Method and system for measuring a variation in the dimension of a mechanical part

Abstract

Method for measuring a variation in dimension (L) of a mechanical part (V) along a so-called longitudinal direction (x), comprising the steps of: a) providing at least a first (TP1) and a second (TP2) piezoelectric transducer arranged at two different positions (ELV, TV) along said longitudinal direction and acoustically coupled via said mechanical part; b) carrying out, at different times, a plurality of electrical measurements to determine at least one value of an electrical parameter dependent on a diagonal term (Z12) of an impedance matrix of an electrical quadrupole modeling the assembly formed by the first and second acoustically coupled piezoelectric transducers; etc.) deducing from the results of said electrical measurements said variation in dimension of the mechanical part. System for implementing this method Figure for the abstract: Fig. 4

Description

Title of the invention: Method and system for measuring a variation in the dimension of a mechanical part

[0001] The invention lies in the field of non-destructive ultrasonic testing. It applies in particular, but not exclusively, to measuring the tightening of screwed fastening systems during the industrial assembly of parts.

[0002] It is known to use acoustic techniques to precisely and dynamically measure the tightening of screwed fastening systems during the industrial assembly of parts by an acoustic method.

[0003] As illustrated in [Fig.l], this can be achieved by attaching a piezoelectric transducer TP to one end ELV of a fixing means (screw) V and exciting it with a sinusoidal voltage. The transducer generates, in the body CV of the fixing means, standing acoustic waves OS which are established in the structure as a function of its length. In turn, these acoustic waves modify the impedance of the piezoelectric transducer TP, which can be measured at the same time as the latter is excited. If the phase and the modulus of the complex impedance of the transducer are represented as a function of frequency, peaks can be observed corresponding to the acoustic resonance frequencies of the structure.

[0004] As the fastening system is tensioned (for example, by screwing a nut E), its length changes, and with it its resonant frequencies. A frequency shift of the transducer impedance peaks is then observed. This is illustrated in [Fig.2], in which 4 impedance curves - corresponding to tensions of 0.6 kN, 12 kN and 18 kN - are superimposed. [Fig.3] shows that the relationship between the applied tension and the frequency shift ôf of the impedance peaks is substantially linear, with an error of the order of 1% (the two curves in the lower part correspond to the deviation from a linear relationship during tensioning and its release).

[0005] The tightening of the nut can be controlled and monitored using this mechanical tension measurement, so as to control the pre-tension more precisely than by using a simple mechanical measurement of the applied tightening torque.

[0006] This technique, known in the English language literature as “Impedence Frequency Shift” (IFS) is described for example in (Heyman 1977), (Smith 1980), (Joshi 1984), (Shao 2016) and (Dreisbach 2023).

[0007] As can be seen, however, the impedance peaks are quite wide relative to their spacing, making it difficult to accurately determine their spacing and, therefore, the variation in length of the fastener. In practice, it is necessary to use frequency analysis which requires acquisition over several periods - therefore with a wide spectral band and a long acquisition time.

[0008] A technique competing with TIFS is that based on measuring the time-of-flight (TOF) of an acoustic wave. As in the case of IFS, a piezoelectric transducer is attached to the structure and excited to generate acoustic waves. Unlike in the case of IFS, however, the transducer is excited by a voltage pulse generating a time-limited acoustic wave. The wave propagates in the structure by making a round trip, before returning to the transducer. The transducer is then used as a sensor by monitoring its voltage which varies when the acoustic wave returns. The time between transmission and reception depends on the speed of the wave (obtained by prior calibration) and the length of the structure. TOF is a very well-known and widely used technique in the state of the art.However, it has a number of drawbacks, including the need to use fast electronics to generate short pulses, and voltage pulses of the order of a few hundred volts (compared to volts in the case of IFS).

[0009] The invention aims to overcome at least in part the aforementioned drawbacks of the prior art. More particularly, it aims to improve the precision and sensitivity of the IFS technique without significantly complicating its implementation.

[0010] According to the invention, this object is achieved by the joint use of two piezoelectric transducers acoustically coupled across the mechanical part whose length is to be measured. The assembly constituted by the two acoustically coupled transducers can be modeled by an electrical quadrupole, characterized by an impedance matrix Z. The measurement of the diagonal element Zn of this matrix (or of another electrical parameter proportional to this element) as a function of the frequency makes it possible to estimate a variation in length of the mechanical part, as in the standard IFS technique. The advantage provided by the invention lies in the fact that, as will be shown later, the impedance peaks are much more pronounced and narrow than in the conventional case in which a single transducer is used.

[0011] An object of the invention is therefore a method for measuring a variation in dimension of a mechanical part along a so-called longitudinal direction, comprising the steps consisting of: a) providing at least a first and a second piezoelectric transducer arranged at two different positions along said longitudinal direction and acoustically coupled via said mechanical part; (b) performing, at different times, a plurality of electrical measurements to determine terminating at least one value of an electrical parameter dependent on a diagonal term of an impedance matrix of an electrical quadrupole modeling the assembly formed by the first and second acoustically coupled piezoelectric transducers; and c) deducing from the results of said electrical measurements said variation in dimension of the mechanical part.

[0012] According to particular embodiments of such a method:

[0013] - Each said electrical measurement of step b) may comprise the sub-steps consisting of: bl) applying to said first piezoelectric transducer a first electrical excitation signal to generate acoustic waves propagating in the mechanical part towards the second transducer, while keeping the second piezoelectric transducer in open circuit, and at the same time measuring the input impedance of said first piezoelectric transducer; b2) applying to said first piezoelectric transducer a second electrical excitation signal to generate acoustic waves propagating in the mechanical part towards the second transducer, while keeping the second piezoelectric transducer short-circuited, and at the same time measuring the input impedance of said first piezoelectric transducer; b3) calculating the value of said electrical parameter from the input impedances thus measured; the order of sub-steps bl) and b2) may be reversed.

[0014] - Said electrical parameter can be the difference of the phases of the impedances input values ​​measured during sub-steps bl) and b2).

[0015] - The second piezoelectric transducer may comprise two electrical terminals connected together by a pair of head-to-tail diodes in parallel, the first excitation signal being sufficiently weak so that the generated acoustic waves induce at the terminals of said second piezoelectric transducer a voltage lower than a threshold of said diodes, which can then be likened to an open circuit, and the second excitation signal being sufficiently strong so that the generated acoustic waves induce at the terminals of said second piezoelectric transducer a voltage higher than a threshold of said diodes, which can then be likened to a short circuit.

[0016] - Step b) may comprise determining the value of said parameter in function of frequency.

[0017] - Step c) may comprise identifying peaks of the value of said parameter electrical as a function of frequency, the variation in dimension of the mechanical part being deduced from a variation in position of said peaks.

[0018] - The first and second piezoelectric transducers may be arranged in pairs opposite ends, along said longitudinal direction, of the mechanical part.

[0019] Another object of the invention is the use of such a method for measuring the tightening of a screw.

[0020] Yet another object of the invention is a system for measuring a variation in dimension of a mechanical part along a so-called longitudinal direction, comprising: - a first and a second piezoelectric transducer, adapted to be fixed at two different positions along said longitudinal direction and acoustically coupled via said mechanical part; - an electronic system configured to determine at least one value of an electrical parameter dependent on a diagonal term of an impedance matrix of an electrical quadrupole modeling the assembly formed by the first and second acoustically coupled piezoelectric transducers; and to deduce from a variation of said at least one value of said parameter said variation in dimension of the mechanical part.

[0021] According to particular embodiments:

[0022] Said electronic system may comprise: - a first electronic device configured to apply electrical excitation signals to said first piezoelectric transducer and at the same time measure its input impedance; - a second electronic device configured to maintain said second piezoelectric transducer successively in open circuit and in short circuit; and - a processor configured to control at least said first electronic device so as to: - applying to said first piezoelectric transducer a first electrical excitation signal to generate acoustic waves propagating in the mechanical part towards the second transducer, while keeping the second piezoelectric transducer in open circuit, and at the same time measuring the input impedance of said first piezoelectric transducer; - applying to said first piezoelectric transducer a second electrical excitation signal to generate acoustic waves propagating in the mechanical part towards the second transducer, while keeping the second piezoelectric transducer short-circuited, and at the same time measuring the input impedance of said first piezoelectric transducer; - calculate the value of said electrical parameter from the input impedances thus measured.

[0023] - The electronic device can be integrated into a screw tightening system, the first and second piezoelectric transducers being adapted to be fixed to a said screw.

[0024] Other characteristics, details and advantages of the invention will emerge on reading the description given with reference to the appended drawings given by way of example and in which:

[0025] [Fig. 1], already described, illustrates a technique for measuring the length of a mechanical part (screw) by impedance phase shift according to the prior art;

[0026] [Fig.2], already described, is a graph illustrating the impedance phase shift at the base of the technique of [Fig.l];

[0027] [Fig.3], already described, is a graph illustrating the quasi-linear relationship between impedance and voltage phase shift in the technique of [Fig.l];

[0028] [Fig.4] is the functional diagram of a measuring system according to a rea ​​mode lization of the invention;

[0029] [Fig.5] illustrates the modeling of a system of two piezoelectric transducers acoustically coupled by an electric quadrupole;

[0030] [Fig.6], a graph illustrating the variation as a function of the frequency of the element of Zn matrix;

[0031] [Fig.7], [Fig.8], [Fig.9], [Fig.10] and [Fig.11], graphs illustrating the variation in function of the frequency of different parameters function of Z[2 which can be used in the implementation of the invention according to different embodiments;

[0032] [Fig. 12] and [Fig. 13] are detailed views of two variants of the system of [Fig.4]

[0033] In the diagram of [Fig.4] the reference V designates a screw, extending into a longitudinal direction x and having, in this direction, a length L. The screw V has at one end a TV head which can cooperate with a tightening tool OS2; the opposite end ELV (free end), threaded, is inserted into a nut E, which can also cooperate with a tightening tool OS1. Two mechanical components Cl and C2 are crossed by the screw body CV and are caught between the TV head and the nut E. By rotating the nut with the tightening tool OS1 while keeping the TV head fixed with the tool OS2, or vice versa, the body of the screw is put under tension, producing a slight elongation of the latter.

[0034] To measure this elongation, the screw V is equipped with two piezoelectric transducers TPI and TP2, arranged respectively at its free end ELV and its head TV.

[0035] The clamping tools OS1 and OS2, which together form a clamping system, are adapted to interact with a respective transducer via electrical contacts. More particularly, in the embodiment of [Fig.4], the first clamping tool OS1 comprises a first electronic device AE1 of the network vector analyzer type, making it possible to excite the first piezo transducer-

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042] electric transducer TPI with a sinusoidal electrical signal of variable frequency, and at the same time measure its complex impedance. The second tightening tool OS2 in turn comprises a second electronic device AE1 for exciting the first piezoelectric transducer TPI with a sinusoidal electrical signal of variable frequency and / or at the same time measuring its complex impedance, or applying a variable load to it (for example, alternately a short circuit and an open circuit). A processor P, for example integrated in the first tightening tool, controls the two electronic devices and receives the impedance measurements made by them. As illustrated in [Fig.5], the assembly consisting of the two transducers acoustically coupled by the body of the screw V can be modeled by an electric quadrupole, characterized by an impedance matrix Z, characterized by four complex elements functions of the frequency: Zn(f), Zi2(f)= Z2i(f), Z22(f). By designating by Vin the voltage applied to the terminals of the transducer TPI, by Iin the current flowing through said transducer, by Vout the voltage which appears at the terminals of the transducer TP2 and by Lut the current flowing through the latter, we have: ^in 1 lAn 4" Z j ^out ( 1 ) . have [ zAn have As mentioned above, one idea behind the invention is to follow the variation of at least one parameter P as a function of Z[2 to deduce a measurement of the variation in the dimension of the mechanical part. This parameter P can be observed at a fixed frequency or on a frequency spectrum. Similarly, the frequency corresponding to a fixed value of this parameter P can be followed over time. Monitoring a P parameter as a function of Z12, instead of the impedance of a single transducer, has several advantages. Such a parameter can indeed present larger variations in terms of modulus and phase (which varies from -180° to +180°) and, in some cases, narrow peaks or abrupt variations, easily identifiable. In addition, Zn is mainly related to the direct path between the two transducers, which makes it possible to avoid standing waves that could be established with other surfaces in the case of a single transducer. [Fig.6], given for reference, illustrates the evolution of the modulus (upper panel) and the phase (lower panel) for three screw tension values: 0 kN (rest), 10 kN and 20 kN. [Fig.7] illustrates the evolution, under the same conditions, of the Zi2 parameter. We note the greater amplitude of relative evolution of the module, and the fact that the phase varies between -180° and +180°, compared to a variation of approximately 60° for that of Zn. [Fig.8] illustrates the evolution, under the same conditions, of the real part of Z[2. We observe the presence of narrow peaks, whose displacement with voltage can be followed by maximum detection algorithms.

[0043] [Fig.9] illustrates the evolution, under the same conditions, of the imaginary part of Zi2. This parameter remains close to zero over a large part of the frequency range of interest, but presents abrupt variations in correspondence with the resonance frequencies. These variations can be detected by signal reversal algorithms.

[0044] [Fig. 10] illustrates the evolution, under the same conditions, of the Zn phase. We have "z>2z"

[0045]

[0046]

[0047]

[0048]

[0049]

[0050]

[0051]

[0052]

[0053]

[0054]

[0055]

[0056]

[0057] notices significant variations, which can be tracked by zero-crossing algorithms, for example. A particularly interesting embodiment is one in which the excitation and the impedance measurement are carried out on one side only, for example that of TPI. In this case, the second electronic device is limited to keeping the second transducer TP2 alternately short-circuited and open-circuited. When AP2 keeps transducer TP2 short-circuited, we have Vout=0. Equation (1) therefore becomes in ~ ^11 kn Zi2IoM ishn "F => ^out = (2) with >- _ zî2 . The input impedance measured under short-circuit conditions y cc — v'm | is therefore worth ~ ZL, w » V 7,, / = 7,,(10).(3) When AP2 keeps transducer TP2 in open circuit, we have Lut=0. The equation (1) therefore becomes [ Wine = ZuIin (4) [ / ,„, = Z,2 / ,„ The input impedance measured under open circuit conditions z.eo: is therefore simply worth 7,, / = 7,,(5) By calculating the difference between the impedance values ​​we find K y _ yy _ y yZ_ y ^12 _ ^12 • (6) '-^in “ ^in ~ 1ZUZ„ “ Z,, We also define |a|z,,,|| = ||z,-"I-|z,-, / ||(7)

[0058] and

[0059] A arg (Zin) = arg (Z^°) - arg(Z^) (8)

[0060] [Fig. 11] shows graphs of | A|Z,„| | (upper part) and | A arg (Zin) | (lower part) as a function of frequency. We can see that the peaks - which correspond to the resonance frequencies allowing the establishment of standing acoustic waves in the screw body - are much narrower than in the cases of [Fig.2] and [Fig.6]; their position can therefore be determined much more precisely. More specifically, in [Fig. 11] the continuous line curve corresponds to a screw body of length L0=l cm at rest and the dotted curve to the same screw body stretched by 0.25% (L=l.0036 cm).

[0061] We note A f the frequency periodicity of these resonance peaks and <5 / the variation of their position following a variation in the length of the screw.

[0062] The wavelength of an acoustic wave of frequency / is given by a = |, with c the speed of the wave. The condition for obtaining the standing wave is given by: nX = 2L, where L is the length of the screw (the half-wavelength and the length are multiples). This gives for the resonance frequencies f = with 11 integer. The [Fig.l 1] shows standing waves for n=18 and n=21, and identifies the corresponding peaks of impedance Zin. The distance between the peaks is therefore given by Af = - — •

[0063] The positions of the peaks f and their distance Af vary with the elongation AL = e L.

[0064] The speed of sound in a homogeneous medium is given, as a first approximation (neglecting the acousto-elastic effect) by: „ _ te 1-v , where E is the modulus L “ y P (l+vXl-2v) Young's ratio of the screw material, v its Poisson's ratio and P its density, which varies also as a function of the elongation (while Young's modulus and Poisson's ratio are considered constant as a first approximation). The variation of volume and therefore of density with the deformation is given by: AV _ _ Ap _ | 2V ) £ where Vo and Pq are, respectively, the volume and the density at rest. We find that the resonance frequencies at rest are given by — 2.31 = nAf0' °where n is an integer index, c0 the speed of sound in the screw at rest and Zf the spacing between the resonance peaks at rest. Under stretched conditions, the frequencies of resonance varies such as:

[0065] £ you

[0066] for a steel with v=0.29. Finally, the variation of the frequency peaks is given by :

[0067] ® ±£*E(8) you

[0068] And

[0069] Af=(l-^E)Af0-(9)

[0070] We find a linear relationship between the variations in position of the impedance peaks and the variation in length of the screw (the latter itself being directly proportional to the applied voltage, as long as the linear elasticity limit is not exceeded).

[0071] [Fig. 12] illustrates a particular embodiment in which the second electronic device AE2 is simply constituted by two diodes DI and D2 mounted head to tail in parallel. When the voltage Vout generated by the transducer TP2 is lower than the threshold voltage of the diodes, this assembly behaves like an open circuit, while when Vout is much higher than the threshold it behaves like a short circuit. It is therefore possible to carry out both types of measurements - in open circuit and in short circuit - simply by varying the amplitude of the excitation signal of the first transducer TPI. The advantage of this embodiment is that the second electronic device AE2 is entirely passive, and only the tightening tool OS1 needs to be equipped with electrical contacts and high-frequency electronics. This device AE2 can even be encapsulated with the transducer TP2 and placed permanently, in which case it is not necessary to have access to the head of the screw.

[0072] [Fig. 13] illustrates yet another embodiment in which a third piezoelectric transducer TP3 is arranged on a collar of the screw head TV. This makes it possible, for example, to determine a variation in length between TPI and TP2 (mainly linked to the deformation of the screw and therefore to the force applied) and a variation in length between TP2 and TP3 (mainly linked to a variation in temperature of the screw).

[0073] The invention has been described with reference to particular embodiments, but variants are possible. For example:

[0074] - Performing measurements by exciting only TPI and changing the state of TP2 (open circuit / short circuit) is an advantageous choice, but not unambiguous. It would also be possible, for example, to excite TPI and perform a voltage or current measurement at TP2, or vice versa. The main thing is to perform measurements that allow access to an electrical parameter dependent on the diagonal term Z12 of the impedance matrix of the electric quadrupole modeling the system formed by the coupled piezoelectric sensors.

[0075] - The parameter of interest is typically complex. We can then take into account takes into account its phase (preferred choice for robustness reasons), its modulus, its real part, its imaginary part or any combination of these values.

[0076] - Instead of a sinusoidal excitation signal of variable frequency (in a manner continuous or stepwise) it is possible to use a signal comprising several frequency components at the same time, for example a pulse signal. This makes it possible to speed up the measurement of length variation, at the cost of more complex electronics, capable of generating and processing intense pulses.

[0077] - Instead of determining the frequency dependence of the parameter, it is possible to measure the value of the parameter of interest at a single frequency, and deduce a variation in the length of the mechanical part from a variation in this value.

[0078] The structure of the clamping system may be different from that of [Fig.4]. For example, it is not essential that the processor P be integrated into a tool OS1. The processor may also be made from a microprocessor, an ASIC or even an FPGA. It is possible to have two separate processors to control the electronic device EA (and, if necessary, the electronic device DE) and to calculate the variation in length of the part from the acquired electrical measurements.

[0079] In the example of [Fig.4] and in that of [Fig. 12], it is the transducer located on the side of the free end of the screw which is excited, and that on the side of the head which is "passive". The opposite choice is also possible. The transducers can also be arranged in different places.

[0080] The part for which a variation in length is measured does not necessarily have to be a screw, or even a fixing element. It can be any mechanical element allowing the establishment of standing acoustic waves and to which piezoelectric transducers can be fixed or attached. The variation in length (more generally, dimension) to be measured does not necessarily have to be caused by a mechanical stress: it can also be, for example, thermal expansion or the effect of corrosion. References

[0081] (Heyman 1977): JS Heyman, “A CW Ultrasonic Bolt-strain Monitor”, Experimental Mechanics (1977)

[0082] (Smith 1980): JFSmith and JD Greiner, “Stress Measurement and Boit Tensioning by Ultrasonic Methods”, Journal of Metals (1980)

[0083] (Joshi 1984): SG Joshi and RG Pathare, “Ultrasonic instrument for measuring water stress”, Ultrasonics (1984)

[0084] (Shao 2016): J. Shao et al., “Boit Looseness Détection Based on Piezoelectric Impédance Frequency Shift”, Applied Sciences, vol. 6 (2016)

[0085] (Dreisbach 2023): A.-L. Dreisbach and C.-P. Fritzen “A Novel Approach for Preload Monitoring in Bolted Connections Using Electro-Mechanical Impedence Spectra” Lecture Notes in Civil Engineering 254, Springer (2023).

Claims

Claims

1. Method for measuring a variation in dimension (L) of a mechanical part (V) along a so-called longitudinal direction (x), comprising the steps of: a) providing at least a first (TPI) and a second (TP2) piezoelectric transducer arranged at two different positions (ELV, TV) along said longitudinal direction and acoustically coupled via said mechanical part; b) performing, at different times, a plurality of electrical measurements to determine at least one value of an electrical parameter dependent on a diagonal term (Zn) of an impedance matrix of an electrical quadrupole modeling the assembly formed by the first and second acoustically coupled piezoelectric transducers; and c) deducing from the results of said electrical measurements said variation in dimension of the mechanical part.

2. Method according to claim 1 wherein each said electrical measurement of step b) comprises the sub-steps consisting of: bl) applying to said first piezoelectric transducer (TPI) a first electrical excitation signal to generate acoustic waves propagating in the mechanical part towards the second transducer (TP2), while keeping the second piezoelectric transducer in open circuit, and at the same time measuring the input impedance of said first piezoelectric transducer; b2) applying to said first piezoelectric transducer (TPI) a second electrical excitation signal to generate acoustic waves propagating in the mechanical part towards the second transducer (TP2), while keeping the second piezoelectric transducer in short circuit, and at the same time measuring the input impedance of said first piezoelectric transducer;b3) calculating the value of said electrical parameter from the input impedances thus measured; the order of sub-steps bl) and b2) may be reversed.;

3. Method according to claim 2 wherein said electrical parameter is the difference in the phases of the input impedances measured during sub-steps b1) and b2).

4. Method according to one of claims 2 and 3 in which the second piezoelectric transducer (TP2) comprises two electrical terminals connected together by a pair of diodes (Dl, D2) head to tail in parallel, the first excitation signal being sufficiently weak so that the generated acoustic waves induce at the terminals of said second piezoelectric transducer a voltage lower than a threshold of said diodes, which can then be likened to an open circuit, and the second excitation signal being sufficiently strong so that the generated acoustic waves induce at the terminals of said second piezoelectric transducer a voltage higher than a threshold of said diodes, which can then be likened to a short circuit.

5. Method according to one of claims 1 to 4 in which step b) comprises determining the value of said parameter as a function of the frequency.

6. Method according to claim 5 in which step c) comprises the identification of peaks of the value of said electrical parameter as a function of the frequency, the variation in dimension of the mechanical part being deduced from a variation in position of said peaks.

7. Method according to one of the preceding claims in which the first (TPI) and the second (TP2) piezoelectric transducer are arranged at two opposite ends (ELV, TV), in said longitudinal direction, of the mechanical part.

8. Use of a method according to one of the preceding claims for measuring the tightening of a screw.

9. System for measuring a variation in dimension (L) of a mechanical part (V) along a so-called longitudinal direction (x), comprising: - a first (TPI) and a second (TP2) piezoelectric transducer, adapted to be fixed at two different positions (ELV, TV) along said longitudinal direction and acoustically coupled via said mechanical part; - an electronic system (AE1, AE2, P) configured to determine at least one value of an electrical parameter dependent on a diagonal term (Zn) of an impedance matrix of an electrical quadrupole modeling the assembly formed by the first and second acoustically coupled piezoelectric transducers; and to deduce from a variation of said at least one value of said parameter said variation in dimension of the mechanical part.

10. A measuring system according to claim 9 wherein said electronic system comprises: - a first electronic device (AE1) configured to apply electrical excitation signals to said first piezoelectric transducer and at the same time measure its input impedance; - a second electronic device (AE2) configured to maintain said second piezoelectric transducer successively in open circuit and in short circuit; and - a processor (P) configured to control at least said first electronic device so as to: - applying to said first piezoelectric transducer (TPI) a first electrical excitation signal to generate acoustic waves propagating in the mechanical part towards the second transducer, while keeping the second piezoelectric transducer (TP2) in open circuit, and at the same time measuring the input impedance of said first piezoelectric transducer; - applying to said first piezoelectric transducer (TPI) a second electrical excitation signal to generate acoustic waves propagating in the mechanical part towards the second transducer, while keeping the second piezoelectric transducer (TP2) short-circuited, and at the same time measuring the input impedance of said first piezoelectric transducer; - calculate the value of said electrical parameter from the input impedances thus measured.

11. Measuring system according to one of claims 9 and 10 in which the electronic device is integrated into a screw tightening system (OS1, OS2), the first and second piezoelectric transducers being adapted to be fixed to a said screw.

Images