TABLE OF CONTENTS |
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.
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.
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 |
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) |
(2) |
(3) |
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) |
f_{1;1}=484 Hz f_{X}=678 Hz f_{O}=867 Hz
f_{2;1}=1080 Hz f_{2;2}=1787 Hz f_{3;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 E_{0} = 40.8 10^{9} N m^{2}. Compared with values of Young's modulus given from the manufacturer which ranges between E_{0} = 43 * 10^{9} N m^{2} and E_{0} = 45 * 10^{9} N m^{2} 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/m^{3} .
f_{1,1}=295 Hz f_{0,2}=691 Hz f_{2,0}=888 Hz
f_{1,2}=1009 Hz f_{2,2}=1462 Hz f_{0,3}=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 |
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
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%.
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