Buzz-o-sonic and a mixed numerical-experimental technique (MNET) drived from FEMtools were used to analyze the resonant frequency data obtained a high density square alumina plate. The plate was found to contain cracks giving rise to an asymmetrical waveform and high internal friction.

To protect the privacy of the client, the product and client name have been withheld

Although a clean spectrum was obtained when testing the square alumina plate using Buzz-o-sonic, the waveform was asymmetrical indicating that the plate contained defects (figure 1).
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Figure 1: screen shot (62% scale) of Buzz-o-sonic showing power spectrum (top) and
waveform (bottom right). Note the asymmetrical waveform


The square plate was sectioned into six rectangular bars for two reasons:

  1. the unusual waveform was indicative of a defective part. By sectioning the plate into bars, any cracks or pores of significant size would be revealed. Cracks were observed in two of the bars (figure 2).

    2.
  2. the equations relating Young's modulus to the resonant frequencies (frequency equations) are more established for bars than for square plates1-7 because rectangular bar shapes are covered in ASTM E1876 and C1259... Also, the available frequency equations for square plates are for very thin plates (width:thickness>100). For thicker plates, the Young's modulus is often underestimated by ~6-10%5.
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Figure 2: bars no. 5 and 6 containing cracks


 

The waveforms for a cracked vs uncracked bar are shown in figure 3.

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Figure 3: comparison of waveforms obtained on an uncracked and cracked bar. Note the asymmetry in the
waveform for the cracked bar that is similar to the waveform obtained on the original plate (figure 1)


 

The internal frictions of the cracked bars (no. 5 and 6) were also much higher in the longitudinal and torsional modes of vibration (table 1). Increases in internal friction in defective parts is quite common.

Table 1: Internal friction of rectangular bars

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The calculated Young's modulus of the plate and bars are shown in tables 2 and 3 respectively.

Table 2: Elastic constants of the square alumina plate

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Table 3: Elastic constants of the alumina rectangular bars

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The results for bars 5 and 6 are shown in italics because these bars contained large cracks. As can be seen, the Young's modulus of the square plate appeared to be ~9% lower than that of the bars. This discrepancy was attributed to the less accurate equations used to calculate the Young's modulus of the square plate, and not to any real significant differences. To confirm this assumption, the MNET was used to calculate the Young's modulus of the plate (figures 4 and 5) more accurately.

A mesh convergence study provided an optimal mesh density (for a 0.01% frequency tolerance) of 13 x 13 x 3 quadratic elements. The resulting mesh is shown in figure 4.

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Figure 4: mesh for the square plate


The material parameters were identified from the resonant frequencies of the three modes shown in figure 5 using the following starting values: E = 300 GPa, µ = 0.25

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Figure 5: first three vibration modes of the plate (sometimes referred to as AA-1, SS-1 and SS-2 respectively7)


The final material parameters were found in three iteration steps. The frequency residuals (the difference between the simulated and experimental resonant frequencies) indicate a good match between the FE-model and the test data (table 4).

Table 4: Elastic constants of the square alumina plate calculated using MNET

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A value of Young's modulus of 332 GPa corresponds well with the values obtained on the bars 1-4 (table 3). A more detailed analysis with MNET using the bar data for the first four bars produced a more precise value of Young's modulus of 330 GPa for the plate and bars as shown in table 5.

Table 5: Elastic constants of the plate and bars calculated using MNET

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Using the mixed numerical-experimental techniques (MNETs) approach, there is no discrepancy between the elastic properties extracted from the plate- and beam-shaped samples, both sample types provide comparable elastic material properties.

  1. M. Alfano and L. Pagnotta, "Measurement of the Dynamic Elastic Properties of a Thin Coating," Review of Scientific Instruments, 77 056107 (2006)
  2. M. Alfano and L. Pagnotta, "An Inverse Procedure for Determining the Material Constants of Isotropic Square Plates by Impulse Excitation of Vibration," Appl. Mech. Mat., 3-4 287-292 (2005)
  3. Tom Lauwagiea, Hugo Solb, Gert Roebbenc, Ward Heylena, Yinming Shib, Omer Van der Biestc, "Mixed numerical–experimental identification of elastic properties of orthotropic metal plates," NDT&E International, 36 487–495 (2003)
  4. A. A. Wereszczak, R. H. Kraft, and J. J. Swab, "Flexural And Torsional Resonances Of Ceramic Tiles Via Impulse Excitation Of Vibration," Ceramic Engineering and Science Proceedings, 24 (2003)
  5. S. Hurlebaus, "Nondestructive Evaluation of Composite Laminates," NDT.net 4 [3] (1999)
  6. Arthur W. Leissa and Y. Narita, "Vibrations of Completely Free Shallow Shells of Rectangular Planform," J. Sound & Vib. 96 [2] 207-218 (1984)
  7. Arthur W. Leissa, "Vibration of Plates," Acoustical Society of America, Published in 1993; Originally issued by NASA in 1973 (published 1969)