METHOD AND SYSTEM FOR MEASURING A DIMENSIONAL CHANGE OF A MECHANICAL PART

Description

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

[0002] It is known to use acoustic techniques to accurately and dynamically measure the tightness of screw fastening systems during industrial assembly of parts using an acoustic method.

[0003] As illustrated on the [ Fig. 1 This can be achieved by attaching a piezoelectric transducer TP to one end ELV of a fastening means (screw) V and exciting it with a sinusoidal voltage. The transducer generates standing acoustic waves OS in the body CV of the fastening means, which are established in the structure according to its length. In turn, these acoustic waves modify the impedance of the piezoelectric transducer TP, which can be measured simultaneously with its excitation. Plotting the phase and magnitude of the complex impedance of the transducer as a function of frequency reveals peaks corresponding to the acoustic resonance frequencies of the structure.

[0004] As the mounting system is tightened (for example, by screwing in a nut E), its length changes, and with it its resonant frequencies. A frequency shift in the transducer's impedance peaks is then observed. This is illustrated in the [ Fig. 2 ], in which 4 impedance curves - corresponding to voltages of 0, 6 kN, 12 kN and 18 kN - are superimposed. The [ Fig. 3 ] shows that the relationship between the applied voltage and the frequency shift δf of the impedance peaks is substantially linear, with an error on the order of 1% (the two curves in the lower part correspond to the deviation from a linear relationship during power-up and release).

[0005] The tightening of the nut can be controlled and regulated 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 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 compared to their spacing, making it difficult to accurately determine their spacing and, consequently, the variation in length of the fixing element. In practice, frequency analysis is required, which necessitates acquisition over several periods—thus requiring a wide spectral bandwidth and a significant acquisition time.

[0008] A competing technique to IFS is based on measuring the time-of-flight (TOF) of an acoustic wave. As with IFS, a piezoelectric transducer is attached to the structure and excited to generate acoustic waves. Unlike IFS, however, the transducer is excited by a voltage pulse, generating a time-limited acoustic wave. The wave propagates through the structure and back, before returning to the transducer. The transducer is then used as a sensor by monitoring its voltage, which varies as the acoustic wave returns. The time between emission and reception depends on the wave speed (obtained through prior calibration) and the length of the structure. TOF is a well-known and widely used technique in state-of-the-art applications.However, it has a number of drawbacks, including the need to use fast electronics to generate short pulses, and voltage pulses on the order of a few hundred volts (as opposed 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 specifically, it aims to improve the accuracy and sensitivity of the IFS technique without significantly complicating its implementation.

[0010] According to the invention, this goal is achieved through the combined use of two acoustically coupled piezoelectric transducers across the mechanical part whose length is to be measured. The assembly of the two acoustically coupled transducers can be modeled by an electrical quadrupole, characterized by an impedance matrix Z. Measuring the off-diagonal element Z12 of this matrix (or another electrical parameter proportional to this element) as a function of frequency allows for the estimation of a variation in the length of the mechanical part, as in the standard IFS technique. The advantage of 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 where a single transducer is used.

[0011] An object of the invention is therefore a method for measuring a variation in the dimensions of a mechanical part along a so-called longitudinal direction, comprising the steps of: a) provide at least one first and a second piezoelectric transducer arranged in two different positions along said longitudinal direction and acoustically coupled through said mechanical part; b) carry out, at different times, a plurality of electrical measurements to determine at least one value of an electrical parameter dependent on an off-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) deduce from the results of said electrical measurements said variation in dimension of the mechanical part.

[0012] According to specific embodiments of such a process: Each said electrical measurement of step b) may include the substeps of: b1) applying a first electrical excitation signal to said first piezoelectric transducer to generate acoustic waves propagating through the mechanical part towards the second transducer, while keeping the second piezoelectric transducer open-circuited, and simultaneously measuring the input impedance of said first piezoelectric transducer; b2) applying a second electrical excitation signal to said first piezoelectric transducer to generate acoustic waves propagating through the mechanical part towards the second transducer, while keeping the second piezoelectric transducer short-circuited.and simultaneously measure the input impedance of said first piezoelectric transducer; b3) calculate the value of said electrical parameter from the input impedances thus measured; the order of substeps b1) and b2) may be reversed. Said electrical parameter may be the phase difference of the input impedances measured during substeps b1) and b2). The second piezoelectric transducer may comprise two electrical terminals connected by a pair of back-to-back diodes in parallel, the first excitation signal being sufficiently weak so that the generated acoustic waves induce a voltage across said second piezoelectric transducer that is lower than a threshold voltage of said diodes, which may then be considered as an open circuit,and the second excitation signal being sufficiently strong that the generated acoustic waves induce a voltage across said second piezoelectric transducer exceeding a threshold of said diodes, which can then be considered equivalent to a short circuit. Step b) may include determining the value of said parameter as a function of frequency. Step c) may include identifying peaks in the value of said electrical parameter as a function of frequency, the variation in the dimensions of the mechanical part being deduced from a variation in the position of said peaks. The first and second piezoelectric transducers may be arranged at two opposite ends, along said longitudinal direction, of the mechanical part.

[0013] Another object of the invention is the use of such a method for measuring the tightening of a screw.

[0014] Another object of the invention is a system for measuring a variation in the dimensions of a mechanical part along a so-called longitudinal direction, comprising: a first and a second piezoelectric transducer, adapted to be fixed in two different positions along said longitudinal direction and acoustically coupled through said mechanical part; an electronic system configured to determine at least one value of an electrical parameter dependent on an off-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.

[0015] In specific embodiments: Said electronic system may include: a first electronic device configured to apply electrical excitation signals to said first piezoelectric transducer and simultaneously measure its input impedance; a second electronic device configured to maintain said second piezoelectric transducer successively in open circuit and short circuit; and a processor configured to control at least said first electronic device so as to: apply 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 simultaneously measure the input impedance of said first piezoelectric transducer; apply 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 simultaneously measure the input impedance of said first piezoelectric transducer; calculate the value of said electrical parameter from the input impedances thus measured.The electronic device can be integrated into a screw clamping system, the first and second piezoelectric transducers being adapted to be fixed to said screw.

[0016] Other features, details and advantages of the invention will become apparent from the description provided with reference to the accompanying drawings given by way of example, in which: [ 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; [ Fig. 2 ], already described, is a graph illustrating the impedance phase shift at the basis of the [ technique Fig. 1 ] ; ] Fig. 3 ], already described, is a graph illustrating the quasi-linear relationship between impedance phase shift and voltage in the technique of the [ Fig. 1 ] ; ] Fig. 4 ] is the functional diagram of a measurement system according to an embodiment of the invention; [ Fig. 5 ] illustrates the modeling of a system of two piezoelectric transducers acoustically coupled by an electrical quadrupole; [ Fig. 6 ], a graph illustrating the variation as a function of frequency of the Z-matrix element 11; [ fig. 7], [Fig. 8 ], [ Fig. 9], [Fig. 10 ] And [ Fig. 11 ], graphs illustrating the variation as a function of frequency of different parameters as a function of Z 12 that can be used in the implementation of the invention according to different embodiments; [ Fig. 12] et [Fig. 13 ] are detailed views of two variants of the [ system Fig. 4 ].

[0017] In the diagram of the [ Fig. 4 The reference V designates a screw, extending in a longitudinal direction x and having a length L in this direction. The screw V has a head TV at one end that can cooperate with a tightening tool OS2; the opposite end ELV (free end), which is threaded, is inserted into a nut E, which can also cooperate with a tightening tool OS1. Two mechanical components C1 and C2 are traversed by the screw body CV and are held between the head TV and the nut E. By rotating the nut with the tightening tool OS1 while holding the head TV fixed with the tool OS2, or vice versa, the screw body is put under tension, producing a slight elongation of the latter.

[0018] To measure this elongation, the screw V is equipped with two piezoelectric transducers TP1 and TP2, arranged respectively at its free end ELV and its head TV.

[0019] The OS1 and OS2 clamping tools, which together form a clamping system, are adapted to interact with a respective transducer via electrical contacts. More specifically, in the embodiment of the [ Fig. 4 The first clamping tool OS1 comprises a primary electronic device AE1 of the vector network analyzer type, which excites the first piezoelectric transducer TP1 with a sinusoidal electrical signal of variable frequency and simultaneously measures its complex impedance. The second clamping tool OS2, in turn, comprises a second electronic device AE2 that excites the first piezoelectric transducer TP1 with a sinusoidal electrical signal of variable frequency and / or simultaneously measures its complex impedance, or applies a variable load to it (for example, alternately a short circuit and an open circuit). A processor P, for example integrated into the first clamping tool, controls both electronic devices and receives the impedance measurements they perform.

[0020] As illustrated on the [ Fig. 5 The assembly consisting of the two transducers acoustically coupled by the body of the screw V can be modeled by an electrical four-terminal network, characterized by an impedance matrix Z, itself characterized by four complex elements that are functions of frequency: Z11(f), Z12(f) = Z21(f), Z22(f). Denoting by Vin the voltage applied across the terminals of transducer TP1, by Iin the current flowing through said transducer, by Vout the voltage appearing across the terminals of transducer TP2, and by Iout the current flowing through the latter, we have: V in = Z 11 I in + Z 12 I out V out = Z 12 I in + Z 22 I out

[0021] As mentioned above, one idea underlying the invention is to monitor the variation of at least one parameter P as a function of Z12 and deduce a measurement of the dimensional change of the mechanical part. This parameter P can be observed at a fixed frequency or over a frequency spectrum. Similarly, the frequency corresponding to a fixed value of this parameter P can be monitored over time.

[0022] Monitoring a parameter P that is a function of Z12, instead of the impedance of a single transducer, offers several advantages. Such a parameter can exhibit greater variations in magnitude and phase (ranging from -180° to +180°) and, in some cases, narrow peaks or abrupt changes that are easily identifiable. Furthermore, Z12 is primarily related to the direct path between the two transducers, thus eliminating standing waves that could develop with other surfaces in the case of a single transducer.

[0023] There [ Fig. 6 ], given as a 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.

[0024] There [ Fig. 7 ] illustrates the evolution, under the same conditions, of the parameter Z 12. We note the greater amplitude of relative evolution of the modulus, and the fact that the phase varies between -180° and +180°, compared to a variation of about 60° for that of Z 11.

[0025] There [ Fig. 8 ] illustrates the evolution, under the same conditions, of the real part of Z 12. We observe the presence of narrow peaks, whose displacement with voltage can be followed by maximum detection algorithms.

[0026] There [ Fig. 9 This illustrates the evolution, under the same conditions, of the imaginary part of Z12. This parameter remains close to zero over a large part of the frequency range of interest, but exhibits abrupt variations corresponding to resonance frequencies. These variations can be detected by signal reversal algorithms.

[0027] There [ Fig. 10 ] illustrates the evolution, under the same conditions, of the phase arg z 12 2 z 22 Z 22 Significant variations are observed, which can be tracked using zero-crossing algorithms, for example.

[0028] A particularly interesting embodiment is one in which the excitation and impedance measurement are performed on only one side, for example that of TP1. In this case, the second electronic device is limited to maintaining the second transducer TP2 alternately in short circuit and open circuit.

[0029] When AP2 keeps the transducer TP2 short-circuited, V out = 0. Equation (1) therefore becomes V in = Z 11 I in + Z 12 I out 0 = Z 12 I in + Z 22 I out ⇒ I out = − z 12 z 22 I in ⇒ V in = Z 11 − z 12 2 z 22 I in = Z 11 1 − ζ 2 I in with ζ = Z 12 Z 11 Z 22 The input impedance measured under short-circuit conditions Z in cc = V in I in V out = 0 therefore Z in cc = Z 11 1 − ζ .

[0030] When AP2 keeps the transducer TP2 in open circuit, I out = 0. Equation (1) therefore becomes V in = Z 11 I in V out = Z 12 I in

[0031] The input impedance measured under open-circuit conditions Z in co = V in I in I out = 0 Therefore, it simply means Z in co = Z 11

[0032] By calculating the difference between the impedance values, we find Δ Z in = Z in co − Z in cc = Z 11 ζ 2 = Z 11 z 12 2 z 11 z 22 = z 12 2 z 22 .

[0033] We also define Δ Z in = Z in co − Z in co And Δ arg Z in = arg Z in co − arg Z in cc

[0034] There [ Fig. 11 ] shows graphs of |Δ| Z in || (upper part) and |Δ arg ( Z in )| (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 body of the screw - are much narrower than in the cases of the [ Fig. 2 ] and the [ Fig. 6 ] ; their position can therefore be determined much more precisely. More specifically, on the [ Fig. 11 The solid line curve corresponds to a screw body of length L0 = 1 cm at rest and the dotted line curve to the same screw body stretched by 0.25% (L=1.0036 cm).

[0035] We denote Δf the frequency periodicity of these resonance peaks and δf the variation in their position following a variation in the length of the screw.

[0036] The wavelength of an acoustic wave of frequency f is given by λ = c f , with c being the wave speed. The condition for obtaining a standing wave is given by: nλ = 2L, where L is the length of the screw (the half-wavelength and the length are multiples). This gives the following for the resonance frequencies f = n 2 c L with n an integer. The [ Fig. 11 ] 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 Δf = 1 2 c L .

[0037] The positions of the peaks f and their gap Δf vary with the elongation ΔL = ε L.

[0038] The speed of sound in a homogeneous medium is given, as a first approximation (neglecting the acousto-elastic effect), by: c = E ρ 1 − ν 1 + ν 1 − 2 ν , where E is the Young's modulus of the screw material, ν Its Poisson's ratio and ρ its density, which also varies with elongation (while Young's modulus and Poisson's ratio are considered constant to a first approximation). The change in volume, and therefore density, with deformation is given by: ΔV V 0 = − Δρ ρ 0 = 1 − 2 ν , where V 0 and ρ 0 are, respectively, the rest volume and density. The rest resonance frequencies are found to be given by f 0 n = n 2 c 0 L 0 = nΔf 0 , where n is an integer index, c0 is the speed of sound in the screw at rest and Δf 0 is the spacing between the resonance peaks at rest. Under stretching conditions, the resonance frequencies vary such that: f f 0 = 1 1 + ε 1 − 1 − 2 ν ε = 1 − 1 + 2 ν 2 ε + o ε 2 ∼ 1 − 0 , 79 ε for a steel with ν =0.29. Finally, the variation of the frequency peaks is given by: δf f 0 ∼ − 1 + 2 ν 2 ε

[0039] And Δf = 1 − 1 + 2 ν 2 ε Δf 0 .

[0040] We do find a linear relationship between the variations in position of the impedance peaks and the variation in length of the screw (the latter being itself directly proportional to the applied tension, as long as the linear elasticity limit is not crossed).

[0041] There [ Fig. 12 [Illustrates a particular embodiment in which the second electronic device AE2 is simply two diodes D1 and D2 connected back-to-back in parallel. When the voltage Vout generated by the transducer TP2 is lower than the diodes' threshold voltage, this configuration behaves as an open circuit, while when Vout is much higher than the threshold, it behaves as a short circuit. It is therefore possible to perform both types of measurements—open circuit and short circuit—simply by varying the amplitude of the excitation signal of the first transducer TP1. The advantage of this embodiment is that the second electronic device AE2 is entirely passive, and only the clamping 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 permanently installed, in which case access to the screw head is not required.]

[0042] There [ Fig. 13 ] illustrates yet another embodiment in which a third piezoelectric transducer TP3 is arranged on a flange of the screw head TV. This makes it possible, for example, to determine a variation in length between TP1 and TP2 (mainly related to the deformation of the screw and therefore to the applied force) and a variation in length between TP2 and TP3 (mainly related to a variation in the temperature of the screw).

[0043] The invention has been described with reference to particular embodiments, but variations are possible. For example: Performing measurements by energizing only TP1 and changing the state of TP2 (open circuit / short circuit) is an advantageous, but not unambiguous, approach. It would also be possible, for example, to energize TP1 and measure voltage or current at TP2, or vice versa. The key is to perform measurements that provide access to an electrical parameter dependent on the off-diagonal term Z12 of the impedance matrix of the four-terminal network modeling the system formed by the coupled piezoelectric sensors. The parameter of interest is typically complex. Its phase (the preferred choice for robustness), its magnitude, its real part, its imaginary part, or any combination of these values ​​can then be considered.Instead of a sinusoidal excitation signal with variable frequency (continuously or in steps), it is possible to use a signal comprising several frequency components simultaneously, for example, a pulsed signal. This allows for faster measurement of length variation, at the cost of more complex electronics capable of generating and processing intense pulses. 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 change in the length of the mechanical part from a change in this value.

[0044] The structure of the clamping system may differ from that of the [ Fig. 4 For example, it is not essential that the processor P be integrated into an OS1 tool. The processor can, in fact, be implemented using a microprocessor, an ASIC, or even an FPGA. It is possible to have two separate processors: one to control the electronic device EA (and, if necessary, the electronic device DE) and the other to calculate the variation in part length from the acquired electrical measurements.

[0045] In the example of the [ Fig. 4 ] and in that of the [ Fig. 12 In this configuration, the transducer located at the free end of the screw is excited, while the one at the head is passive. The reverse configuration is also possible. Furthermore, the transducers can be arranged in different locations.

[0046] The part whose length variation is being measured does not necessarily have to be a screw, or even a fastener. It can be any mechanical element that allows the establishment of standing acoustic waves and to which piezoelectric transducers can be attached or attached. The change in length (more generally, dimension) to be measured does not necessarily have to be caused by mechanical stress: it can also be, for example, due to thermal expansion or the effect of corrosion. Références

[0047] (Heyman 1977): J.S. Heyman, "A CW Ultrasonic Bolt-strain Monitor", Experimental Mechanics (1977) (Smith 1980): J. F.Smith and J. D. Greiner, "Stress Measurement and Bolt Tensioning by Ultrasonic Methods", Journal of Metals (1980) (Joshi 1984): S. G. Joshi and R. G. Pathare, "Ultrasonic instrument for measuring bolt stress", Ultrasonics (1984) (Shao 2016): J. Shao et al., "Bolt Looseness Detection Based on Piezoelectric Impedance Frequency Shift", Applied Sciences, vol. 6 (2016) (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

1. A method for measuring a dimension variation (L) of a mechanical component (V) in a direction (x) referred to as the longitudinal direction, comprising the steps consisting of: a) obtaining at least a first (TP1) and a second (TP2) piezoelectric transducer which are arranged at two different positions (ELV, TV) in said longitudinal direction and which are coupled acoustically by means of said mechanical component; b) carrying out, at different times, a plurality of electrical measurements in order to determine at least one value of an electrical parameter which is dependent on an off-diagonal term (Z12) of an impedance matrix of an electrical quadrupole which models the group formed by the first and the second piezoelectric transducer which are acoustically coupled; and c) deriving from the results of said electrical measurements said dimension variation of the mechanical component.

2. Method according to claim 1, wherein each said electrical measurement of step b) comprises the sub-steps consisting of: b1) applying to said first piezoelectric transducer (TP1) a first electrical excitation signal in order to generate sound waves which propagate in the mechanical component in the direction of the second transducer (TP2), whilst maintaining the second piezoelectric transducer in an open circuit, and measuring at the same time the input impedance of said first piezoelectric transducer; b2) applying to said first piezoelectric transducer (TP1) a second electrical excitation signal in order to generate sound waves which propagate in the mechanical component in the direction of the second transducer (TP2), whilst maintaining the second piezoelectric transducer in a short-circuit, and measuring at the same time the input impedance of said first piezoelectric transducer; b3) calculating the value of said electrical parameter from the input impedances measured in this manner; the order of the sub-steps b1) and b2) being able to be reversed.

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

4. Method according to either of claims 2 and 3, wherein the second piezoelectric transducer (TP2) comprises two electrical terminals which are connected to each other by means of a pair of diodes (D1, D2) which are arranged head-to-tail in parallel, the first excitation signal being sufficiently weak for the sound waves generated to bring about at the terminals of said second piezoelectric transducer a voltage less than a threshold of said diodes, which may then be likened to an open circuit, and the second excitation signal being sufficiently strong for the sound waves generated to bring about at the terminals of said second piezoelectric transducer a voltage greater than a threshold of said diodes, which may then be likened to a short-circuit.

5. Method according to any one of claims 1 to 4, wherein step b) comprises the determination of the value of said parameter as a function of the frequency.

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

7. The method according to any one of the preceding claims, wherein the first (TP1) and the second (TP2) piezoelectric transducer are arranged at two opposing ends (ELV, TV) in said longitudinal direction of the mechanical component.

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

9. System for measuring a dimension variation (L) of a mechanical component (V) in a direction (x) referred to as the longitudinal direction, comprising: - a first (TP1) and a second (TP2) piezoelectric transducer which are capable of being fixed at two different positions (ELV, TV) in said longitudinal direction and which are coupled acoustically by means of said mechanical component; - an electronic system (AE1, AE2, P) which is configured to determine at least one value of an electrical parameter which is dependent on an off-diagonal term (Z12) of an impedance matrix of an electrical quadrupole which models the group formed by the first and the second piezoelectric transducer which are acoustically coupled; and in order to derive from a variation of said at least one value of said parameter said dimension variation of the mechanical component.

10. Measurement system according to claim 9, wherein said electronic system comprises: - a first electronic apparatus (AE1) which is configured to apply electrical excitation signals to said first piezoelectric transducer and at the same time to measure its input impedance; - a second electronic apparatus (AE2) which is configured to maintain said second piezoelectric transducer successively in an open circuit and in a short-circuit; and - a processor (P) which is configured to control at least said first electronic apparatus in order: - to apply to said first piezoelectric transducer (TP1) a first electrical excitation signal in order to generate sound waves which propagate in the mechanical component in the direction of the second transducer, whilst maintaining the second piezoelectric transducer (TP2) in an open circuit, and to measure at the same time the input impedance of said first piezoelectric transducer; - to apply to said first piezoelectric transducer (TP1) a second electrical excitation signal in order to generate sound waves which propagate in the mechanical component in the direction of the second transducer, whilst maintaining the second piezoelectric transducer (TP2) in a short-circuit, and to measure at the same time the input impedance of said first piezoelectric transducer; - to calculate the value of said electrical parameter from the input impedances measured in this manner.

11. Measurement system according to either of claims 9 and 10, wherein the electronic apparatus is integrated in a screw tightening system (OS1, OS2), the first and the second piezoelectric transducer being capable of being fixed to a said screw.