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Adaptation of the near-field acoustic sensor to the measurement of the media properties
Sensors and Actuators A 120 (2005) 390–396
Adaptation of the near-field acoustic sensor to the measurement of the media properties Mohamed Rguiti, Jean-Yves Ferrandis∗ Laboratoire d’Analyse des Interfaces et de Nanophysique, LAIN, Universit´e Montpellier2, Place Eug`ene Bataillon, 34095 Montpellier, France Received 19 July 2004; received in revised form 10 December 2004; accepted 8 January 2005 Available online 19 February 2005
Abstract In this work, the geometry of the near-field acoustic (NFA) sensor has been modified and adapted to the media properties to be characterized. The NFA method consists in studying the alterations of resonance curve of a horn whose tip, the probe, is immersed in the medium to be characterized. The sensor design has been changed by variation of cylinders lengths and their sections ratio. So sensitivity in the measuring of the liquids properties as well as in the characterization of the cement setting has been increased. © 2005 Elsevier B.V. All rights reserved. Keywords: Near-field acoustic; Viscoelastic properties; Resonator; Coupling
1. Introduction
2. The near-field acoustic technique
The NFA technique has been previously used to characterize the properties evolution of fluid media [1,2], polymers [3], silica gels [4], mango [5] and cement paste [6,7]. The sensor has different resonance frequencies because of its geometry. These resonance frequencies correspond to the mechanical coupling between the tip and the body resonances. The first resonance (the lower frequency peak) is sensitive to properties of elastic media in which the tip is immersed [6] and the second resonance (the higher frequency peak) is sensitive to both density and viscosity of the medium [4,8,9]. This technique controls without discontinuity the proprieties’ variations of materials changing from the liquid to solid state. The aim of this work is to achieve the improvement of the sensor sensitivity to the viscous properties as well as to the elastic properties by varying its geometry. This improvement will increase the load range measured by this technique.
The NFA sensor is an oscillating system that uses the alterations of the resonance state of an acoustic horn as a function of the evolution of the rheological properties of the medium in which the terminal part, the tip, is immersed (Fig. 1). The elastic wave is transmitted to the horn by a piezoelectric transducer fed by a low-frequency sinusoidal electrical signal (around 50 kHz). Because the tip is much smaller than the wavelength, the sensor operates in nearfield conditions. This resonating system is very sensitive to the medium in contact with the probe. For the horn, the external medium constitutes a mechanical load which modifies its resonance state so the acoustic signal is altered by the interaction between the probe and the medium. This involves a variation in the electrical signal at the transducer. In return, the study of the electrical resonance state provides information on the medium immersing the probe. Two characteristics are detected by a frequency scan: the resonance frequency f and the maximum of the real part of the electric impedance Zelec . The parameters measured are: f (frequency in the analyzed medium minus frequency in air) and (1/Z) (inverse of the value of Zelec measured when the
∗
Corresponding author. Tel.: +33 467 144909; fax: +33 467 521584. E-mail address: [email protected] (J.-Y. Ferrandis).
0924-4247/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2005.01.005
M. Rguiti, J.-Y. Ferrandis / Sensors and Actuators A 120 (2005) 390–396
Fig. 1. Sensor geometry and electric circuit.
tip is in the medium minus inverse of the value of Zelec in air). When the probe is free or immersed in a liquid medium (simulating weak loads), the resonance curve shows two peaks. The stronger one (referred to mode 1) corresponds to the resonance of the large cylinder disturbed by the coupling with the tip. The second peak (referred to mode 2) corresponds to the resonance of the tip (immersed in the sample) disturbed by the coupling with the large cylinder. When the load increases (simulating an increase in the viscosity of the sample for instance), the amplitudes of both peaks decrease (Fig. 2 ).
3. Theoretical study 3.1. Coupling phenomena The piezoelectric (PZT) excitations produce longitudinal oscillation in the large cylinder, which is free in the upper side and coupled to a much lighter resonator, the tip, at the lower side. The fundamental mode of this large cylinder corresponds approximately to a λ/2 mode. On the other hand the tip, which is free at the lower end and strongly coupled to a greater mass on the other end, oscillates with a λ/4 mode. The horn operates then like two coupled resonators. Neither
Fig. 2. Attenuation of the experimental resonance curve as a function of the medium properties.
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of the two cylinders vibrates at its proper frequency, the vibration mode of each one being modified by the coupling with the other. The sections ratio of the two cylinders is then a main parameter that permits the variation of this coupling. As l1 (large cylinder length) is roughly equal to 2l2 (l2 = tip length), the two oscillators are strongly coupled and vibrate in the same frequency range. In other words, there are two ways to change the fundamental resonance frequencies: by changing the sections ratio or by varying cylinders lengths. For the simulation of the sensor response, the equations of the rod axial oscillations are used and described by plane waves inside the cylinders. The medium properties are taken into account via the boundary condition for the tip end. The PZT element acts as a sensor of the acoustical load impedances (Za,1 and Za,2 ) exerted on its two sides and gives the following electrical impedance [10,11]: Zelec =
1 jωC0
j(Za,1 + Za,2 ) sin βe − 2Zp (1 − cos βe)
1 + k2 Zp T 2 βe (Zp + Za,1 Za,2 ) sin βe − j(Za,1 + Za,2 )Zp cos βe
with C0 the clamped capacitance of the PZT, ω the angular frequency, kT the piezoelectric coupling constant, β the wave vector, Zp the characteristic impedance of the PZT and e its thickness. The values of parameters used in the theoretical study are given as: C0 = 1 nF, kT = 0.55, e = 1 mm, β = ω/Vp (rad/m), Zp = ρp Vp (Pa s m−1 ), ρp = 7.7 g/cm3 , Vp = 4400(1 + 10−2 j) m/s. ρp and Vp are the density and the sound velocity of the PZT, respectively. Some parameters are in the form of a complex number in order to take account the losses in the device. Za,1 is a function of the load exerted (by air) on the upper end of the large cylinder. Za,2 depends on the acoustic load exerted on the tip end. These two loads are transferred up to the piezoelectric element with a conventional model of impedance transfer acoustic horn [11]. The experimental variations f and (1/Z) can be simulated by, for a given medium, knowing its load impedance Zload . Therefore, the electrical impedance Zelect can be calculated for given experimental conditions (kind of material, horn design and immersion depth of the probe). In the case of liquids, a relation between electrical impedance and rheological properties (density and viscosity) has been established [8]. For the viscoelastic media, the vibrations are complex and cannot be represented by a simple analytical expression. A numerical analysis has been developed using a finite element calculation [7]. In this paper, for the modeling of the sensor impedance, the mechanical load on the immersed part of the tip is simulated by using an exponential variation of the medium properties. This exponential expression is obtained by fitting the experimental values of G (storage modulus) and G (loss modulus) measured, in the reference [7], during the cement setting.
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Fig. 3. Modeling of the variation f1 and (1/Z1 ) as a function of the load for two sections ratios.
3.2. Variation of the sensitivity as a function of the horn geometry The variation of the sensor’s sensitivity is theoretically studied according to its geometry (sections ratio, large cylinder length and small cylinder length). To show the effect of each part, several impedance curves are compared. The mechanical loads used in modeling are complex and vary from a weak load to a strong load (this increase of the load corresponds to the tip interaction with a medium changing from the viscoelastic state to the elastic state for a 15 mm immersion depth of the tip). A better sensitivity is defined when the values of f and (1/Z) are higher in the same load range (same medium). 3.2.1. Variation of the sections ratio To modify the sections ratio, we change only the tip diameter d2 . The large cylinder diameter is fixed (d1 = 10 mm). The large and small cylinders lengths are l1 = 50 mm and l2 = 25 mm, respectively. Figs. 3 and 4 represent the varia-
tions of f and (1/Z) of the two resonance modes according to the acoustic load at the tip-end for two different sections ratios (S2 /S1 = 0.01 and S2 /S1 = 0.04). f1 and (1/Z1 ) are the variations relating to the first mode. f2 and (1/Z2 ) are those concerning the second mode. The second mode sensitivity increases slightly when the sections ratio decreases. In order to have a better sensitivity in the tip end, weak coupling (S2 /S1 ) is needed for weak loads. 3.2.2. Variation of the tip length In Fig. 5 the variations f1 and (1/Z1 ) of a sensor are compared for two different tip lengths: l2 = 22.5 mm and l2 = 25 mm. The large cylinder length is equal to 50 mm and the sections ratio is 0.01. The sensitivity of the first mode, to the load variation, is better when the tip length is equal to 25 mm. These curves show that the values of f and (1/Z) can be modified for the same load. The sensitivity at a specific load can be then improved by adaptation of the sensor geometry.
Fig. 4. Modeling of the variation f2 and (1/Z2 ) as a function of the load for two sections ratios.
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Fig. 5. Modeling of the variation f1 and (1/Z1 ) as a function of the load for two tip lengths.
4. Experimental results To verify the influence of each sensor part on the sensitivity, several sensors of different geometries have been manufactured. The material used is stainless steel. Fig. 6 shows three sensors with different tips diameters (1, 1.4 and 2 mm). These sensors are called C1, C2 and C3, respectively. The large cylinder diameter is the same for all sensors and it is equal to 10 mm. The cylinders lengths are also approximately equal. 4.1. Variation of the sections ratio In air, the device vibrates at its fundamental frequencies and is not damped by exterior medium. Fig. 7 shows the experimental resonance curves obtained in air for the sensors C1 and C3 with different sections ratios (S2 /S1 = 0.01 for C1 and S2 /S1 = 0.04 for C3) and approximately the same lengths. For a weak sections ratio (sensor C1), the cylinders are slightly coupled: each one has little influence on the other and the sensor resonance frequencies are not very different from the fundamental cylinders frequencies. The tip, a low-
Fig. 6. Three sensors manufactured with different diameters tips.
size one, only modifies the free/free condition of the large cylinder. For the tip, a large cylinder brings it closer to the free/blocked condition. The raise of the sections ratio (sensor C3) increases the influence of each resonator on the other and the second resonance frequency moves away from the first one. To compare the C1 and C3 sensibilities, a medium, in which viscoelastic properties vary according to the time, is used. So measurements were taken in a cement prepared with 38% water/cement ratio. The curves in Figs. 8 and 9 represent the experimental variations f and (1/Z), respectively, for the first and the second resonance mode. If the load applied by the immersion medium is increased, the dissipated energy increases. This decreases the energy stored in the horn and involves a lessening of the resonance peaks of both cylinders ((1/Z) increases). When the immersion medium is stiff, the probe is jammed and the second mode disappears. That corresponds approximately to the maximum of (1/Z1 ). Thus only one resonance peak is detected. Its frequency is that of a resonator equivalent to the whole horn having one end fixed. The energy remains mainly
Fig. 7. Real parts of the admittance of the sensors C1 and C3.
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Fig. 8. Evolution of f1 and (1/Z1 ) according to the cement setting for different sections ratio.
Fig. 9. Evolution of f2 and (1/Z2 ) according to the cement setting for different sections ratio.
stored. The dissipation is weak and decreases when the stiffness of the medium increases. The value of the resonance frequency becomes close to the natural frequency of the equivalent resonator. The disappearance of the second mode is faster for the tip with the diameter 1 mm. That can be explained by the fact that this tip requires less load than the tip of 2 mm so that its oscillations disappear. Concerning the first resonance mode the sensitivity is significant. The increase of the coupling moves apart the two resonance peaks and increases the range of measured loads. The sensor of low coupling coefficient saturates quickly for the same load. We note that the accuracy of the equipment is about 10 Hz and its Q factor depends on the cylinders lengths and their coupling. For example, for a free sensor (without external load) with the lengths l1 = 50 mm, l2 = 25mm and the section ratio S2 /S1 = 0.01, the Q factor is equal to 180 and 460 for the first and the second mode, respectively.
and a part of it was cut to obtain a length of 22.5 mm. The large cylinder length is fixed to 50 mm and the sections ratio is equal to 0.01. The reduction of tip length involves an increase in its resonance frequency and therefore the raise of the margin between the resonance peaks.
4.2. Variation of the tip length The curves represented in Fig. 10 correspond to a sensor with two different tip lengths. The initial length was 25 mm
Fig. 10. Variation of the real part of the impedance for two different tip lengths.
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Fig. 11. Variation f1 as a function of
√
ρµ for two tip lengths.
The sensitivity for these two tip lengths is compared by taking measures in liquids with various viscosities and also in a viscoelastic medium (cement). The liquids of various viscosities are obtained by water–glycerol mixtures with various glycerol percentages (60, 70, 90 and 98%). The results obtained for the first resonance mode are displayed in Fig. 11 for the two tip lengths. The variations f1 are represented √ according to ρµ where ρ and µ are, respectively, the density and the viscosity of the solution. For the first mode, the curve corresponding to l2 = 25 mm varies quickly than curve of l2 = 22.5 mm. It means that the sensitivity is better when l2 = 25 mm. The results obtained during the cement setting are reported in Fig. 12 for the first resonance mode. The sensitivity to the variations of the medium properties is better for the tip with length 25 mm in particular between 10 and 30 min. The big change on f1 can be theoretically explained by the variation of the coupling (between the two cylinders) caused by the change of the tip length. Indeed, when this length increases, its resonance frequency comes near that of the large cylinder. Thus, the influence of the tip on the large-cylinder response increases. This phenomenon is theoretically shown in Fig. 5.
Fig. 12. Evolution of f1 as a function of the time for two tip lengths.
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Fig. 13. Variation of the admittance magnitude as a function of frequency for different large cylinder length of the sensor C2.
4.3. Variation of the large cylinder length We have measured the resonance evolution of the sensor C2 according to the large cylinder length as we cut out a part of it (Fig. 13). The initial lengths were 52.5 mm for the large cylinder and 25 mm for the tip.
Fig. 14. Evolution of f1 as a function of the cement setting for some large cylinder lengths.
Fig. 15. Evolution of (1/Z2 ) as a function of the cement setting for some large cylinder lengths.
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When the large cylinder length decreases, the frequency of the first mode comes near that of the second mode and the margin between their amplitudes decreases. In the theory of the coupled oscillators (mechanical and electrical oscillators), this phenomenon is observed when the proper frequencies of the oscillators get closer. To compare the sensitivity according to the large cylinder length, some measures have been carried out according to the cement setting as a function of the time for a 37% water/cement ratio. In Figs. 14 and 15 the variations f1 and (1/Z2 ) are respectively presented. The first mode sensitivity increases and the second mode sensitivity decreases when the large cylinder length decreases.
5. Conclusion The choice of the sensor dimensions depends on required application and the load range to be measured. If the sensitivity was sought on the tip, its length must be smaller than half of the large cylinder and the sections ratio must be weak (thin tip). This geometry is desirable to measure the weak variations loads (variation of liquids properties for example). But the measure of high loads is possible without discontinuity using the first mode. So, to have a best sensitivity on this peak, the large cylinder must be smaller than 2l2 . The results show that the horn operates according to two modes. The first one appears with the viscoelastic media; the second mode is involved when the sample is fluid. The sensor response not only depends on the rheological properties of the medium but also on the horn geometry. This geometry can be adapted to the material properties by changing cylinders lengths and their sections ratio. These results are very important because they provide a basic tool for a future sensor optimization. For the same load, various responses can be obtained depending on the sensor geometry. That gives the possibility to measure a wide range of material properties. The second mode sensitivity remains, generally, larger for weak loads. This mode disappears when the load is important, i.e., when the medium starts to have elastic properties. These results confirm the theoretical results which were used for this optimization.
References [1] J. Van Est, C. Prugne, B. Cros, J. Attal, B. Cretin, Local measurements of viscoelastic properties of fluids using near-field ultrasonics, Appl. Phys. A 66 (1998) S357–S360. [2] C. Prugne, J. Van Est, B. Cros, G. L´evˆeque, J. Attal, Measurements of the viscosity of liquids by near-field acoustics, Meas. Sci. Technol. 9 (1998) 1894–1998. [3] B. Cros, J.Y. Ferrandis, M. Pauthe, Study of the polymer processing by acoustic near field, La Revue de M´etallurgie, Science et G´enie des Mat´eriaux 96 (1999) 1537–1545. [4] B. Cros, M. Pauthe, M. Rguiti, J.Y. Ferrandis, On-line characterization of silica gels by acoustic near field, Sens. Actuators B 76 (2001) 115–123. [5] M. Valente, J.Y. Ferrandis, Evaluation of textural properties of mango tissue by a near-field acoustic method, Postharvest Biol. Technol. 29 (2003) 219–228. [6] B. Cros, J.Y. Ferrandis, Characterization of the cement setting by acoustic near field, Sens. Actuators B 60 (1999) 200–207. [7] J.Y. Ferrandis, G. L´evˆeque, In situ measurement of elastic properties of cement by an ultrasonic resonant sensor, Cement Concrete Res. 33 (2003) 1183–1187. [8] G. L´evˆeque, J.Y. Ferrandis, J. Van Est, B. Cros, An acoustic sensor for simultaneous density and viscosity measurements in liquids, Rev. Sci. Instrum. 71 (3) (2000) 1433–1440. [9] J.Y. Ferrandis, H. Nounah, B. Cros, Sensitivity of the second resonance mode of a near-field acoustic sensor to viscoelastic media: investigation of bitumens, Eur. Phys. J. AP 10 (2000) 149–155. [10] W. Mason, An electromechanical representation of a piezoelectric crystal used as a transducer, Inst. Radio Eng. 23 (1935) 1252. [11] G.S. Kino, Acoustic Waves: Devices Imaging and Analog Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1987.
Biographies Mohamed Rguiti was born in 1973. He studied physics and materials and received a master’s degree in physics in 1996 and a PhD degree in 2002 in the near-field acoustic technique. His major research interests include the development and characterization of PZT materials and their applications as sensors and actuators. He is currently a assistant professor.
Jean-Yves Ferrandis was born in 1973. He studied physics and engineering and received a solid state physics master’s degree in 1996 and a PhD degree in 2000. His major research interests include the development of acoustic sensors, their applications in materials characterization and the theoretical study of near-fields acoustics. He is currently a research engineer at the CNRS.
Adaptation of the near-field acoustic sensor to the measurement of the media properties Mohamed Rguiti, Jean-Yves Ferrandis∗ Laboratoire d’Analyse des Interfaces et de Nanophysique, LAIN, Universit´e Montpellier2, Place Eug`ene Bataillon, 34095 Montpellier, France Received 19 July 2004; received in revised form 10 December 2004; accepted 8 January 2005 Available online 19 February 2005
Abstract In this work, the geometry of the near-field acoustic (NFA) sensor has been modified and adapted to the media properties to be characterized. The NFA method consists in studying the alterations of resonance curve of a horn whose tip, the probe, is immersed in the medium to be characterized. The sensor design has been changed by variation of cylinders lengths and their sections ratio. So sensitivity in the measuring of the liquids properties as well as in the characterization of the cement setting has been increased. © 2005 Elsevier B.V. All rights reserved. Keywords: Near-field acoustic; Viscoelastic properties; Resonator; Coupling
1. Introduction
2. The near-field acoustic technique
The NFA technique has been previously used to characterize the properties evolution of fluid media [1,2], polymers [3], silica gels [4], mango [5] and cement paste [6,7]. The sensor has different resonance frequencies because of its geometry. These resonance frequencies correspond to the mechanical coupling between the tip and the body resonances. The first resonance (the lower frequency peak) is sensitive to properties of elastic media in which the tip is immersed [6] and the second resonance (the higher frequency peak) is sensitive to both density and viscosity of the medium [4,8,9]. This technique controls without discontinuity the proprieties’ variations of materials changing from the liquid to solid state. The aim of this work is to achieve the improvement of the sensor sensitivity to the viscous properties as well as to the elastic properties by varying its geometry. This improvement will increase the load range measured by this technique.
The NFA sensor is an oscillating system that uses the alterations of the resonance state of an acoustic horn as a function of the evolution of the rheological properties of the medium in which the terminal part, the tip, is immersed (Fig. 1). The elastic wave is transmitted to the horn by a piezoelectric transducer fed by a low-frequency sinusoidal electrical signal (around 50 kHz). Because the tip is much smaller than the wavelength, the sensor operates in nearfield conditions. This resonating system is very sensitive to the medium in contact with the probe. For the horn, the external medium constitutes a mechanical load which modifies its resonance state so the acoustic signal is altered by the interaction between the probe and the medium. This involves a variation in the electrical signal at the transducer. In return, the study of the electrical resonance state provides information on the medium immersing the probe. Two characteristics are detected by a frequency scan: the resonance frequency f and the maximum of the real part of the electric impedance Zelec . The parameters measured are: f (frequency in the analyzed medium minus frequency in air) and (1/Z) (inverse of the value of Zelec measured when the
∗
Corresponding author. Tel.: +33 467 144909; fax: +33 467 521584. E-mail address: [email protected] (J.-Y. Ferrandis).
0924-4247/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2005.01.005
M. Rguiti, J.-Y. Ferrandis / Sensors and Actuators A 120 (2005) 390–396
Fig. 1. Sensor geometry and electric circuit.
tip is in the medium minus inverse of the value of Zelec in air). When the probe is free or immersed in a liquid medium (simulating weak loads), the resonance curve shows two peaks. The stronger one (referred to mode 1) corresponds to the resonance of the large cylinder disturbed by the coupling with the tip. The second peak (referred to mode 2) corresponds to the resonance of the tip (immersed in the sample) disturbed by the coupling with the large cylinder. When the load increases (simulating an increase in the viscosity of the sample for instance), the amplitudes of both peaks decrease (Fig. 2 ).
3. Theoretical study 3.1. Coupling phenomena The piezoelectric (PZT) excitations produce longitudinal oscillation in the large cylinder, which is free in the upper side and coupled to a much lighter resonator, the tip, at the lower side. The fundamental mode of this large cylinder corresponds approximately to a λ/2 mode. On the other hand the tip, which is free at the lower end and strongly coupled to a greater mass on the other end, oscillates with a λ/4 mode. The horn operates then like two coupled resonators. Neither
Fig. 2. Attenuation of the experimental resonance curve as a function of the medium properties.
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of the two cylinders vibrates at its proper frequency, the vibration mode of each one being modified by the coupling with the other. The sections ratio of the two cylinders is then a main parameter that permits the variation of this coupling. As l1 (large cylinder length) is roughly equal to 2l2 (l2 = tip length), the two oscillators are strongly coupled and vibrate in the same frequency range. In other words, there are two ways to change the fundamental resonance frequencies: by changing the sections ratio or by varying cylinders lengths. For the simulation of the sensor response, the equations of the rod axial oscillations are used and described by plane waves inside the cylinders. The medium properties are taken into account via the boundary condition for the tip end. The PZT element acts as a sensor of the acoustical load impedances (Za,1 and Za,2 ) exerted on its two sides and gives the following electrical impedance [10,11]: Zelec =
1 jωC0
j(Za,1 + Za,2 ) sin βe − 2Zp (1 − cos βe)
1 + k2 Zp T 2 βe (Zp + Za,1 Za,2 ) sin βe − j(Za,1 + Za,2 )Zp cos βe
with C0 the clamped capacitance of the PZT, ω the angular frequency, kT the piezoelectric coupling constant, β the wave vector, Zp the characteristic impedance of the PZT and e its thickness. The values of parameters used in the theoretical study are given as: C0 = 1 nF, kT = 0.55, e = 1 mm, β = ω/Vp (rad/m), Zp = ρp Vp (Pa s m−1 ), ρp = 7.7 g/cm3 , Vp = 4400(1 + 10−2 j) m/s. ρp and Vp are the density and the sound velocity of the PZT, respectively. Some parameters are in the form of a complex number in order to take account the losses in the device. Za,1 is a function of the load exerted (by air) on the upper end of the large cylinder. Za,2 depends on the acoustic load exerted on the tip end. These two loads are transferred up to the piezoelectric element with a conventional model of impedance transfer acoustic horn [11]. The experimental variations f and (1/Z) can be simulated by, for a given medium, knowing its load impedance Zload . Therefore, the electrical impedance Zelect can be calculated for given experimental conditions (kind of material, horn design and immersion depth of the probe). In the case of liquids, a relation between electrical impedance and rheological properties (density and viscosity) has been established [8]. For the viscoelastic media, the vibrations are complex and cannot be represented by a simple analytical expression. A numerical analysis has been developed using a finite element calculation [7]. In this paper, for the modeling of the sensor impedance, the mechanical load on the immersed part of the tip is simulated by using an exponential variation of the medium properties. This exponential expression is obtained by fitting the experimental values of G (storage modulus) and G (loss modulus) measured, in the reference [7], during the cement setting.
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Fig. 3. Modeling of the variation f1 and (1/Z1 ) as a function of the load for two sections ratios.
3.2. Variation of the sensitivity as a function of the horn geometry The variation of the sensor’s sensitivity is theoretically studied according to its geometry (sections ratio, large cylinder length and small cylinder length). To show the effect of each part, several impedance curves are compared. The mechanical loads used in modeling are complex and vary from a weak load to a strong load (this increase of the load corresponds to the tip interaction with a medium changing from the viscoelastic state to the elastic state for a 15 mm immersion depth of the tip). A better sensitivity is defined when the values of f and (1/Z) are higher in the same load range (same medium). 3.2.1. Variation of the sections ratio To modify the sections ratio, we change only the tip diameter d2 . The large cylinder diameter is fixed (d1 = 10 mm). The large and small cylinders lengths are l1 = 50 mm and l2 = 25 mm, respectively. Figs. 3 and 4 represent the varia-
tions of f and (1/Z) of the two resonance modes according to the acoustic load at the tip-end for two different sections ratios (S2 /S1 = 0.01 and S2 /S1 = 0.04). f1 and (1/Z1 ) are the variations relating to the first mode. f2 and (1/Z2 ) are those concerning the second mode. The second mode sensitivity increases slightly when the sections ratio decreases. In order to have a better sensitivity in the tip end, weak coupling (S2 /S1 ) is needed for weak loads. 3.2.2. Variation of the tip length In Fig. 5 the variations f1 and (1/Z1 ) of a sensor are compared for two different tip lengths: l2 = 22.5 mm and l2 = 25 mm. The large cylinder length is equal to 50 mm and the sections ratio is 0.01. The sensitivity of the first mode, to the load variation, is better when the tip length is equal to 25 mm. These curves show that the values of f and (1/Z) can be modified for the same load. The sensitivity at a specific load can be then improved by adaptation of the sensor geometry.
Fig. 4. Modeling of the variation f2 and (1/Z2 ) as a function of the load for two sections ratios.
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Fig. 5. Modeling of the variation f1 and (1/Z1 ) as a function of the load for two tip lengths.
4. Experimental results To verify the influence of each sensor part on the sensitivity, several sensors of different geometries have been manufactured. The material used is stainless steel. Fig. 6 shows three sensors with different tips diameters (1, 1.4 and 2 mm). These sensors are called C1, C2 and C3, respectively. The large cylinder diameter is the same for all sensors and it is equal to 10 mm. The cylinders lengths are also approximately equal. 4.1. Variation of the sections ratio In air, the device vibrates at its fundamental frequencies and is not damped by exterior medium. Fig. 7 shows the experimental resonance curves obtained in air for the sensors C1 and C3 with different sections ratios (S2 /S1 = 0.01 for C1 and S2 /S1 = 0.04 for C3) and approximately the same lengths. For a weak sections ratio (sensor C1), the cylinders are slightly coupled: each one has little influence on the other and the sensor resonance frequencies are not very different from the fundamental cylinders frequencies. The tip, a low-
Fig. 6. Three sensors manufactured with different diameters tips.
size one, only modifies the free/free condition of the large cylinder. For the tip, a large cylinder brings it closer to the free/blocked condition. The raise of the sections ratio (sensor C3) increases the influence of each resonator on the other and the second resonance frequency moves away from the first one. To compare the C1 and C3 sensibilities, a medium, in which viscoelastic properties vary according to the time, is used. So measurements were taken in a cement prepared with 38% water/cement ratio. The curves in Figs. 8 and 9 represent the experimental variations f and (1/Z), respectively, for the first and the second resonance mode. If the load applied by the immersion medium is increased, the dissipated energy increases. This decreases the energy stored in the horn and involves a lessening of the resonance peaks of both cylinders ((1/Z) increases). When the immersion medium is stiff, the probe is jammed and the second mode disappears. That corresponds approximately to the maximum of (1/Z1 ). Thus only one resonance peak is detected. Its frequency is that of a resonator equivalent to the whole horn having one end fixed. The energy remains mainly
Fig. 7. Real parts of the admittance of the sensors C1 and C3.
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Fig. 8. Evolution of f1 and (1/Z1 ) according to the cement setting for different sections ratio.
Fig. 9. Evolution of f2 and (1/Z2 ) according to the cement setting for different sections ratio.
stored. The dissipation is weak and decreases when the stiffness of the medium increases. The value of the resonance frequency becomes close to the natural frequency of the equivalent resonator. The disappearance of the second mode is faster for the tip with the diameter 1 mm. That can be explained by the fact that this tip requires less load than the tip of 2 mm so that its oscillations disappear. Concerning the first resonance mode the sensitivity is significant. The increase of the coupling moves apart the two resonance peaks and increases the range of measured loads. The sensor of low coupling coefficient saturates quickly for the same load. We note that the accuracy of the equipment is about 10 Hz and its Q factor depends on the cylinders lengths and their coupling. For example, for a free sensor (without external load) with the lengths l1 = 50 mm, l2 = 25mm and the section ratio S2 /S1 = 0.01, the Q factor is equal to 180 and 460 for the first and the second mode, respectively.
and a part of it was cut to obtain a length of 22.5 mm. The large cylinder length is fixed to 50 mm and the sections ratio is equal to 0.01. The reduction of tip length involves an increase in its resonance frequency and therefore the raise of the margin between the resonance peaks.
4.2. Variation of the tip length The curves represented in Fig. 10 correspond to a sensor with two different tip lengths. The initial length was 25 mm
Fig. 10. Variation of the real part of the impedance for two different tip lengths.
M. Rguiti, J.-Y. Ferrandis / Sensors and Actuators A 120 (2005) 390–396
Fig. 11. Variation f1 as a function of
√
ρµ for two tip lengths.
The sensitivity for these two tip lengths is compared by taking measures in liquids with various viscosities and also in a viscoelastic medium (cement). The liquids of various viscosities are obtained by water–glycerol mixtures with various glycerol percentages (60, 70, 90 and 98%). The results obtained for the first resonance mode are displayed in Fig. 11 for the two tip lengths. The variations f1 are represented √ according to ρµ where ρ and µ are, respectively, the density and the viscosity of the solution. For the first mode, the curve corresponding to l2 = 25 mm varies quickly than curve of l2 = 22.5 mm. It means that the sensitivity is better when l2 = 25 mm. The results obtained during the cement setting are reported in Fig. 12 for the first resonance mode. The sensitivity to the variations of the medium properties is better for the tip with length 25 mm in particular between 10 and 30 min. The big change on f1 can be theoretically explained by the variation of the coupling (between the two cylinders) caused by the change of the tip length. Indeed, when this length increases, its resonance frequency comes near that of the large cylinder. Thus, the influence of the tip on the large-cylinder response increases. This phenomenon is theoretically shown in Fig. 5.
Fig. 12. Evolution of f1 as a function of the time for two tip lengths.
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Fig. 13. Variation of the admittance magnitude as a function of frequency for different large cylinder length of the sensor C2.
4.3. Variation of the large cylinder length We have measured the resonance evolution of the sensor C2 according to the large cylinder length as we cut out a part of it (Fig. 13). The initial lengths were 52.5 mm for the large cylinder and 25 mm for the tip.
Fig. 14. Evolution of f1 as a function of the cement setting for some large cylinder lengths.
Fig. 15. Evolution of (1/Z2 ) as a function of the cement setting for some large cylinder lengths.
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When the large cylinder length decreases, the frequency of the first mode comes near that of the second mode and the margin between their amplitudes decreases. In the theory of the coupled oscillators (mechanical and electrical oscillators), this phenomenon is observed when the proper frequencies of the oscillators get closer. To compare the sensitivity according to the large cylinder length, some measures have been carried out according to the cement setting as a function of the time for a 37% water/cement ratio. In Figs. 14 and 15 the variations f1 and (1/Z2 ) are respectively presented. The first mode sensitivity increases and the second mode sensitivity decreases when the large cylinder length decreases.
5. Conclusion The choice of the sensor dimensions depends on required application and the load range to be measured. If the sensitivity was sought on the tip, its length must be smaller than half of the large cylinder and the sections ratio must be weak (thin tip). This geometry is desirable to measure the weak variations loads (variation of liquids properties for example). But the measure of high loads is possible without discontinuity using the first mode. So, to have a best sensitivity on this peak, the large cylinder must be smaller than 2l2 . The results show that the horn operates according to two modes. The first one appears with the viscoelastic media; the second mode is involved when the sample is fluid. The sensor response not only depends on the rheological properties of the medium but also on the horn geometry. This geometry can be adapted to the material properties by changing cylinders lengths and their sections ratio. These results are very important because they provide a basic tool for a future sensor optimization. For the same load, various responses can be obtained depending on the sensor geometry. That gives the possibility to measure a wide range of material properties. The second mode sensitivity remains, generally, larger for weak loads. This mode disappears when the load is important, i.e., when the medium starts to have elastic properties. These results confirm the theoretical results which were used for this optimization.
References [1] J. Van Est, C. Prugne, B. Cros, J. Attal, B. Cretin, Local measurements of viscoelastic properties of fluids using near-field ultrasonics, Appl. Phys. A 66 (1998) S357–S360. [2] C. Prugne, J. Van Est, B. Cros, G. L´evˆeque, J. Attal, Measurements of the viscosity of liquids by near-field acoustics, Meas. Sci. Technol. 9 (1998) 1894–1998. [3] B. Cros, J.Y. Ferrandis, M. Pauthe, Study of the polymer processing by acoustic near field, La Revue de M´etallurgie, Science et G´enie des Mat´eriaux 96 (1999) 1537–1545. [4] B. Cros, M. Pauthe, M. Rguiti, J.Y. Ferrandis, On-line characterization of silica gels by acoustic near field, Sens. Actuators B 76 (2001) 115–123. [5] M. Valente, J.Y. Ferrandis, Evaluation of textural properties of mango tissue by a near-field acoustic method, Postharvest Biol. Technol. 29 (2003) 219–228. [6] B. Cros, J.Y. Ferrandis, Characterization of the cement setting by acoustic near field, Sens. Actuators B 60 (1999) 200–207. [7] J.Y. Ferrandis, G. L´evˆeque, In situ measurement of elastic properties of cement by an ultrasonic resonant sensor, Cement Concrete Res. 33 (2003) 1183–1187. [8] G. L´evˆeque, J.Y. Ferrandis, J. Van Est, B. Cros, An acoustic sensor for simultaneous density and viscosity measurements in liquids, Rev. Sci. Instrum. 71 (3) (2000) 1433–1440. [9] J.Y. Ferrandis, H. Nounah, B. Cros, Sensitivity of the second resonance mode of a near-field acoustic sensor to viscoelastic media: investigation of bitumens, Eur. Phys. J. AP 10 (2000) 149–155. [10] W. Mason, An electromechanical representation of a piezoelectric crystal used as a transducer, Inst. Radio Eng. 23 (1935) 1252. [11] G.S. Kino, Acoustic Waves: Devices Imaging and Analog Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1987.
Biographies Mohamed Rguiti was born in 1973. He studied physics and materials and received a master’s degree in physics in 1996 and a PhD degree in 2002 in the near-field acoustic technique. His major research interests include the development and characterization of PZT materials and their applications as sensors and actuators. He is currently a assistant professor.
Jean-Yves Ferrandis was born in 1973. He studied physics and engineering and received a solid state physics master’s degree in 1996 and a PhD degree in 2000. His major research interests include the development of acoustic sensors, their applications in materials characterization and the theoretical study of near-fields acoustics. He is currently a research engineer at the CNRS.