect of fluid properties on the nonlinear vibration of an elastic beam has several interest,

challenges and applications. The partially immersed nature of the beam allows us to carry out

remote sensing of the fluid properties even at higher temperatures. The schematic in Fig. 2.1 shows

the setup used in this study. The length of the beam immersed in fluid experiences a different

boundary condition compared to rest of the beam. Typically this has been shown to affect the

linear vibration response, for example the natural frequency and damping ratio [32] [38]. However,

nonlinear vibration has not been explored in the literature.

In general, nonlinear vibrational

response can arise from two sources: (a) Geometric nonlinearity, where the beam displacement

amplitude is very large, and can be defined through a nonlinear strain-displacement equations of

the beam. (b) Material nonlinearity: where the displacements are relatively small compared to

geometric nonlinearity, however, the stress-strain response of the elastic beam is nonlinear. In terms

of vibrational response, both of these nonlinearities will result in frequency shift and amplitude

dependent damping [36]. Geometric nonlinearity results in increase in resonant frequency as a

function of forcing amplitude, also termed as the nonlinear frequency shift. Whereas, material

nonlinearity results in decrease in the resonant frequency. As shown earlier, the vibration response

of a beam as function of strain amplitude can be divided into three parts: (a) linear range where no

frequency shift occurs, (b) material nonlinearity regime where the resonant frequency decreases,

and (c) geometric nonlinearity regime which occurs at very high strain amplitudes that results in

increase of the resonant frequency of the beam. From a mechanics perspective, there is a lot of

interest in understanding the role each of these nonlinearities play in a partially immersed beam.

If we choose the torsional mode and force it to vibrate at higher strain amplitudes, i.e. geometric

nonlinearity regime, then we can potentially approximate it to a traditional viscometer. However,

there are several challenges from beam geometry, and other experimental considerations.

If a

similar response can be extracted in the lower strain range, i.e. material nonlinearity regime, then

6

it could be useful for several applications. Therefore, we limit this study to only focus on material

nonlinearity of the elastic beam.

2.2 Beam frequency response hypotheses

Figure 2.1 schematics

Figure 2.1 illustrates a schematic of the setup used in this study. From the schematic, we can

notice that the boundary conditions on the elastic beam will change based on the following factors:

(a) the length of the beam immersed in the fluid, (b) volume of the fluid, and (c) fluid properties

such as density and viscosity. We can hypothesize that the length of immersion, 𝐿, will also change

the hydrostatic pressure on the length of the beam that is immersed, thus affecting the nonlinear

response directly. This will also result in nonlinear damping, i.e. amplitude dependent damping

factor, that will change with strain amplitude. We can also hypothesize that the beam vibration

could result in possible resonances that are setup in the fluid. If these resonance do form in the fluid

as shown earlier using numerical analysis [8], they will be a function of the volume. Finally, the

7

damping and resonant frequency of the beam are known to be functions of the viscosity and density

of the fluid, however, their dependencies on the nonlinear vibration have not been understood. The

first objective of this study is to create a matrix of different combinations of these parameters, test

these hypotheses, and establish a statistical relationship between these parameters.

8

CHAPTER 3

METHODS AND MATERIALS

3.1 Experimental setup

Figure 3.1 NRAS Experimental setup

Figure 3.1 illustrates the experimental (NRAS - Nonlinear resonant acoustic spectroscopy) setup

utilized in this study. The setup consists of a magnetostrictive linear actuator controlled with a

lock-in amplifier. The linear actuator was rigidly clamped, and its actuation head was coupled with

an AISI 1080 low carbon steel cantilever beam of length 165.1 mm, thickness t = 1.6 mm and width

w = 3.2 mm, using a 3D printed coupler, such that 152.4 mm of its length hangs under. The opposite

end of the cantilever beam is partially immersed in a test fluid. The actuation signal for the linear

actuator was generated by a lock-in amplifier, which was programmed to output a sinusoidal signal

that was subsequently amplified by an audio power amplifier. The amplified output, in turn, drives

the actuator. An accelerometer was used to measure the amplitude of vibrations induced in the beam

by mounting it on the surface of the beam above its geometric center. The output signal generated

by the accelerometer is passed through a signal conditioner and subsequently passed back into the

lock-in amplifier. A MATLAB script was used to transmit data over a USB connection using VISA

protocol to program the lock-in amplifier. An internal reference frequency and amplitude were

9

programmed, which the lock-in amplifier uses to generate the sinusoidal signal for the actuator.

The vibrational amplitude corresponding to set internal reference frequency is precisely tracked in

the output signal from the accelerometer by the Lock-in amplifier, and the measurement is logged.

This measured amplitude and its corresponding frequency were transmitted to a computer over a

USB-VISA connection and then processed using a MATLAB script. As illustrated in the figure

3.1, to study the effects of the rheological properties of a fluid on the frequency response of the

partially immersed beam, its immersion length L is varied from 5 to 50 mm in fixed increments.

The volume of the fluid is also varied as described in Table 3.1. The frequency response of the

system between 150 Hz and 950 Hz for three different fluids: Deionized water, SAE 10W30 engine

oil and SAE 85W140 gear oil, at a fixed volume and immersion length, was studied. From the

resonant modes observed, the mode in the 750 Hz to 850 Hz window for different fluids is selected

as it was found to be the most sensitive mode to the variation in viscosity and density of the tested

fluids. Figure 3.2 illustrates the selected resonant mode, and the changes in its response to fluids in

increasing order of viscosity from (a) to (c).

Figure 3.2 Sensitive resonant mode (a) water (b) SAE 10W30 (c) SAE 85W140

Subsequently an investigation of shift in the selected mode’s resonant frequency with variation

in fluid: density, viscosity, volume and length of beam immersion was carried out. The tested

immersion lengths (depths) for the beams were: 5mm, 15mm, 30mm, 45 mm and 50mm. And the

tested fluid volumes are listed in table 3.1. The frequency response of the chosen mode with increase

in excitation signal amplitude for each fluid is logged. The highest excitation corresponding to a

displacement of ± 5.5 µm.

10

Table 3.1 Properties of tested fluids

Fluid
Water
SAE 10W 30
SAE 85W 140

Density, 𝜌 (𝑘𝑔/𝑚3) Dynamic viscosity, 𝜂 (𝑚𝑃𝑎𝑠) Volume 𝑉 (𝑚𝑙)
170,237,295
173,240,301
171,239,298

963.5
799.2
855.7

0.6
53.9
351.7

3.2 Linear and nonlinear parameters

Figure 3.3 (a) Frequency response of the chosen mode, (b) Damping ratio calculation

A logged frequency response of the mode in the 750 Hz to 850 Hz window, with increase in

excitation signal amplitude is illustrated in the figure 3.3 (a) for an SAE 10W30 motor oil sample,

with the beam immersed up to 30 mm in the fluid and a volume of 301 ml.

To quantify the obtained frequency response plots, the following parameters are defined.

• 𝑓0: Linear (low amplitude) resonant frequency

• Δ 𝑓 / 𝑓0: Relative nonlinear shift in resonant frequency with increase in excitation amplitude.

This is given by: Δ 𝑓 / 𝑓0 = ( 𝑓0 − 𝑓𝑝)/ 𝑓0

• 𝑄0

-1 : Damping ratio as illustrated in Figure 3.3(b). This is given by:

• 𝑄 𝑝-1 : Damping ratio of the highest excitation resonance curve.

• Δ𝑄-1 : Change in Damping ratio with increase in excitation amplitude. Given by: Δ𝑄-1 =

𝑄 𝑝-1 − 𝑄0

-1

11

These parameters were recorded for each experimental run and subsequently used for establish-

ing statistical relationships with fluid properties.

12

CHAPTER 4

EMPIRICAL STUDY OF FLUID PROPERTY EFFECT

Three fluids with distinct dynamic viscosities and mass densities were chosen to study the empirical

relationships with the frequency response of the beam. The three chosen fluids were filled in separate

beakers of different volumes and tested using the NRAS setup shown in Fig. 3.1. First, the fluid was

filled to a volume of 170 ml, and the beam was immersed with an immersion length of 5 mm, then

the frequency sweep was carried out over a 100 Hz window for various excitation amplitudes. Next,

the immersion length was increased to 15mm and the process was repeated for all the immersion

lengths (depths) listed in section 3.1 upto 50 mm. Once all the immersion lengths were tested,

the volume was increased to 240 ml, and the entire process was repeated for different volumes

listed in table 3.1. For each dataset, the raw frequency-amplitude data was processed further to

obtain the linear and nonlinear parameters listed in Sec.3.2. This allows us to determine the effect

of immersion length and volume on the different linear and nonlinear parameters of the vibrating

beam. Furthermore, as the experiment was carried out for three different fluids, it allowed us to

determine the effect of fluid viscosity and density on the nonlinear vibrations of the beam. Section

4.1 presents the variation in measured linear and non-linear parameters for the three chosen fluid:

(a) Deionized water, (b) SAE 10W 30 oil and (c) SAE 85W 140 oil.

4.1 Proportionality analysis

4.1.1 Linear frequency ( 𝑓0)

The results of linear resonant frequency ( 𝑓0) as a function of different volume and immersion

length is shown in the figure 4.1. It can be observed that 𝑓0 decreases with increase in the immersion

length of the beam. A relative decrease in 𝑓0 (≈ 2 − 3 Hz) can also be observed with increase

in viscosity of the fluid. Interestingly, there seems to be no Apparent change in 𝑓0 with increase

in volume of the fluid. Finally, a decrease in 𝑓0 at higher immersion lengths seems to be more

prominent at higher values of viscosity. This suggests some type of coupling between viscosity of

the fluid and immersion length.

13

4.1.2 Nonlinear frequency shift (Δ 𝑓 / 𝑓0)

Figure 4.1 Linear resonant frequency

Figure 4.2 Relative nonlinear shift in resonant frequency

A relative shift was observed in the selected resonant mode with increase in excitation amplitude.

This relative shift Δ f/f0 appears to be independent of volume, which is similar to 𝑓0. A reduction in

Δ 𝑓 / 𝑓0 in with increase in viscosity of the tested fluid is observed. Overall, the Δ 𝑓 / 𝑓0 changes with

immersion length, similar to 𝑓0. However, there seems to be a strong coupling to the viscosity and

immersion length as well. It can be observed that the Δ 𝑓 / 𝑓0 increases with increase in immersion

length for lower viscosity fluids, however, this effect is diminished for higher viscosity.

4.1.3 Damping ratio (𝑄−1
0 )

It was observed that the lowest excitation signal damping ratio 𝑄−1
0

increases with the increase

in viscosity. It also increases with the increase in immersion length. Similar to 𝑓0 and Δ 𝑓 / 𝑓0, there

is a strong coupling between immersion length and viscosity of the fluid. The increase in damping

ratio at greater lengths is more pronounced with an increase in viscosity. No apparent change was

14

observed with the increase in the volume of the fluid.

Figure 4.3 Damping ratio for the minimum excitation amplitude

4.1.4 Nonlinear Damping ratio 𝑄−1
𝑝

Figure 4.4 Damping ratio for the maximum excitation amplitude

For the highest excitation signal damping ratio 𝑄−1

𝑝 it was observed that: There is no apparent

change with the increase in volume of the fluid The nonlinear damping ratio 𝑄−1

𝑝 also increases with

the increase in immersion length similar to lowest excitation signal damping ratio 𝑄−1

0 . Though the

𝑄−1

increase for lowest viscosity is not as prominent compared to 𝑄−1

0 . The non linear damping ratio
0 . And, similar to 𝑓0 and Δ 𝑓 / 𝑓0,
a strong coupling was also observed between the immersion length and viscosity of the fluid. The

𝑝 also increased with the increase in viscosity. Similar to 𝑄−1

increase in non linear damping ratio 𝑄−1
𝑝 .

is more pronounced with increase in viscosity, again

similar to the minimum excitation damping ratio 𝑄−1
0 .

15

4.2 Regression analysis

To investigate the dimensional relationship between the parameters described in Sec. IV: 𝑓0,

Δ 𝑓 / 𝑓0, 𝑄−1

0 , and 𝑄−1

𝑝 , with respect to the rheological and extensive properties of the fluids tested,

a regression analysis was performed.

A multivariate exponential relationship is assumed for the tested parameters.

𝑓0 ∝ 𝜌 𝐴1𝜂𝐵1 𝐿𝐶1𝑉 𝐷1𝑒𝐸1

Δ 𝑓 / 𝑓0 ∝ 𝜌 𝐴2𝜂𝐵2 𝐿𝐶2𝑉 𝐷2𝑒𝐸2
0 ∝ 𝜌 𝐴3𝜂𝐵3 𝐿𝐶3𝑉 𝐷3𝑒𝐸3
𝑄−1
𝑝 ∝ 𝜌 𝐴4𝜂𝐵4 𝐿𝐶4𝑉 𝐷4𝑒𝐸4
𝑄−1

(4.1)

To estimate the constants of proportionality, a natural log function is applied to the set of

equations 4.1. Where 𝜌 is fluid density, 𝜂 is fluid viscosity, 𝐿 is the length of beam immersion and

𝑉 is the volume of the fluid. To fit the linear model obtained by taking the natural log of proposed

equations 4.1, the method of ordinary least square is applied. The considered linear model was of

the form:

𝑌 = 𝑋 𝛽 + 𝜖,

(4.2)

where 𝑦 = (𝑦1, 𝑦2, ..., 𝑦𝑛)′ is an n × 1 response vector associated with parameters 𝑓0, Δ 𝑓 / 𝑓0, 𝑄−1
0 ,
𝑛]′ is an n × (p + 1) incidence matrix associated with fluid

𝑝 to be modelled, 𝑋 = [𝑥′
1

and 𝑄−1

, ..., 𝑥′

, 𝑥′
2

density, dynamic viscosity, volume and length of beam immersion. The incidence matrix is for a

vector of effects 𝛽 = (𝜇, 𝛽1, ..., 𝛽𝑝)′, which yield the proportionality constants. Here, n (=135) is

the number of measured data points obtained from the experimental setup, and p (=4) is the number

of independent variables.

The natural log values of the experimentally measured parameters are min-max normalized and

the linear model is fit.

Ordinary Least Squares (OLS) estimates are obtained by minimizing the Residual Sum of

Squares (RSS) with the solution.

ˆ𝛽 = [𝑋′𝑋]−1𝑋′𝑦

(4.3)

16

The resulting proportionality exponential co-efficient are listed in table 4.1. and their corre-

sponding P-values are listed in table 4.2. The F-significance of the fit for 𝑓0, Δ 𝑓 / 𝑓0, 𝑄−1

0 , and

𝑄−1

𝑝 , are 2.13e-43, 3.43e-34, 4.40e-47 and 3.21e-62 respectively.

Indicating a strong statistical

correlation.

parameter
f0
Δ 𝑓 / 𝑓0
Q−1
0
Q−1
𝑝

Table 4.1 Model co-efficients

length

viscosity

Density
Volumes
A1 = −0.3 B1 = −0.4 C1 = −0.5 D1 = 0
A2 = −0.2 B2 = −0.6 C2 = 0.1
D2 = 0
C3 = 0.3
B3 = 0.6
A3 = 0.3
D3 = 0
C4 = 0.3
B4 = 0.7
A4 = 0.3
D4 = 0

proportionality
e𝐸1 = 3.4
e𝐸2 = 2.7
e𝐸3 = 0.8
e𝐸4 = 0.7

Table 4.2 Model P-values

parameter Density
1.05e-11
1.56e-9
1.68e-13
9.18e-21

f0
Δ 𝑓 / 𝑓0
Q−1
0
Q−1
𝑝

viscosity
3.78e-19
6.77e-31
4.56e-37
3.45e-52

length
1.41e-40
0.033
5.06e-25
1.42e-31

Volumes
0.51
0.98
0.40
0.96

proportionality
9.24e-62
8.32e-48
1.08e-10
1.3e-22

From table 4.1 and 4.2, the results of initial observations are validated. Volume has no statistical

significance in predicting any of the measured linear or non-linear resonance frequency parameters.

While, fluid density, viscosity, and length of immersion are strong predictors. Establishing a

statistical correlation between non-linear resonance response of the partially immersed beam and

fluid properties was the focus of the current study. Hence, other dependencies, such as the material

properties of the cantilever beam and its geometry, ambient and fluid temperature, other higher-

order resonance modes, either flexural or torsional, which may be sensitive to rheological properties,

will be explored in future work. A limitation of the approach proposed in this study is that the

resonant mode selected in the window between 750 and 850 Hz, gets damped and merges with

another resonance mode near 1 KHz, as illustrated in figure 4.5, for fluids with high viscosity such

as honey. greater than 10 mPas. Hence, parameters such as resonance frequency and damping ratio

cannot be meaningfully derived from their resonance curves.

For future work, building a physics-based analytical model that can capture and validate the

presented experimental observations, along with exploring variation in other previously listed

17

dependencies is required.

Figure 4.5 Frequency response of honey: 150 to 1500 Hz

18

CHAPTER 5

RHEOLOGICAL PROPERTY ESTIMATION THROUGH
MACHINE LEARNING MODELS

5.1 Training data acquisition

After an empirical relationship has been established between the frequency response parameters

of the immersed beam and the rheological properties of the fluid under test in section 4.2, the

experimental setup illustrated in figure 3.1 is modified. To contain the test fluids, a 12 mm diameter

test tube of length 75 mm was utilized. Sections 4.1 and 4.2 established that change in nonlinear

frequency response of a cantilever beam, with respect to forcing amplitude, is independent of the

volume of the fluid in which the beam is immersed. The length of immersion of the beam is fixed

at 50 mm, and different fluids were tested by swapping the test tubes containing them.

Table 5.1 lists the fluids tested along with their associated mass density, dynamic viscosity and a

count of experimental frequency sweep runs. The mass density varies from 848 to 1261.3 𝐾𝑔/𝑚3,

and dynamic viscosity varies from 0.000657 to 0.370311 𝑃𝑎𝑠. Note that there are fewer training

samples of fluids : SAE 80W90 oil, anhydrous glycerin solution, 75% glycerin solution and 50%

glycerin solution.

Table 5.1 Tested Fluids Rheological properties

Fluid
Distilled water
SAE 0W20
SAE 5W20
SAE 10W30
SAE 30 chain oil
SAE 80W90
SAE 85W140
Anhydrous glycerin
Vegetable oil
Glycerin 75%
Glycerin 50%

Density,𝜌 (Kg/m3) Dynamic viscosity, 𝜂 (Pas) Count

0.000657
0.03816
0.03842
0.05906
0.09165
0.12329
0.37031
0.27244
0.02725
0.0182
0.0018

39
40
40
40
40
35
40
40
40
20
20
394

998
848
850
875
875.4
887
901
1261.3
866.48
1180
1100
TOTAL COUNT

19

Figure 5.1 Frequency response curve parametrization

5.1.0.1 Feature extraction

To train the machine learning models, measured parameters 𝑓0 and 𝑄−1 as described in section

3.2, for both the highest and the lowest excitation amplitudes were used as features. In addition

to these, the frequency response within a 100 Hz window of the selected resonance mode is also

parametrized by fitting a log-normal distribution with three terms as illustrated in Figure 5.1. The

fitted function takes the form:

𝑓 (𝑥) = 𝑎1 · 𝑒

(cid:32)

−

1)

(cid:16) ( 𝑥−𝑏
𝑐
1

(cid:17) 2 (cid:33)

(cid:32)

−

2)

(cid:16) ( 𝑥−𝑏
𝑐
2

(cid:17) 2 (cid:33)

(cid:32)

−

3)

(cid:16) ( 𝑥−𝑏
𝑐
3

(cid:17) 2 (cid:33)

+ 𝑎3 · 𝑒

+ 𝑎2 · 𝑒

(5.1)

The coefficients of the function a1,b1,c1,a2,b2,c2,a3,b3 and c3, for both the lowest and the

highest excitation amplitudes are used as features for the ML models. For the resonance curve

obtained from higher amplitude (500 mV) excitation, the coefficients of the curve fit are labeled

with capital letters: A1,A2,A3; B1,B2,B3 and C1,C2,C3 respectively. From the curve fit using

equation 5.1, 18 features are extracted, 9 each for low and high amplitude response, and 5 features

are obtained from the frequency response plots by quantifying resonance mode frequency and

quality factor as described in section 3.2. Thus totaling 23 features.

5.1.0.2 Exploratory data analysis and pre-processing

For the obtained features, a correlation matrix is illustrated in figure 5.2. The non-linear

resonance parameters have the highest correlation to viscosity and density.

From logged frequency response vs. amplitude, for both high and low excitations, the mode

resonance frequency and quality factors are first calculated. Then a log-normal curve fit on the

20

Figure 5.2 Pearson’s feature correlation matrix

data, and coefficients of the fit are extracted. The 394 count training dataset is divided into 70-30%

train-test split, and the pre-processed dataset is then shuffled. This data is subsequently used to

train multiple standard and ensemble machine learning (ML) models. A 10 fold cross validation

strategy is employed during training, with coefficient of determination 𝑅2 as scoring metric.

5.2 Machine Learning methods

Classical as well as ensemble ML methods were tested for both viscosity and density. To prevent

information leakage for the predictive models, normalization was not performed on the training and

testing data. Table 5.2 lists the predictive performance of 11 ML models for viscosity, and table

5.3 lists it for density. Extra tree regressor was found to be the best predictor for both.

For the trained Extra tree regressor models, the feature importance is evaluated by calculating

21

Table 5.2 Viscosity: Tested Classical and ensemble ML models

Model
Extra Trees Regressor
Light Gradient Boosting Machine
Gradient Boosting Regressor
AdaBoost Regressor
Random Forest Regressor
Extreme Gradient Boosting
Decision Tree Regressor
K Neighbors Regressor
Lasso Least Angle Regression
Lasso Regression
Linear Regression

RMSE
MAE MSE
0.0207
0.0142
0.0004
0.0260
0.0170 0.0007
0.0271
0.0008
0.0157
0.0008
0.0179
0.0263
0.0009 0.0285
0.0167
0.0283
0.0162
0.0009
0.0159 0.0015
0.0368
0.0067 0.0801
0.0526
0.1158
0.0137
0.0930
0.0137
0.0930
0.1158
0.0139 0.1163
0.0931

R2
0.9648
0.9454
0.9377
0.9354
0.9320
0.9290
0.8910
0.5008
0.0040
0.0040
-0.0062

RMSLE
0.0178
0.0223
0.0231
0.0229
0.0243
0.0244
0.0312
0.0690
0.0989
0.0989
0.0995

Table 5.3 Density: Tested Classical and ensemble ML models

Model
Extra Trees Regressor
AdaBoost Regressor
Light Gradient Boosting Machine
Extreme Gradient Boosting
Gradient Boosting Regressor
Random Forest Regressor
Decision Tree Regressor
K Neighbors Regressor
Orthogonal Matching Pursuit
Linear Regression
Dummy Regressor

MAE
32.1344
39.1869
35.6577
33.6578
37.8397
34.8088
37.8538
65.2732
105.8219
105.8219
105.8449

MSE
3044.4059
3471.9668
3558.3204
4239.6198
4603.1409
4837.0159
9065.0057
9770.0391
17481.0572
17481.0572
17603.9681

RMSE
53.3965
56.7112
57.5824
61.1819
64.3483
63.2773
84.9151
97.3512
130.8749
130.8749
131.3255

R2
0.8138
0.7854
0.7783
0.7419
0.7191
0.7106
0.4612
0.4203
-0.0216
-0.0216
-0.0286

RMSLE
0.0528
0.0578
0.0578
0.0604
0.0641
0.0625
0.0844
0.0970
0.1273
0.1273
0.1277

a SHAP value associated with each feature. The resulting feature importance is illustrated in the

figure 5.3.

Note that non-linear damping ratio 𝑄 𝑝 is the highest value feature for both viscosity and density

models. Non-linear relative shift in resonance mode Δ 𝑓 / 𝑓0 is also a feature of high importance.

To further improve predictive performance blended and stacked ensemble models were con-

structed from the top 3 best performing models for viscosity and density respectively.

22

Figure 5.3 SHAP values: (a) Viscosity (b) Density

5.2.1 Ensemble models

5.2.1.1 Blended model

Here blending models involve training a Voting regressor for selected top 3 𝑅2 score best

performing models, whose Predictions are the average of contributing models. For viscosity the

blended models are: Extra tree regressor, light gradient boosting machine and gradient boosting

regressor. Whereas, for density the models are: Extra tree regressor, AdaBoost regressor and light

gradient boosting machine.

Figure 5.4 illustrates a prediction error plot (a), and a 10-fold cross validation learning curve

(b), for the blended model estimating viscosity, and figure 5.5 illustrates it for the model estimating

density.

5.2.1.2 Stacked model

For Stacking, a meta estimator, linear regressor here, was trained on the output of the selected

top 3 𝑅2 score best performing models. For both viscosity and density the stacked models are the

23

Figure 5.4 Blended model viscosity: (a) Prediction error, (b) Learning curve

Figure 5.5 Blended model density: (a) Prediction error, (b) Learning curve

same as the ones used in blending.

Figure 5.6 illustrates a prediction error plot (a), and a 10-fold cross validation learning curve

(b), for the stacked model estimating viscosity, and figure 5.7 illustrates it for the model estimating

density.

24

Figure 5.6 Stacked model viscosity: (a) Prediction error, (b) Learning curve

Figure 5.7 Stacked model density: (a) Prediction error, (b) Learning curve

5.2.1.3 Multi-output Extra Tree model

As fluid viscosity and density are correlated features, a multi output regression approach is also

applied. A Multi output Extra Tree Regressor with target variables viscosity and density is trained

to predict both simultaneously. Each target variable is modeled separately, and the predictions are

combined to make the final output.

Figure 5.8 shows the prediction error plot of the multi-output regressor for both the target

25

variables viscosity and density. and figure 5.9 shows the learning curve.

Figure 5.8 Multi-output model: (a) Viscosity and (b) density Prediction error

Figure 5.9 Multi-output model: Viscosity and density learning curve

5.3 Result and discussion

The performance of the Extra tree regressor as a standalone viscosity and density estimator,

as well as a multi-output estimator is represented in table 5.4. Blended models perform better at

density prediction, and Stacked models perform marginally better at both density and viscosity

predictions.

26

Table 5.4 Result: extra tree regressor models

Extra tree regressor R2(70%𝑇𝑟𝑎𝑖𝑛, 10 𝑓 𝑜𝑙𝑑𝐶𝑉) R2(30%𝑇 𝑒𝑠𝑡) RMSLE(CV)

Viscosity
Density
Both

0.9648
0.8138
0.8958

0.914
0.814
0.860

0.0178
0.0528
0.0358

Table 5.5 Result: Blended models
Blended model R2(70%𝑇𝑟𝑎𝑖𝑛, 10 𝑓 𝑜𝑙𝑑𝐶𝑉) R2(30%𝑇 𝑒𝑠𝑡) RMSLE(CV)

Viscosity
Density

0.9651
0.8207

0.910
0.832

0.0179
0.0520

Table 5.6 Result: stacked models
Stacked model R2(70%𝑇𝑟𝑎𝑖𝑛, 10 𝑓 𝑜𝑙𝑑𝐶𝑉) R2(30%𝑇 𝑒𝑠𝑡) RMSLE(CV)

Viscosity
Density

0.9665
0.8139

0.914
0.858

0.0174
0.0529

The 10 fold Cross Validation 𝑅2 score and Root mean square log error (RMSLE) are listed as

performance metrics.

From the tabulated results it can be discerned that the machine learning based approach pre-

sented in this work is capable of predicting the dynamic viscosity and mass density of fluids with

considerable accuracy.

Thus, the described measurement setup is capable of assessing a relatively broad range of fluid

densities and viscosities, in contrast to micro-acoustic sensors which are limited to lower viscosities.

With fewer than 275 training data instances, the presented models achieve prediction accuracy as

high as 0.9665 (10-fold cross-validation R²) for viscosity, and 0.8207 (10-fold cross-validation

R²) for density, and 0.8958 for both density and viscosity. Hence, this work demonstrates a proof

of concept for a rheological property sensing setup that leverages machine learning for real-time

viscosity and density measurement.

5.4 Future work

A limitation of the presented machine learning based approach is that the training data collection

is a time consuming process, and requires a trained individual to operate the setup. Depending on

the number of data point collected per frequency sweep from 750 to 850 Hz, and the total number

27

of incremental amplitudes between minimum and maximum excitation amplitudes, a single plot,

yielding one data point, can take between 5 to 20 minutes to log. Hence, a considerable amount of

time is required to collect the entire data set when manual labor time of switching different fluids

and aligning their vessel is accounted for.

Figure 5.10 Auto-sampler setup

However, increasing the number of training data points has proven to increase performance

when tested. Hence, an auto-sampler system capable of 2-axis movement, and controlled by same

script that is responsible for logging data, could act as viable solution. Figure 5.10 presents one

such setup, capable of holding 5 sample tubes, and moving both laterally and vertically, the setup

can be used to reduce time required for manual loading.

Additionally, though the resonance mode in the 750 to 850 Hz was found to be the most sensitive

to rheological changes, other modes in sub kilo Hertz range also displayed marginal sensitivity.

Hence, future work on deep learning based approach, which utilizes a broad frequency response

with multiple resonance modes may improve the prediction performance further.

28

CHAPTER 6

SUMMARY

In this study, we introduce a vibrating cantilever system to measure the mass density and dynamic

viscosity of fluids. This system leverages the non-linear resonance of a partially fluid-immersed

cantilever, and its frequency response changes in relation to change in excitation amplitude. We

first aimed to establish an empirical relationship between experimental parameters such as fluid

volume and the length of immersion for the beam, density and viscosity of the fluid under test, and

non-linear resonant properties of the cantilever: resonance frequency 𝑓0 and quality 𝑄−1 factor of

a selected sensitive mode.

Following this, we introduce a machine learning based approach to model the relationship

between a fluid’s rheological properties and the non-linear resonance frequency response of the

cantilever beam. The measured linear and non-linear resonance parameters serve as features for

training ensemble regression models to predict the density and viscosity of the tested fluids. Extra

tree regressor was found to be the best performing ensemble regression model for both viscosity

and density prediction. To improve the prediction performance, model blending and stacking

were tested, which yielded better performance. The models were more accurate at predicting

viscosity compared to density. A multi-output extra tress regressor was also trained which yielded

a prediction accuracy that lies between individual viscosity and density predictions.

Thus, the described measurement setup was capable of assessing a broad range of fluid densities

and viscosities, in contrast to micro-acoustic sensors which are limited to lower viscosities. Using

fewer than 275 training data points, the model achieves a viscosity prediction with a 96.65%

(𝑅2, 10-fold cross-validation) score and both density and viscosity with an 89.58% (𝑅2, 10-fold

cross-validation) score.

6.1 Conclusion

The regression models presented in the study are capable of predicting the viscosity and density

of fluids with considerable accuracy, even with limited training data. The ML models are light

weight compared to more complex deep learning based models described in the literature, and

29

hence the models presented in the study can be easily deployed on low power, edge embedded

devices. This, along with the fact that sub 1 KHz resonant frequency modes are utilized in the

models, and hence, the cantilever can also be driven by small low-frequency linear actuator; a

compact sensor module for online process monitoring can be designed using a powerful enough

digital signal processor. Thus, providing a minimally intrusive and compact method of estimating

rheological properties of fluids, with a wide range of viscosities and densities.

30

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