6th International Conference on Vibration Measurements by Laser Techniques (Proc. SPIE vol. 5503), Ancona (Italy), 22-25 June 2004, 169-176. (http://www.aivela.org/general.htm)

EFFECTIVE PARAMETER IDENTICAFATION OF 2D STRUCTURES FROM MEASUREMENTS USING A SCANNING LASER VIBROMETER

Jean-Claude Pascal (a), Jing-Fang Li (b), Xavier Carniel (c) (a)

Laboratoire d'Acoustique de l'Université du Maine (UMR CNRS 6613) and Ecole Nationale Supérieure d'Ingénieurs du Mans (ENSIM) Université du Maine, rue Aristote, 72000 Le Mans, France. Email: [email protected] (b)

Visual VibroAcoustics, 51 rue d'Alger, 72000 Le Mans, France Email: [email protected] (c)

Centre Technique des Industries Mécaniques (CETIM) 52 avenue Felix-Louat, BP 80067, 60300 Senlis, France, Email: [email protected]

ABSTRACT For specific materials, it is necessary to know the dispersion curve associated with the flexural waves, particularly in the case of anisotropic composite materials whose parameters are not known a priori. The method described here is based on energy concepts. It uses the vibration velocity measured by scanning laser Doppler vibrometer. The Spatial Fourier Transform (SFT) of vibration velocity field is used to compute the spatial derivatives of velocity which are then used to obtain energy quantities. The dispersion curve and the loss factor are computed from these quantities. It is shown that the use of these energy quantities leads to results independent to the analysis resolution in the wavenumber space, contrary to the direct extraction obtained by looking for the maximum values in the phase spectrum. The pre-processing allows one to eliminate the disturbances brought by the mechanical excitation zones and plate edges. Furthermore, the mechanical parameters for plane structures such as dynamic stiffness and loss factor, are estimated by this technique. This method can be extended to orthotropic and anisotropic structures. Experimental results are presented. Keywords: Optical measurements, Scanning laser vibrometry, dispersion curve, loss factor, wavenumber processing, structural intensity

1. INTRODUCTION In medium and high frequency domains, analyses based on wave approaches are often replaced the modal methods. Energy approaches, which use basically wave representation, are more and more employed to predict the dynamics of structures, which leads to local (WIA, conductivity) or global methods (SEA). In addition, the techniques of measurements of structural intensity are used to characterize behaviours between substructures in an assembly or applied in active control procedure for limiting vibration transfer. To use all these methods, it is imperative to know the dispersion curve associated with the flexural waves, particularly if it deals with anisotropic composite materials because their parameters are not known a priori. For a long time the finite-difference-approximation method has been proposed to estimate the flexural wavenumbers in one-dimensional structures such as beams from vibrating measurements by the use of three accelerometers1. This method is directly based on the wave equation associated with the far-field approximation and the

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finite difference method makes it possible to compute the second order spatial derivative of transverse displacement. This method was validated under various experimental conditions 1, 2, 3. The main disadvantage of this method is a too high sensitivity to phase differences between sensors due to the use of the finite difference technique. Another method was proposed to determine the maximum of wavenumber spectrum in beams, which was then used to identify the value of natural flexural wavenumber. To reduce the distortions brought by Spatial Fourier Transform (SFT) a regressive method was proposed 2. The comparison 3 of several techniques on beams shows that both the finite-difference method using 3 sensors and the regressive method for computations of wavenumber give equivalent results whereas the use of the direct Fourier Transform results significant errors in the computations because of truncated signals. Recently, a method has been published, which allows one to extract flexural wavenumbers in 2D structures by spatial correlation using a wave model 4, 5. The method described in this work is quite different from the existing methods since it is based on an energy concept. The vibration velocity measured by a scanning laser vibrometer is processed in this method. The Spatial Fourier Transform of vibration velocity provides one with the possibilities to compute the spatial derivatives of order m ( m = 1,2,3,L) that are necessary to calculate energetic quantities. It is shown that the use of the energetic quantities allows an estimation that is independent of the resolution of the analysis in wavenumber domain, contrary to the direct extraction obtained by searching the maximum values in the spectrum of phase. A pre-processing makes it possible to eliminate the disturbances brought by the zones of mechanical excitations and edge distortions. By this technique, the mechanical parameters such as dynamic stiffness, loss factor, for isotropic structures can be estimated. The extension of this technique to orthotropic and anisotropic structures thus authorizes the construction of an equivalent model usable for calculations in medium frequency range. Examples are given of the results of measurements.

2. ESTIMATION OF FLEXURAL WAVENUMBER IN TWO-DIMENSIONAL STRUCTURES The idea is to express the effective wavenumber and loss factor from an energetic analysis of flexural vibrations. In this section the expressions for estimations of flexural wavenumber are derived from the equation of plate-like structure. It is shown that the parameters in structures can be expressed in complex vibrating velocity and its derivatives. The complex vibrating velocity can easily be measured using the optical measurement techniques such as the scanning laser vibrometry 6 and holographic interferometry 7. Therefore the parameters in structures can be estimated by using only the complex velocity measurements. 2.1 Energetic expressions In a plate that is excited by one or more mechanical forces (shakers for example), it is possible at the first approximation to neglect structural dissipation and the losses by radiation. Indeed, these quantities are always smaller than the power injected by mechanical excitations. Considering a thin isotropic plate, the equation of Kirchhoff is expressed by

(

) ∑ F δ (r − r ) ,

1 D ∇4 v − ω 2 ρ h v = jω

i

i

(1)

i

with v is the flexural velocity and Fi point mechanical forces applied on the plate. D is the bending stiffness of plate, ∇ 4 is the bi-harmonic operator (double Laplacian), ω the angular frequency, ρ h the surface mass density of the plate. By using the expression of natural flexural wavenumber in vacuum k B4 = ω 2 ρ h D , Eq. (1) can be written in the following form

(

) ∑ F δ (r − r ) .

D 4 ∇ v − k B4 v = jω

i

i

(2)

i

An energetic equation was obtained if both sides of the equation (2) are multiplied the complex conjugate of the velocity

(

)

D 2 ∇ 4 vv ∗ − k B4 v = j 2ω

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∑ Π i δ (r − ri ) , i

(3)

with Π i =

1 2

Fi v ∗ (ri ) = Wi + jQi , the complex power associated to mechanical forces. Taking the real parts on the left- and

right-hand-sides of Equation (3) results in the following expression:

{

} ∑W

D Im ∇ 4 vv ∗ = 2ω

δ (r − ri ) .

i

(4)

i

The right-hand side of Equation (4) shows the mechanical powers (active) which can be injected by a force or absorbed by a sink of energy formed by passive damper or active absorber using a shaker. Whereas the left-hand side of Eq. (4) is recognised to be the divergence ∇ ⋅ I s of the structural intensity in the plate 8. For a non-dissipation plate, outside the force excitation and energy injecting or absorbing zones ri , the divergence is zero, that is: ∇ ⋅ I s = 0 for r ∈ Ω 0 . In the zero-divergence zones, the following equation is obtained by considering the imaginary parts of the both sides of Eq. (3):

{

}

Re ∇ 4 vv ∗ − k B2 v

2

=0,

(5)

from which estimators for measuring the wavenumber of flexural waves k B is given by

γa

In this paper, the brackets zones.

{

 Re ∇ 4 v v = 2  v 

}

   

{

 Re ∇ 4 v v  γb = 2  v 

1/ 4

or

}    

1/ 4

.

(6a, 6b)

denote the spatial average over the points r ∈ Ω 0 , that is, outside the mechanical excitation

2.2 Determination of flexural wavenumber Equations (6a) and (6b) show that the estimators of the bending wavenumber contain the double Laplacian ∇ 4 v of the vibrating velocity. It was shown in the previous work 6 that the double Laplacian can efficiently be computed by the use of the wavenumber processing. The basic expression of this technique is to perform Spatial Fourier Transform (SFT) to the vibrating velocity v . The partial derivatives of any order for v are computed in the wavenumber domain by

∂ m + n v ( x, y ) ∂m x ∂n y

SFT

a

( − jK x ) m ( − jK y ) n V ( K x , K y ) , with m = 1,2,3L M , n = 1,2,3L N .

(7)

where V ( K x , K y ) is the Spatial Fourier Transform of v (x , y ) . The Fourier transform of double Laplacian of the velocity is then given by:

∇4v

SFT

a

(K

2 x

+ K 2y

)

2

V (K x , K y ) .

(8)

As demonstrated in a previous paper 6, large distortions cannot be avoided when using SFT to compute the derivatives of high orders because of amplifying the components of high wavenumber values due to truncated signals. The higher is the order of the derivatives, the larger is the distortions brought by the use of SFT. In order to cancel or reduce these distortions, special measures should be taken in wave number processing. An efficient technique – mirror method – has been proposed and applied in wavenumber processing 6. Meanwhile a k-space filter is applied to the spectrum of velocity V ( K x , K y ) in order to reduce the components contaminated by noises during experimental measurements. These techniques together with SFT are used in the estimations of the bending wavenumber and loss factor from the normal vibrating velocity in the plate, which is measured by the use of a scanning laser vibrometer.

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2.3 Method of excluding unwanted data As stated in the previous section, the formulas for estimations of the effective wavenumbers are based on the assumptions of no mechanical power injected into the plate. However in reality there are often ‘sinks’ and ‘sources’ zones produced by mechanical excitations. These zones should not be included in the computation of the bending wavenumber if equations (5) and (6) are used. In the following is shown an example of a brass plate of 350 × 200 × 3 mm. The plate is excited by a shaker. The energy in it is dissipated by a damper positioned in its centre. The normal vibrating velocity is measured by

{

}

using a scanning laser vibrometer over a grid of 95 × 54 points. Figure 1(a) shows the map of Im ∇ 4 vv ∗ in the plate, from

{

which one can noticed that Im ∇ vv 4

∗

} is proportional to the external power in the plate towards outside where the ‘hotpots’

due to the forces of excitation and dissipation can be seen. Figure 1(b) shows the histogram of γ a4 calculated over the entire plate, meanwhile the unwanted data, in particular negatives values are shown in this Figure 1(b).

{

}

Figure 1 (a) – Map of Im ∇ 4 vv ∗ proportional to external power flow due to forces acting on the brass plate (left). (b) – Histogram of the bending wavenumbers γ a4 for all points (right). The frequency is 1500 Hz.

{

Figure 2 (a) – Vector of cumulative energy obtained from the values of Im ∇ 4 vv ∗ of γ a (right). Excluded points are represented by green circles.

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} setting in increasing order (left). (b) Map of values

In fact, those values correspond to energy transfers due to external forces, as it shows in Equation (4) and in Figure 1(a). In

{

}

order to remove the undesirable values, a vector is constructed by sorting the values Im ∇ 4 vv ∗ of the map in Figure 1(a) in ascending order. The cumulative values of this vector, which presents the dissipative power, are plotted in Figure 2(a). The steep increase in the curve corresponds to the contributions of the external forces. A threshold value is selected with the help of the curve and marked by a point in Figure 2(a). The values greater than the threshold value are removed from the computations. Therefore, exclusion of these points from the computations of the bending wavenumber results in removing the force excitation zones in the plate. Good results are then obtained by using the estimator γ b . They are a little bit improved by removing the points corresponding to the values in the bins that are too far from the median bins in the histogram γ a . The results obtained by this method are shown in Figure 3 for the frequencies of 1500 Hz, 2500 Hz and 3500 Hz. Figure 3 (a) shows the histograms of γ a at each of these frequencies. The values of ω 2 γ b4 that correspond to the mechanical characteristics D ρ h of the plate are shown in Figure 3 (b) and compared with those shown in Tables (horizontal line).

Figure 3 – (a) Histograms of γ a using the selected points r ∈ Ω 0 for three frequencies (left). (b) Values of ω 2 γ b4 compared to the table value for a brass plate of 3 mm thickness (right).

3. ANALYSIS CONSIDERING DISSIPATIVE TERMS In the previous analysis, the dissipated terms have been neglected. In reality there are two kinds of dissipations : losses due to structural dissipation and losses by radiations. When considering these losses, Equation (1) can be written in the following form

(

)

1 D ∇ 4 v − ω 2 ρ h v = − pa + jω

∑ Fi δ (r − ri ) , i

(9)

( ) ( )

where D = D (1 + jη ) is the complex bending stiffness, with η the structural loss factor. p a = p z + − p z − denotes the acoustic radiation pressure on the two sides of the plate. Performing the similar processing to Eq. (9) as to equation (3), the following energetic relationship is obtained:

173

(

)

1 2 D ∇ 4 vv ∗ − ω 2 ρ h v = − I n − j J n + j 2ω

∑ Π i δ (r − ri ) ,

(10)

i

p a v ∗ , where I n and J n are the normal components of active and reactive acoustic intensity (exterior to the two faces of the plate). Decomposing the real and imagery part of equation (10) gives the following two equations with I n + j J n =

1 2

{

}

{

}

Im ∇ 4 vv ∗ + η Re ∇ 4 vv ∗ =

{

}

{

}

Re ∇ 4 vv ∗ − η Im ∇ 4 vv ∗ − k B4 v

   − I n + Wi δ (r − ri )   i    2ω  = J n − Q i δ (r − ri )  D  i 

2ω D 2

∑

∑

(11a)

(11b)

It can be shown from Eq. (11b) that the estimator γ b can be used to obtain the wavenumbers with a negligible bias if the points selected for computations are outside the excitation zones. In fact, the term 2ω J n D v

{

∗

}

2

can be neglected for a light

that is proportional to the loss factor have been removed with the fluid such as the air and the large values of Im ∇ vv rejected points. An estimator of the loss factor can be got from Equation (11a) if the points outside of the excitation zones due to external mechanical forces are removed:

ηt =

4

{ } = η +η Re{∇ v v }

− Im ∇ 4 v v * 4

*

a

.

(12)

Eq. (12) shows the total loss factor includes (i) the structural loss factor η and (ii) the loss factor due to acoustic radiations ηa which can, according to Eq.(11a), be expressed by

ηa =

In 2ω D Re ∇ 4 vv ∗

{

}

.

(13)

Expressing the flexural wavenumber from the estimator Eq. (6) yields the following equation

Re{∇ 4 vv ∗ } ≅ k B4 v

{

Substituting the expression for Re ∇ 4 vv ∗

}

= (ω 2 ρ h D ) v

2

2

into Eq.(13) yields

ηa =

In ωρ h v

2

=2 2

ρ 0c σ , ωρh

(14)

 ρ c v 2 2  (the factor 2 denotes the radiation of  0    the two edges of the plate). The ratio ρ 0 c (ω ρ h ) = β is known as the fluid loading factor 9. The maximum-magnitude order for the radiating loss factor can be estimated by evaluating the expression (14) at the critical frequency of the plate,

with the consideration of the radiation efficiency coefficient 2σ = I n

when σ ≈ 1 . For the brass plate in the air, we have η a < 9.4 ×10 −4 . By varying the number of the selected points, the total loss factor estimated by Eq. (12) is 7 × 10 -4 < η T < 5 × 10 −3 .

174

4. EXPERIMENTAL RESULTS OF EFFECTIVE PARAMETERS This section is mainly to show the typical results of the effective wavenumbers and loss factor computed from the complex velocity by using the method developed in the paper. The complex velocity employed in the calculations is experimentally measured by the use of the optical technique, vibrometer scanner, which was described in previous paper 10. Briefly describing, a scanning vibrometer, together with LMS software were used for the data acquisition. It uses a OFV 300 optical head from Polytec, which comprises a single point interferometer with a motor-driven control focusing device. Two galvodriven mirrors direct the laser beam horizontally and vertically. It allows one to obtain easily the 1024 (32 × 32) measurement points with a high resolution in a large frequency range. The test assembly shown in Figure 4 consists of two steel plates of thickness 1 mm. The size of each plate is 850 mm × 475 mm. These two plates are connected in a such way so that the length of the joint overlap is 20 mm, resulting in the size of the test assembly being 850 mm × 930 mm . The two opposite edges parallel to the joint line are clamped. The two other edges of the plate are free. A normal point force is acting on the plate to produce flexural wave spreading all over the plate assembly. The normal complex vibration velocity obtained from cross-spectrum between output signal of the vibrometer and the excitation force signal is collected at each prescribed position on the test assembly at 512 frequencies sampling in the range from 512 to 1022 Hz. The processing algorithm in wavenumber domain to calculate the effective wavenumber and loss factor in the test plate from the knowledge of the vibrating velocity on the plate v ( x , y ) is implemented as follows:

20

930 mm

free edge

•

Input the measured complex vibrating velocity data v(x,y) in a frequency range and plate geometry.

•

clamped edge

Perform the Spatial Fourier Transform V ( K x , K y ) of v ( x , y ) using mirror technique described in Ref. 6.

y

•

Apply k-space window to the velocity spectral and obtain the windowed spectral V ( K x , K y )W ( K x , K y ) to remove

850 mm x

the noise contaminated components. Figure 4 – Configuration of test plate assembly.

•

Calculate the double Laplacian the velocity in wavenumber domain by the use of Eq. (8). Then compute the inverse Fourier transform to obtain the ∇ 4 v in space domain.

•

Calculate the energetic quantities ∇ 4 vv ∗ and v

•

Use these quantities to compute γ a that is then used to remove the points in excitation or loss zones.

•

Calculate the estimators γ b for the effective wavenumber of Eq. (6b) and ηT for the total loss factor of Eq. (12).

2

The procedure described in the previous section is taken in getting the results shown in Figure 5. The curve in dashed line is the wavenumber expression

ω (ρ h De )1 4 obtained from the effective bending stiffness that is calculated by using the following

1 De = ω 2 − ω1

ω2

ω 2 ρh

∫ γ b4 (ω ) dω

ω1

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Figure 5 – (a) Dispersion curve and (b) effective loss factor of the test plate shown in Figure 4.

5.

REMARKS AND CONCLUSIONS

The effective parameters – wavenumbers, loss factors and bending stiffness – can be estimated from the knowledge of the complex velocity, which can be measured by the use of a scanning laser vibrometer. The formulas for computing these parameters are obtained from energetic concepts. However they are based on the assumption of no mechanical power injected into the plate. Therefore if there are injecting or excitation zones, the computations of the effective parameters should not include in these zone. An efficient method is developed to remove the values of the excitation zones to make the formulations being widely useful. Very accurate results are given of the dispersion curves respectively for the brass plate with free-free boundary conditions and for an assembly plate-like structure. The main interests of these techniques are that they are independent of the resolution in wavenumber domain, which is in contrary to other methods. The total loss factor can be obtained and given a good order of magnitude in low dissipative structures. The first results are very encouraging to be used for measuring viscoelastic material. Therefore, this method can be extended to the measurements of material characteristics of orthotropic and anisotropic structures.

REFERENCES 1.

X. Carniel, J.-C. Pascal, "Caractéristiques de propagation et mesure du flux d'énergie vibratoire dans les barres (Progation characteristics and vibrational power flow measurements in beams)", Proceedings of the 2nd International Congress on Acoustic Intensity, (CETIM publications, ISBN 2-855400-054-4), Senlis, France, 23-26 September 1985, pp. 211-217. 2. J.R.F. Arruda, J.P.R. Campos, J.I. Piva, "Measuring flexural power flow using in beams using a spatial domain regressive discrete Fourier series", Proceedings of ISMA 21 Conference, Leuven, Belgium, 18-20 September 1996, pp. 641-652. 3. M. Noël, F. Simon, J.-C. Pascal, "One dimensional theoretical and experimental wave number extraction for structural intensity", paper N°385, Inter-Noise 2001, The Hague, Netherlands, 27-30 August 2001. 4. N.S. Ferguson, C.R. Halkyard, B.R. Mace, K.H. Heron, "The estimation of vawenumbers in two-dimensional structures", Proceedings of ISMA 2002, Leuven, 16-18 September 2002, pp. 799-806. 5. J. Berthaut, M.N. Ichchou, L. Jezequel, "Identification large bande des paramètres de structures 2-D anisotropes dans l'espace des nombres d'ondes", Mécaniques & Industries 4, pp. 377-384, 2003. 6. J.-C. Pascal, J.-F. Li, X. Carniel, "Wavenumber processing techniques to determine structural intensity and its divergence from optical measurements without leakage effects", Shock and vibration 9 (1-2), pp. 57-66, 2002. 7. J.-C. Pascal, X. Carniel, V. Chalvidan, P. Smigielski, Determination of phase and amplitude of vibration for energy flow measurements in a plate using holographic interferometry, Optics and Lasers in Engineering 24, pp.1-18, 1996. 8. J.-C. Pascal, T. Loyau, X. Carniel, "Complete determination of structural intensity in plates using laser vibrometers", J. Sound Vib. 161(3), pp. 527-531, 1993. 9. W.K. Blake, Mechanics of flow-induced sound and vibration, Academic Press, 1986. 10. J.-C. Pascal, X. Carniel, J.-F. Li, "Characterisation of dissipative assembly by structural intensity and optical measurements", 5th International Conference on Vibration Measurements by Laser Techniques, Ancona, Italy, 18-21 June 2002, pp. 180-191

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