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INSTITUTE OF PHYSICS PUBLISHING Smart Mater. Struct. 14 (2005) 247–256

SMART MATERIALS AND STRUCTURES doi:10.1088/0964-1726/14/1/025

A magnet-based vibrating wire sensor

: design and simulation

Fr? ed? eric Bourquin and Michel Joly

Laboratoire Central des Ponts et Chauss? ees and Laboratoire Lagrange, 58 boulevard Lefebvre, 75015 Paris, France E-mail: frederic.bourquin@lcpc.fr

Received 7 July 2004 Published 23 December 2004 Online at stacks.iop.org/SMS/14/247 Abstract Vibrating strings help in measuring relative displacements in a mechanical system. Since the ground natural frequency of a string increases when it is stretched, monitoring the ground frequency yields the current length of the string. Therefore a wire able to vibrate between two anchor points of a system acts as a relative displacement sensor. Excitation is usually achieved by means of an active coil, which is very close to the vibrating iron wire. Vibrating wire sensors (VWS) based on this excitation may prove obtrusive and one is limited to wires of small length. The new VWS takes advantage of distributed passive magnets, which force the wire to vibrate mainly in its fundamental mode. The sensor proves scalable and much less obtrusive when fully embedded, since it can be made ?at and very ?exible. On the basis of a simpli?ed electromechanical modelling of the measurement process, a suitable distribution of magnets is proposed, which is proved numerically and experimentally to make the measurement robust with respect to mechanical uncertainties. Moreover, numerical simulations suggest measuring not the voltage in the vibrating wire but the current in an auxiliary circuit. (Some ?gures in this article are in colour only in the electronic version)

1. Introduction and motivation

Since their underlying principle is very general, vibrating wire sensors (VWS) have been implemented in numerous ways and applied in various areas, including civil engineering, where the technique is nowadays widely used. As a matter of fact, density and viscosity of ?uids, water pressure, pore water pressure, tides, strain inside structures, settlement and tilt in buildings, dams and excavation walls can be monitored in this way and speci?c VWS do exist to that purpose. Vibrating wire load cells are put to work in soils whereas vibrating wire inclinometers help monitor natural slopes and embankments. See e.g. [8, 7, 17, 11, 10, 12, 14]. But the principle proves also of interest when it comes to measuring magnetic susceptibility over large temperature ranges with high sensitivity [3]. Wire scanners are also widely used in particle accelerators. When colliding with the wire, particles deposit energy that is mostly converted

0964-1726/05/010247+10$30.00 ? 2005 IOP Publishing Ltd

into heat. Since the fundamental frequency of the vibrating wire strongly depends on the temperature, monitoring the frequency yields the temperature and then the intensity of the particle beam [1, 2]. Vibrating wires of metric scale also help with checking the alignment of focusing elements in particle accelerators, where optical tools cannot operate. The magnetic ?eld distribution along the wire is recovered after the fundamental mode as well as harmonics have been excited by a tuned driving current [15, 16]. Civil engineering professionals claim that VWS are known to perform well especially when long-term reliability, minimum zero drift and signal immunity to electrical noise are of prime importance [13, 4]. In?uence factors on longterm stability and reliability have been studied. The effect of temperature changes and of mechanical impacts during installation have been investigated by manufacturers. Just like musical string instruments, the VWS must be excited. In the past, a synthetic excitation was manually 247

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performed by means of a hammer, for example, and the ground vibration frequency was measured by means of an oscillograph and a stroboscope. Nowadays, fully automatic excitation and measurement processes have emerged. Our contribution consists of an enhancement of the current ones. Since metallic strings are mainly used, excitation and detection usually rely on electromagnetic coupling. An electrical current in a coil creates a magnetic ?eld, which attracts the metallic string possessing an inner magnetic ?eld. On the other hand, the movement of the magnetized string in front of the coil induces in the latter an electrical voltage, whose time variation accounts for the movement of the string. Various implementations taking advantage of these very classical effects exist: The permanent excitation method consists in supplying an impulse of electrical energy to the string each time the voltage vanishes. The impulse maintains suf?cient energy in the string for it to vibrate forever. The period between two successive impulses yields the frequency to be determined. The resonance method is very often encountered in VWS with two electromagnets. The ?rst one acts as an actuator, whereas the second one serves as a sensor. When inserted in an electrical circuit, both act as oscillators tuned on the main resonance frequency of the string. In the impulse method, the electromagnet serves as a shock actuator and as a velocity sensor. An electrical impulse through the coil of the electromagnet applies a very brief point force on the string. Then, assuming no signi?cant electromechanical coupling between the string and the electromagnet, the string oscillates freely. Assuming a minimal damping of all modes, the ?rst mode component of the free response dominates after a while, depending mainly on the difference between the ?rst frequency and the second one. Then the resulting electrical signal in the coil of the electromagnet oscillates at the same frequency. However, the above frequency difference is proportional to the inverse of the string length. Therefore, the longer the string, the longer the second and subsequent modes will remain in the response. The electromagnet-based impulse method has allowed VWS to be fully automated and their use to become widely widespread, but suffers from some drawbacks: The level of intrusion is usually very high: the electromagnet consists of a protuberant appendage of nonnegligible size, located at the middle of the string and connected to the electrical supply cable. This results in a signi?cant intrusion of the sensor into the surrounding material, whose deformation the VWS is supposed to measure. Especially for short strings, the appendage drastically modi?es the stress and strain ?elds. The quality of the sensor proves very sensitive to the ?ne positioning of the electromagnet. Any electromagnet of reasonable size should be placed very close to the vibrating string—of the order of 0.2 mm away in current VWS based on this technique (see ?gure 1). Hence, any bending of the sensor due to internal strain of the surrounding material may lead to a contact between the wire and the coil, or to a physical disconnection and, in any case, to a meaningless signal. Therefore, the sensor should be made extremely stiff, thus perturbing the surrounding material, and the amplitude of 248

Figure 1. The usual electromagnetic actuation of the wire.

vibration should be kept very small. This severely limits the length of the string, since relatively very small amplitudes of vibration prove dif?cult to realize in a controlled way, and the useful signal where a single mode dominates appears after the amplitude has somewhat decayed, especially for long strings. In conclusion, this technique seems to be limited to short strings. The need for a quasi-monochromatic signal is currently dictated by the economical advantage of using simple signal processing. The zero-counting technique, for example, requires the string to vibrate on its fundamental mode exclusively. However, this is not the case because the excitation takes place in the middle of the string. The situation becomes worse when the string becomes longer, because the unwanted modes are not damped suf?ciently fast. Our contribution aims at overcoming the above dif?culties and dimensional limitations. This paper is organized as follows. Section 2 is devoted to background information on the mechanical behaviour of the VWS. Section 3 describes the new method of excitation. Using Laplace and Lenz law, the resulting voltage in the wire is expressed in terms of the magnet distribution. A suitable distribution is introduced and tested both numerically and experimentally. Numerical simulations are shown to be qualitatively in good agreement with experiments. They con?rm the soundness of the whole measurement process.

2. The basic principle of the vibrating wire sensor

This section owes much to [11]. 2.1. The wire as a string Let us consider the free vibrations of a stretched string. Just writing down the simplest wave equation governing its small amplitude transverse vibrations around a given static equilibrium state, in addition to suitable boundary conditions, the ground frequency reads f = 1 2L σ 1 = ρ 2L τ ? (2.1)

where L denotes the length of the string, ρ its mass density, σ the normal stress of the reference static equilibrium state, ? its density per unit length and τ its tension. Of course, if S stands for the cross-sectional area, we have ? = ρ S and τ = σ S . All of these quantities are assumed to be constant along the string.

A magnet-based vibrating wire sensor

Figure 2. The end displacement curve in microns plotted against the frequency shift in hertz: experiments (small circles), theory for the string (dashed curve) and beam theory (solid curve).

Figure 3. The curve of end displacement against frequency shift for a 85 mm wire.

On the other hand, considering the reference con?guration as a natural state and assuming a linearly elastic stress–strain constitutive law as well as the absence of longitudinal waves, we may write σ = Eε (2.2) where E denotes the Young’s modulus and ε = L / L the global strain. In this setting, the ground frequency associated with the small vibrations around a given static equilibrium state reads 1 Eε . (2.3) f = 2L ρ Now, let L 1 and L 2 denote the lengths associated with two different stretched states and L 0 the length of the string at rest. Moreover, let f 1 and f 2 denote the corresponding ground frequencies; then ρ Li ? L0 = 4 L2 f2 L0 E i i which, in turn, yields the closed-form formula L2 ? L1 ρ ≈ 4 L2 ( f 2 ? f 12 ) L0 E 0 2 (2.5) (2.4)

coef?cient called K exp . For f1 = 1000 Hz and f2 varying in the frequency band [400, 1800] Hz, the relative difference between K str and K exp was shown to be approximately 7%. Another model predicts the real behaviour of the wire better [11, 14] as explained in the next section. 2.2. The wire as a beam under tension To some extent the wire resists bending. Therefore a beam model including both longitudinal tension and bending was introduced [11, 14]. Since the real wire is clamped at both ends, it was reasonable to assume the corresponding beam to be clamped. In this setting, the following closed-form formula was derived in [11, 14]: f = 1 2L π2 E I τ EI + 4 + 1+2 ? τ L2 2 τ L2 (2.7)

whenever both initial strains ε1 and ε2 are suf?ciently small. Therefore, measuring a shift of the ground frequency immediately leads to the length variation of the string. Rewriting the above formula as L 2 ? L 1 ≈ K str ( f22 ? f 12 ) (2.6)

where I denotes the cross-sectional inertia of the beam and the other notation remains the same as in the previous subsection. Clearly, the frequency to length curve will no longer be parabolic except locally and if bending is very small compared to tension. In this case, the experimental locally parabolic-like curve (?gure 2) ?ts the curve that we get from above closedform formula (2.7). It can be noted that the above curve gets close to the initial parabolic one when the ratio E I /τ L 2 becomes small—that is to say, when bending becomes negligible in comparison to stretching. This is why a high tension is chosen: in view of the easy interpretation of the frequency shifts in terms of length variations. Typical curves showing the behaviour of the stretched wire are shown in ?gures 3 and 4.

we may plot the length variation as a parabolic function of f2 where f 1 is given (see ?gure 2). More than 70 VWS of the same type have been tested at LCPC. Their mechanical and geometric characteristics were perfectly known. Therefore the experimental response could be compared with the above formula (2.6) (see ?gure 2). It turns out that the experimental length variation as a function of the frequency variation exhibits locally the same parabolic shape as (2.6) but with a slightly different multiplicative

3. The proposed magnet-based VWS

3.1. Excitation and detection principles The new technique, based on permanent magnets, yields a permanent physical tuning of the wire vibration on the fundamental mode, independently of the wire length. The dif?culties mentioned in the last subsection are thus engineered away. Downsizing of electronic devices, high 249

F Bourquin and M Joly

Figure 4. The curve of end displacement against frequency shift for a 3000 mm wire.

Figure 6. The magnetic actuation.

Figure 5. A typical small scale (250 mm) magnet-based vibrating wire sensor.

active in the vicinity of any given point. The assumption that the length of each magnet is negligible with respect to the length of the wire is not fundamental but helps with keeping the presentation as simple as possible. The assumption does not simplify the mathematical analysis, since in particular high frequency modes that do not vanish at a given point xk will show up in the resulting voltage whereas they would be cancelled out if we considered magnets of positive length in the analysis. The force g (x ) is equivalent to the one produced by an electromagnet whose length is very small, except that now the wire will undergo displacements in a plane that will be parallel to the plane of the magnet. Therefore, relatively large amplitudes are now allowed. Assume that a permanent voltage is applied; then the wire will vibrate and reach, after a while, due to internal damping, a permanent deformation state u 0 (x ) such that ?τ ?x x u 0 (x ) = g (x ) on [0, L ] (3.3)

power permanent magnets, dynamics of structures and the combination of electromagnetic principles are the key technologies here. In order to make clear how the VWS works, consider a straight wire hosting an electrical current i , close to a permanent magnet of size L , polarized in a direction orthogonal to the wire (see ?gure 6). Then, according to Laplace’s law, the force applied by the magnet on the wire reads F = Bi L . (3.1) Assume that N such magnets of magnetic induction per unit N length B and of lengths ( L k )k =1 are placed at some locations N (xk )k =1 , and that the lengths are very small compared to the length of the wire. Let g (x ) denote the distribution of corresponding forces; then we have approximately

N

if we consider the basic vibrating string equation as a reference model. Note that the force is supposed to act pointwise; therefore the displacement u 0 will be piecewise linear. Moreover, equation (3.3) holds in a weak sense only. Besides that, the right-hand side is not even a function. Once the string is at rest, the current is shut down in the wire and the wire starts vibrating. On the basis of the same modelling assumption, and if T denotes the possibly in?nite time horizon where the vibration is considered, the transverse displacement satis?es ??tt u ? τ ?x x u = 0 u (x , 0) = u 0 (x ), on [0, L ] × [0, T ] [0, T ] [0, L ]. (3.4) ?t u (x , 0) = 0

u (0, t ) = u ( L , t ) = 0

g (x ) =

k =1

i B L k δx k ( x )

(3.2)

where δx k denotes the Dirac distribution at point x = xk —that L is to say, in some sense to be made precise, 0 δx k (x )w(x ) d x = w(xk ) for any continuous displacement ?eld w . In other words, the force i B L k δx k (x ) acts only at point xk . Thus, at most one term of the expansion of the electromagnetic force g (x ) is 250

It is assumed that the impedance is so large that no current develops in the wire and therefore no damping of this origin attenuates the free vibrations of the wire. This is a very important feature since the ?rst mode would be damped much more than the other ones since the magnets are designed to excite the ?rst mode predominantly, as we shall see later.

A magnet-based vibrating wire sensor

Expanding the solution u in a Fourier series leads to u (x , t ) =

+∞ j =1

1 0.8

α0 j cos(ω j t )θ j (x )

(3.5)

0.6 0.4 0.2 voltage 0

where ν j = ω j /2π and θ j (x ) stand for the eigenfrequencies and corresponding normalized mode shapes, arranged in increasing order of the frequencies, and α0 j =

0 L

u 0 (x )θ j (x )? d x .

(3.6)

– 0.2 – 0.4 – 0.6 – 0.8 –1

Of course, the modes are normalized so as to have unit mass, L that is to say, 0 (θi (x ))2 ? d x = 1. The above series expansion converges in the sense of energy. Since Fourier analysis also works for statics, equation (3.3) can be solved explicitly, leading to α0 j = iB λj

N

0

0.5

1

1.5

2 time

2.5

3

3.5

4

L k θ j (xk ).

k =1

(3.7)

Figure 7. The ideal voltage, for a continuum of magnets.

The acquisition starts after a while. The electrical current is produced in the wire as a result of Lenz’s law that, in this setting, basically reads

N

Remark 2. The resulting excitation of the wire could also be achieved by means of distributed coils, but in practice it would be dif?cult to synchronize the signals. Moreover, the sensor would become really large and even more intrusive. Remark 3. The proposed technique relies on the availability of small powerful magnets. Such magnets are now available off the shelf. 3.2. Magnet distribution Hence, the problem of magnet design can be expressed in N a simple way: ?nd the magnet placement (xk )k =1 and the N in such a way as to maximize the length distribution ( L k )k =1 contribution of the ?rst mode relatively to the contribution of all others, or to the most signi?cant other modes. Again, damping or enhanced modelling should lead to negligible contributions of high frequency modes, which is observed in practice. The maximization could be achieved by imposing that the force g be proportional to the ?rst eigenmode θ1 if we had a continuous distribution of magnets at our disposal. In this case, all but the ?rst participation factors in the series expansion (3.9) would vanish. But this is not possible since only a discrete set of magnets are available. Nevertheless, choosing equally spaced magnets with lengths L k = C θ1 (xk ) leads to an initial displacement u 0 that will be closer to the ?rst eigenmode when the number of magnets gets larger, as can be shown by using standard interpolation theory. Other choices based on a ?nite element representation of the ?rst eigenmode with a consistent mass matrix extend to cases when the modes are not known explicitly. The corresponding analysis is explained in [5, 6] in some detail. Anyway, the above choice is very simple and yields interesting results in terms of participation factors for the higher modes, as shown on ?gures 7–10. 3.3. Numerical simulation of the VWS The voltage e is simply computed with the closed-form solution (3.9). As shown in the left parts of ?gures 7–10, which represent the voltage in (3.9) computed with 17 modes, if the wire were exactly described by the wave equation, the number of magnets would have no impact on the frequency estimate 251

e=C

k =1

Lk

?u (xk , t ) ?t

(3.8)

where e denotes the electrical force in the circuit and C some positive constant related to the magnetic induction at each magnet location and initial current. The velocity of the wire is supposed to be constant in the vicinity of each of these locations and the lengths of the magnets are involved in the de?nition of the ?ux crossing the wire. Remark 1. From a mathematical point of view, the voltage e(t ) is not de?ned pointwise but in the sense of distributions, because we assumed no damping, and pointwise excitation by the magnets. This is why the series expansion in equations (3.5) and (3.9) may not converge pointwise. As a consequence of equations (3.5), (3.7) and (3.8), the voltage e reads e=C

+∞ j =1 1 ω? j sin ω j t N 2

L k θ j (xk )

k =1

.

(3.9)

The frequency ν1 is recovered by a simple and very cheap zero-counting technique which can be implemented far from the sensor itself if needed. Of course, a modern modal analysis technique such as the subspace method would certainly yield better results. In any case, magnet placement and optimization should improve whatever signal post-processing is put to work. In practice, the frequency slightly decreases at the beginning of the detection phase, probably due to the nonlinearity of the wire dynamics. This non-linearity shows up for suf?ciently high levels of transverse displacement. As a matter of fact, the global stiffness of the wire consists of the sum of a constant term and the mean square of the displacement along the wire. This VWS requires a constant current generator, in view of the initial shaping of the wire, an electronic box, in view of the need for amplifying the electrical signal due to the vibration of the wire in front of the magnets, and a frequency counter.

F Bourquin and M Joly

unperturbed harmonicity, nb of magnets : 1 0.3 0.5 0.4 0.2 0.3 0.2 0.1 0.1 voltage voltage 0 0 – 0.1 – 0.1 – 0.2 – 0.3 – 0.2 – 0.4 – 0.3 0 – 0.5 0 perturbed harmonicity, nb of magnets : 1

0.5

1

1.5

2 time

2.5

3

3.5

4

0.5

1

1.5

2 time

2.5

3

3.5

4

Figure 8. The voltage with one magnet: the case of pure harmonicity left, distorted harmonicity right.

unperturbed harmonicity, nb of magnets : 3 1.5

1.5 perturbed harmonicity, nb of magnets : 3

1

1

0.5

0.5

voltage

0

voltage

0.5 1 1.5 2 time 2.5 3 3.5 4

0

– 0.5

– 0.5

–1

–1

– 1.5 0

– 1.5 0

0.5

1

1.5

2 time

2.5

3

3.5

4

Figure 9. The voltage with three magnets: the case of pure harmonicity left, distorted harmonicity right.

unperturbed harmonicity, nb of magnets : 7 6 6 perturbed harmonicity, nb of magnets : 7

4

4

2

2

voltage

0

voltage 0.5 1 1.5 2 time 2.5 3 3.5 4

0

–2

–2

–4

–4

–6 0

–6 0

0.5

1

1.5

2 time

2.5

3

3.5

4

Figure 10. The voltage with seven magnets: the case of pure harmonicity left, distorted harmonicity right.

252

A magnet-based vibrating wire sensor

desired current 1 0.8 0.6 0.4 0.2 current 0

– 0.2 – 0.4

in parallel with the main circuit instead of the voltage in the main circuit would amount to time integrating the voltage (3.9). More sophisticated analogue integrators exist. Nevertheless, the current is computed and plotted in ?gures 11–14 for a wire described by the wave equation, on the left-hand side, and for a wire with the same distortion from harmonicity as previously, on the right-hand side. As expected, ?gures 11–14 prove the interest of using the current instead of the voltage as regards extracting the fundamental frequency. 3.4. Experiments

– 0.6 – 0.8 –1 0

0.5

1

1.5 time

2

2.5

3

Figure 11. The ideal current, for a continuum of magnets.

from the zero-counting technique. However, the number and distribution of magnets does matter when the modes of the wire are no longer harmonic, as in the case of the stretched beam model, which was proved to conform the experiments. See also section 3.4. The right parts of ?gures 8–10 display the response of the wire in the case of a few per cent distortion from harmonicity of the eigenfrequencies. Increasing the number of magnets has a very positive impact on the voltage which looks increasingly closer to the reference signal, and easier to handle by means of a zero-counting technique. Hence this elementary signal processing used to extract the ground frequency proves robust with respect to uncertainties regarding the mechanical behaviour of the wire because of the special magnet distribution. Now measuring the voltage may not be the best thing to do. Other related physical quantities may yield signals from which it is easier to extract the fundamental frequency of the wire. For example, measuring the current in a small coil put

The real behaviour of the sensor has been checked by means of an experiment on a 3 m vibrating wire. The voltage e(t ) has been monitored after the steady current was shut down. Figures 15 and 16 correspond to the case of a single magnet and of ?ve magnets respectively. Note that simulations match experiments: the lack of harmonicity shows up in the experiment and the pollution of the fundamental mode by higher modes is of the same order of magnitude. Furthermore, increasing the number of magnets produces a similar enhancement of the resulting signal in the experiment and in the numerical simulation. Note that the observed signal exhibits high order harmonics despite the positive length of the actual magnets and the damping of the wire. It turns out that the modelling described in section 3.1 is not too oversimpli?ed from this point of view. 3.5. Comparison with other techniques The magnet-based technology provides several enhancements: First, the smallness of the magnets and the use of the vibrating wire as a conducting medium leads to a volume reduction and increased slenderness. The sensor itself is nothing but a magnet carrier and can be made ?at and ?exible, and thus much less intrusive. This feature proves important for several applications where the surrounding material should

current, unperturbed harmonicity, nb of magnets : 1 0.15

0.15

current, perturbed harmonicity, nb of magnets : 1

0.1

0.1

0.05

0.05

current

0

current 0 – 0.05

– 0.05

– 0.1

– 0.15

0

0.5

1

1.5 time

2

2.5

3

– 0.1 0

0.5

1

1.5 time

2

2.5

3

Figure 12. The current with one magnet: the case of pure harmonicity left, distorted harmonicity right.

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current, unperturbed harmonicity, nb of magnets : 3 0.5 0.4 0.3

0.2 0.4 current, perturbed harmonicity, nb of magnets : 3

0.3

0.2 0.1 current 0 – 0.1 – 0.2

– 0.2 0.1

current

0

– 0.1

– 0.3 – 0.4 – 0.5

– 0.3

0

0.5

1

1.5 time

2

2.5

3

– 0.4 0

0.5

1

1.5 time

2

2.5

3

Figure 13. The current with three magnets: the case of pure harmonicity left, distorted harmonicity right.

current, unperturbed harmonicity, nb of magnets : 7 2 1.5 current, perturbed harmonicity, nb of magnets : 7

1.5 1 1 0.5 0.5 current current 0 0.5 1 1.5 time 2 2.5 3

0

0

– 0.5 – 0.5 –1 –1 – 1.5

–2

– 1.5

0

0.5

1

1.5 time

2

2.5

3

Figure 14. The current with seven magnets: the case of pure harmonicity left, distorted harmonicity right.

not be perturbed too much. From this point of view, the magnet-based VWS sensor is thus quali?ed for embedded monitoring. Second, the new VWS proves fully scalable (see ?gure 17) without additional complexity or volume increase. The same conditioner will be used for excitation and measurement. The only parameter which will be tuned is the timescale, which is closely related to the length. To this end, the initial tension of the wire will be adjusted so as to keep the ground frequency suf?ciently high. Third, the magnet-based VWS features enhanced robustness with respect to standard ones: in the case of electromagnetic actuation, the coil must be very close to the wire (0.2 mm), as explained in section 1. In contrast, the distance between the wire and the magnet is here about 2 mm. Moreover, the displacement takes place in a plane that is perpendicular to the axis of the magnets. Therefore the amplitude of vibration is no longer limited and the global electromechanical behaviour proves relatively insensitive to the deformation of the surrounding medium. This increased 254

robustness also contributes to the relevance since slenderness and ?exibility are now allowed.

4. Large scale implementation

At the outset, the magnet-based VWS was designed to match special research needs in civil engineering. This is why the main implementation took place in the framework of research aiming at investigating the weight and deformation of a structure due to chemical reactions [10, 12]. The experiment (see ?gure 18) was carried out at a constant temperature of 38 ? C, and in a highly humid atmosphere. It utilized 216 10 cm VWS, that are fully embedded in concrete, 36 25 cm VWS, 36 50 cm VWS and 24 300 cm VWS placed at the surface of the structures to be tested. The failure rate was around 2% after several years of experiments in very severe climate conditions. Most of the failures were due to repeated manipulations of the structures that were inherent to the test programme. This shows that the technology is readily available. At this stage, it can be fully operated for laboratory use, at least.

A magnet-based vibrating wire sensor

Figure 17. Vibrating wire sensors of various lengths: from 85 to 500 mm. Figure 15. Experiment: the voltage with one magnet.

Figure 18. A practical implementation.

Figure 16. Experiment: the voltage with ?ve magnets.

More than 500 sensors of this type, ranging from 10 to 300 cm, have been designed, manufactured and successfully used.

5. Concluding remarks

Ongoing civil engineering research stimulated the emergence of a new displacement sensor. The resulting technique yielded several functional enhancements with respect to standard electromagnet-based technologies: First, the magnet-based VWS proves easily scalable because large displacements of the wire are admissible and the structure of the sensor can be ?exible since it is reduced to a tube carrying the magnets whose position with respect to the wire does not in?uence the magnetic driving force signi?cantly (see ?gure 6). Moreover, the quality of the signal is very good even for long wires because of the suitable magnet distribution. Second, the ?exibility of the sensor combined with the lack of protuberant appendages makes the magnet-based VWS much less obtrusive than classical ones. This VWS can thus be embedded without perturbing the surrounding material too much. The technology itself consists of combining electromechanical principles, dynamics of structures as well as actuator placement optimization. It relies on magnet downsizing.

Numerical simulations showed the robustness of the whole process with respect to mechanical uncertainties. They also pointed out the interest of using auxiliary circuits which would act as signal integrators. The proposed technique remains very cheap. It can be enhanced in several ways, including signal processing and thermal compensation. The last point has been addressed for classical VWS. Extensions of the method such as sensor networking and upscaling are considered. The smart measurement of relative displacements at two points very far away from one another seems of interest, especially in view of applications in the cheap monitoring of large historical monuments [9].

References

[1] Arutunian S, Avetisyan A, Dobrovolski N, Mailian M, Sinenko I G and Vasiniuk I 1999 Vibrating wire for beam pro?le scanning Phys. Rev. Special Topics—Accelerators Beams 2 122801 [2] Arutunian S, Avetisyan A, Dobrovolski N, Mailian M, Vasiniuk I, Wittenburg K and Reetz R 2002 Problems of installation of vibrating wire scanners into accelerator vacuum chamber Proc. EPAC 2002 (Paris) pp 1837–9 [3] Asti G, Solzi M and Podini P 2000 Phase and frequency control in the vibrating wire magnetic susceptometer Sensors Actuators A 81 343–5 [4] Bordes J L and Debreuille P J 1985 Some facts about long-term reliability of vibrating wire instruments Transp. Res. Record 1004 20–6

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[5] Bourquin F and Joly M 2002 Analysis and optimization of a magnet-based vibrating wire sensor Proc. 1st European Workshop on Structural Health Monitoring [6] Bourquin F and Joly M 2003 Optimisation d’un corde vibrante I2M 3 (1–2) 115–26 [7] Choquet P, Dupuis M, Dadoun F and Klebba J-M 1998 Integrated automatic data acquisition systems for dam monitoring Proc. Symp. on Rehabilitation of Dams pp 163–72 [8] Coutts D R, Wang J and Cai J G 2001 Monitoring and analysis of results for two strutted deep excavations using vibrating wire strain gauges Tunnelling Underground Space Technol. 16 87–92 [9] Holnicki-Szulc J 2001 private communication [10] Larive C 1998 Combined Contribution of Experiments and Modelling to the Understanding of Alkali–Aggregate Reaction and its Mechanical Consequences (Paris: Laboratoire Central des Ponts et Chaussees) [11] Larive C, Joly M, Dubois J P and Choquet P 1995 Theoretical and experimental validation of vibrating wire principle. Application to a low stiffness strainmeter for embedment in

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