AbstractIn this contribution we discuss a generalized, reduced order model for resonant viscosity and mass density sensors which considers also the devices’ cross sensitivities to temperature. The applicability of the model is substantiated by experimental results from measurements obtained with a circular steel tuning fork in various liquids and temperatures. Advantages of this model are its simplicity, its general applicability for resonant mass density and viscosity sensors which furthermore facilitates the comparison of different sensors

Dufour, Isabelle

Heinisch, Martin

Jakoby, Bernhard

Reichel, Erwin K.

Elsevier - Publisher Connector 

 Procedia Engineering  87 ( 2014 )  472 – 475 
1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/3.0/).
Peer-review under responsibility of the scientific committee of Eurosensors 2014
doi: 10.1016/j.proeng.2014.11.391 
ScienceDirect
Available online at www.sciencedirect.com
EUROSENSORS 2014, the XXVIII edition of the conference series
Modeling and Experimental Investigation of Resonant Viscosity and
Mass Density Sensors Considering their Cross-Sensitivity to
Temperature
Martin Heinischa,∗, Erwin K. Reichela, Isabelle Dufourb, Bernhard Jakobya
aInstitute for Microelectronics and Microsensors, Johannes Kepler University, Linz, Austria
bLaboratoire IMS, Universite´ de Bordeaux, Talence, France
Abstract
In this contribution we discuss a generalized, reduced order model for resonant viscosity and mass density sensors which
considers also the devices’ cross sensitivities to temperature. The applicability of the model is substantiated by experimental
results from measurements obtained with a circular steel tuning fork in various liquids and temperatures. Advantages of this
model are its simplicity, its general applicability for resonant mass density and viscosity sensors which furthermore facilitates
the comparison of diﬀerent sensors.
c© 2014 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the scientiﬁc commitee of Eurosensors 2014.
Keywords: General model, temperature, viscosity, mass density, viscosity sensor, resonator
1. Introduction
Miniaturized viscosity sensors are attractive devices for condition monitoring applications involving ﬂuid media. Re-
cently introduced devices utilize vibrating resonant mechanical structures interacting with the ﬂuid where the resonance
frequency fr and the quality factor Q are aﬀected by the ﬂuid’s viscosity [1]. These devices have been of great interest
during the past two decades and the amount of reported principles seems to be still increasing. This interest results not only
from the obvious advantages such as the devices’ applicability for miniaturized (handheld) devices which furthermore allows
low-cost production rather than the possibility for measurements which cannot be performed with conventional laboratory
experiments, e.g. the rheometric characterization at frequencies higher than 100 Hz. Many principles are aiming to achieve
similar or even better accuracies than provided by high precision laboratory instruments.
An extensive literature review of resonant devices for viscosity and mass density showed, that in most cases, the devices’
cross-sensitivities of fr and Q to temperature are not investigated. However, such an investigation and a precise model-
ing of this temperature dependence is extremely important, particularly when aiming at high accurate measurement results.
∗Corresponding author. Tel.: +43-732-2468-6266 ; fax: +43-732-2468-6250
E-mail address: martin.heinisch@jku.at
 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/3.0/).
Peer-review under responsibility of the scientific committee of Eurosensors 2014
473 Martin Heinisch et al. /  Procedia Engineering  87 ( 2014 )  472 – 475 
436 438
f / Hz
437 439 440
A
m
pl
. /
 A
.U
.
Ph
as
e. 
/ 
A
.U
.
Increasing Temperature 0 °C70 °C
(a) Frequency responses
0 10 30 50 70436
437
438
439
440
/°C
f /
H
z
0 10 30 50 704000
4100
4200
4300
/°C
Q
/1
r
(b) Evaluated resonance frequency and quality factor
Figure 1: a) Cross-sensitivity of the frequency response to temperature (in air): The frequency responses of a steel tuning fork have been recorded in a
temperature range from 0 ◦C to 70 ◦C in 10 ◦C steps. After the temperature has been controlled to the desired value, the resonator was kept at this temperature
for an additional hour to ensure stable measuring conditions. The measurements were performed back and forth, i.e. going from 0 ◦C to higher temperatures,
and once 70 ◦C were reached the temperatures were gradually degreased (in 10 ◦C steps) until 0 ◦C were reached again. b) Evaluated resonance frequencies
and quality factors from the frequency responses
This statement is explained by the following example: In [2] a device featuring a mass density resolution of 0.01 kg/cm3
with a sensitivity of 16 Hz/(kg/m3) at fr ≈ 130 kHz is reported. Considering the relative cross-sensitivity of similar de-
vices which typically is Δ fr/( frΔT ) ≈ (30 . . . 100) · 106 /K (see [3], [4]), implies a cross-sensitivity to temperature of
Δ fr/ΔT = (3.9 . . . 13)Hz/K for this particular device. This reveals that for facilitating the aforementioned sensitivity, an
accuracy of (0.01 . . . 0.04)K of the temperature measurement would be necessary, considering the modeling of the tempera-
ture dependence to be exact at 100 %. Furthermore, a thorough investigation of the temperature characteristics of resonant
viscosity and mass density sensors is indispensable, as all liquids’ viscosities (and mass densities) are signiﬁcantly tem-
perature dependent [5]. Thus measuring these quantities without precisely measuring the ambient temperature and without
knowing the sensors temperature characteristics yield non-reliable results.
For a ﬁrst attempt to evaluate the resonator’s temperature dependence, frequency responses in air have been recorded at
various temperatures. Figure 1 shows frequency responses as well as evaluated resonance frequencies and quality factors in
a temperature range from 0 ◦C to 70 ◦C in air which have been recorded using a steel tuning fork with circular cross-section.
The almost linear dependence of fr on temperature ϑ can be explained considering thermal expansion as well as the change
of the Young’s modulus. However, the supposedly parabolic dependence of Q is diﬃcult to explain. For this reason further
investigations in air were not reﬁned, but frequency responses in four diﬀerent liquids (ethanol, isopropanol, DI-water and
a mixture of 61 % glycerol in DI-water) at ﬁve temperatures (15, 20, 25, 30 and 35 ◦C) have been performed and analyzed.
The results obtained from these experiments in liquids allowed to formulate an empiric temperature related extension of a
previously introduced model.
2. Model
2.1. Temperature-independent model
Recently, we devised a reduced order model applicable for immersed resonant viscosity and mass density sensors relating
a liquid’s viscosity η and mass density ρ to the resonator’s resonance frequency fr = ω0/(2 π) and quality factor Q wich are
determined from its frequency response, see [6]. This model has been successfully applied for several types of resonant
viscosity and mass density sensors found in literature and sensors of our own work, see [7] and reads
1
ω20
=
m
k0
≈ 1
ω0m
= m0k + mρk ρ + m∗ηρk
√
η ρ, and (1)
1
Q
=
c√
k0 m
≈ 1
Qm
=
c0k + cηk η + c∗ηρk
√
η ρ
√
m0k + mρk ρ + m∗ηρk
√
η ρ
,
474   Martin Heinisch et al. /  Procedia Engineering  87 ( 2014 )  472 – 475 
Recorded
frequency responses
Immersed resonator
in different liquids at 25 °C 
1 2 3Evaluated resonance
frequencies & quality factors
5 Evaluated error with 
temperature independent model
Parameter fit for model at 25 °C
Immersed resonator in different 
liquids at different temperatures
6Resonance frequencies & quality 
factors at different temperatures
Formulate temperature
dependent model
4
at 25 °C
Figure 2: Modeling aproach: The frequency responses of a mechanical resonator are recorded upon immersion in liquids (denoted with the index i) with
viscosity ηi and mass density ρi at 25 ◦C. The angular resonance frequencies ω0,i and quality factors Qi are determined ﬁtting the frequency response of a
second order resonator into the recored frequency response. ω0,i and Qi from at least three liquids are necessary to identify the parameters of a temperature-
independent model relating ω0 and Q to η and ρ. The evaluation of relative deviations δω0m,i(ϑ j) and δQm,i(ϑ j) of the modeled temperature-independent
quantities ω0m,i and Qm,i from measured quantities ω0,i(ϑ j) and Qi(ϑ j) at temperatures ϑ j reveals the failing of the temperature-independent model. With
the knowledge of this error, a temperature dependence can be introduced in the existing model.
where m0k, mρk, m∗ηρk, c0k, cηk and c
∗
ηρk are parameters, which can be determined through a ﬁt with at least three diﬀerent
resonance frequencies and quality factors which have been evaluated from recorded frequency responses in diﬀerent liquids.
Measurements in ethanol, isopropanol, DI-water and a mixture of 61 % glycerol in DI-water at 25 ◦C were performed to
determine these parameters with a linear least square ﬁt, see also Fig. 2. The viscosities ηi(ϑ j) and mass densities ρi(ϑ j) of
the aforementioned liquids were evaluated with an Anton Paar SVM 3000 viscometer at temperatures ϑ j of 15, 20, 25, 30 and
35 ◦C. With the knowledge of the temperature dependent viscosities ηi(ϑ j) and mass densities ρi(ϑ j), resonance frequencies
ω0m,i and quality factors Qm,i were calculated using the identiﬁed model, see Eq. 1. With these parameters (resulting from the
temperature-independent model) the relative deviations δω0,i(ϑ j) and δQi(ϑ j) from measured resonance frequencies ω0,i(ϑ j)
and quality factors Qi(ϑ j) were evaluated for every liquid at each temperature and are depicted in Fig. 3(a).
2.2. Temperature-dependent model
The evaluation of δ fr depicted in Fig. 3(a) suggests that the inﬂuence of ϑ on fr is (almost) linear and independent of
the liquid’s viscosity or mass density. This behavior is considered by an additive term m0kϑ · (ϑ − 25 ◦C), wehre m0kϑ is a
parameter which is ﬁtted such that the relative deviation δ fr reaches a minimum. (Note: the notation of the factor m0kϑ might
be misleading as it suggests an increased mass instead of an decreased spring constant. In fact, the result is the same although
not intuitive.)
The interpretation of δQ shows that for low viscous liquids (ethanol) the temperature gradient on δQ is negative, whereas
it is positive in case of higher viscous liquids (61 % Glycerol in DI-water). The value for δQ for water at 35 ◦C is considered as
an outlier and thus, is not respected for interpretation. To model this behavior, two terms c0kϑ·(ϑ−25 ◦C) and cηkϑ·(ϑ−25 ◦C)·η
are added in the the expression for Q in Eq. 1. Thus the equations for the temperature dependent model read:
1
ω20(ϑ)
=
m
k0(ϑ)
≈ m0k + m0kϑ(ϑ − 25 ◦C) + mρk ρ + m∗ηρk
√
η ρ, and (2)
1
Q(ϑ)
=
c(ϑ)√
k0(ϑ)m
≈
c0k + c0kϑ(ϑ − 25 ◦C) + (cηk + cηkϑ(ϑ − 25 ◦C)) η + c∗ηρk
√
η ρ
√
m0k + m0kϑ(ϑ − 25 ◦C) + mρk ρ + m∗ηρk
√
η ρ
.
The relative deviations of temperature dependent, modeled results from the measured values are depicted in Fig. 3(b).
475 Martin Heinisch et al. /  Procedia Engineering  87 ( 2014 )  472 – 475 
-1
0
1
x 10
-3
 f r
15 20 25 30 35
-0.05
0
0.05
 Q
/°C
b
b
Ethanol
Isopropanol
DI-water
61% Glyc
(a) Temperature-independent model
-1
0
1
x 10
-3
 f r
15 20 25 30 35
-0.05
0
0.05
b Q
(
)
/°C
b
(
)
(b) Temperature-dependent model
Figure 3: Relative deviation δQ and δ fr between measured and modeled results. The value for water at 35 ◦C is considered to be an outlier. a) deviation
between measured values and results calculated with the temperature-independent model. b) deviation between measured values and results calculated with
the temperature-dependent model.
In both cases (i.e. for δ fr and δQ) the extension of the equations by the aforementioned temperature terms shows a clear
improvement of the model (except for the outlier). The main advantage of such a model which can be used for many resonant
mass density and viscosity sensors is that it allows the estimation of achievable sensitivities, and comparability of diﬀerent
sensor designs but also maximum achievable accuracies. The inaccuracy in the temperature measurement (which is present
in any case) directly limits the sensor’s accuracy in viscosity and mass density. With this model this accuracy limit can be
estimated.
3. Conclusion and Outlook
The parameters of a recently developed, reduced order model relating an immersed mechanical resonator’s resonance
frequency and quality factor to the liquid’s mass density and viscosity were determined through measurements obtained in
four diﬀerent liquids at 25 ◦C. With this identiﬁed model, quality factors and resonance frequencies were calculated for the
liquids’ viscosities and mass densities at 15, 20, 30 and 35 ◦C. By evaluating the relative deviations of these calculated results
from the measured values, a temperature dependence was empirically introduced in the model.
Regarding future work, further measurements have to be performed (in a larger temperature and viscosity range) to
experimentally analyze these ﬁndings. Furthermore, a physically motivated model accounting for the cross-sensitivity to
temperature has to be developed.
Acknowledgment
We are indebted to the Austrian COMET program (Austrian Centre of Competence in Mechatronics, ACCM) for the
ﬁnancial support.
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Modeling and Experimental Investigation of Resonant Viscosity and Mass Density Sensors Considering their Cross-Sensitivity to Temperature 

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Modeling and Experimental Investigation of Resonant Viscosity and Mass Density Sensors Considering their Cross-Sensitivity to Temperature

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