TABLE OF CONTENTS

Abstract 1 Introduction 2 Experimental Setup: 3 Determination of the Stiffness Parameters 3.1 Isotropic Square Plate 3.2 Orthotropic Square Plate Experimental Results and Discussion 4.1 Isotropic Square Plate 4.2 Orthotropic Square Plate Conclusions References

Abstract

Often composites consist of a matrix filled with fibers. These fibers may be aligned in one direction due to production and therefore the material properties are no longer isotropic. The present study aimes to identify the material properties of square isotropic as well as orthotropic plates by a nondestructive method.

The eigenfrequencies of a free suspended plate are measured by Electronic-Speckle-Pattern-Interferometry (ESPI) first. Theory of plate vibrations allows to determine Young's modulus and Poisson's ratio from the measured eigenfrequencies.

1 Introduction:

Vibrations of plates have been frequently studied. Rayleigh [8] and Iguchi [7] laid the theoretical foundation for the study of plate vibrations, while Chladni [2] pioneered experimental investigations of vibrations of plates and on longitudinal and torsional vibrations of rods. Caldersmith [1] presented a practical suitable theory of rectangular orthotropic plate vibrations which was tested by measuring the eigenfrequencies of several wooden plates.

The object of this paper is to present a simple method to determine the material properties using a nondestructive ESPI-technique. First of all a short description of the experimental setup is given. Moreover the next section summarizes the most important plate results which are needed to determine the material properties from measured data. A detailed derivation of these equations is given by Gaul et. al [4] or Hurlebaus and Willner [6]. A series of experiments is performed in order to test the suggested method. These measurements are carried out with an isotropic and an orthotropic carbon fibre composite square plate, respectively. Finally, the experimental results are compared with values found in the literature and by a static bending test, respectively.

2 Experimental Setup:

The experimental procedure consists of measuring the eigenfrequencies and the eigenmodes of a plate. The plate is excited by a shaker that is driven with an harmonic signal from a function generator. Laser detection of these vibrattions is accomplished with an Electronic-Speckle-Pattern-Interferometer (ESPI) that is described in detail in Ref. [5] (Fig. 1).

A single frequency laser light passes an acoustic-optical modulator and is later split into two beams; the reference and the object beam. These beams are travelling two paths of the interferometer. The first contains the sample plate being monitored. The second is the reference beam. The two beams are recombined at a CCD-sensor of the camera and produce the interference pattern. This pattern is recorded on a personal computer and is visualized on a video monitor. Two different methods are used to examine the plate vibrations. Firstly the time-average-method is utilized to determine the eigenfrequencies of the plate. Secondly the eigenmodes of the plate are evaluated by applying the stroboscope-technique.

Figure 1: ESPI setup with shaker driven plate

Figure 2: Suspension of the plate by wires

In reality the free suspension of a plate is impossible. Therefore a compromise has to be found of the free boundary condition on the one side and the avoidance of the rigid body motion on the other side. Earlier experiments showed that there is no significant difference of eigenfrequencies between supporting the plate on thin needles or suspending the plate with soft wires. Therefore the soft wire was chosen (Fig. 2).

3 Determination of the Stiffness Parameters:

In order to get the relation between the eigenmodes and the stiffness parameters the eigensolution of the plate has to be determined. It is obvious that a closed form solution is not available. Therefore an approximate solution is used which is based on the concept of sinusoidal equivalent length. This concept provides suitable relations between eigenfrequencies and material properties. The results are based on linear elastic material behavior.

3.1 Isotropic Square Plate
First the O- and X-mode of the isotropic square plate have to be determined. Poisson's ratio v can be obtained by ([1])

(1)

The next step is to calculate the fundamental frequency f_2,0 = f_0,2 with

(2)

and to specify the Young's modulus

(3)

where l is the side length, h is the thickness of the square plate and is the material density, respectively.

3.2 Orthotropic Square Plate
For the orthotropic square plate it is possible to determine the frequencies of higher modes from the fundamental mode frequencies f_2,0, f_0,2 and f_1,1 (or vice versa) using the following frequency equation:

(4)

It is important that the expressions in the brackets of equation (4) have to be greater or equal to unity. Otherwise the expressions have to vanish. Suppose the fundamental modes are specified, one obtains the stiffness parameter
order to calculate the five unknown constitutive parameters E_x, E_y, G_xy v_xand v_y, respectively, from the last four equations (8) - (11) one more condition is necessary. First of all one obtains the ratio ff of the Young's moduli from eq. (8) and (9) which is equal to the ratio of the Poission's ratios (10) and (11)

For the missing condition it can be assumed that v_x v_y = v₂ , where v is Possion's ratio of a corresponding isotropic plate. This is possible because v is usually in the range between 0 and 0.5. Moreover the Poisson's ratios are different and therefore it follows that v_x v_y has to be smaller than 0,25. With this assumption the Young's moduli can be calculated by using eq. (8) and (9) of the orthotropic plate Furthermore with eq. (12) it is possible to determine a and Posson's ratios together with (8):
The shear modulus can be calculated by using eq. (10)
Therefore all required relations are known.

4 Experimental Results and Discussion:

4.1 Isotropic Square Plate
A carbon fibre composite laminate is used as material for a quasi-isotropic plate because it is possible to compare the measured properties with values published in literature. The length of the plate is l = 150 mm, the thickness is h = 3 mm and the density is = 1440 kg/m³ .

f1;1=484 Hz fX=678 Hz fO=867 Hz f2;1=1080 Hz f2;2=1787 Hz f3;0=2006 Hz

Figure 3: Eigenmodes and eigenfrequencies of the quasi-isotropic plate

The eigenfrequencies and the eigenmodes of the quasi-isotropic plate are shown in Fig. 3. Using equation (1) and the eigenfrequencies of the X-mode and O-mode one can get Poisson's ratio as v = 0.34. Applying equation (2) and (3) one obtains Young's modulus as E₀ = 40.8 10⁹ N m². Compared with values of Young's modulus given from the manufacturer which ranges between E₀ = 43 * 10⁹ N m² and E₀ = 45 * 10⁹ N m² the deviation of the measured parameter is 7.3 % of the arithmetic means of the given value.

This is a relatively small deviation as has been shown in a technical report by Frederiksen [3]. The determination of material properties by vibration measurements of plates is critical, especially Poisson's ratio depends very sensitive on errors in measurements and ratio as well as uncertainties of other parameters. A possible reason for such an error is for example the deviation of exact isotropy. Moreover the actual vallues, might be different from the values presented in the literature.

4.2 Orthotropic Square Plate
As an representative of an orthotropic plate again a carbon fibre composite laminate is examined. The difference between the previous described plate is that the numbers of carbon fibre laminates in x-direction are different from the y-direction. 45 % of the fibres are aligned unidirectional while the remaining fibres are in the 0° and 90° direction. The geometric magnitudes are the same as the quasi-isotropic plate, except that the orthotropic plate has a thickness of h = 3.2 mm. The density of this material is = 1416.6 kg/m³ .

f1,1=295 Hz     f0,2=691 Hz    f2,0=888 Hz f1,2=1009 Hz    f2,2=1462 Hz    f0,3=1864 Hz

Figure 4: Eigenmodes and eigenfrequencies of the orthotropic plate

The eigenfrequencies and the eigenmodes of the orthotropic plate are shown in Fig. 4. With the values of the twisting mode f_1,1 = 295 Hz and the bending modes of f_0,2 = 691 Hz and f_2,0 = 888 Hz, respectively, one obtains from equations (5), (6) and (7) the following stiffness parameters:

Figure 5: Experimental setup of the static bending test

D _x = 53.1 10⁹ N m², D _y = 32.2 10⁹ N m², D _xy = 14.8 10⁹ N m² . Together with the assumption that v_x v_y is the same as v² of the quasiisotropic plate (v² = 0.12) and equations (12) to (17) the material properties are determined as



E_x = 46.6 10⁹ N m²,
E_y = 28.3 10⁹ N m²,
G_xy = 0.23 10⁹ N m²,
v_x = 0.45 ,
v_y = 0.27 .


It is obvious that the parameters describe an orthotropic material. The reason is the reinforcement by carbon fibers. However, in comparison with the quasiisotropic plate, Young's modulus is decreased in one direction while it is increased in the other direction. A search for the material properties of orthotropic carbon fibre composite plates in the literature produced no result. In order to compare the measured values, a simple static bending test was done. The setup of this experiment is shown in Fig. 5.

Briefly, the orthotropic plate is supported on two parallel sides and is loaded in the middle of the plate parallel to the line of support by a small weight (line force). The deflection due to the weigth is measured by a vibrometer. The advantages of applying a vibrometer is that the plate is not loaded by a sensor and there is no need to integrate the signal, because it gives directly a signal of displacement. Utilizing the equation of bending of a simply supported beam leads to the displacement w at the middle of the beam



where l_x is the distance between the suspensions, I is the moment of inertia of area, I = ly h³/12, and F is the force at the center of the beam. In order to extend the previous equation to plates the flexural stiffness B_x = E_xh³ =12(1 ody and g = 9.81m/s² is the gravitational acceleration, one obtains the stiffness parameter as

For a mass of m = 0.087 kg and the distance l_x = 0.145 m one obtains a displacement of w_x = 2.66 µm and w_y = 3.8 µm, respectively. Applying the relation given above leads to the stiffnesses of
D_x = 49.8 * 10⁹ N/m² ;
D_y = 34.8 * 10⁹ N/m² .

By comparing the stiffness parameters of the bending experiment with the one obtained from the eigenfrequencies one can easily see that the results differ only in the range of about 8%.

5 Conclusion:

The present paper shows that Young's moduli and Poisson's ratios of isotropic and orthotropic square plates can be obtained by measuring the eigenfrequencies using the ESPI-technique. The study investigates approximations of free suspended plates. The presented method allows to get further information about the frequency dependency of the material properties.

References

1. Caldersmith, G.W. : Vibrations of orthotropic rectangular plates. In: Acoustica 56 (1984), S. 144--152
2. Chladni, E.E.F. : Die Akustik. Leipzig: , 1802
3. Frederiksen, P.S. : Parameter uncertainity and design of optimal experiments for the estimation of elastic constants. Technical Report 527, DCAMM, Technical University of Denmark, Lingby, Denmark, 1996
4. Gaul, L. , Willner, K. , ; Hurlebaus, S. : Determination of Material Properties of Plates from Modal ESPI Measurements. In: Proc. of the 17th IMAC Orlando, Florida. Bd. I. 1999
5. Hurlebaus, S. : . Holographische Schwingungsanalyse mit Hilfe der Electronic-Speckle- Pattern-Interferometry - Grundlagen und Anwendungen. Institute A of Mechanics, University of Stuttgart, 1995. Studienarbeit
6. Hurlebaus, S. ; Willner, K. : Bestimmung von Materialdaten aus Eigenschwingungen am Beispiel der orthotropen Rechteckplatte. Institute A of Mechanics, University of Stuttgart: Technical Report, 1996
7. Iguchi, S. : Die Eigenschwingungen und Klangfiguren der vierseitig freien rechteckigen Platte. In: Ingenieur-Archiv 21 (1953), Nr. 5/6, S. 303--322
8. Rayleigh, J.W.S. : The theory of sound. New York: Dover, 1945