Vibronic sensor

Abstract

Vibronic sensor and method of operation for monitoring the density and/or the viscosity of a medium in a container, comprising a mechanically oscillatable unit, a driving/receiving unit and an electronics unit, wherein the driving/receiving unit is embodied, using an electrical exciter signal, to excite the mechanically oscillatable unit to execute mechanical oscillations, and to receive the mechanical oscillations and to convert them into an electrical, received signal, wherein the electronics unit is embodied to produce the exciter signal such that a predeterminable phase shift is present between the exciter signal and received signal, wherein the electronics unit is embodied to set a first predeterminable phase shift and a second predeterminable phase shift, and to ascertain a first frequency and a second frequency corresponding to the predeterminable phase shifts, and to determine from the two frequencies the density and/or the viscosity of the medium using a first and/or second analytical formula.

Claims

1. A vibronic sensor for monitoring at least the density and/or the viscosity of a medium in a container, the sensor comprising: a mechanically oscillatable unit; a driving/receiving unit structured to excite the mechanically oscillatable unit using an electrical exciter signal to execute mechanical oscillations, and to receive the mechanical oscillations of the mechanically oscillatable unit and convert them into an electrical, received signal; and an electronics unit embodied to produce the exciter signal based on the received signal such that a predeterminable phase shift is present between the exciter signal and received signal, wherein the electronics unit is configured to alternately set at least a first phase shift and a second phase shift at a time interval, to ascertain both a first frequency and a second frequency corresponding to the first phase shift and the second phase shift, respectively, and to calculate the density and viscosity of the medium independently of each other using the first frequency and the second frequency, wherein the density of the medium is calculated using a first analytical formula and the viscosity of the medium is calculated using a second analytical formula, and wherein the first phase shift essentially amounts to +90 or 90 and the second phase shift essentially to +45 or 135, wherein the first phase shift of essentially +90 corresponds to the second phase shift of essentially +45, and wherein the first phase shift of essentially 90 corresponds to the second phase shift of essentially 135.

2. The vibronic sensor of claim 1, wherein the oscillatable unit is arranged in a defined position within the container such that it extends to a determinable immersion depth in the medium.

3. The vibronic sensor of claim 1, wherein the electronics unit is configured to determine and/or to monitor a predetermined fill level of the medium in the container.

4. The vibronic sensor of claim 1, wherein the oscillatable unit is a membrane, single rod or oscillatory fork.

5. The vibronic sensor of claim 1, wherein the driving/receiving unit includes a piezoelectric element.

6. The vibronic sensor of claim 1, wherein the driving/receiving unit is an electromagnetic driving/receiving unit.

7. The vibronic sensor of claim 1, wherein the first analytical formula and the second analytical formula are each based on a solution of an equation of motion for an oscillatory movement of the oscillatable unit, the equation of motion including interaction of the oscillatable unit with the medium in the form of a compressive force and a frictional force, which arise from the medium surrounding the oscillatable unit, and as a frictional force that arises from an equally formed movement of the oscillatable unit within the medium.

8. The vibronic sensor of claim 1, wherein the first analytical formula is: $= - \frac{T_{2} + T_{6} + \sqrt{T_{3} + T_{4}}}{{\hat{B}}_{a} T_{5}},$ and the second analytical formula is: $= \frac{T_{1} +_{135} (T_{2} - \sqrt{T_{3} + T_{4}})}{{\hat{C}}_{a} T_{5}},$ wherein:
T.sub.1=2{circumflex over (B)}.sub.a.sub.a.sub.90 .sup.2(.sub.135.sub.90 .sup.2+D.sub.r(T)(.sub.135 .sup.2+.sub.90.sup.2)),
T.sub.2=.sub.a.sup.2D.sub.r(T).sub.90.sup.3.sub.a.sup.2.sub.135 .sub.90 .sup.3.sub.a.sup.2.sub.135 .sup.2.sub.90.sup.3,
T.sub.3=.sub.a.sup.2.sub.90 .sup.3.sub.a.sup.2(D.sub.r(T)+.sub.135(.sub.135)).sup.2.sub.90.sup.2,
T.sub.4=.sub.a.sup.2.sub.904{circumflex over (B)}.sub.a.sub.a.sub.135(D.sub.r(T).sub.90.sup.2)(.sub.135.sub.90.sup.2+D.sub.r(T) (.sub.135.sup.2.sub.90.sup.2)),
T.sub.5=2.sub.135.sub.90.sup.3(.sub.a.sup.2.sub.135+{circumflex over (B)}.sub.a.sub.a.sub.90), and
T.sub.6=2{circumflex over (B)}.sub.a.sub.a.sub.135.sub.90.sup.2(D.sub.r(T)+.sub.90.sup.2), wherein is the mass moment of inertia of the oscillatory rods of the oscillatable unit in the state not covered by medium, wherein .sub.0 is the angular frequency of the oscillatable unit in the undamped case, wherein D.sub.r(T) is the temperature dependent torsional stiffness of the oscillatable unit, wherein is the damping of the oscillatable unit not covered with medium, wherein .sub.a, {circumflex over (B)}.sub.a, and .sub.a are geometry dependent parameters, and wherein .sub.90 is the first frequency corresponding to the first phase shift of essentially +/90, and .sub.135 is the second frequency corresponding to the second phase shift of essentially 45 or 135 between the exciter signal and the received signal.

9. A method for determining density and/or viscosity of a medium in a container using a vibronic sensor, the method comprising: providing a vibronic sensor including a mechanically oscillatable unit, a driving/receiving unit, and an electronics unit; exciting the oscillatable unit using the driving/receiving unit to execute mechanical oscillations using an electrical exciter signal produced by the electronics unit; receiving and converting the mechanical oscillations of the mechanically oscillatable unit into an electrical, received signal using the driving/receiving unit, wherein the exciter signal is produced starting from the received signal such that a predeterminable phase shift is present between the exciter signal and the received signal; using the electronics unit, alternately setting at least a first phase shift and a second phase shift at a time interval; ascertaining both a first frequency and a second frequency corresponding to the first phase shift and the second phase shift, respectively; calculating the density of the medium from the first frequency and the second frequency using a first analytical formula; and calculating the viscosity of the medium, independently of the density, from the first frequency and the second frequency using a second analytical formula, wherein the first phase shift is set to essentially +90 or 90 and the second phase shift to essentially +45 or 135, wherein the first phase shift of essentially +90 corresponds to the second phase shift of essentially +45, and wherein the first phase shift of essentially 90 corresponds to the second phase shift of essentially 135.

10. The method of claim 9, the method further comprising: monitoring a predetermined fill level of the medium in the container.

11. The method of claim 9, wherein that the first analytical formula and/or the second analytical formula are each based on a solution of an equation of motion for an oscillatory movement of the oscillatable unit, the equation of motion including interaction of the oscillatable unit with the medium as expressed as a compressive force and a frictional force, which result from the medium surrounding the oscillatable unit, and as a frictional force that arises as a result of an equally formed movement of the oscillatable unit within the medium.

12. The method of claim 9, wherein the first analytical formula is $= - \frac{T_{2} + T_{6} + \sqrt{T_{3} + T_{4}}}{{\hat{B}}_{a} T_{5}},$ and second analytical formula is $= \frac{T_{1} +_{135} (T_{2} - \sqrt{T_{3} + T_{4}})}{{\hat{C}}_{a} T_{5}},$ wherein
T.sub.1=2{circumflex over (B)}.sub.a.sub.a.sub.90 .sup.2(.sub.135.sub.90 .sup.2+D.sub.r(T)(.sub.135 .sup.2+.sub.90.sup.2)),
T.sub.2=.sub.a.sup.2D.sub.r(T).sub.90.sup.3.sub.a.sup.2.sub.135 .sub.90 .sup.3.sub.a.sup.2.sub.135 .sup.2.sub.90.sup.3,
T.sub.3=.sub.a.sup.2.sub.90 .sup.3.sub.a.sup.2(D.sub.r(T)+.sub.135(.sub.135)).sup.2.sub.90.sup.2,
T.sub.4=.sub.a.sup.2.sub.904{circumflex over (B)}.sub.a.sub.a.sub.135(D.sub.r(T).sub.90.sup.2)(.sub.135.sub.90.sup.2+D.sub.r(T) (.sub.135.sup.2.sub.90.sup.2)),
T.sub.5=2.sub.135.sub.90.sup.3(.sub.a.sup.2.sub.135+{circumflex over (B)}.sub.a.sub.a.sub.90), and
T.sub.6=2{circumflex over (B)}.sub.a.sub.a.sub.135.sub.90.sup.2(D.sub.r(T)+.sub.90.sup.2), wherein is the mass moment of inertia of the oscillatory rods of the oscillatable unit in the state not covered by medium, wherein .sub.0 is the angular frequency of the oscillatable unit in the undamped case, wherein D.sub.r(T) is the temperature dependent torsional stiffness of the oscillatable unit, wherein is the damping of the oscillatable unit not covered with medium, wherein .sub.a, {circumflex over (B)}.sub.a and .sub.a are geometry dependent parameters, and wherein .sub.90 is the first frequency corresponding to the first phase shift of essentially +/90, and .sub.135 is the second frequency corresponding to the second phase shift of essentially 45 or 135 between the exciter signal and the received signal.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention as well as advantageous embodiments thereof will now be described in greater detail based on the appended drawing, the figures of which show as follows:

(2) FIG. 1 shows a schematic drawing of a vibronic sensor according to state of the art,

(3) FIG. 2 shows a schematic drawing of an oscillatory fork, and

(4) FIG. 3 shows an illustration of the approximation of the geometry of oscillatory forks using elliptical cylinders.

DETAILED DESCRIPTION

(5) FIG. 1 shows a vibronic sensor 1. Included in the vibronic sensor 1 is an oscillatable unit 4 in the form of an oscillatory fork, which extends partially into a medium 2, which is located in a container 3. The oscillatable unit is excited to execute mechanical oscillations by means of the exciter/receiving unit 5, which can be, for example, a piezoelectric stack- or bimorph drive. It is understood, however, that also other embodiments of a vibronic sensor fall within the scope of the invention. Furthermore, an electronics unit 6 is shown, by means of which the signal registration, -evaluation and/or feeding occurs.

(6) FIG. 2 shows an oscillatable unit 4 in the form of an oscillatory fork, such as is integrated, for example, into the vibronic sensor 1 sold by the applicant under the mark, LIQUIPHANT. Oscillatory fork 4 includes two oscillatory rods 7a, 7b, also called fork tines, mounted on a membrane 8. In order to cause the oscillatory rods 7a, 7b to execute mechanical oscillations, a driving/receiving unit 5 mounted by material bonding on the side of the membrane 8 facing away from the rods 7a, 7b exerts a force on the membrane 8. The driving/receiving unit 5 is an electromechanical transducer unit, and comprises, for example, a piezoelectric element 9, or also an electromagnetic drive. The drive unit 5 and the receiving unit are constructed as two separate units, or as a combined driving/receiving unit. In the detail drawing on the right side of FIG. 2, the driving/receiving unit 5 is shown in detail. A piezoelectric element 9 is arranged on a steatite disk 10 and equipped with electrodes 11 for applying the exciter signal as well as for tapping the received signal.

(7) In the case, in which the driving/receiving unit 5 includes a piezoelectric element 9, the force applied to the membrane 8 is generated by applying an exciter signal U.sub.E, for example, in the form of an electrical, alternating voltage. A change of the applied electrical voltage effects a change of the geometric shape of the driving/receiving unit 5, thus a contraction or expansion of the piezoelectric element 9, in such a manner that the applying of an electrical, alternating voltage in the form of exciter signal U.sub.E brings about an oscillation of the membrane 8 connected by material bonding with the driving/receiving unit 5.

(8) As indicated above, a goal of the present invention is to expand the range of applications for determining density and/or viscosity by means of a vibronic sensor 1. Previously, the methods for determining density and/or viscosity have been based on empirically ascertained assumptions. According to the invention, these empirical models are replaced by an analytical model for describing the oscillatory movements of a vibronic sensor 1 in a viscous medium. Such a model has not previously been available. Therefore, the bases of this model will now be briefly explained.

(9) Under the assumption of an ideal drive unit 5, the oscillatable unit 4 can be described as a harmonic, single-mass oscillator. The oscillatory movements of the two oscillatory rods 7a, 7b, or fork tines, correspond largely to the deflections of a bending beam. Due to the usually much more complex geometric structure of the oscillatable unit 4 of a vibronic sensor 1, it is, however, helpful to approximate the oscillatory movements of the two oscillatory rods 7a, 7b by a rotational movement. Differences between the mathematical description and the actual oscillatory movements resulting from the approximation can be eliminated a posteriori by taking suitable correction terms into consideration.

(10) The equation of motion for a free, forced oscillation of an oscillatable unit 4 such as in FIG. 2 in the form of a rotational movement is, in principle, known from the state of the art and can be derived based on the moments acting on the two oscillatory rods 7a, 7b. If, supplementally, also the interactions between the fluid and the oscillatable unit are taken into consideration, there results as equation of motion a second order differential equation:
M.sub.S+M.sub.D+M.sub.R+M.sub.F=M.sub.E
{umlaut over ()}(t)+{dot over ()}(t)+D.sub.r(t)+M.sub.F=.sub.0 D.sub.rsin(t).

(11) In such case, describes M.sub.F the moment due to the fluid-structure interaction, M.sub.E the exciter moment, M.sub.R the moment from the stiffness of the membrane, M.sub.D the moment due to the inner damping of the oscillatable unit and M.sub.S the moment due to the mass moment of inertia of the fork tines. Furthermore, is the deflection, or in the here considered approximation the rotational angle, of the oscillatory rods 7a,7b of the oscillatable unit 4 from the rest position, the mass moment of inertia brought about by the mass of the oscillating oscillatory rods 7a,7b, the damping coefficient resulting from the inner damping of the oscillatable unit 4 and D.sub.r the torsional stiffness due to the stiffness of the membrane 8.

(12) A particular solution results by means of the ansatz (t)=.sub.0Vsin(t+), with the amplification function

(13) $V () = \frac{1}{\sqrt{4 {D^{2} (\frac{}{_{0}})}^{2} + {(1 - {(\frac{}{_{0}})}^{2})}^{2}}},$
which describes the amplitude behavior of the oscillatable unit 4, and with the phase of the oscillatable unit 4 obeying

(14) $\tan = \frac{2 D_{0}}{^{2} -_{0}^{2}} .$

(15) In such case,

(16) $D = \frac{}{2_{0}},$
the damping ratio or Lehr, which represents a characteristic variable for the quality of the oscillatory system, and

(17) $_{0} = \sqrt{\frac{D_{r}}{}},$
the eigenfrequency of the corresponding undamped oscillator.

(18) The moment M.sub.F due to the fluid-structure interaction is dependent on the geometry of the oscillatable unit 4 and describes, in principle, the interaction between the oscillatable unit 4 and the relevant medium 2. The case M.sub.F=0 describes the case of an oscillation of the oscillatable unit 4 outside of the medium 2.

(19) An analytical solution of the equation of motion can only be obtained by assuming simple geometrical structures for the oscillatable unit 4, such as, for example, a sphere (see also W. Y. Shih, X. Li, H. Gu, W. H. Shih and I. A. Aksay: Simultaneous liquid viscosity and density determination with piezoelectric unimorph cantilevers, published in Journal of Applied Physics, 89(2):1497-1505, 15. 1 in the year 2001, a cuboid-shaped structure (see also J. E. Sader: Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope, published in the Journal of Applied Physics, 84(1):64-76,1. 7 in the year 1998 or an infinitely extended circular cylinder (see also the Dissertation of W. Zhang: Energy Dissipations in MEMS Resonators: Fluid Damping of Flexura Resonators and Thermoelastic Damping, published December 2006 at the University of California). These geometries do not really suit the much more complex geometry of the oscillatable unit 4, for example, in the form of an oscillatory fork of a vibronic sensor 1. Much better suited for an oscillatory fork is the geometry of an elliptical cylinder with the semi axes a and b, such as, for example, in the dissertation of J. Friedmann, Untersuchungen ber das Verhalten der Schwingfrequenz bei Stimmgabelgasdichtemessgerten (Investigations concerning the behavior of oscillation frequency in the case of tuning fork gas density measuring devices, at the University Fridericiana Karlsruhe (Technical Hochschule) at the Faculty for Elektrotechnik, January 1976. Using this approximation, the real dimensions of the oscillatory rods 7a, 7b of the oscillatable unit 4 enter advantageously into the analytical solution. This approximation of the geometry for the oscillatable unit 4 is illustrated in FIG. 3. For the oscillatory fork 4 applied in the LIQUIPHANT instrument, a paddle 12a, 12b is formed terminally on each of the two oscillatory rods 7a, 7b, this being shown in FIG. 3a. In order to take this into consideration, the geometry of the oscillatable unit is approximated by two elliptical cylinders for each oscillatory rod, such as illustrated in FIG. 3b. The first elliptical cylinder 13 of length l.sub.1, width a.sub.1 as well as thickness b.sub.1 serves to represent the paddles 12a, 12b, while the second elliptical cylinder 14 of length l.sub.2, width a.sub.2 as well as thickness b.sub.2 represents the oscillatory rods 7a, 7b. FIG. 3c shows a fork tine 7a of the oscillatory fork 4 as well as the approximated geometry with the two elliptical cylinders 13,14 in side view.

(20) The interaction between the oscillatable unit 4 and the medium 2 arises as a result of the medium 2 being moved by the movement of the oscillatable unit 4 in the immersed state. This has the result that forces F.sub.F occur opposing the movement of the oscillatable unit 4. These can be subdivided into compressive forces and frictional forces, i.e.:
F.sub.F=F.sub.C+F.sub.RF.sub.SP.

(21) In such case, F.sub.C is the compressive force, F.sub.R the frictional force acting due to the fluid surrounding the oscillating, elliptical cylinder, and F.sub.SP a frictional force acting supplementally as a result of the equally formed movement of an elliptical cylinder. In order to calculate these forces, the velocity distribution of the medium 2 in the environment of the oscillatable unit 4 must be known. For this, reference is made to the Lehrbuch der Hydrodynamik (hydrodynamics textbook) by H. Lamb, Vol. 26 of B. G. Teubner's collection of mathematical science textbooks including applications, B. G. Teubner, publisher, Leipzig and Berlin, 3rd edition of 1907.

(22) Based on the textbook Hydrodynamik (hydrodynamics), Vol. 6, of the series, Lehrbuch der Theoretischen Physik (textbooks of theoretical physics), by L. D. Landau and E. M Lifschitz, Akademie Verlag, publishers, Berlin, 5th edition of 1991, the velocity distribution can be divided into a normal and a tangential component. While the normal velocity component is not influenced by the viscosity of the medium 2, it can be determined based on the model of an ideal fluid. The tangential component, in contrast, in the region around the oscillatable unit 4 is influenced by the viscosity of the medium. In the limiting case of infinite distance from the oscillatable unit 4, in contrast, the normal component transforms into the tangential velocity component.

(23) Taking into consideration the geometry of an elliptical cylinder 13,14 and using the basic hydrodynamic equations, Euler's equation, the continuity equation, as well as Thomson's theorem and d'Alembert's paradox, the compressive force F.sub.C per unit length, which acts on an elliptical cylinder 13,14, can be determined as follows:

(24) 0 $F_{C} = a^{2} \frac{du}{dt},$
wherein is the density of the medium 2 and u the velocity of the oscillatable unit 4.

(25) The frictional force F.sub.R, which acts due to the fluid surrounding the oscillating, elliptical cylinder, can be derived starting from the frictional force F.sub.y, which acts on an infinitely extended, planar area

(26) ${F_{y} = \frac{\overset{->}{v_{y}}}{x} .Math.}_{x = 0},$
and results as:

(27) $F_{R} = 2 \sqrt{2} bX (\frac{b}{a}) (\sqrt{} u + \sqrt{\frac{}{}} \frac{du}{dt}),$
with the oscillation frequency , and the function

(28) $X (\frac{b}{a}) = \frac{{EI}_{1} [1 - \frac{1}{{(b / a)}^{2}}] - {EI}_{2} [1 - \frac{1}{{(b / a)}^{2}}]}{\frac{b}{a} ({(\frac{b}{a})}^{2} - 1)},$
wherein EI.sub.1 and EI .sub.2 refer to the complete elliptic integrals of first and second order.

(29) The function

(30) $X (\frac{b}{a})$
can be approximated by an exponential function.

(31) The force F.sub.SP acting supplementally as a result of the equally formed movement of an elliptical cylinder is a result of the Stoke's frictional force. It is independent of the oscillation frequency and proportional to the viscosity of the medium. Based on the dissertation, Energy Dissipations in MEMS Resonators: Fluid Damping of Flexural Resonators and Thermoelastic Damping of W. Zhang as well as the Lehrbuch der Hydrodynamik of H. Lamb, there results for the frictional force F.sub.SP

(32) $F_{SP} = \frac{4}{\frac{b}{a + b} -_{E} - \log (\frac{Re .Math. (a + b)}{16 a})} .Math. u,$
with the Euler constant .sub.E0.577 and the Reynolds number Re, a dimensionless, characteristic variable for the flow of a medium 2.

(33) In order to obtain the equation of motion describing the oscillatory movement of an oscillatable unit in a viscous medium, the entire force per length unit F.sub.F acting on the oscillatable unit due to the interaction between the oscillatable unit and the medium must be converted into the associated moment M.sub.F. In such case, it must be taken into consideration that each of the oscillatory rods was approximated by two elliptical cylinders of different dimensions, such as described in connection with FIG. 3.

(34) There then results for the equation of motion for the oscillatory movement of the oscillatable unit 4 in a viscous medium 2
(+{circumflex over ()}.sub.F){umlaut over ()}(t)+(+{circumflex over ()}.sub.F){dot over ()}(t)+D.sub.r(t)=M.sub.Esi n(t),
wherein {circumflex over ()}.sub.F is the supplementally coupling, mass moment of inertia and {circumflex over ()}.sub.F the supplementally acting, torsional damping due to the interaction of the oscillatable unit 4 with the medium 2. For these terms, the already mentioned, numerically ascertainable, correction terms are available for matching the equation of motion to a bending oscillation. The correction terms can be calculated, for example, from a comparison of the deflections of the oscillatable unit 4 in the case of a bending oscillation and in the case of a torsional oscillation by means of the ANSYS simulation tool.

(35) A particular solution for this differential equation of second order can be won of the form
(t)=.sub.0V()sin(t+())
wherein V() is the so-called amplification function, which represents the amplitude behavior of the oscillatable unit 4, and () the phase difference of the oscillatable unit 4. The equation of motion for an oscillatable unit 4 oscillating in a viscous medium 2 differs, thus, clearly from that in the uncovered case, such as described above. The oscillatory movements of the oscillatable unit 4 depend in the case of the immersion in a viscous medium 2 on, besides the density and viscosity , also the oscillation frequency of the oscillatable unit 4.

(36) The oscillatory movement of a vibronic sensor 1 in a viscous medium 2 is that of a time variable oscillatory system. The mass coupling by the viscous medium 2 depends, in such case, on changes of the density and/or viscosity . The mass coupling is additionally dependent on the oscillation frequency of the oscillatable unit 4. It is thus, strictly speaking, not possible to characterize a vibronic sensor 1 using a constant eigenfrequency or a constant Lehr's damping ratio.

(37) By evaluation of the oscillation frequency at a phase shift between exciter signal U.sub.E and received signal U.sub.R of essentially +/90 and 45 or 135, the already mentioned analytical formulas for the density and the viscosity can be ascertained:

(38) $= \frac{T_{1} +_{135} (T_{2} - \sqrt{T_{3} + T_{4}})}{{\hat{C}}_{a} T_{5}}, and$ $= - \frac{T_{2} + T_{6} + \sqrt{T_{3} + T_{4}}}{{\hat{B}}_{a} T_{5}} .$

(39) In such case, the following relationships hold:
T.sub.1=2{circumflex over (B)}.sub.a.sub.a.sub.90 .sup.2(.sub.135.sub.90 .sup.2+D.sub.r(T)(.sub.135 .sup.2+.sub.90.sup.2)),
T.sub.2=.sub.a.sup.2D.sub.r(T).sub.90.sup.3.sub.a.sup.2.sub.135 .sub.90 .sup.3.sub.a.sup.2.sub.135 .sup.2.sub.90.sup.3,
T.sub.3=.sub.a.sup.2.sub.90 .sup.3.sub.a.sup.2(D.sub.r(T)+.sub.135(.sub.135)).sup.2.sub.90.sup.2,
T.sub.4=.sub.a.sup.2.sub.904{circumflex over (B)}.sub.a.sub.a.sub.135(D.sub.r(T).sub.90.sup.2)(.sub.135.sub.90.sup.2+D.sub.r(T) (.sub.135.sup.2.sub.90.sup.2)),
T.sub.5=2.sub.135.sub.90.sup.3(.sub.a.sup.2.sub.135+{circumflex over (B)}.sub.a.sub.a.sub.90), and
T.sub.6=2{circumflex over (B)}.sub.a.sub.a.sub.135.sub.90.sup.2(D.sub.r(T)+.sub.90.sup.2),

(40) Here, is the mass moment of inertia of the oscillatory rods of the oscillatable unit in the state, not covered by medium, as calculated, for example, by means of the ANSYS software. Furthermore, .sub.0 and D.sub.r(T) can be measured. The damping y of the oscillatable unit not covered with medium can, finally, be calculated by measuring Lehr's damping ratio and is, in given cases, even negligible. The geometry dependent parameters .sub.a, {circumflex over (B)}.sub.a and .sub.a can, finally, be calculated, for example, by means of a so-called parameter estimation method, such as, for example, described in DE102012113045A1 or in the previously unpublished application DE102013106172.9. The frequencies .sub.90 and .sub.135 are then the frequencies measured during operation of the vibronic sensor at a predeterminable phase shift of essentially +/90 and 45 or 135 between the exciter signal and the received signal.

LIST OF REFERENCE NUMBERS

(41) 1 vibronic sensor 2 medium 3 container 4 oscillatable unit 5 electromechanical transducer unit 6 electronics unit 7a, 7b oscillatory rods of the oscillatable unit 8 membrane 9 piezoelectric element 10 steatite disk 11 electrodes 12a, 12b paddles of the oscillatable unit 13 first elliptical cylinder 14 second elliptical cylinder U.sub.E exciter signal U.sub.R received signal rotational angle of the oscillatory rods of the oscillatable unit from the resting position mass moment of inertia resulting from the mass of the oscillating oscillatory rods damping coefficient resulting from the inner damping of the oscillatory system D.sub.r torsional stiffness due to the stiffness of the membrane a,b semi axes of an elliptical cylinder l.sub.i length of the elliptical cylinder a.sub.i width of the elliptical cylinder d.sub.i thickness of the elliptical cylinder F.sub.C compressive force F.sub.R frictional force due to the fluid surrounding the oscillating, elliptical cylinder F.sub.SP supplemental frictional force acting as a result of the equally formed movement of an elliptical cylinder density of the medium u velocity of the oscillatable unit oscillation frequency of the oscillatable unit .sub.0 angular frequency of the oscillatable unit in the undamped case .sub.E Euler's constant Re Reynolds number {circumflex over ()}.sub.F supplementally coupling, mass moment of inertia resulting from interaction with the medium {circumflex over ()}.sub.F supplementally acting, torsional damping due to interaction of the oscillatable unit with the medium V() amplification function, which represents the amplitude behavior of the oscillatable unit () phase difference of the oscillatable unit .sub.a, {circumflex over (B)}.sub.a, .sub.a geometry dependent parameters .sub.90, .sub.135 frequencies corresponding to predeterminable phase shifts of 90 and 45 .sub.45, .sub.90 predeterminable phase shifts of 45 and 90

Vibronic sensor

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G01N11/10

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G01N2291/02818

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G01N2011/0073

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Abstract

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