Patent Publication Number: US-11022509-B2

Title: Measurement transducer for measuring a force

Description:
TECHNICAL FIELD 
     The invention relates to a measurement transducer for measuring a force according to the preamble of the independent claim. 
     PRIOR ART 
     Each of EP0065511A1 and U.S. Pat. No. 4,441,044, which is hereby incorporated herein by this reference for all purposes, teaches a measurement transducer for measuring a dynamic force comprising a piezoelectric sensor element. The piezoelectric sensor element comprises a plurality of disc-shaped plates made of piezoelectric material wherein a disc diameter of the plates is substantially larger than a thickness of the plates. The direct piezoelectric effect in the form of the longitudinal effect is used for measuring the force. The spatial orientation of the piezoelectric material with respect to the force is such that the force acts normally on disc surfaces of the discs and generates electrical polarization charges on the disc surfaces. The electrical polarization charges are received by electrodes and transmitted as charge signals to an evaluation unit. The electrical polarization charges are proportional to the force acting on the discs. Since leakage currents practically always exist, the direct piezoelectric effect can only measure a dynamic force with rates of change in the range of several Hz up to several MHz while quasi-static force measurements with a duration of several minutes are possible. 
     In contrast, a static force shows no change even over long time periods of hours, weeks and years. The measurement transducer according to EP0065511A1 teaches to utilize the inverse piezoelectric effect for measuring a static force. Another piezoelectric sensor element comprises a plurality of disc-shaped plates made of piezoelectric material to which an alternating electrical field is applied as frequency signals via electrodes. The alternating electrical field stimulates the plates by means of the inverse piezoelectric effect to mechanically oscillate. The alternating electrical field is tunable and is generated by the evaluation unit. Resonance will occur when an excitation frequency of the alternating electrical field is equal to a mechanical natural frequency of the plates, the corresponding frequency is called the resonance frequency. The impact of the static force changes the resonance frequency of the plates, which frequency change is detected by an oscillator circuit of the evaluation unit. According to the teaching of EP0065511A1, measurements of the dynamic force and the static force are done simultaneously. 
     BRIEF OBJECTS AND SUMMARY OF THE INVENTION 
     It is an object of the present invention to improve the prior art measurement transducer. 
     This object is achieved by the features described herein. 
     The invention relates to a measurement transducer for measuring a force comprising a resonator element which can be excited to at least one resonance frequency; and comprising at least one force application element on which the force is applied and which transmits the force to the resonator element; wherein said force application element is a hollow body having a top surface, a lateral surface and a cavity, the top surface and lateral surface being mechanically connected to each other and enclosing the cavity; wherein the resonator element is disposed in the cavity; said resonator element being mechanically connected to the lateral surface; wherein the force acts on said top surface from which the force is transmitted into the lateral surface; wherein said lateral surface comprises at least one recessed area extending into the cavity and preventing the transmission of the force within the lateral surface; and wherein the lateral surface comprises at least one non-recessed area and it is only the non-recessed area that transmits the force. 
     Upon being excited by a resonance frequency, the resonator element responds to the force to be measured with a change in frequency of the resonance frequency. However, this frequency change is proportional to a force sensitivity of the resonator element which itself is dependent on a force application angle. Depending on the position of the force application angle with respect to a maximum force sensitivity and depending on the size of a force application angle area, the total frequency change will be measured by an average force sensitivity which is only a fraction of the maximum force sensitivity. 
     This is one problem that the present invention seeks to solve. In the invention, the force application angle area is adjusted to have, in total, an average force sensitivity that is close to the maximum of the force sensitivity of the resonator element. Optimal adjustment of the force application angle area may be achieved by means of the force application element. Here, the recessed area of the lateral surface is used to limit the force application angle area to be close to the maximum of the force sensitivity of the resonator element. Transmission of the force to be detected occurs only in the non-recessed area of the lateral surface. The smaller the force application angle area, the higher will be the average force sensitivity, however, the higher will also be a local pressure acting on the resonator element. With an optimum limitation of the force application angle area, the average force sensitivity will be as high as possible, while the local pressure on the resonator element will be below a permissible pressure resistance. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In the following the invention will be explained by way of example with reference to the figures in which 
         FIG. 1  is a schematic exploded view of a portion of a measurement transducer; 
         FIG. 2  is a perspective view of a portion of a first embodiment of the measurement transducer according to  FIG. 1  comprising one piezoelectric transducer element; 
         FIG. 3  shows a section through a portion of the first embodiment of a measurement transducer according to  FIG. 2 ; 
         FIG. 4  shows a perspective view of a portion of a second embodiment of the measurement transducer according to  FIG. 1  comprising two piezoelectric transducer elements; 
         FIG. 5  shows a section through a portion of the second embodiment of a measurement transducer according to  FIG. 4 ; 
         FIG. 6  is a perspective view of a portion of a third embodiment of the measurement transducer according to  FIG. 1  comprising one piezoelectric transducer element; 
         FIG. 7  shows a section through a portion of the third embodiment of a measurement transducer according to  FIG. 6 ; 
         FIG. 8  is a perspective view of a portion of a fourth embodiment of the measurement transducer according to  FIG. 1  comprising two piezoelectric transducer elements; 
         FIG. 9  shows a section through a portion of the fourth embodiment of a measurement transducer according to  FIG. 8 ; 
         FIG. 10  shows a first illustration of a mean force sensitivity of a resonator element of the measurement transducer according to  FIGS. 1 to 9  as a function of a force application angle; 
         FIG. 11  shows a second illustration of the mean force sensitivity of a resonator element of the measurement transducer according to  FIGS. 1 to 9  as a function of force application angle areas; 
         FIG. 12  shows a perspective view of a first embodiment of a force application element of the measurement transducer according to  FIGS. 1 to 9 ; 
         FIG. 13  shows a perspective view of a second embodiment of a force application element of the measurement transducer according to  FIGS. 1 to 9 ; 
         FIG. 14  shows a plan view of a third embodiment of a force application element of the measurement transducer according to  FIGS. 1 to 9  with a first force application angle area; 
         FIG. 15  shows a plan view of a fourth embodiment of a force application element of the measurement transducer according to  FIGS. 1 to 9  with a second force application angle area; 
         FIG. 16  shows a section through a portion of the measurement transducer according to  FIG. 2 or 3  comprising a force application element according to  FIG. 14 or 15 ; 
         FIG. 17  shows a section through a portion of the measurement transducer according to  FIG. 8 or 9  comprising two force application elements according to  FIG. 14 or 15 ; 
         FIG. 18  shows a first illustration of an average coefficient of linear thermal expansion of a resonator element of the measurement transducer according to  FIGS. 1 to 9  as a function of a force application angle; 
         FIG. 19  is a second illustration of the mean coefficient of linear thermal expansion of a resonator element of the measurement transducer according to  FIGS. 1 to 9  as a function of force application angle areas; and 
         FIG. 20  is a view of an embodiment of a mounted ready-to-use measurement transducer according to  FIGS. 1 to 9  comprising a force application element according to  FIG. 14 or 15 . 
     
    
    
     DESCRIPTION OF EXEMPLARY EMBODIMENTS OF THE PRESENT INVENTION 
     The measurement transducer  1  is configured for simultaneously measuring a force F which may be both dynamic and static. A dynamic force F changes in short periods of time with rates of change in the range of several Hz up to several MHz. A static force F shows no change over long periods of time of hours, weeks and years and has rates of change in the mHz to nHz range. Whether the force F to be detected is dynamic or static solely depends on its rate of change. Those skilled in the art knowing the present invention may also configure the measurement transducer for simultaneous measurement of a dynamic pressure and a static pressure. Furthermore, those skilled in the art may configure the measurement transducer for measuring an acceleration. 
     Measurement transducer  1  has a longitudinal axis AA′, a transverse axis BB′ and a diagonal axis CC′ as schematically shown in  FIGS. 1-9, 12 and 13 . The axes extend at an angle to one another, preferably they are perpendicular to one another. For the purposes of the present invention, the term “at an angle” refers any angle that is different from zero at which the axes extend to one another. The longitudinal axis AA′ is perpendicular to a transverse plane BC.  FIGS. 1 to 9  show several embodiments of a measurement transducer  1  comprising at least one piezoelectric transducer element  10 ,  10 ′ which measures a force F by means of the direct piezoelectric effect, and comprising a resonator element  20  which measures a transverse expansion Q by means of a frequency change of a resonance frequency. The force F and transverse expansion Q are shown schematically as arrows, which in the case of the force F are pointing parallel to the longitudinal axis AA′. The arrows schematically representing the transverse expansion Q are pointing radially from and perpendicular to the longitudinal axis AA′. 
     The force F is introduced into the measurement transducer  1  in a direction of force that extends parallel to the longitudinal axis AA′. For the purposes of the present invention, this is called the direction of force AA′. The force F acts on the piezoelectric transducer element  10 ,  10 ′ in at least one transducer contact area K 1 , K 1 ′, the transducer contact area K 1 , K 1 ′ being located in the load path. The force F may act directly or indirectly on the piezoelectric transducer element  10 ,  10 ′. The force F acts on the resonator element  20  in at least one resonator contact area K 2 , K 2 ′, said resonator contact area K 2 , K 2 ′ also being located in the load path. The force F may act directly or indirectly on the resonator element  20 . In the embodiments as shown in  FIGS. 1 to 9 , the piezoelectric transducer element  10 ,  10 ′ is in mechanical contact to the resonator element  20  over the resonator contact area K 2 , K 2 ′. 
     The piezoelectric transducer element  10 ,  10 ′ measures the force F directly in the load path while the resonator element  20  measures the transverse expansion Q caused by the force F in an oscillation region which lies outside the load path. The force F causes an elastic deformation of the resonator element  20 . The elastic deformation of the resonator element  20  is a function of the force F and is preferably proportional to the magnitude of the force F. A change in length occurs in the resonator element  20  along the longitudinal axis AA′, while the transverse expansion Q occurs in the transverse plane BC. A magnitude of the transverse expansion Q is a function of the force, and it is preferably proportional to the magnitude of the force F. In the embodiments shown in  FIGS. 1-5 , the force F is a compressive force while in the embodiments according to  FIGS. 6-9  the force F is a tensile force. The transverse expansion Q may be isotropic or anisotropic. Especially in the case of piezoelectric material, the transverse expansion Q is anisotropic. 
     The piezoelectric transducer element  10 ,  10 ′ is hollow cylindrical in shape (ring, sleeve, etc.). The resonator element  20  is cylindrical in shape (disc, round rod, etc.). The longitudinal axis AA′ extends through the centers of the piezoelectric transducer element  10 ,  10 ′ and the resonator element  20 . However, those skilled in the art knowing the present invention may also adopt other known shapes of piezoelectric transducer elements and of resonator elements such as polyhedral, cubic, cuboidal, hollow truncated conical, half-discoidal, quarter-discoidal shapes, etc. 
     The piezoelectric transducer element  10 ,  10 ′ is made from piezoelectric material  11 ,  11 ′. The piezoelectric material  11 ,  11 ′ is a crystal material such as quartz (SiO 2  single crystal), calcium gallo-germanate (Ca 3 Ga 2 Ge 4 O 14  or CGG), langasite (La 3 Ga 5 SiO 10  or LGS), tourmaline, gallium orthophosphate, and piezoceramics, etc. Preferably, the piezoelectric material  11 ,  11 ′ consists of crystal material belonging to the m, 32, 3 m, 42 m, 2 m or 23 symmetry classes. Preferably, quartz is used as the piezoelectric material  11 ,  11 ′ with the orthogonal axes x, y, z being crystal axes and the axis z being the optical axis. An elasticity coefficient s λμ =(δS λ /δT μ ) of quartz is of the same magnitude for different orientations of the piezoelectric material  11 ,  11 ′ in a plane xy. S λ  refers to a mechanical strain tensor in matrix notation while T μ  denotes a mechanical stress tensor in matrix notation with the tensor indices λ, μ=1 to 6. Furthermore, a coefficient of linear thermal expansion α of quartz is also the same for different orientations of the piezoelectric material  11 , 11 ′ in the plane xy. 
     The force F acts on the piezoelectric transducer element  10 ,  10 ′ and generates electrical polarization charges on element surfaces of the piezoelectric transducer element  10 ,  10 ′. For achieving the direct piezoelectric effect in the form of the longitudinal effect, the piezoelectric transducer element  10 ,  10 ′ is oriented in such a way that electrical polarization charges are generated on those end faces on which the force F acts. Thus, the end faces are the element surfaces. The end faces are oriented in the transverse plane BC. For achieving the direct piezoelectric effect, a piezoelectric coefficient d iμ =(δD i /δT μ ) must be different from zero. Wherein D i  refers to an electrical displacement field and T μ  denotes a mechanical stress tensor in matrix notation with the Latin indices i=1 to 3 and with the tensor indices μ=1 to 6. A sensitivity indicates how strongly the first piezoelectric material  11 ,  11 ′ responds to the force F, i.e. how many electrical polarization charges the force F is able to generate. The sensitivity is specified by the piezoelectric coefficient d iμ =(δD i /δT μ ). The piezoelectric material  11 ,  11 ′ is oriented with respect to the force F to have high sensitivity, preferably maximum sensitivity for the direct piezoelectric effect. Preferably, quartz is used as the piezoelectric material  11 ,  11 ′, with the orthogonal axes x, y, z being the crystal axes and the axis z being the optical axis. The piezoelectric coefficient d 11  exhibits maximum sensitivity for the longitudinal effect when the piezoelectric transducer element  10 ,  10 ′ is cut as an x-ring having an axis x that is normal to a plane yz. The x axis of the x-ring is parallel to the longitudinal axis AA′. The optical axis z of the x-ring is located in the transverse plane BC. However, those skilled in the art knowing the present invention may also use the transverse effect in which case the piezoelectric transducer element is oriented to generate electrical polarization charges on element surfaces that are perpendicular to those end faces on which the force F acts. 
     Resonator element  20  may also consist of piezoelectric material  21 . Preferably, piezoelectric transducer element  10 ,  10 ′ and resonator element  20  are made of the same piezoelectric material  11 ,  11 ′,  21  so that the production of the measurement transducer  1  will be cost-effective and the piezoelectric transducer element  10 ,  10 ′ and the resonator element  20  will have substantially similar (in a range of plus or minus 10%) physical properties. However, those skilled in the art knowing the present invention may fabricate the resonator element from a composite of a piezoelectric material and a non-piezoelectric material such as a metal, a non-metal, a semiconductor, an organic material, an inorganic non-metallic material, etc. Preferably, the two materials are bonded to each other by material bonding such as thermocompression bonding, adhesive bonding, etc. Excitation of oscillation of the piezoelectric material results in mechanical oscillation of the composite. 
     The resonator element  20  can be excited by an excitation frequency of the alternating electrical field to exhibit at least one resonance frequency with a fundamental tone and n overtones. Resonance occurs if the excitation frequency is equal to a mechanical natural frequency of the resonator element  20 . To obtain the inverse piezoelectric effect a piezoelectric modulus e iλ =δD i /δS λ ) must be different from zero wherein D i  is the electric displacement field, S λ  refers to the mechanical strain tensor in matrix notation with the Latin indices i=1 to 3 and with the tensor indices λ=1 to 6. The resonator element  20  is oriented to oscillate as a thickness oscillator or as a longitudinal mode or expansion oscillator or as a flexural mode oscillator or as a face shear mode oscillator or as a thickness shear mode oscillator. Preferably, the resonator element  20  is operated as a thickness shear mode oscillator. Functioning as a thickness shear mode oscillator, the resonator element  20  is oriented in such a way that two end faces of the cylindrical resonator element  20  are displaced in opposite directions in the transverse plane BC. The end faces are located in the transverse plane BC. 
     The transverse expansion Q causes a frequency change Δf of the resonance frequency f of the resonator element  20 . The frequency change Δf is a function of the force F. Preferably, the frequency change Δf is proportional to the magnitude of the transverse expansion Q. Thus, the frequency change Δf=Q*K f *(f 2 *η/n*D) may be a function of a plurality of parameters. The frequency change Δf may, thus, also be proportional to a force sensitivity K f . The force sensitivity K f  itself is dependent on a force application angle area θ. D denotes a dimension parameter such as a diameter in the transverse plane BC of the piezoelectric material  21  while η represents a device parameter. Preferably, quartz is used as the piezoelectric material  21  of the resonator element  20  with the orthogonal axes x, y, z being the crystal axes and the axis z being the optical axis. The resonator element  20  is cut as a y-disc in which the y axis is the normal to a plane zx and having a piezoelectric modulus e 26  and an elastic modulus c 66 . The y axis of the y-disc is parallel to the longitudinal axis AA′. The optical axis z of the y-disc extends in the transverse plane BC. It is also possible to use an AT disc instead of a y-disc as the resonator element  20 . The AT disc is cut in a plane z′x having an angle of 35.25° between the z axis and an axis z′. Those skilled in the art knowing the present invention may use other known cuts in piezoelectric crystal material such as a CT cut, GT cut, BT cut, DT cut, FT cut, etc. 
     In the embodiments as shown in  FIGS. 2, 3, 6, 7 , the measurement transducer  1  comprises a single piezoelectric transducer element  10 . The resonator element  20  is arranged in the piezoelectric transducer element  10  as seen along the longitudinal axis AA′. The resonator element  20  is spatially arranged between a first piezoelectric material  11  and a second piezoelectric material  11 ′ of the piezoelectric transducer element  10 . 
     In the embodiments as shown in  FIGS. 4, 5, 8, 9 , the measurement transducer  1  comprises two piezoelectric transducer elements  10 ,  10 ′. The resonator element  20  is arranged between a first piezoelectric transducer element  10  and a second piezoelectric transducer element  10 ′ as seen along the longitudinal axis AA′. The resonator element  20  is arranged spatially between a first piezoelectric material  11  of the first piezoelectric transducer element  10  and a second piezoelectric material  11 ′ of the second piezoelectric transducer element  10 ′. Thus, the sensitivity for the direct piezoelectric effect of the measurement transducer  1  comprising two piezoelectric transducer elements  10 ,  10 ′ is higher, preferably twice as high, as compared to that of the measurement transducer  1  comprising only one piezoelectric transducer element  10 . 
     To introduce the force F into the piezoelectric transducer element  10 ,  10 ′ and the resonator element  20  in a substantially symmetrical manner the measurement transducer  1  has a symmetrical construction along the longitudinal axis AA′. In the embodiments as shown in  FIGS. 2 to 9 , the resonator element  20  is arranged between a first piezoelectric material  11  and a second piezoelectric material  11 ′ in the direction of the longitudinal axis AA′. Preferably, the first piezoelectric material  11  and the second piezoelectric material  11 ′ are identical parts which renders the production of the measurement transducer  1  cost-effective. In the embodiments as shown in  FIGS. 2, 3, 6, 7 , the second piezoelectric material  11 ′ is mechanically connected to the piezoelectric material  21  of the resonator element  20 . This mechanical connection is preferably achieved by means of material bonding such as welding, diffusion bonding, thermocompression bonding, soldering, etc. Such material bond can be made in a cost-effective manner and exhibits high long-term mechanical stability. 
     Resonator element  20  is cylindrical in shape along the longitudinal axis AA′ and is arranged between a hollow cylindrical first piezoelectric material  11  and a hollow cylindrical second piezoelectric material  11 ′. A first cavity  33  is formed within the hollow cylinder of the first piezoelectric material  11  and a second cavity  33 ′ is formed within the hollow cylinder of the second piezoelectric material  11 ′. Cavities  33 ,  33 ′ are positioned above and below the resonator element  20 . A spatial dimension of the cavities  33 ,  33 ′ along the longitudinal axis AA′ and in the transverse plane BC is greater than the amplitudes of the mechanical oscillations of the resonator element  20 . Thus, the resonator element  20  has sufficient space to oscillate in the cavities  33 ,  33 ′ with a high Q factor. The mechanical oscillations may occur axially along the longitudinal axis AA′. The mechanical oscillations may occur radially in the transverse plane BC. The mechanical oscillations may occur in the form of a thickness shear mode oscillation in the transverse plane BC. Finally, the mechanical oscillations may occur in the form of a combination of an axial and a radial oscillation or a combination of a radial oscillation and a thickness shear mode oscillation. 
     For measuring the force F electrical polarization charges of the piezoelectric material  11 ,  11 ′ of the piezoelectric transducer element  10 ,  10 ′ are received by electrodes  12 ,  12 ′,  13 ,  13 ′ schematically shown in  FIG. 5  for example and are transmitted as charge signals to an evaluation unit  50 . Electrodes  12 ,  12 ′,  13 ,  13 ′ are referred to as the charge-receiving electrodes  12 ,  12 ′,  13 ,  13 ′. For measuring the transverse expansion Q an alternating electrical field is applied as frequency signals to the piezoelectric material  21  of the resonator element  20  via electrodes  22 ,  23 . The electrodes  22 ,  23  are also referred to as frequency electrodes  22 ,  23 . Electrodes  12 ,  12 ′,  13 ,  13 ′,  22 ,  23  are thin layers of electrically conductive material such as pure metals, nickel alloys, cobalt alloys, iron alloys, etc. Electrodes  12 ,  12 ′,  13 ,  13 ′,  22 ,  23  are connected to the piezoelectric material  11 ,  11 ′,  21  mechanically and electrically. Preferably, the mechanical and electrical connection is achieved by material bonding such as welding, diffusion bonding, thermocompression bonding, soldering, etc. Such material bond can be made in a cost-effective manner and exhibits high long-term mechanical and electrical stability. 
     The charge-receiving electrodes  12 ,  12 ′,  13 ,  13 ′ are hollow cylindrical in shape such as annular rings and extend substantially completely over end faces of the piezoelectric transducer element  10 ,  10 ′. The frequency electrodes  22 ,  23  are cylindrical in shape and extend into an oscillation area over end faces of the resonator element  20 . The oscillation area is in a central region of the end faces of the resonator element  20 , and this region is close to the longitudinal axis AA′ that extends through the center of the end faces. Only narrow electrical leads to the frequency electrodes  22 ,  23  are arranged in a peripheral region of the end surfaces of the resonator element  20  and are spaced apart from the longitudinal axis AA′ in a radial direction. In the oscillation area of the resonator element  20 , the piezoelectric material  21  is excited by the electric currents supplied via the frequency electrodes  22 ,  23  to generate mechanical oscillations, and the resonator element  20  oscillates with a relatively higher amplitude than the amplitude of oscillations of the resonator element  20  in the more peripheral regions of the resonator element  20 . In the peripheral region of the resonator element  20  is the resonator contact area K 2 , K 2 ′ where the force F is introduced into the resonator element  20 . In the peripheral region of the resonator element  20 , the piezoelectric material  21  is not excited by the frequency electrodes  22 ,  23  to mechanically oscillate and the resonator element  20  oscillates at a relatively lower amplitude than the amplitude of oscillations of the resonator element  20  in the more central regions of the resonator element  20 , and thus a magnitude of the amplitude of oscillations of the resonator element  20  decreases with increasing radial distance from the longitudinal axis AA′. A radial dimension of the peripheral region is chosen so that the mechanical oscillations are not or only to a negligible extent dampened by a mechanical connection of the piezoelectric transducer element  10 ,  10 ′ and the resonator element  20 . 
     The first and third embodiments according to  FIG. 2, 3, 6, 7  comprise four electrodes  12 ,  13 ,  22 ,  23 , the second and fourth embodiments of  FIG. 4, 5, 8, 9  comprise six electrodes  12 ,  12 ′,  13 ,  13 ′,  22 ,  23 . In the first embodiment as shown in  FIGS. 2, 3 , a charge-receiving electrode  12  is mechanically and electrically connected both to the first piezoelectric material  11  of the piezoelectric transducer element  10  and to the piezoelectric material  21  of the resonator element  20 . The charge-receiving electrode  12  is also called the counter electrode  12 . A charge-receiving electrode  13  is arranged on the first piezoelectric material  11  of the piezoelectric transducer element  10 ; and it serves as the electrical ground and is also called the ground electrode  13 . The grounding element electrode  13  is electrically connected to a grounded housing  41  ( FIG. 20 ) of the measurement transducer  1 . In exactly one contacting zone Z close to the diagonal axis CC′ the counter electrode  12  and frequency electrodes  22 ,  23  are easily accessible for electrical and mechanical contacting with electrical conductors. The measurement transducer  1  requires only three electrical conductors for four electrodes  12 ,  13 ,  22 ,  23 . 
     In the second embodiment as shown in  FIGS. 4, 5 , two charge-receiving electrodes  12 ,  12 ′ are mechanically connected to the piezoelectric material  21  of the resonator element  20 . The two charge-receiving electrodes  12 ,  12 ′ are arranged on different end faces of the piezoelectric material  21  of the resonator element  20 . The two charge-receiving electrodes  12 ,  12 ′ are electrically connected to one another and serve as a common counter electrode  12 ,  12 ′. The first piezoelectric material  11  of the first piezoelectric transducer element  10  and the second piezoelectric material  11 ′ of the second piezoelectric transducer element  10 ′ are arranged with opposite polarization direction with respect to each other so that the two counter electrodes  12 ,  12 ′ receive electrical polarization charges of the same polarity. Two charge-receiving electrodes  13 ,  13 ′ are arranged on the piezoelectric material  11 ,  11 ′ of the piezoelectric transducer elements  10 ,  10 ′ and serve as electrical ground and are also called the ground electrodes  13 ,  13 ′. The two grounding element electrodes  13 ,  13 ′ are electrically connected to the grounded housing  41  ( FIG. 20 ) of the measurement transducer  1 . In exactly one contacting zone Z in the proximity of the diagonal axis CC′ the common counter electrode  12 ,  12 ′ and the frequency electrodes  22 ,  23  are easily accessible for electrical and mechanical contacting with electrical conductors. The measurement transducer  1  requires only three electrical conductors for six electrodes  12 ,  12 ′,  13 ,  13 ′,  22 ,  23 . 
     In the third embodiment as shown in  FIGS. 6, 7 , a charge-receiving electrode  13  is mechanically and electrically connected to both the first piezoelectric material  11  of the piezoelectric transducer element  10  and the piezoelectric material  21  of the resonator element  20 . Charge-receiving electrode  13  is electrically connected to a frequency electrode  23 . The two electrically connected electrodes  13 ,  23  serve as a common electrical ground and are also called the ground electrodes  13 ,  23 . A charge-receiving electrode  12  is arranged on the piezoelectric material  11 ,  11 ′ of the piezoelectric transducer element  10  and is also called the counter electrode  12 . In exactly one contacting zone Z in the proximity of the diagonal axis CC′ the ground electrodes  13 ,  23 , counter electrode  12  and frequency electrode  22  are easily accessible for electrical and mechanical contacting with electrical conductors. The measurement transducer  1  requires only three electrical conductors for four electrodes  12 ,  13 ,  22 ,  23 . 
     In the fourth embodiment as shown in  FIGS. 8, 9 , two charge-receiving electrodes  12 ′,  13  are mechanically connected to the piezoelectric material  21  of the resonator element  20 . The two charge-receiving electrodes  12 ′,  13  are arranged on different end faces of the piezoelectric material  21 . The two charge-receiving electrodes  12 ,  13 ′ are arranged on the piezoelectric material  11 ,  11 ′ of the piezoelectric transducer elements  10 ,  10 ′. A charge-receiving electrode  12 ′ that is mechanically connected to the piezoelectric material  21  of the resonator element  20  and a charge-receiving electrode  12  that is arranged on the first piezoelectric material  11  of the first piezoelectric transducer element  10  are electrically connected to each other and serve as a common counter electrode  12 ,  12 ′. The first piezoelectric material  11  of the first piezoelectric transducer element  10  and the second piezoelectric material  11 ′ of the second piezoelectric transducer element  10 ′ are arranged with the same direction of polarization so that the two counter electrodes  12 ,  12 ′ receive electrical polarization charges of the same polarity. A charge-receiving electrode  13  that is mechanically connected to the piezoelectric material  21  of the resonator element  20  and a charge-receiving electrode  13 ′ that is arranged on the second piezoelectric material  11 ′ of the second piezoelectric transducer element  10 ′ are electrically connected to each other and serve as an electrical ground and are also called the ground electrodes  13 ,  13 ′. The two grounding element electrodes  13 ,  13 ′ are electrically connected to the grounded housing  41  ( FIG. 20 ) of the measurement transducer  1 . In exactly one contacting zone Z in the proximity of the diagonal axis CC′ the ground electrode  13 , counter electrode  12 ′ and frequency electrodes  22 ,  23  are easily accessible for electrical and mechanical contacting with electrical conductors. The measurement transducer  1  requires only three electrical conductors for six electrodes  12 ,  12 ′,  13 ,  13 ′,  22 ,  23 . 
     Construction of the measurement transducer  1  comprising three electrical conductors for four or six electrodes  12 ,  12 ′,  13 ,  13 ′,  22  and  23  is space-saving and provides for cost-effective production of the measurement transducer  1 . The electrical and mechanical contacting of the electrodes  12 ,  12 ′,  13 ,  13 ′,  22 ,  23  with electrical conductors in exactly one contacting zone Z is achieved by means of material and/or friction bonding. The material bond is made by welding, diffusion welding, thermocompression, soldering, etc. while the friction bond is achieved by screw connection, clamp connection, etc. However, those skilled in the art knowing the present invention may also provide for electrical and mechanical contacting of the electrodes with electrical conductors in more than one contacting zone. 
     In the embodiments according to  FIGS. 6 to 9 , the measurement transducer  1  comprises an electrical insulation element  14 . The electrical insulation element  14  is hollow cylindrical in shape. The electrical insulation element  14  is made of electrically insulating and mechanically rigid material such as ceramics, Al 2 O 3  ceramics, sapphire, etc. The electrical insulation element  14  is mechanically connected to a counter electrode  12 . Preferably, the mechanical connection is made by means of material bonding such as welding, diffusion bonding, thermocompression bonding, soldering, etc. Such material bond can be made in a cost-effective manner and exhibits high long-term mechanical stability. 
     An amplification and evaluation of the charge signals of the piezoelectric material  11  of the piezoelectric transducer element  10 ,  10 ′ is performed in an evaluation unit  50 . For this purpose, as schematically shown in  FIGS. 3, 4, 7 and 9 , the charge-receiving electrodes  12 ,  12 ′,  13 ,  13 ′ of the piezoelectric transducer element  10 ,  10 ′ are electrically connected to an amplifier circuit of the evaluation unit  50 . The piezoelectric material  21  of the resonator element  20  is excited by frequency signals from the evaluation unit  50 . For this purpose, the frequency electrodes  22 ,  23  of the resonator element  20  are electrically connected to an oscillator circuit  51  of the evaluation unit  50 . The frequency change Δf is determined by the oscillator circuit  51 . From the detected frequency change Δf the evaluation unit  50  determines the transverse expansion Q. Preferably, the oscillator circuit  51  is a Colpitts circuit. Typical resonance frequencies are in the range of a few kHz up to several MHz while typical frequency changes Δf are in the range of 2 Hz/N to 100 Hz/N. 
       FIGS. 10 and 11  show the force sensitivity K f  as a function of the force application angle θ. This force sensitivity K f  applies to a resonator element  20  according to the embodiments as shown in  FIGS. 1 to 9  oscillating in the cavity  33 ,  33 ′. For a force application angle θ along the transverse axis BB′ the force sensitivity K f  is at a maximum and normalized to 100%. For a force application angle θ of 45° to the transverse axis BB′ the force sensitivity K f  is zero or 0%. For a force application angle θ along the diagonal axis CC′ the force sensitivity K f  is at a minimum or 60%. 
     Referring to  FIG. 10 , if a force application angle area Ω extends over 90° from the transverse axis BB′ to the diagonal axis CC′ a sum of the force sensitivities K f  results in a total force sensitivity K fG  of 10%. A force application angle area Ω is limited in  FIG. 11 . For a first force application angle area Ω 1  that covers 45° from the transverse axis BB′ to the diagonal axis CC′, then the sum of the force sensitivities K f  results in a first mean force sensitivity K fM1  of 50%. For a second force application angle area Ω 2  that covers 30° from the transverse axis BB′ to the diagonal axis CC′, then the sum of the force sensitivities K f  results in a second mean force sensitivity K fM2  of 75%. 
     Optimum adjustment of the force application angle area Ω can be achieved by means of at least one force application element  30 ,  30 ′. The force F acts on the force application element  30 ,  30 ′ and is transmitted by the force application element  30 ,  30 ′ into the piezoelectric transducer element  10 ,  10 ′ and the resonator element  20 .  FIGS. 12 and 13  show a perspective view of first and second embodiments of a hollow spherical segment-shaped force application element  30 ,  30 ′.  FIGS. 14 and 15  show a plan view of third and fourth embodiments of a hollow truncated cone-shaped force application element  30 ,  30 ′. The force application element  30 ,  30 ′ may also be hollow cylindrical, hollow truncated pyramidal in shape, etc. The force application element  30 ,  30 ′ consists of mechanically rigid material such as pure metals, nickel alloys, cobalt alloys, iron alloys, ceramics, etc. 
     The force application element  30 ,  30 ′ is a hollow body comprising a top surface  31 ,  31 ′, a lateral surface  32 ,  32 ′ and a cavity  33 ,  33 ′. The resonator element  20  is arranged in the cavity  33 ,  33 ′. The cavity  33 ,  33 ′ lies both in the force application element  30 ,  30 ′ and in the piezoelectric transducer element  10 ,  10 ′. The top surface  31 ,  31 ′ and lateral surface  32 ,  32 ′ are mechanically connected to each other and enclose the cavity  33 ,  33 ′. The force F acts along the longitudinal axis AA′ onto the top surface  31 ,  31 ′ and is transmitted from the top surface  31 ,  31 ′ into the lateral surface  32 ,  32 ′. The lateral surface  32 ,  32 ′ comprises at least one recessed area  34 ,  34 ′ which extends into the cavity  33 ,  33 ′ and prevents the transmission of the force F in the lateral surface  32 ,  32 ′. In addition, the lateral surface  32 ,  32 ′ comprises at least one non-recessed area  35 ,  35 ′ and it is only this non-recessed area  35 ,  35 ′ that transmits the force F. The hollow body comprises a base  36 ,  36 ′ at an opposite end from the top surface  31 ,  31 ′. The top surface  31 ,  31 ′ and base  36 ,  36 ′ are mechanically connected to each other by the lateral surface  32 ,  32 ′. The top surface  31 ,  31 ′, lateral surface  32 ,  32 ′ and base  36 ,  36 ′ are preferably made of one piece. The top surface  31 ,  31 ′ and base  36 ,  36 ′ are parallel to each other in the transverse plane BC. As schematically shown in  FIGS. 14 and 15  for example, the non-recessed area  35 ,  35 ′ of the lateral surface  32 ,  32 ′ forms at least one force transmission angle area Ω, Ω 1 , Ω 2  with the base  36 ,  36 ′ in the transverse plane BC. 
     The recessed area  34 ,  34 ′ is preferably formed like a window. In the embodiment according to  FIG. 12 , the recessed area  34 ,  34 ′ is arranged in a lower region of the lateral surface  32 ,  32 ′ as seen along the longitudinal axis AA′. In the embodiment according to  FIG. 13 , the recessed area  34 ,  34 ′ is arranged in a central region of the lateral surface  32 ,  32 ′ as seen along the longitudinal axis AA′. In the two embodiments according to  FIGS. 14 and 15 , the recessed area  34 ,  34 ′ corresponds to a large portion of the lateral surface  32 ,  32 ′. In the third embodiment according to  FIG. 14 , the force application element  30 ,  30 ′ comprises four first force application angle areas Ω 1  each covering 45°. In the fourth embodiment according to  FIG. 15 , the force application element  30 ,  30 ′ comprises four second force application angle areas Ω 2  each covering 30°. In both cases two first or second force application angle areas Ω 1 , Ω 2  are delimited by a recessed area  34 ,  34 ′ so that the force F is transmitted only by the first or second force application angle areas Ω 1 , Ω 2 . Preferably, each force application element  30 ,  30 ′ comprises two recessed areas  34 ,  34 ′ that are rotated by an angle of 180° with respect to each other as seen along the longitudinal axis AA′. 
     The first or second force application angle area Ω 1 , Ω 2  is limited in certain areas so that, in total, a mean force sensitivity K fM1 , K fM2  is close to the maximum of 100% of the force sensitivity K f , The smaller the first or second force application angle area Ω 1 , Ω 2 , then the greater is a local pressure in the piezoelectric transducer element  10 ,  10 ′ and in the resonator element  20 . The size of the first or second force application angle area Ω 1 , Ω 2  and, thus, the magnitude of the local pressure in the piezoelectric transducer element  10 ,  10 ′ and in the resonator element  20  may be selected as a function of the compressive strength of the piezoelectric transducer element  10 ,  10 ′ and the resonator element  20 . The more pressure-resistant the piezoelectric transducer element  10 ,  10 ′ and the resonator element  20  are, then the smaller the first or second force application angle area Ω 1 , Ω 2  may be selected, and the higher will be the mean force sensitivity K fM1 , K fM2 . 
       FIG. 16  shows a cross-section of the measurement transducer  1  according to  FIGS. 2 and 3  comprising one force application element  30  according to  FIG. 14 or 15 .  FIG. 17  shows a cross-section of the measurement transducer  1  according to  FIGS. 8 and 9  comprising two force application elements  30 ,  30  according to  FIG. 14 or 15 . Preferably, the two force application elements  30 ,  30 ′ are identical parts, said identical parts transmitting the force F symmetrically to the piezoelectric transducer element  10 ,  10 ′ and the resonator element  20 . 
     The piezoelectric transducer element  10 ,  10 ′ and the resonator element  20  are mechanically connected to the lateral surface  32 ,  32 ′, preferably by an indirect mechanical connection. Thus, in the embodiments as shown in  FIGS. 16 and 17 , the piezoelectric transducer element  10 ,  10 ′ is mechanically connected to the force application element  30 ,  30 ′ indirectly by the transducer contact area K 1 , K 1 ′ and the base  36 ,  36 ′. Furthermore, in the embodiments as shown in  FIGS. 16 and 17  there is another indirect mechanical connection via the resonator contact area K 2 , K 2 ′ of the resonator element  20  to the piezoelectric transducer element  10 ,  10 ′ which is in turn mechanically connected to the force application element  30 ,  30 ′ by the transducer contact area K 1 , K 1 ′ and the base  36 ,  36 ′. Preferably, the mechanical connection is achieved by means of material bonding such as welding, diffusion bonding, thermocompression bonding, soldering, etc. 
     Referring to  FIG. 16 , the force F acts both on a first force application element  30  and on the piezoelectric transducer element  10 . In this case, the force F acts on the piezoelectric transducer element  10  on the one hand indirectly via the first force application element  30 , while the force F acts on the piezoelectric transducer element  10  on the other hand directly via a transducer contact area K 1 ′. Referring to  FIG. 17 , the force F acts on the first force application element  30  and is transmitted via a first base  36  and the transducer contact area K 1  into the first piezoelectric transducer element  10 ; and the force F acts on a second force application element  30 ′ and is transmitted via a second base  36 ′ and the transducer contact area K 1 ′ into the second piezoelectric transducer element  10 ′. In this case shown in  FIG. 17 , the force F acts on both piezoelectric transducer elements  10 ,  10 ′ indirectly over force application elements  30 ,  30 ′. In this way, the force F is transmitted into the resonator element  20  by the piezoelectric transducer element  10 ,  10 ′. 
     Top surface  31 ,  31 ′ and base  36 ,  36 ′ are parallel to each other in the transverse plane BC. The longitudinal axis AA′ is perpendicular to the transverse plane BC. The top surface  31 ,  31 ′ is preferably circular having a constant outer radius or maximum distance B 1  of the top surface  31 ,  31 ′ to the longitudinal axis AA′. Base  36 ,  36 ′ is preferably annular having a constant outer radius or maximum distance B 2  of the base  36 ,  36 ′ to the longitudinal axis AA′. The maximum distance B 2  is greater than the maximum distance B 1 . Thus, in the lateral surface  32 ,  32 ′ the force F is transmitted away from the longitudinal axis AA′. Transmission of the force F occurs in a curved section of the non-recessed area  35 ,  35 ′ of the lateral surface  32 ,  32 ′. Transmission of the force F occurs from the top surface  31 ,  31  towards the resonator contact area K 2 , K 2 ′. A 2  designates a length along the longitudinal axis AA′ from the top surface  31 ,  31 ′ to the resonator contact area K 2 , K 2 ′. The force F acting on the top surface  31 ,  31 ′ causes an increased transverse expansion Q in the resonator element  20 . The greater a difference between the maximum distance B 2  and the maximum distance B 1 , then the more the transverse expansion Q increases. Because the transverse expansion Q is increased, also the frequency change that is generated by the force F increases. When the static force is measured using the inverse piezoelectric effect, thus, the high sensitivity which is a market requirement is achieved. 
     The fact that the force sensitivity K f  is a function of the force application angle θ also has an impact on a magnitude of a coefficient of linear thermal expansion α of the piezoelectric material  11 ,  11 ′,  21 . As shown in  FIG. 18 , at maximum force sensitivity K f  with a force application angle θ along the transverse axis BB′ the coefficient of linear thermal expansion α is at a maximum and is normalized to 100%. When the force sensitivity K f  is at a minimum with a force application angle θ along the diagonal axis CC′, then the coefficient of linear thermal expansion α of the piezoelectric material  11 ,  11 ′,  21  is at a minimum and is normalized to 55%. According to  FIG. 18 , in the whole force application angle area Ω of 90° between the transverse axis BB′ and the diagonal axis CC′ transverse axis BB′ towards the diagonal axis CC′ results, in total, a total coefficient of linear thermal expansion (K f *α) G . The total coefficient of linear thermal expansion (K f *α) G  is the product of the force sensitivity K f  and the coefficient of linear thermal expansion α summed up over all force application angles θ. However, if the force application angle area Ω is limited, in total, a resulting mean coefficient of linear expansion (K f *α) M1 , (K f *α) M2  of the piezoelectric material  11 ,  11 ′,  21  will vary depending on the size of a first or second force application angle area Ω 1 , Ω 2 . According to  FIG. 19 , for a first force application angle area Ω 1  covering 45° from the transverse axis BB′ towards the diagonal axis CC′ results in total a first mean coefficient of linear thermal expansion (K f *α) M1 . For a second force application angle area Ω 2  covering 30° from the transverse axis BB′ towards the diagonal axis CC′ results in total a second mean thermal coefficient of linear thermal expansion (K f *α) M2 . 
     In operation the measurement transducer  1  is permanently exposed to temperatures of 200° C. and higher. In this case, the linear expansions of the piezoelectric material  11 ,  11 ′,  21  and the force application element  30 ,  30 ′ may be different. The force application element  30 ,  30 ′ has a coefficient of linear thermal expansion α 30 . For comparing the coefficient of linear thermal expansion an to the mean coefficients of linear thermal expansion (K f *α) M1 , (K f *α) M2 , the mean coefficients of linear thermal expansion (K f *α) M1 , (K f *α) M2  according to  FIG. 19  are weighted with the mean force sensitivities K fM1 , K fM2  according to  FIG. 11 . For the first force application angle area Ω 1 , the piezoelectric material  11 ,  11 ′,  21  has a first weighted coefficient of linear thermal expansion α M1 =(K f *α) M1 /K fM1 . For the second force application angle area Ω 2 , the piezoelectric material  11 ,  11 ′,  21  has a second weighted coefficient of linear thermal expansion α M2 =(K f *α) 2 /K fM2 . To keep the differential linear expansions of the piezoelectric material  11 ,  11 ′,  21  and the force application element  30 ,  30 ′ as small as possible, the first or second weighted coefficient of linear thermal expansion α M1 , α M2  of the piezoelectric material  11 ,  11 ′,  21  will be substantially equal to the coefficient of linear thermal expansion α 30  of the force application element  30 ,  30 ′. Preferably, quartz is used as the piezoelectric material  11 ,  11 ′,  21  which has a maximum coefficient of linear thermal expansion α of 13.7*10 −6  K −1  that is normalized to 100% in  FIGS. 18 and 19 , and a minimum coefficient of linear thermal expansion α of 7.5*10 −6  K −1  corresponding to the normalization to 55% in  FIGS. 18 and 19 . A force application element  30 ,  30 ′ made of steel has a coefficient of linear thermal expansion α 30  of 13.0*10 −6  K −1 . For the first force application angle area Ω 1 =45°, the piezoelectric material  11 ,  11 ′,  21  has a first weighted coefficient of linear thermal expansion α M1 =12*10 −6  K −1  while for the second force application angle area Ω 1 =30°, the piezoelectric material  11 ,  11 ′,  21  has a second weighted coefficient of linear thermal expansion α M2 =13*10 −6  K −1  which is substantially equal to the coefficient of linear thermal expansion α 30  of the force application element  30 ,  30 ′. 
     The resonator element  20  measures a temperature T by means of the inverse piezoelectric effect. The resonance frequency of the piezoelectric material  21  of the resonator element  20  is dependent on a temperature T according to the following equation: f(T)=½d(T)*(e iλ (T)/ρ(T)) 1/2  wherein the piezoelectric modulus e iλ , a material thickness d of the piezoelectric material  21  and a density ρ are dependent on the temperature T. The parameter Δf T  denotes a temperature-dependent frequency change. The frequency change Δf T  is detected by means of the oscillator circuit of the evaluation unit  50 . The evaluation unit  50  determines the temperature T from the detected temperature-dependent frequency change Δf T . Preferably, quartz is used as the piezoelectric material  21  of the resonator element  20  having resonance frequencies in the range of a few kHz to several MHz and a temperature-dependent frequency change Δf T  in the range of 20 Hz/K to 200 Hz/K. 
       FIG. 20  shows the mounted ready-to-use measurement transducer  1  according to  FIGS. 1 to 9  comprising a force application element  30 ,  30 ′ according to  FIGS. 14 to 17 . The measurement transducer  1  is mounted in a housing  41  under vacuum. The housing  41  is cylindrical. At least one force application element  30 ,  30 ′ protrudes from the housing  41  at an end face of the housing  41  and has a top surface  31 ,  31 ′. Preferably, two force application elements  30 ,  30 ′ protrude from the housing  41  at two end faces of the housing  41 , each having a top surface  31 ,  31 ′. Only one end face is shown in the representation in  FIG. 20 . Each end face comprises a hollow cylindrical cover  42 . The housing  41  and cover  42  protect the measurement transducer  1  from harmful external conditions such as dirt, moisture, mechanical impacts, etc. The housing  41  and cover  42  are made of mechanically resistant material such as pure metals, nickel alloys, cobalt alloys, iron alloys, etc. The force application element  30 ,  30 ′ is mechanically connected to the housing  41  by the cover  42 . The cover  42  is mechanically connected to the housing  41  at a radially outer edge with respect to the longitudinal axis AA′. Preferably, the mechanical connection is performed by material bonding such as welding, diffusion bonding, thermocompression bonding, soldering, etc. Cover  42  is mechanically connected to the force application elements  30 ,  30 ′ via a radially inner edge with respect to the longitudinal axis AA′. Preferably, the mechanical connection is made by material bonding such as welding, diffusion bonding, thermocompression bonding, soldering, etc. The cover  42  is thin, the thickness of the cover  42  being 0.3 mm, preferably 0.2 mm, preferably 0.1 mm, to achieve a small force shunt towards the housing  41 . The thinner the cover  42 , the smaller is the force shunt towards the housing  41 . 
     Charge signals of the charge-receiving electrodes  12 ,  12 ′,  13  and  13 ′ are transmitted by electrical conductors to an electrical connection  43  to the evaluation unit  50 . Frequency signals from the evaluation unit  50  are transmitted via the electrical connection  43  by electrical conductors to the frequency electrodes  22 ,  23 . The evaluation unit  50  may be electrically connected to the electrical connection  43  by a plug and cable (not shown). The electrical connection  43  is arranged in an opening of a side wall of the housing  41 . The electrical connection  43  is mechanically connected to the housing  41 . Preferably, the mechanical connection is made by means of material bonding such as welding, diffusion bonding, thermocompression bonding, soldering, etc. 
     Electrical and mechanical contacting of the electrodes  12 ,  12 ′,  13 ,  13 ′,  22  and  23  and the electrical conductors is easily accomplished in exactly one contacting zone Z. For this purpose, the measurement transducer  1  according to  FIGS. 1 to 9  is electrically and mechanically contacted with stripped ends of the electrical conductors before it is inserted in the housing  41 . Afterwards, the measurement transducer  1  is inserted in the housing  41  and the electrical conductors that are electrically and mechanically contacted with the measurement transducer  1  are led outside of the housing through the opening in the side wall of the housing  41  and are electrically and mechanically contacted to the electrical connection  43  by a material and/or frictional connection. The material bond is achieved by welding, diffusion welding, thermocompression, soldering, etc. The frictional connection is achieved by a screw connection, clamp connection, etc. 
     The mechanical connections between the housing  41  and the covers  42 , between the covers  42  and the force application elements  30 ,  30 ′ as well as between the housing  41  and the electrical connection  43  are pressure-tight. 
     LIST OF REFERENCE NUMERALS 
     
         
         
           
             A 2  length 
             B 1 , b 2  maximum distance 
             AA % BB′ CC′ axis 
             BC transverse plane 
             K f  force sensitivity 
             K fG  total force sensitivity 
             K fM1 , K fM2  mean force sensitivity 
             (K f *α) G  total coefficient of linear thermal expansion 
             (K f *α) M1 ; (K f *α) M2  mean coefficient of linear thermal expansion 
             F force 
             Q transverse expansion 
             K 1 , K 1 ′ transducer contact area 
             K 2 , K 2 ′ resonator contact area 
             Z contacting zone 
             α coefficient of linear expansion 
             θ force application angle 
             Ω, Ω 1 , Ω 2  force application angle area 
               1  measurement transducer 
               10 ,  10 ′ piezoelectric transducer element 
               11 ,  11 ′,  21  piezoelectric material 
               12 ,  12 ′,  13 ,  13 ′ charge-receiving electrode 
               14  electrical insulator element 
               20  resonator element 
               22 ,  23  frequency electrode 
               30 ,  30 ′ force application element 
               31 ,  31 ′ top surface 
               32 ,  32 ′ lateral surface 
               33 ,  33 ′ cavity 
               34 ,  34 ′ recessed area 
               35 ,  35 ′ non-recessed area 
               36 ,  36 ′ base 
               41  housing 
               42  cover 
               43  electrical connection