Patent Application
20120166135
The invention relates to a method for determining a physical parameter representative of a point P of a plate, a device for implementing said method, and a transducer for emitting and/or receiving a wave on said plate.
More particularly, the invention relates to a method for determining a physical parameter representative of a point P of a plate, wherein the physical parameter is chosen from among the thickness of the plate h, the propagation speed of a wave in the plate VP, and the product VPh of the thickness and the propagation speed of a wave in the plate, and said method is implemented by a device comprising:
Methods of this type are known. In particular, in one known method, a vibration wave is generated or emitted at a first point of the plate, the received wave is measured at a second point of the plate, and the time-of-flight of the wave between the first and second points is estimated. The distance separating the first and second points then allows estimating the propagation speed of the wave in the plate.
The accuracy of the time measurement is important to the accuracy of this estimate, and this method requires a high frequency wave sensitive to the condition of the plate surface.
In addition, this method requires a model of the plate in order to determine the plate thickness. The model incorporates the connections and boundary conditions, which increases the complexity of the method and, above all, adds uncertainty related to the values of the parameters for the model used.
The aim of the invention is to offer an alternative to the known methods, and in particular to allow accurately estimating physical parameters of a plate.
For this purpose, the method of the invention is characterized by said method comprising the following steps:
W
contour({right arrow over (r)})=fshape(f)Gplate({right arrow over (r)}−{right arrow over (r)}s)
where
With these arrangements, it is possible to determine a physical parameter of the plate at point P, in a simple and precise manner.
In various embodiments of the method of the invention, one or more of the following may be used:
J
0(Z)=J0(a√{square root over (f)});
Δφ=φ(S2(f)−S1(f))
|bJ0(a√{square root over (f)})|,
where b is another scale parameter, and
|.| is the modulus function,
best approaches:
|S2(f)/S1(f)|
for a set of test frequencies fn;
φ(bJ0(a√{square root over (f)})),
where b is another scale parameter, and
φ(.) is the phase function,
best approaches:
φ(S2(f)/S1(f))
for a set of test frequencies fn;
where
where
where
a test device comprising:
a test method comprising the following test steps:
Δφ(f)=φ(S11(f)S12(f)*)−φ(S21(f)S22(f)*)
where
where
The invention also relates to an imaging method, wherein an image of a plate is constructed, said image comprising a plurality of pixels, each pixel corresponding to a point of the plate and representing a physical parameter of the plate at said point of the plate, said physical parameter of said point being determined by the method defined above.
The invention also relates to a device for implementing the method for determining a physical parameter representative of a point P of a plate according to any of the above, wherein the physical parameter is chosen from among the thickness of the plate h, the propagation speed of a wave in the plate VP, and the product VPh of the thickness and the propagation speed of a wave in the plate, said device comprising:
W
contour({right arrow over (r)})=fshape(f)Gplate({right arrow over (r)}−{right arrow over (r)}S)
where
In various embodiments of the device of the invention, one or more of the following may be applied:
Other features and advantages of the invention will be apparent from the following description of some of its embodiments, provided as non-limiting examples, with reference to the attached drawings.
In the drawings:
a and 4b are a second embodiment of a device of the invention;
a and 6b are a fourth embodiment of a device of the invention;
a, 7b and 7c are a fifth embodiment of a device of the invention;
In the rest of this document, the term “vibration” will be understood as indicating a vibration wave, an acoustic wave, or an ultrasound wave. The wave in question has a frequency, for example, of between 100 Hz and 50 kHz, and preferably between 1000 Hz and 20 kHz, such that inexpensive materials can be used to measure such a wave.
The invention relates to a method for determining a physical parameter representative of a point P of a plate 1, wherein the physical parameter is chosen in particular from among the thickness of the plate h, the propagation speed of a wave in the plate VP, and the product VPh of the thickness and the propagation speed of a wave in the plate.
The method is implemented by a device comprising:
A point P is defined on the plate, corresponding to the location where said physical parameter is to be determined, and a closed contour C surrounding said point P is also defined.
The method is based on the emitting, by an emitter, of a wave on the contour C or, reciprocally, the receiving, by a receiver, of a wave on the contour C. The principle of reciprocity for the propagation of an acoustic or vibration wave in a structure leads to various ways of embodying this method, in which the elements are either emitters or receivers.
The wave can be a vibration, acoustic, or ultrasound wave. It propagates in the plate or on a surface of the plate. This wave can be measured by a receiver, or generated by an emitter.
An emitter or receiver of a wave can be a transducer, for example a device of piezoelectric material attached to the plate. In a receiver mode, the transducer converts a displacement, deformation, stress, or pressure into a voltage, representing a measurement of said displacement, deformation, stress or pressure. The voltage can be converted by an analog-to-digital converter to provide a digital value to a calculation unit. Reciprocally, in an emitter mode, the transducer converts a voltage into a displacement, deformation, stress, or pressure. The voltage can be produced by a digital-to-analog converter of a calculation unit, possibly followed by a voltage amplifier.
Alternatively, the emitter or receiver may have no contact with the plate. For example, an electromagnetic transducer or high power laser operating in pulse mode can be used. Reciprocally, an optical receiver can be used such as a laser vibrometer, adapted for measuring, remotely and without direct contact, a vibration of a point or a multitude of points on a plate.
An emitter or receiver on a contour C may also be realized in multiple ways.
A first possibility is to make use of piezoelectric transducers. A predetermined number T of transducers Ti are attached or placed on the plate, where is a positive integer index between 1 and T. These transducers are placed, possibly regularly, along the contour C, as represented in
In a second possibility, represented in
Such a device therefore integrates a first and second receiver R1, R2 in a manner that may be used to implement the method of the invention. In addition, this device may be low in cost and easy to implement.
In a third possibility, a vibrometer (not represented) is used. The vibrometer advances along the contour C by predetermined increments to produce the measurement of each point on said contour C. The measurement on the contour C will then be calculated digitally by summing the signals from each point.
The same vibrometer can be used to obtain measurements of one or more other points on the plate which are not part of the contour C, such that a single measurement device is used to perform all necessary measurements.
When it is stated in this patent application that a wave is emitted or received on a contour C, it is understood that a predetermined number of interconnected transducers can be positioned on the plate, for measuring or generating this wave for the entire contour C, or that a no-contact sensor such as a scanning vibrometer can be used to perform this function, or any other known means can be used.
In a fourth possibility (not represented), piezoelectric transducers are used of the type presented in the first possibility in
The film may have one side coated with an adhesive for directly attaching all the transducers to the plate. The adhesive will, however, not be present on the entire film but only under the transducers, and the film must have a low elasticity so that the assembly does not locally modify the vibration response of the plate.
As a variant, the transducers may have a side coated with an adhesive for directly attaching them to the plate.
The transducers Ti are connected to each other with flexible electrical conductors formed on the film, and according to the same principle as for the first possibility. The transducers Ti, the transducer P, the film, and the flexible conductors together form a product comprising a first and second receiver R1, R2 for implementing the method. In addition, this assembly is ready for fast and easy placement on a plate.
The theoretical foundation that provides an understanding of the various embodiments of the device and method of the invention is described below.
Using a thin-plate mechanics approach in the frequency domain, the radiation from a point in the plate is governed by the following equation:
D∇
4
w({right arrow over (r)})+ρhω2w({right arrow over (r)})=δ({right arrow over (r)}) (1)
where
where
For an infinite plate, the solution of this equation is a Green's function Gfree for a point of coordinates F and with a vibration or acoustic source placed at point S of the plate and coordinates {right arrow over (r)}s:
H0 is the Hankel function of the first kind.
For a plate of finite dimensions, the vibration wave also results from the interference with multiple waves reflected at the edges of the plate, such that the Green's function Gplate for the point of coordinates F on a plate of finite dimensions, can be written as:
G
plate(|{right arrow over (r)}−{right arrow over (r)}s|)=Gfree(|{right arrow over (r)}−{right arrow over (r)}s)+Crefl(Gfree(|{right arrow over (r)}−{right arrow over (r)}s|)) (5)
where
The radiation from the contour C can then be calculated by summing the localized radiations along this contour, each one calculated using the above formula. In the case of a contour C that is a circle, applying the addition theorem for cylindrical harmonics yields the Green's function Wcircle for a contour C having a circular shape on a plate of finite dimensions:
W
circle({right arrow over (r)})=2πAJ0(kR)Gplate(|{right arrow over (r)}−{right arrow over (r)}s), (6)
where
In addition:
where:
The following product kR is therefore obtained:
Thus the product VPh, multiplying the thickness of the plate h by the propagation speed of the wave in the plate VP, is written:
The Bessel function J0(Z) is an oscillating function, represented in
Z
0≈2.4048
Z
1≈5.5201
Z
2≈8.6537
Z
3≈111.7915
Z
4≈14.9309
Z
5≈18.0711 (10)
The zeros of the Bessel function J0 are spaced apart in a periodic manner, such that, when the frequency of a wave is known, one can determine the products kR corresponding to each zero Zn, and from this can determine the product VPh which multiplies the thickness h by the propagation speed of the wave in the plate VP. Thus a physical parameter of the plate is determined.
Various embodiments of the device are possible, each having multiple variants.
In a first embodiment of the device represented in
In a second embodiment of the device, the device comprises a single receiver R1 and a single emitter E1. If the receiver or sensor R1 is adapted to measure a wave on the contour C of the plate, the emitter E1 is adapted to generate a wave at any point of the plate that is not part of the contour C (
In a third embodiment of the device, the device comprises two receivers R1, R2, but no emitter.
In a fourth embodiment of the device, the device comprises two receivers R1, R2 and one emitter E1.
In a fifth embodiment of the device, the device comprises two emitters E1, E2 and one receiver R1.
Various embodiments of the method are possible, each adapted to one or more of the embodiments of the device. These embodiments of the method are described below.
In a first embodiment of the method, particularly suitable for the first and second embodiments of the device comprising a single receiver R1, said method for determining the physical parameter then comprises the following steps:
The signal s1(t) has zero amplitude for certain frequencies, particularly for antiresonances of the plate structure, but also for particular frequencies of the shape function.
In the case of a contour in the shape of a circle and of a substantially isotropic material and according to equations (6) and (8), the shape function is a Bessel function of the first kind J0. This Bessel function is a function of a scale parameter a multiplied by the square root of the frequency f of the wave:
J
0(Z)=J0(a√{square root over (f)}), and
it is canceled out for the zeros Zn, Zn=a√{square root over (f)}
In the other cases in which the material is not isotropic or the contour C is not a circle, a shape function can be determined numerically, also having zeros for certain specific frequencies.
A set of test frequencies fn is considered, n being a zero or positive integer index of between zero and N, N also being a positive integer, for example equal to five. The test frequencies fn are defined as proportional to the square of the zeros Zn of the Bessel function of the first kind J0. If, for the test frequencies fn, the signal s1(t) has a low or zero amplitude, and for example less than a predetermined threshold S, then these test frequencies f1 correspond to the zeros of the Bessel function J0 and the scale parameter a of the Bessel function can be calculated by:
for any n between zero and N.
A physical parameter can then be calculated.
According to equation (9), the product VPh which multiplies the thickness h and the propagation speed VP of a wave in the plate at point P, can be calculated by:
for n between 0 and N,
where
fn is the test frequency of index n of the set,
Zn is the zero of index n of the Bessel function of the first kind J0, said zero Zn corresponding to said test frequency fn of the same index, and
R is the radius of the contour C.
If the value of the propagation speed VP of a wave in the material of the plate is known, the thickness of the plate at point P can be calculated by:
for n between 0 and N.
If the value of the plate thickness h is known, the propagation speed of a wave in the plate can be calculated by:
for n between 0 and N.
In a second embodiment of the method, particularly suitable for the third and fourth embodiments of the device of the invention comprising at least two receivers R1, R2 and possibly an emitter E1, one of them being on the contour C (
As a result, if the first signal is of the type
s
1(t)=cos(2πft),
then according to equation 6, the second phase-shifted signal s*2(t) can be written:
s*
2
=AJ
0(kR)sin(2πft)
Meaning s(t) is the sum of s1(t) and s*2(t).
Postulating that tan(φ)=AJ0(kR), we obtain:
As a result, for the zeros of the Bessel function of the first kind J0, tan(φ)=0. Therefore φ=0. Under these conditions, the summed signal s(t) is in phase with the first signal s1(t).
To determine whether one signal is in phase with another, any technique may be used in the time or frequency domain.
The second embodiment then also comprises a step in which test frequencies fn are defined as proportional to the square of the zeros Zn of the Bessel function of the first kind J0. If, for the test frequencies fn, the signal s1(t) is substantially in phase with a summed signal s(t) corresponding to the sum of the first signal and the second signal phase-shifted by π/2, then these test frequencies fn, correspond to the zeros of the Bessel function J0.
A physical parameter of the plate can then be calculated, using the formulas 11 to 13 defined above.
This second embodiment of the method is also usable with the fifth embodiment of the device (FIGS. 7a, 7b and 7c) comprising two emitters E1, E2 and a single receiver R1.
In this case:
Then the second signal s2(t) is phase shifted by π/2 to form a second phase-shifted signal s*2(t). The rest of the method is then identical to what is described above.
The two described variants of the second embodiment of the method therefore use phase shifting of the second signal at reception.
In a third embodiment of the method, phase shifting is performed at emission. This third embodiment of the method is particularly suitable for the fifth embodiment of the device comprising two emitters E1, E2 and one receiver R1 (
A summed signal s(t) is then calculated, which is the sum of the first signal s1(t) and the second signal s2(t).
s(t)=s1(t)+s2(t).
If the first signal s1(t) is in phase with the summed signal s(t), for a set of test frequencies fn, the test frequencies fn being proportional to the square of the zeros Zn of the Bessel function of the first kind J0, then these frequencies correspond to the specific frequencies desired.
A physical parameter of the plate can then be calculated, using the formulas 11 to 13 defined above.
As a variant, the emissions from the first and second emitters are simultaneous. In this case, the method for determining the physical parameter then comprises the following steps:
The second emission signal e2(t) is phase shifted by π/2 relative to the first emission signal e1(t).
If the first signal s1(t) is in phase with the first emission signal e1(t), for a set of test frequencies fn, the test frequencies fn being proportional to the square of the zeros Zn of the Bessel function of the first kind J0, then these frequencies correspond to the specific frequencies desired.
A physical parameter of the plate can then be calculated, using the formulas 11 to 13 defined above.
In a fourth embodiment of the method, particularly suitable for the third and fourth embodiments of the device of the invention comprising at least two receivers R1, R2 and possibly an emitter E1, one of them being on the contour C (
The first signal s1(t) and the second signal s2(t) are in phase or in phase opposition for specific frequencies corresponding to the abscissas for which the Bessel function J0 presents a zero.
These frequencies can then be identified:
In the first case, the sign of the real part of the first Fourier transform S1(f) is compared to the sign of the real part of the second Fourier transform S2(f). One must observe the frequency bands in which the signs are identical and the frequency bands in which the signs are opposite. The transition frequencies between these frequency bands allow identifying the specific frequencies of the zeros of the Bessel function.
For example, a test function ƒtest(f) is calculated which has a first value V1 if the signs are the same and a second value V2 if the signs are different:
where V1 and V2 can have any differing values. For example, V1=1 and V2=0.
Specific frequencies fn at which the test function ƒtest(f) changes value are looked for, either from the first value V1 to the second value V2, or conversely from the second value V2 to the first value V1.
A physical parameter of the plate can then be calculated, using the formulas 11 to 13 described above.
In the second case, a phase difference Δφ between the first Fourier transform S1(f) and the second Fourier transform S2(f) is calculated, by:
Δφ=φ(S2(f)−S1(f));
The phase difference Δφ then presents phase jumps between 0 and π or between π and 0, for the specific frequencies desired.
Specific frequencies fn of the phase difference Δφ are looked for, at which said phase difference Δφ has such a jump, said specific frequencies fn being proportional to the square of the zeros Zn, of the function of the first kind J0.
Any technique may be employed for detecting a jump in a function, such as the phase difference. In particular, one can detect an increase or decrease that crosses an intermediate threshold, near π/2, with or without hysteresis.
Once the frequencies are identified, it is then possible to calculate a physical parameter of the plate, using the formulas 11 to 13 described above.
This fourth embodiment of the method is also usable with the fifth embodiment of the device (
In this case:
A first Fourier transform S1(f) of the first signal s1(t) and a second Fourier transform S2(f) of the second signal s2(t) are calculated.
Similarly to the third embodiment of the method, the first signal s1(t) and the second signal s2(t) are in phase or in phase opposition for specific frequencies corresponding to the abscissas for which the Bessel function J0 presents a zero.
In the rest of the method, the specific frequencies are identified in the same manner, either directly by comparing the sign of the real part of the first Fourier transform to the sign of the real part of the second Fourier transform, or indirectly by calculating a phase difference dip, with the rest of the method being identical.
Having determined the specific frequencies, a physical parameter of the plate can be calculated using the formulas 11 to 13 described above.
In a fifth embodiment of the method, particularly suitable for the third and fourth embodiments of the device of the invention comprising at least two receivers R1, R2 and possibly an emitter E1, one of them being on the contour C (
the receiver R2 is used to measure a second signal s1(t), and a second Fourier transform S2(f) of this second signal is calculated.
When the contour C is a circle and the material of the plate is a substantially isotropic material, equation 6 can be applied to identify, for a set of test frequencies fn, the form of the Bessel function of the first kind J0.
This identification can be done:
In the first case, one looks for the parameters a and b of a parametric function |bJ0(a√{square root over (f)})| which draw nearest to |S2(f)/S1(f)| for a set of test frequencies fn, |.| being the modulus function.
In the second case, one looks for the parameters a and b of a parametric function φ(bJ0(a√{square root over (f)})) which draw nearest to φ(S2(f)/S1(f)) for a set of test frequencies fn, φ(.) being the phase function.
Once this identification of the scale parameter a has been made, the link between the abscissa a√{square root over (f)} of the Bessel function J0 and the physical parameter of the plate is established by equation 8.
A physical parameter of the plate can be calculated, using the formulas 11 to 13 described above.
This fifth embodiment of the method is also usable with the fifth embodiment of the device (
In this case:
A first Fourier transform S1(f) of the first signal s1(t) and a second Fourier transform S2(f) of the second signal s2(t) are calculated.
Then, when the contour C is a circle and when the material of the plate is a substantially isotropic material, one can also apply equation 6 to identify, for a set of test frequencies fn, the form of the Bessel function of the first kind J0.
This identification is done either on the moduli, or on the phases, as above, to obtain a scale parameter a.
A physical parameter of the plate can be calculated, using the formulas 11 to 13 described above.
In this manner, one or more embodiments of the method can be applied to each embodiment of the device, and a physical parameter of the plate can be determined in all embodiments of the method.
The above methods can also be applied if the material of the plate is anisotropic. In this case, the contour C will have a shape that is not a circle.
In a first case, the propagation speed of a wave is dependent on the direction according to a law of ellipses of the type:
V
p
2[cos2θ/Vpx2+sin2θ/Vpy2]=1
where
Equation (6) can then be written for a contour that has the shape of an ellipse:
W
ellipse({right arrow over (r)})=2πAJ0(kR)Gplate(|{right arrow over (r)}−{right arrow over (r)}s|)
If two receivers (R1, R2) are used at points of the plate at coordinates {right arrow over (r)} and {right arrow over (r)}2 which are not part of the ellipse-shaped contour, the above relation can be written twice, to determine:
W′
ellipse({right arrow over (r)}1)W′ellipse({right arrow over (r)}2)*=|2πAJ0(kR)|2Gplate(|{right arrow over (r)}1−{right arrow over (r)}s|)Gplate(|{right arrow over (r)}2−{right arrow over (r)}s|)*
where
As a result, the phase of W′ellipse({right arrow over (r)}1)W′ellipse({right arrow over (r)}2)* must be equal to the phase of Gplate(|{right arrow over (r)}1−{right arrow over (r)}s|)Gplate(|{right arrow over (r)}2−{right arrow over (r)}s|)*.
A test method is deduced from this in which one or more emitters (E2) are used, adapted to generate a wave on a test contour Cn having a predetermined ellipse shape, where n is the positive integer index. A test method is applied to each of them to determine the best test contour Cn, meaning the shape of the ellipse, comprising the following steps:
Δφ(f)=φ(S11(f)·S12(f)*)−φ(S21(f)·S22(f)*)
where
The ellipse shape of the test contour Cn then corresponds to an optimum contour, such that the first wave is propagated and spatially superimposed on the plate substantially on the second wave, when the phase difference function Δφ(f) is minimal for a set of test contours Cn to which the above test steps are applied.
With this test method, the optimum shape is determined for the contour C to be used in the method of the invention for which all the equations established for a circle are now usable for the ellipse.
In a more general case, in which the propagation speed of a wave is dependent on the direction according to a law for a predetermined shape, the contour C to be used in all embodiments of the method of the invention will have this same predetermined shape, in order to be able to apply the case of the circular contour C as was done above for the ellipse. In particular, equation 6 will be satisfied and the shape function used will be a Bessel function of the first kind J0.
In other words, the ideal shape of the contour C can be determined by the angular variation of the phase velocity of the first bending mode of the plate. For an isotropic plate, the contour C is circular. For an orthotropic plate, the contour C is elliptical. For any plate, an analysis of the first bending mode may enable predetermining the ideal shape to be used for the contour C.
The contour C may also be of a shape not corresponding to the profile of the speeds in the plate material.
For example, the contour C may be a rectangle as shown in
The method comprises the following steps:
One can then apply one of the previously described methods to the virtual contour CI of radius R. In particular:
|bJ0(a√{square root over (f)})|,
where b is another scale parameter, and
|.| is the modulus function,
best approaches:
|S2(f)/S1(f)|
for a set of test frequencies fn.
A physical parameter is then determined as has already been described.
As shown in
In a first variant applied to a contour C that is square in shape, the contour points Cj are considered to be close to a circle CI of radius R calculated by:
where
One can then apply one of the methods described above to the circle CI with the above calculated radius.
In a second variant represented in
Equation (6) is now written as two equations:
W
circle1(r)=2πAJ0(kd1)Gplate(r),
and
W
circle2(r)=2πAJ0(kd2)Gplate(r).
Using one of the methods described above, one then compares:
W
circle1
J
0(kd2)+Wcircle2J0(kd1),
and
J
0(kd2)J0(kd1)Gplate(r)
to determine a physical parameter of the plate at point P.
In addition, the method using contour points positioned on a rectangular contour C may advantageously be implemented with a scanning vibrometer. The scanning vibrometer will provide vibration measurements for the wave propagating on the plate 1 for a set of points, distributed over a matrix grid on the plate 1.
Thus an image of the plate representing the physical parameter can be calculated, successively using each point of the grid as a reference point P where the physical parameter is to be determined, and the other points immediately surrounding this last point as points belonging to a closed contour C.
The image comprises a plurality of pixels. Each pixel:
In particular, it is possible, for example, to provide an image of the thickness of a plate, remotely and without direct contact, said image having a spatial precision equal to the distance between the measured points. Such an image therefore allows detecting, determining, and localizing a difference in thickness in the plate.
Such products and methods may be implemented for measuring the thicknesses of plates or sheets on large structures (boat hull, aircraft fuselage, storage tank, buildings) or small structures.
They can be used with flat, curved, or tubular plates.
In the case of curved plates or tubes, the waves propagate at speeds varying with the direction of propagation. The device will then advantageously have a closed contour C of an elliptical shape adapted to the curve of the structure, as in the case of a flat plane consisting of an anisotropic material.
Such products and products have numerous industrial applications:
| Number | Date | Country | Kind |
|---|---|---|---|
| 0955629 | Aug 2009 | FR | national |
| Filing Document | Filing Date | Country | Kind | 371c Date |
|---|---|---|---|---|
| PCT/FR10/51618 | 7/29/2010 | WO | 00 | 3/13/2012 |
