Application of Resonant Steel Tuning Forks with Circular and Rectangular Cross Sectionsfor Precise Mass Density and Viscosity Measurements

M. Heinischa, T. Voglhuber-Brunnmaiera,b, E.K. Reichela, I. Dufourc, B. Jakobya

aInstitute for Microelectronics and Microsensors, Johannes Kepler University, Linz, AustriabCenter for Integrated Sensors Systems, Danube University, Krems, Austria

cUniversité de Bordeaux, Laboratoire de l’Intégration du Matériau au Système, Pessac, France

Abstract

The feasibility of using commercially available steel tuning forks for viscosity and mass density sensing is investigated. For thistask, the tuning forks are electromagnetically driven and read out to record their frequency responses containing the fundamentalresonant mode upon immersion in a sample liquid. Evaluated resonance frequencies and quality factors are then related to theliquids’ mass density and viscositiy. The used electromagnetic actuation and readout principle allows that only the tuning forkwhich is placed in the center of a glass tube gets wetted with the liquid to be examined. All excitation and read out relatedstructures and electronics are placed outside the glass tube and thus, are not affected or influenced by the liquid. A generalizedmodel relating evaluated quality factors and resonance frequencies to viscosity and mass density is used to describe the tuningforks’ sensitivities and furthermore to estimate required stabilities of apparent quality factors and resonance frequencies to achievemeasurement accuracies similar to those of laboratory instruments. It is shown that relative accuracies in the order of 1 % inviscosity and and 0.1 mg/cm3 in mass density are achievable.

Keywords: Tuning fork, sensor, viscosity, mass density, circular, rectangular

1. Introduction

Lately, we examined the applicability of mechanical res-onators for sensing a liquid’s (complex) viscosity and mass den-sity, see [1]. A very promising approach for this task is the useof electrodynamically driven and read out mechanical oscilla-tors. The investigated principles featuring fundamental reso-nance frequencies in the range from some hundreds of hertzto several kilohertz include, amongst others, oscillating mem-branes [2, 3], in-plane oscillating platelets [4, 5], straight wires[6] and U-shaped wires [7, 8]. Similar miniaturized devices aresilicon cantilevers [9, 10], quartz crystal tuning forks [11], dou-bly clamped silicon beams [12] and vibrating diaphragms [13]just to name a few examples of the relatively large variety ofprinciples reported in literature.

The resonant principles mentioned above are all potentialcandidates for mass density and viscosity sensors. In somecases they were especially designed for specific applications,such as, e.g., the use of miniaturized devices for liquids whereonly tiny amounts of sample volumes are available [14]. Asan alternative to these devices usually aiming at low viscos-ity measurements and furthermore serving as reference devices,conventional steel tuning forks showing a fundamental reso-nance frequency at nominally 440 Hz have been investigatedin this contribution. A first investigation of such tuning forkbased setups has already been presented in [15]. The motiva-tion for such steel tuning fork setups is based on several ar-guments. First, due to the momentum balanced motion in thefundamental mode, the resonant behavior of tuning forks is less

sensitive to clamping issues as it is the case e.g. for singly ordoubly clamped beams and membranes. Second, the relativelylarge and solid structure is less prone to deteriorations, suchas not perfectly cleaned surfaces, air bubbles, etc. Third, incomparison to doubly clamped structures in general, the crosssensitivity of their resonance frequency to temperature is small.Such cross sensitivities of the resonance frequency to tempera-ture can be characterized, determined and modeled on the onehand but on the other hand, they limit the sensor’s accuracy andthus, should be kept as low as possible.

In this contribution, the basic setup for ferromagnetic tuningforks used for viscosity and mass density sensing is explained.Furthermore, measurements showing the response to viscosityand mass density for circular and rectangular cross-sectionedtuning forks are presented. The sensitivities of both tuning forksare discussed in detail and required stabilities for the resonancefrequency and quality factor to achieve measurement accura-cies of 1 % for viscosity and 1 mg/cm3 for mass density areestimated. It is furthermore shown, that with the investigatedsteel tuning forks accuracies in the order of 1 % in viscosityand and 0.1 mg/cm3 in mass density are achievable.

2. Measurement setup

Figure 1 shows a basic sketch of the setup for viscosityand mass density measurements using commercially availabletuning forks with circular and rectangular cross sections. Themeasurement procedure is depicted in Fig. 1(a) and the geome-tries of the used steel tuning forks, both resonating at nominally

Preprint submitted to Sensors and Actuators A: Physical February 11, 2015

Stand

TuningFork

Electromagnet

Qfr

Pick-Up

Recorded frequency responses

Evaluatedresonance frequencyand quality factor

Calculated viscosityand mass density

(a) Sensor principle

Top view

Front view Dimensions in mm

Circular Rectangular

(b) Geometries of investigated tuning forks

Figure 1: Sensor principle: a) A ferromagnetic steel tuning fork is actuated and read out with an electromagnet and an electromagnetic pick-up, respectively.Frequency responses are recorded for the completely immersed tuning fork and evaluated resonance frequencies and quality factors are related to the sample liquid’sviscosity and mass density. b) in the table above, the geometrical dimensions of the circular and rectangular tuning fork are given.

440 Hz in air in their fundamental mode are given in Fig. 1(b).Figure 2 shows a photograph of the circular tuning fork setup.

The steel tuning forks were welded to a solid stainless steelstand and put into a glass tube (not depicted in Fig. 1(a)) whichwas sealed at both sides. To avoid corrosion, the tuning forkswere gold-coated by electro-plating. An electromagnet, usedfor excitation, is placed (outside of the tube) close to the endof one of the ferromagnetic tuning fork’s prongs. At the end ofthe opposed prong, an electrodynamic pick-up is placed, con-sisting of a permanent magnet in the center of a copper coil. Asinusoidal Voltage Vin = V̂in sin(ω t) + Vin,offs with a DC offsetVin,offs ≥ V̂in/2 is used as input signal, exciting harmonic os-cillations of the tuning fork (ω is the angular frequency and tis the time). These oscillations effect an induced voltage in thepick-up serving as the read out signal. By sweeping the excita-tion current’s frequency (containing the frequency of a resonantmode), the frequency response of the tuning fork is recorded.

The measured frequency response Vout(ω) is composed ofthree effects. First, a motion-induced voltage VM in the pick-up coil resulting from the movement of the tuning fork’s fer-romagnetic prong. This voltage is proportional to the prong’svelocity. The second measured signal component is an inducedoffset voltage Voffs due to electrical cross talk from the exci-

Pick-upElectro-magnet

Tuningfork

Glasstube

Thermo-meter

Figure 2: Photograph of the measurement setup with a gold coated circularcross-sectioned tuning fork.

tation into the read out coil. And third, the influence of themeasurement setup’s total phase shift ϕm which results fromthe phase response of the measurement instruments and transittimes affected by the wiring. Thus, the measurable voltage canbe written using complex notation as follows

Vout(jω) =(VM(jω) + Voffs(jω)

)ejϕm(ω) (1)

where the motion induced voltage response can be described asa second order resonator

VM(jω) =Vmax

1 + jQ(ωω0−

ω0ω

) (2)

whereω0 and Q are the resonator’s angular resonance frequencyand quality factor, respectively. For the resonator’s velocity,the angular eigenfrequency (of undamped vibrations) and thefrequency where the amplitude reaches its maximum value areidentical. Thus, we may call ω0 resonance frequency. For theresonator’s deflection, the frequency of the peak is smaller thanthe eigenfrequency. A characteristic resonance curve of the mo-tion induced voltage and the effect of an offset voltage as wellas the phase shift on the measured signal are qualitatively de-picted in Fig. 3. These additional offset signals can signif-icantly deform the resonance curve and yield asymmetries inthe latter. For highly damped resonators, i.e. Q < 100, as itmight be the case for resonant viscosity and mass density sen-sors when examining high viscous liquids, asymmetries in both,the resonator’s velocity and deflection frequency response be-come large. Due to these deformations, searching the maxi-mum peak frequency and the frequencies where the amplitudedecreased to the 1/

√2 of the peak values or methods based on

a Lorentzian fit [16, 17] are not appropriate methods for evalu-ating resonance frequency fr and Q. In this work, an algorithmpresented in [18], which was especially developed for highlydamped resonators is used. This algorithm separates a secondorder resonance of the form of Eq. 2 from spurious offset sig-nals and determines the resonant parameters fr and Q by fittingthe resonance circle in a Nyquist plot of the response function.

Figure 4 shows the circular tuning fork’s frequency responsein air from 100 Hz to 3.5 kHz. There, amplitude and phase

2

00

(a) Motion induced voltage VM

0

0

(b) VM plus offset voltage Voffs

00

(c) (VM + Voffs) phase shifted by ϕm

Figure 3: Illustration of split signal components: a) shows the motion induced voltage which is proportional the resonator’s velocity. b) illustrates the effect ofsignals Voffs resulting from electrical cross talk. c) depicts the total measurable signal including phase shifts resulting from the measuring system and transit timesfrom the wiring.

responses as well as the Nyquist plots are illustrated. In therecorded frequency range the first and second mode at 441 Hzand 2775 Hz, respectively, are measurable and depicted in de-tail. For both resonances, the effect of electrical cross talk and

0 0.5 10

0.5

1

-5 0 5 10

-10

-5

0

0

5

10

0 0.5 1 1.5 2 2.5 3 3.5

-180

0

180

kHz

1.1

1.2

1.3

1.4

2773 2775 277710

15

20

0

5

10

439 441 443-180

0

180

2773 2775 2777439 441 443

Bode plot

Nyquist plot

Resonance data

1st mode 2nd mode

1st mode 2nd mode

Figure 4: Frequency response of the circular tuning fork in air containing thefirst and the second mode

phase shifts from the measurement system is significant. Theparameters necessary for reproducing the measured resonancesusing Eqs. 1 and 2 are given on the bottom of Fig. 4. In thiscase, the offset and phase shift spectra, i.e., Voffs(jω) and ϕ(ω)are approximated by their mean values.

For investigating the effect of the liquid’s viscosity and massdensity on the resonant behavior, i.e., resonance frequency andquality factor, the tuning fork is completely immersed into thesample liquid. As a rule of thumb, higher viscosities yieldhigher damping and higher mass densities yield lower reso-nance frequencies. However, as it will be experimentally shownin Sec. 3 and discussed in Sec. 4.1, viscosity and mass densityaffect both quantities, resonance frequency and quality factor.In this contribution, the examination of the first resonant modeand achievable accuracies using the latter are discussed for acircular and rectangular cross-sectioned steel tuning fork. In[19] the second mode was investigated in liquids for the circulartuning fork. Similar sensitivities of the first and second modewere achieved and thus examining the second mode instead ofthe first or recording both modes at once for η and ρ measure-ments did not show a significant advantage. As spurious offsetsignals and the motion induced voltage are significantly lowerand larger respectively, for the first mode, higher measurementaccuracies can be obtained examining and evaluating the latter.

3. Measurements in liquids

3.1. Viscosity measurements

The response to viscosity was investigated by recording bothtuning forks’ frequency responses in five acetone-isopropanolsolutions covering a viscosity range of 0.2 mPa·s to 2 mPa·s formass densities of roughly 0.78 g/cm3 at 25 ◦C. After mixing,the viscosity and mass density of these solutions were mea-sured with an Anton Paar SVM 3000 for reference purposes.In every liquid, 100 frequency responses have been recordedfor both tuning forks in a Weiss WKL 100 climate chamber attemperatures of 25 ± 0.1 ◦C. 100 measurements appeared to bea reasonable number of measurement points particularly when

3

0.5 1 1.5 2200

300

400

500

0.8 0.9 1.0240

250

260

0.5 1 1.5 2417

417.2

417.4

417.6

0.8 0.9 1.0412

414

416

412 416 420 424

0.1

0.2

0.3

405 410 415 420

0.05

0.1

0.15

Viscosity Series

Mass Density Series

(a) Circular

0

0.1

0.2

0.3

0.4

395 400 405 410 415

Viscosity Series

Mass Density Series

404 408 412 4160

0.2

0.4

0.6

0.8

0.5 1 1.5 2200

300

400

500

0.8 0.9 1.0260

270

280

409.2

409.4

409.6

402

404

406

408

0.5 1 1.5 2

0.8 0.9 1.0

(b) Rectangular

Figure 5: Recorded frequency responses and therefrom evaluated resonance frequencies and quality factors for the circular and rectangular tuning fork.

considering the measuring time. For determining the liquids’stemperature more accurately than it is possible with the climatechamber’s internal thermometer, a Dostmann electronic GmbHP795 thermometer with an accuracy of 0.01 ◦C was used. Theprobe head was put inside of the glass tube and is visible inFig. 2.

The upper parts in Figs. 5(a) and 5(b) show the recordedfrequency responses of measurements in the viscosity series aswell as the evaluated resonance frequencies and quality factorsfor the circular and rectangular tuning fork, respectively. Res-onance frequency and quality factor have been evaluated usingthe fitting algorithm presented in [18]. In Tab. 4, the measuredvalues for viscosity and mass density, as well as evaluated res-onance frequencies and quality factors are given.

3.2. Mass density measurements

For investigating the response to mass density, five solu-tions using acetone, isopropanol, ethanol, DI-water and glyc-erol were prepared. The liquids were mixed to obtain constantviscosities of 1 mPa·s but mass densities between 0.78 g/cm3

and 1 g/cm3. The values for viscosity and mass density of thesesolutions measured with a SVM 3000 at 25 ◦C are given inTab. 5.

The recorded frequency responses as well as evaluated qual-ity factors and resonance frequencies versus mass density areshown in the lower parts of Figs. 5(a) and 5(b). The values forviscosity and mass density measured with a SVM 3000 as wellas evaluated resonance frequencies and quality factors are givenin Tab. 5.

4. Data interpretation

For both, the circular and the rectangular cross-sectionedtuning fork, clearly, higher viscosities yield higher damping andlower resonance frequencies. The evaluation of the resonancefrequency versus viscosity, see Fig. 5, shows an (almost) equalresonance frequency for the first and second liquid in case ofthe round tuning fork and an even higher resonance frequency

in case of the rectangular tuning fork. This behavior can be ex-plained by the fact, that the mass density of the second liquidis about 5.1 mg/cm3 lower than of the first liquid, which shiftsthe resonance frequency upwards. The quality factor, however,significantly decreases for both tuning forks, as the viscosity ofthe second liquid is more than twice as high as the viscosity ofthe first liquid. (The first liquid is acetone and the second is asolution of 51 % mass isopropanol in aceteone.) This behav-ior is more distinct for the rectangular tuning fork, as the lat-ter shows a higher sensitivity to mass density than the circularcross-sectioned tuning fork. This finding is also substantiatedby the results obtained with the mass density measurements.

4.1. Generalized model

In [20], Sauerbrey introduced an equation for the resonancefrequency shift of a thickness shear mode quartz resonator whichis induced by the rigid attachment of a film of mass on thequartz’s surface. This equation was further developed in [21] byKanazawa and Gordon for such a quartz crystal microbalance(QCM) interacting with a liquid, considering its mass densityand viscosity. In [22], Martin et al. extended the Butterworth–Van Dyke equivalent circuit, see also [23, 24], to describe theimpedance of a liquid loaded QCM. This equivalent circuit en-abled evaluating not only the resonance frequency but also thequality factor of a liquid loaded QCM. However, these mod-els consider only one-dimensional shear-wave propagation inthe fluid, which is valid for an infinitely extended in-plane os-cillating plate. The obtained equations (which consider one-dimensional shear waves only) do not allow separating the ef-fect of η and ρ on the resonator’s fr and Q. On the one hand,due to non-uniform shear displacement of the QCM, spuriouscompressional waves are also excited, see [25]. Such spuri-ous effects which are not considered in the models mentionedabove, however, allow separating the effect of η and ρ on theresonator’s fr and Q.

For this purpose and furthermore to provide simple equa-tions for fr and Q, a recently developed model was introducedin [26] which is generally applicable to resonant viscosity and

4

m0k/s2 mρk/m3 ·s2

kg mηρk/m2

kg · s2 c0k/s cηk/

m·s2

kg cηρk/m2

kg · s2

Circular 1.31 · 10−7 1.74 · 10−11 2.71 · 10−8 2.79 · 10−7 7.82 · 10−5 2.41 · 10−8

Rectangular 1.32 · 10−7 2.45 · 10−11 2.50 · 10−8 2.97 · 10−7 7.16 · 10−5 2.24 · 10−8

Table 1: Fitted model parameters for the generalized model, Eqs. 3 and 4 for the circular and the rectangular tuning fork

m0k/s2 mρk/m3 ·s2

kg m∗ηρk/

m2

kg · s5/2 c0k/s cηk/

m·s2

kg c∗ηρk/

m2

kg · s3/2

Circular 1.31 · 10−7 1.74 · 10−11 5.31 · 10−10 2.92 · 10−7 9.38 · 10−5 1.20 · 10−6

Rectangular 1.31 · 10−7 2.45 · 10−11 4.94 · 10−10 3.13 · 10−7 9.09 · 10−5 1.10 · 10−6

Table 2: Fitted model parameters for the simplified model Eqs. 5 and 6 for the circular and the rectangular tuning fork

mass density sensors relating η and ρ to fr = ω0/(2 π) and Q.The model reads

ω0 =1√

m0k + mρk ρ + mηρk

√η ρ

ω0

(3)

andQ =

1ω0·

1c0k + cηk η + cηρk

√ω0 η ρ

, (4)

where mxk and cxk are coefficients and mρk as well as cηk arezero in case of pure one-dimensional shear waves. Similar ap-proaches and expressions have been found in [27, 28, 29].

4.1.1. Simplified generalized equationsEquation 3 is an implicit equation for ω0, which makes an

exact evaluation of ω0 for given η and ρ difficult. Therefore,the equations are simplified by considering that the frequencydependence of certain parameters occurring in the analysis isnegligible as they are virtually constant within the bandwidthof the resonant system. Doing so, we obtain the following ex-pressions for the angular resonance frequency

ω0 =1√

m0k + mρk ρ + m∗ηρk√η ρ

(5)

and the quality factor

Q =

√m0k + mρk ρ + m∗

ηρk√η ρ

c0k + cηk η + c∗ηρk√η ρ

. (6)

Thus, if ω0 and Q for given η and ρ using Eqs. 3 and 4have to be calculated, numerical (e.g., iterative) methods can beused. Alternatively, the above simplified expressions give rem-edy. Conversely, if η and ρ have to be determined from mea-sured fr = ω0/(2 π) and Q, Eqs. 3 and 4 can be used directly.

The parameters m0k, mρk, mηρk, m∗ηρk, c0k, cηk, cηρk and

c∗ηρk are determined by a parameter fit using the values for η,ρ, fr and Q given in Tab. 4 and 5. The fitted model parametersfor the circular and rectangular tuning fork are given in Tabs. 1and 2. The derivation of the equations for the generalized andthe simplified model is based on the equations for resonancefrequency and quality factor of a lumped element mechanicaloscillator, considering the interaction with in-plane oscillating

platelets [22, 30], oscillating spheres [31] and laterally oscil-lating cylinders [32, 33]. If only one-dimensional shear wavepropagation should be considered, Eqs. 3 to 6 can be simplifiedby neglecting the terms multiplied by η and ρ, i.e. substitutingmρk = 0 and cηk = 0.

The complete derivation of Eqs. 3 to 6 is presented in[26]. There, the model has been applied to the measurementsobtained by eight different sensor setups showing good perfor-mance for all investigated sensors including tuning forks, sili-con cantilevers [34], silicon platelets [35], U-shaped wires [36],and spiral spring sensors [37]. The investigation of the appli-cability of the model evaluating the relative root mean squaredeviations of modeled and measured resonance frequencies andquality factors yielded results better than 1.7 · 10−3 for the res-onance frequency and 3.3 · 10−2 for the quality factor, respec-tively.

With the identified model for fr and Q it is now possible toevaluate the sensors’ sensitivities to viscosity and mass density.Furthermore, different sensors (in this case a circular and rect-angular cross-sectioned tuning fork) can be compared and anestimation for required accuracies and stabilities of resonancefrequency and quality factor to achieve a certain accuracy forviscosity and mass density measurements can be made.

4.2. Measurement accuracy

Equations 3 and 4 can be used to calculate η and ρ for evalu-ated fr and Q. The fitted model parameters using a linear fittingprocedure described in [26, 38] are given in Tab. 1. The meanvalues for fr and Q given in Tabs. 4 and 5 are used to calculatethe values for viscosity and mass density for both liquid series.The evaluation of absolute and relative errors given in Tabs. 4and 5 shows that with the present setup, absolute and relativeaccuracies for viscosity and mass density as given in Tab. 3 canbe achieved.

Circular Rectangular[|∆ η|min, |∆ η|max

]/µPa·s [3.34, 16.22] [3.02, 21.44][

|∆ ηrel |min, |∆ ηrel |max]/10−3 [3.16, 21.57] [2.10, 33.44][

|∆ ρ|min, |∆ ρ|max]/mg/(cm3) [14.06, 610.50] [6.08, 150.51][

|∆ ρrel |min, |∆ ρrel |max]/10−6 [17.93, 694.43] [7.76, 152.49]

Table 3: Achieved accuracies with the circular and rectangular tuning fork

5

Viscosity Series T = 25◦CCircular tuning fork

η/(mPa·s) ρ/(g/cm3) fr/Hz Q η̂/(mPa·s) ∆ η/(µPa·s) ∆ ηrel/10−3 ρ̂/(g/cm3) ∆ ρ/(mg/cm3) ∆ ρrel/10−6

0.207 0.7841 417.653 ± 2.89 · 10−3 477.4 ± 1.88 0.211 4.46 21.57 0.7841 0.014 17.930.433 0.7790 417.650 ± 2.92 · 10−3 371.8 ± 1.44 0.427 -6.02 -13.90 0.7791 0.067 86.410.980 0.7793 417.418 ± 3.47 · 10−3 265.2 ± 0.34 0.986 5.31 5.41 0.7792 -0.101 -129.901.576 0.7803 417.224 ± 3.08 · 10−3 214.5 ± 0.33 1.591 14.76 9.36 0.7798 -0.541 -694.432.054 0.7804 417.083 ± 7.08 · 10−3 190.8 ± 0.52 2.048 -6.50 -3.16 0.7808 0.409 523.50

Rectangular tuning fork

η/(mPa·s) ρ/(g/cm3) fr/Hz Q η̂/(mPa·s) ∆ η/(µPa·s) ∆ ηrel/10−3 ρ̂/(g/cm3) ∆ ρ/(mg/cm3) ∆ ρrel/10−6

0.207 0.7841 409.517 ± 3.82 · 10−3 499.0 ± 3.03 0.214 6.92 33.44 0.7841 -0.006 -7.760.433 0.7790 409.576 ± 6.27 · 10−3 394.6 ± 1.91 0.427 -6.05 -13.95 0.7789 -0.055 -70.320.980 0.7793 409.365 ± 0.93 · 10−3 285.7 ± 0.29 0.977 -3.08 -3.14 0.7794 0.054 69.261.576 0.7803 409.173 ± 1.80 · 10−3 232.2 ± 0.39 1.582 5.30 3.36 0.7803 0.039 49.682.054 0.7804 409.067 ± 3.09 · 10−3 206.2 ± 0.74 2.059 4.32 2.10 0.7804 -0.020 -25.79

Table 4: Results for viscosity measurements. The plus-minus values are evaluated typical errors (single standard deviations). x̂ are the calculated values for viscosityand mass density using Eqs. 3 and 4 and evaluated mean values for fr and Q. ∆ x = x̂ − x and ∆ xrel = ∆ x/x are absolute and relative deviations from the values forviscosity and mass density, respectively.

Density Series T = 25◦CCircular tuning fork

η/(mPa·s) ρ/(g/cm3) fr/Hz Q η̂/(mPa·s) ∆ η/(µPa·s) ∆ ηrel/10−3 ρ̂/(g/cm3) ∆ ρ/(mg/cm3) ∆ ρrel/10−6

1.006 0.7849 417.278 ± 1.53 · 10−3 264.1 ± 0.26 0.989 -16.22 -16.13 0.7847 -0.220 -280.530.994 0.8411 415.851 ± 8.36 · 10−3 259.7 ± 1.00 0.979 -14.79 -14.88 0.8413 0.179 212.311.010 0.8931 414.523 ± 0.92 · 10−3 250.0 ± 0.68 1.023 13.51 13.38 0.8937 0.611 683.571.006 0.9870 412.220 ± 7.24 · 10−3 243.6 ± 0.88 1.013 7.40 7.36 0.9872 0.177 179.330.998 1.0073 411.750 ± 8.03 · 10−3 244.1 ± 0.62 0.995 -3.34 -3.34 1.0067 -0.597 -592.31

Rectangular tuning fork

η/(mPa·s) ρ/(g/cm3) fr/Hz Q η̂/(mPa·s) ∆ η/(µPa·s) ∆ ηrel/10−3 ρ̂/(g/cm3) ∆ ρ/(mg/cm3) ∆ ρrel/10−6

1.006 0.7849 409.176 ± 1.78 · 10−3 284.3 ± 0.31 0.984 -21.44 -21.32 0.7849 0.016 20.410.994 0.8411 407.301 ± 1.04 · 10−3 279.9 ± 0.29 0.976 -17.46 -17.57 0.8412 0.125 148.901.010 0.8931 405.578 ± 0.87 · 10−3 269.9 ± 0.65 1.023 13.39 13.26 0.8932 0.067 74.421.006 0.9870 402.551 ± 2.62 · 10−3 262.7 ± 0.93 1.023 17.58 17.48 0.9868 -0.151 -152.490.998 1.0073 401.911 ± 3.41 · 10−3 264.4 ± 0.75 0.995 -3.02 -3.03 1.0072 -0.761 -75.51

Table 5: Results for mass density measurements. The same notation as in Tab. 4 is used.

4.3. Relative sensitivity

For resonant viscosity and mass density sensors, absolutesensitivities as, e.g., the sensitivity of the resonance frequencyto mass density in Hz/(g/cm3) are in general not very descrip-tive. First of all, by evaluating absolute values, the comparisonof sensors operated in a different frequency range is hardly pos-sible. Second, it is difficult to compare the sensitivities to massdensity and viscosity, as usually the investigable range of vis-cosities is much larger than the range of mass densities. Forcommon liquids, the range of mass densities is narrow, hardlyexceeding the range between 0.6 to 1.8 g/cm3, whereas therange of viscosities covers several orders of magnitudes. E.g.,in [36] a viscosity range of 0.2 to 216 mPa·s has been inves-tigated with a single resonant sensor. Due to these reasons werefrain from evaluating absolute rather than relative sensitivi-ties. A difficulty in interpretation which arises for absolute aswell as for relative sensitivities is, that both types of sensitivitiesdepend on η and ρ.

As it can be observed in the measured results as well as inEqs. 3 to 6, fr and Q are both dependent on η and ρ. Thus,to completely describe a resonant viscosity and mass densitysensor’s sensitivity, four sensitivities have to be evaluated. For

this, we define the relative sensitivity of a quantity X(yi) to oneof its variables yi as

S X,yi =

∣∣∣∣∣∂X∂yi·

yi

X

∣∣∣∣∣ (7)

where in this case X stands either for fr or Q and yi for η and ρ.The evaluation of relative sensitivities, Eq. 7 (i.e. relative

change of fr or Q versus relative change of η or ρ) is shownin Fig. 6 for the circular and rectangular case for the experi-mentally investigated viscosity and mass density range. Both,sensitivity to viscosity and mass density increase for higher vis-cosities and mass densities, respectively. The relative sensitiv-ity of the quality factor to viscosity is only slightly higher thanthat to mass density. In case of the sensitivity of the resonancefrequency, the dependence on viscosity is significantly smallerthan the sensitivity to mass density. The comparison of bothtuning forks (i.e. the comparison of Figs. 6(a) and 6(b)) shows,that they show similar sensitivities except for the relative sensi-tivity of the resonance frequency to mass density. In this case,the sensitivity of the rectangular tuning fork is higher. In otherwords, both tuning forks show similar sensitivities to viscos-ity, in case of sensitivity to mass density, the rectangular tuning

6

4

6

8

10

12

x 10-4

0.3

0.35

0.4

0.45

0.5

0.05

0.06

0.07

0.08

0.2

0.3

0.4

0.5

0.5 1 1.5 2 0.8 0.9 1.0

(a) Circular

4

6

8

10

12

x 10-4

0.3

0.35

0.4

0.45

0.5

0.05

0.06

0.07

0.08

0.2

0.3

0.4

0.5

0.5 1 1.5 2 0.8 0.9 1.0

(b) Rectangular

Figure 6: Sensitivities to viscosity and mass density: The sensitivities of fr and Q to η and ρ are not constant but both dependent on η and ρ. For this reason thesensitivities are depicted as bands, where the given values at each band indicate the boundary of evaluated values.

4

6

8

10

12

x 10-6

3

3.5

4

4.5

x 10-3

6

7

8

x 10-5

2.5

3

3.5

4

x 10-4

RectangularCircular

4

6

8

10

12x 10

-6

3

3.5

4

4.5

x 10-3

6

7

8

x 10-5

2.5

3

3.5

4

x 10-4

0.5 1 1.5 2 0.8 0.9 1.00.5 1 1.5 2 0.8 0.9 1.0

(a) Relative

RectangularCircular

2

3

4

5x 10

-3

0.8

1

1.2

1.4

0.024

0.028

0.032

0.05

0.1

0.15

2

3

4

5x 10

-3

0.8

1

1.2

1.4

0.024

0.028

0.032

0.05

0.1

0.15

0.5 1 1.5 2 0.8 0.9 1.0 0.5 1 1.5 2 0.8 0.9 1.0

(b) Absolute

Figure 7: Required accuracies for fr and Q to achieve a relative accuracy of ∆η/η = 10−2 and an absolute accuracy ∆ρ = 1 mg/cm3.

fork is more sensitive.

4.4. Estimation of required accuracies for fr and Q

Manufacturers of viscosity and mass density meters usuallyspecify the performance of their instruments with absolute ac-curacy in mass density and relative accuracy in viscosity. Forexample, the high precision laboratory instrument Anton PaarSVM 3000 features a reproducibility of 0.35 % in viscosity and0.0005 g/cm3 for mass density and a repeatability of 0.1 % and0.0002 g/cm3, for η and ρ respectively. The reproducibility intemperature of the SVM 3000 is 0.02 ◦C and its repeatabilityis 0.005 ◦C. To get in the accuracy range of such laboratoryinstruments, at this point, we target a relative accuracy in vis-cosity ∆ η/η = 10−2 and absolute accuracy in mass density∆ ρ = 1 mg/cm3 which corresponds to a relative accuracy of∆ ρ/ρ = 10−3 for aqueous liquids. Repeated measurements andevaluation of fr and Q showed instabilities yielding a certainspread for both quantities, which yields an inaccuracy in the ηand ρ determination.

Using the sensitivities evaluated from Eq. 7, the change offr and Q upon change of η and ρ can be expressed in matrixnotation as follows:

∆ fr

fr

∆ QQ

=

S fr, η S fr, ρ

S Q, η S Q, ρ

·

∆ ηη

∆ ρρ

. (8)

With this equation, maximum tolerable variations in the frand Q evaluation can be estimated for achieving the desiredaccuracies in η and ρ. For this estimation it is assumed that η is(exactly) known, if ρ is evaluated from fr or Q and vice versa.The evaluated relative and absolute changes for fr and Q aredepicted in Fig. 7.

For a rough estimation for the resonance stability it is as-sumed that ρ is evaluated from fr and η from Q, respectively. Itfollows for the case of the presented tuning forks that variancessmaller than 10−2 Hz and 1 for fr and Q have to be obtained toachieve the claimed accuracies of ∆ η/η = 1 % for viscosity and∆ ρ = 1 mg/cm3 for mass density. The accordingly requiredrelative accuracies are in the order of 10−5 for the resonancefrequency and 10−3 for the quality factor.

These values are estimations for the resonance frequencyand quality factor stability which are required in any case. I.e.,if the instability is larger than these values, the addressed accu-racies cannot be achieved on no account. A further source formeasurement inaccuracies is the cross-sensitivity of the reso-

7

nance frequency to temperature, which limits the accuracies inη and ρ as the temperature measurement accuracy is also limitedto a certain extent. For measurements presented in this contri-bution, the fluids’ temperatures were controlled at 25 ◦C andthus, the tuning forks’ cross-sensitivities were not consideredfor these accuracy estimations. The validity of this approxima-tion is substantiated by a first investigation of the circular tun-ing fork’s temperature behavior presented in [39]. There, it wasfound that the circular tuning fork’s cross sensitivity to temper-ature is about −0.049 Hz/◦C. In the investigated viscosity andmass density range, the sensitivity of the resonance frequencyto mass density is about −0.024 Hz/(mg/cm3). Thus, it followsfor the required temperature measurement accuracy to be about∆T = 0.5 ◦C to distinguish mass density variations of 1 mg/cm3

from temperature variations. This guideline is fulfilled by theused climate chamber and the thermometer which directly mea-sures the liquid’s temperature.

4.5. Error propagationAs it was already mentioned and and as it can be observed

in Fig. 6, the sensitivity parameters S X,yi (which have been cal-culated using Eq. 7 and the simplified model for fr and Q, i.e.,Eqs. 5 and 6) depend on density and viscosity. For proper mea-surements of the resonance characteristics processed with theestimation procedure from [18] it was shown in [40] that thereis a relation between the relative standard deviations of fr and Qwhich is determined by the signal-to-noise ratio1 (SNR) of theacquired frequency spectra and the number of frequency pointsM:

std { fr}fr

=1

2Qstd {Q}

Q≈

√2M

1Q SNR

(9)

It was furthermore shown, that fr and Q are uncorrelated incase of reasonably sampled resonance curves e.g., as shown inFig. 5. If deviations from Eq. 9 are observed, this is an indicatorthat unmodeled influences such as parameter drifts persist andthat there is still potential for setup improvements. Relativeerrors on ρ and η for given relative deviations on fr and Q canbe determined by inverting Eq. 8, i.e.,

∆ηη

∆ρρ

=

S f r,η S f r,ρ

S Q,η S Q,ρ

−1

·

∆ frfr

∆QQ

. (10)

The error propagation depends on the invertibility of thematrix in Eq. 10. E.g., for pure shear resonators, the matrixis singular and ρ and η can not be separated. However, this isnot the case for the presented tuning fork sensors.

Fig. 8 shows the error propagation for the round and rectan-gular tuning fork for the liquids of the viscosity and the densityseries. Errors in fr and Q cause much higher relative devia-tions in viscosity than in density. Therefore, the requirementson the frequency accuracy are much stricter when low errorson viscosity shall be achieved. With the current setups the rel-ative standard deviations in frequency are 10−6 approximately,

1It is defined as the ratio of Nyquist circle diameter to standard deviation ofnoise on the Nyquist circle diameter.

20

30

40

50

0.5 1 1.5 21800

2200

2600

0.8 0.9 1.0

Viscosity Series Mass Density Series

circular TF

rectangular TF

Figure 8: The error propagation for rectangular and circular tuning forks eval-uated for the viscosity and the density series.

and thus accuracies around ±1% (±3 standard deviations) areachievable for η and 0.01% (i.e. 0.1 mg/cm3 for aqueous liq-uids) for ρ, approximately. These results meet the requirementsfrom Sec. 4.4.

4.6. Experimental estimation of achievable accuracies with thepresent setup

To furthermore investigate experimentally the accuracy andthe resolution of this viscosity and mass density measurementsetup, the dissolving of rubber in ethanol was recorded during150 hours, see Fig. 9. The viscosity and mass density havebeen determined with a SVM 3000 before and after the exper-iment. The change of the liquid’s viscosity and mass densitywas 0.05 mPa·s (i.e. 5.3 %) and 0.0039 g/cm3 (i.e. 0.5 %) re-spectively, which can be clearly detected. With an appropriatedata analysis similar accuracies as they are achieved with highprecision laboratory instruments might be obtainable with suchcomperatively tuning fork-based sensors.

0 50 100 150

236

238

240

242

244

246

248

250

252

t/h

Q

t/h

0 50 100 150416.08

416.1

416.12

416.14

416.16

416.18

416.2

416.22

Figure 9: Long term measurements for dissolving rubber in ethanol. Thechanges of the liquid properties are 5.3 % in viscosity and 0.5 % in mass density,respevtively. The gray shaded areas indicate the span of recorded resonance fre-quencies and quality factors. The black dots indicate the values averaged over100 measurements.

8

5. Conclusion

Viscosity and mass density show a clear and significant in-fluence on the frequency responses of electromagnetically drivenand read out steel tuning forks with circular and rectangularcross-sections. The application and the discussion of a fittedmodel for fr and Q revealed that both tuning forks have a sim-ilar sensitivity to viscosity. However, in case of sensitivity tomass density, the rectangular cross-sectioned tuning fork showsa higher sensitivity. An estimation of the required stabilities ofthe sensors’ resonance frequencies and quality factors to achieveaccuracies for viscosity and mass density in the order of com-mercially available measurement instruments showed that theaccuracy of the resonance frequency has to be in the order of10−2 Hz (i.e. 10−5 relative stability) and for the quality factorroughly 1 (i.e. 10−3 relative stability).

With the present setups, accuracies in the order of 1 % inviscosity and and 0.01 % in mass density are achievable.

Acknowledgment

This work has been partially supported by the Linz Cen-ter of Mechatronics (LCM) in the framework of the AustrianCOMET-K2 program. We also want to thank Bernhard Mayrhoferand Johann Katzenmayer for their help and excellent assistance.

References

[1] B. Jakoby, M. Vellekoop, Physical sensors for liquid properties, IEEEsensors journal 11 (12) (2011) 3076–3085.

[2] T. Voglhuber-Brunnmaier, M. Heinisch, E. K. Reichel, B. Weiss,B. Jakoby, Derivation of reduced order models from complex flow fieldsdetermined by semi-numeric spectral domain models, Sensors and Actu-ators A: Physical 202 (2013) 44–51.

[3] M. Heinisch, E. K. Reichel, I. Dufour, B. Jakoby, A resonating rheome-ter using two polymer membranes for measuring liquid viscosity andmass density, Sensors and Actuators A: Physical 172 (1) (2011) 82–87.doi:10.1016/j.sna.2011.02.031.

[4] E. K. Reichel, C. Riesch, F. Keplinger, C. E. A. Kirschhock, B. Jakoby,Analysis and experimental verification of a metallic suspended plate res-onator for viscosity sensing, Sensors and Actuators A: Physical 162(2010) 418–424. doi:10.1016/j.sna.2010.02.017.

[5] A. Abdallah, M. Heinisch, B. Jakoby, Measurement error estimation andquality factor improvement of an electrodynamic-acoustic resonator sen-sor for viscosity measurement, Sensors and Actuators A: Physical 199(2013) 318–324.

[6] M. Heinisch, E. K. Reichel, I. Dufour, B. Jakoby, Tunable resonators inthe low khz range for viscosity sensing, Sensors and Actuators A: Physi-cal 186 (2012) 111–117. doi:http://dx.doi.org/10.1016/j.sna.2012.03.009.

[7] M. Heinisch, E. K. Reichel, B. Jakoby, U-shaped wire based resonatorsfor viscosity and mass density sensing, Proc. of. SENSOR 2013 OPTO2013 IRS2 2013.

[8] E. Lemaire, M. Heinisch, B. Caillard, B. Jakoby, I. Dufour, Comparisonand experimental validation of two potential resonant viscosity sensors inthe kilohertz range, Measurement Science and Technology 24 (8) (2013)084005.

[9] I. Dufour, A. Maali, Y. Amarouchene, et al., The microcantilever: Aversatile tool for measuring the rheological properties of complex fluids,Journal of Sensors 2012. doi:10.1155/2012/719898.

[10] A. Maali, C. Hurth, R. Boisgard, C. Jai, T. Cohen-Bouhacina, J. P. Aimé,Hydrodynamics of oscillating atomic force microscopy cantilevers in vis-cous fluids, Journal of Applied Physics 97 (7) (2005) 074907–074907.

[11] J. Toledo, T. Manzaneque, J. Hernando-García, J. Vázquez, A. Ababneh,H. Seidel, M. Lapuerta, J. Sánchez-Rojas, Application of quartz tuningforks and extensional microresonators for viscosity and density measure-ments in oil/fuel mixtures, Microsystem Technologies (2014) 1–9.

[12] I. Etchart, H. Chen, P. Dryden, J. Jundt, C. Harrison, K. Hsu, F. Marty,B. Mercier, Mems sensors for density–viscosity sensing in a low-flow mi-crofluidic environment, Sensors and Actuators A: Physical 141 (2) (2008)266–275.

[13] X. Huang, S. Li, J. Schultz, Q. Wang, Q. Lin, A capacitive mems vis-cometric sensor for affinity detection of glucose, Journal of Microelec-tromechanical Systems 18 (6) (2009) 1246–1254.

[14] B. A. Bircher, L. Duempelmann, K. Renggli, H. P. Lang, C. Gerber,N. Bruns, T. Braun, Real-time viscosity and mass density sensors re-quiring microliter sample volume based on nanomechanical resonators,Analytical chemistry 85 (18) (2013) 8676–8683.

[15] M. Heinisch, A. Abdallah, I. Dufour, B. Jakoby, Resonant steel tuningforks for precise inline viscosity and mass density measurements in harshenvironments, Procedia Engineering 87 (2014) 1139–1142.

[16] P. J. Petersan, S. M. Anlage, Measurement of resonant frequency andquality factor of microwave resonators: Comparison of methods, J. Appl.Phys.

[17] M. C. Sanchez, E. Martin, J. M. Zamarro, Unified and simplified treat-ment of techniques for characterising transmission, reflection or absorp-tion resonators, Microwaves, Antennas and Propagation, IEE ProceedingsH 137 (1990) 209 – 212.

[18] A. O. Niedermayer, T. Voglhuber-Brunnmaier, J. Sell, B. Jakoby, Meth-ods for the robust measurement of the resonant frequency and qualityfactor of significantly damped resonating devices, Measurement Scienceand Technology 23 (8) (2012) 085107.

[19] M. Heinisch, T. Voglhuber-Brunnmaier, E. K. Reichel, I. Dufour,B. Jakoby, Investigation of higher mode excitation of resonant mass den-sity and viscosity sensors, Proceedings IEEE Sensors 2014.

[20] G. Sauerbrey, Verwendung von schwingquarzen zur wägung dünnerschichten und zur mikrowägung, Zeitschrift für Physik 155 (1959) 206–222.

[21] K. K. Kanazawa, J. G. Gordon, Frequency of a quartz microbalance incontact with liquid, Analytical Chemistry 57 (8) (1985) 1770–1771.

[22] S. J. Martin, V. E. Granstaff, G. C. Frye, Characterization of a quartzcrystal microbalance with simultaneous mass and liquid loading, Anal.Chem. 63 (1991) 2272–2281.

[23] S. Butterworth, On electrically-maintained vibrations, Proceedings of thePhysical Society of London 27 (1) (1914) 410.

[24] K. Van Dyke, The piezo-electric resonator and its equivalent network,Radio Engineers, Proceedings of the Institute of 16 (6) (1928) 742–764.

[25] R. Beigelbeck, B. Jakoby, A two-dimensional analysis of spurious com-pressional wave excitation by thickness-shear-mode resonators, Journalof applied physics 95 (9) (2004) 4989–4995.

[26] M. Heinisch, T. Voglhuber-Brunnmaier, E. K. Reichel, I. Dufour,B. Jakoby, Reduced order models for resonant viscosity and mass den-sity sensors, Sens. Actuators A: Physical 220 (2014) 76–84.

[27] W. Y. Shih, X. Li, H. Gu, W.-H. Shih, I. A. Aksay, Simultaneous liq-uid viscosity and density determination with piezoelectric unimorph can-tilevers, Journal of Applied Physics 89 (2) (2001) 1497–1505.

[28] C. Riesch, E. K. Reichel, F. Keplinger, B. Jakoby, Characterizing vibrat-ing cantilevers for liquid viscosity and density sensing, Journal of Sensors2008 (Article ID 697062) (2008) 9 pages.

[29] T. Manzaneque, V. Ruiz-Díez, J. Hernando-García, E. Wistrela,M. Kucera, U. Schmid, J. L. Sánchez-Rojas, Piezoelectric memsresonator-based oscillator for density and viscosity sensing, Sensors andActuators A: Physical 220 (2014) 305–315.

[30] M. Heinisch, E. K. Reichel, B. Jakoby, A suspended plate in-plane res-onator for rheological measurements at tunable frequencies, in: Proc.Sensor + Test, 2011, pp. 61– 66.

[31] L. Rosenhead, Laminar Boundary Layers: Fluid Motion Memoirs,Clarendon, Oxford, 1963.

[32] E. O. Tuck, Calculation of unsteady flows due to small motions of cylin-ders in a viscous fluid, Journal of Engineering Mathematics 3 (1) (1969)29–44.

[33] J. E. Sader, Frequency response of cantilever beams immersed in viscousfluids with applications to the atomic force microscope, Journal of Ap-plied Physics 84 (1998) 64–76.

9

[34] S. Boskovic, J. W. M. Chon, P. Mulvaney, J. E. Sader, Rheological mea-surements using microcantilevers, Journal of Rheology 46 (4) (2002)891–899.

[35] C. Riesch, E. K. Reichel, A. Jachimowicz, J. Schalko, P. Hudek,B. Jakoby, F. Keplinger, A suspended plate viscosity sensor featuring in-plane vibration and piezoresistive readout, J. Micromech. Microeng. 19(2009) 075010. doi:10.1088/0960-1317/19/7/075010.

[36] M. Heinisch, E. K. Reichel, I. Dufour, B. Jakoby, A u-shaped wire forviscosity and mass density sensing, Sens. Actuators A: Phys. 214 (2014)245–251.

[37] M. Heinisch, S. Clara, I. Dufour, B. Jakoby, A spiral spring resonatorfor mass density and viscosity measurements, Proc. Eurosensors XXVIII(2014).

[38] E. K. Chong, S. H. Zak, An introduction to optimization, 2nd Edition,John Wiley & Sons, 2001.

[39] M. Heinisch, E. K. Reichel, I. Dufour, B. Jakoby, Modeling and experi-mental investigation of resonant viscosity and mass density sensors con-sidering their cross-sensitivity to temperature, Proc. Eurosensors XXVIII(2014).

[40] T. Voglhuber-Brunnmaier, A. Niedermayer, R. Beigelbeck, B. Jakoby,Resonance parameter estimation from spectral data: Cramér–rao lowerbound and stable algorithms with application to liquid sensors, Measure-ment Science and Technology 25 (10) (2014) 105303–105313.

Biographies

Martin Heinisch obtained his Dipl.-Ing. (M.Sc.) in Mecha-tronics from Johannes Kepler University Linz, Austria, in 2009.After his Master studies he went to the University of Califor-nia, Los Angeles (U.C.L.A.) as a Marshall Plan Scholarshipgrantee, where he did research in the field of microfluidic appli-cations and self assembling systems. In 2010 he started a Ph.D.program at the Institute for Microelectronics and Microsensorsof the Johannes Kepler University Linz, Austria where he iscurrently working on resonating liquid sensors.

Thomas Voglhuber-Brunnmaier received the Dipl.-Ing.(M.Sc.) degree in Mechatronics in 2007 and the Dr.techn. (Ph.D.)in May 2013 at the Institute for Microelectronics and Microsen-sors (IME) at the Johannes Kepler University (JKU) in Linz,Austria. From May 2013 he holds a PostDoc position at theCenter for Integrated Sensor Systems (CISS) at the DanubeUniversity Krems (DUK), where he works in close cooperationwith IME on fluid sensors. His fields of interest are the mod-eling of micro-sensors, statistical signal processing, numericalmethods and analog electronics.

Erwin K. Reichel was born in Linz, Austria, in 1979. Hereceived the Dipl.-Ing. (M.Sc.) degree in mechatronics fromJohannes Kepler University, Linz, Austria, in 2006. From 2006to 2009 he was working on the Ph.D. thesis at the Institute forMicroelectronics and Microsensors of the Johannes Kepler Uni-versity, Linz and graduated in October 2009. Afterwards heheld a post-doctoral position at the Centre for Surface Chem-istry and Catalysis as well as at the Department for ChemicalEngineering, KU Leuven, Belgium until June 2012. Since thenhe holds a position as university assistant at the Johannes KeplerUniversity Linz. The main research fields are the modeling, de-sign, and implementation of sensors for liquid properties, andmonitoring of phase transition in complex solutions.

Isabelle Dufour graduated from Ecole Normale Supérieurede Cachan in 1990 and received the Ph.D. and H.D.R. degreesin engineering science from the University of Paris-Sud, Orsay,

France, in 1993 and 2000, respectively. She was a CNRS re-search fellow from 1994 to 2007, first in Cachan working onthe modelling of electrostatic actuators (micromotors, microp-umps) and then after 2000 in Bordeaux working on microcantilever-based chemical sensors. She is currently Professor of elec-trical engineering at the University of Bordeaux and her re-search interests are in the areas of microcantilever-based sen-sors for chemical detection, rheological measurements and ma-terials characterisation.

Bernhard Jakoby obtained his Dipl.-Ing. (M.Sc.) in Com-munication Engineering and his doctoral (Ph.D.) degree in elec-trical engineering from the Vienna University of Technology(VUT), Austria, in 1991 and 1994, respectively. In 2001 heobtained a venia legendi for Theoretical Electrical Engineeringfrom the VUT. From 1991 to 1994 he worked as a ResearchAssistant at the Institute of General Electrical Engineering andElectronics of the VUT. Subsequently he stayed as an ErwinSchrödinger Fellow at the University of Ghent, Belgium, per-forming research on the electrodynamics of complex media.From 1996 to 1999 he held the position of a Research Associateand later Assistant Professor at the Delft University of Tech-nology, The Netherlands, working in the field of microacousticsensors. From 1999 to 2001 he was with the Automotive Elec-tronics Division of the Robert Bosch GmbH, Germany, wherehe conducted development projects in the field of automotiveliquid sensors. In 2001 he joined the newly formed IndustrialSensor Systems group of the VUT as an Associate Professor.In 2005 he was appointed Full Professor of Microelectronics atthe Johannes Kepler University Linz, Austria. He is currentlyworking in the field of liquid sensors and monitoring systems.

List of Figures

1 Sensor principle: a) A ferromagnetic steel tun-ing fork is actuated and read out with an elec-tromagnet and an electromagnetic pick-up, re-spectively. Frequency responses are recordedfor the completely immersed tuning fork andevaluated resonance frequencies and quality fac-tors are related to the sample liquid’s viscosityand mass density. b) in the table above, the ge-ometrical dimensions of the circular and rect-angular tuning fork are given. . . . . . . . . . . 2

2 Photograph of the measurement setup with agold coated circular cross-sectioned tuning fork. 2

3 Illustration of split signal components: a) showsthe motion induced voltage which is propor-tional the resonator’s velocity. b) illustrates theeffect of signals Voffs resulting from electricalcross talk. c) depicts the total measurable signalincluding phase shifts resulting from the mea-suring system and transit times from the wiring. 3

4 Frequency response of the circular tuning forkin air containing the first and the second mode . 3

5 Recorded frequency responses and therefrom eval-uated resonance frequencies and quality factorsfor the circular and rectangular tuning fork. . . 4

10

6 Sensitivities to viscosity and mass density: Thesensitivities of fr and Q to η and ρ are not con-stant but both dependent on η and ρ. For thisreason the sensitivities are depicted as bands,where the given values at each band indicate theboundary of evaluated values. . . . . . . . . . . 7

7 Required accuracies for fr and Q to achieve arelative accuracy of ∆η/η = 10−2 and an abso-lute accuracy ∆ρ = 1 mg/cm3. . . . . . . . . . 7

8 The error propagation for rectangular and cir-cular tuning forks evaluated for the viscosityand the density series. . . . . . . . . . . . . . . 8

9 Long term measurements for dissolving rubberin ethanol. The changes of the liquid propertiesare 5.3 % in viscosity and 0.5 % in mass den-sity, respevtively. The gray shaded areas indi-cate the span of recorded resonance frequenciesand quality factors. The black dots indicate thevalues averaged over 100 measurements. . . . . 8

11