Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates

1. Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report

2. Purpose • To use Finite Element Analysis (FEA) to perform a modal analysis on functionally graded materials (FGM) to determine modes and mode shapes.

3. Background • FGMs • FGMsare defined as an anisotropic material whose physical properties vary throughout the volume, either randomly or strategically, to achieve desired characteristics or functionality • FGMs differ from traditional composites in that their material properties vary continuously, where the composite changes at each laminate interface. • FGMs accomplish this by gradually changing the volume fraction of the materials which make up the FGM. • Modal Analysis • Modal analysis involves imposing an excitation into the structure and finding when the structure resonates, and returns multiple frequencies, each with an accompanying displacement field.

4. Each Case Frequencies (4) Mode Shapes (4) Plates 1m x 1m, 0.025m and 0.05m thick Case A Compare to theoretical values Case B-D Compare to isotropic Select Cases Compare to Efraim formula Problem Description

5. Methodology • FEA • Modal analysis performed by COMSOL eigenfrequency module.

6. Mori-Tanaka estimation of material properties for FGMs Gives accurate depiction of material properties at certain point in the thickness, dependent on volume fractions and material properties of the constituent materials Get density (ρ), shear (K) and bulk (μ) moduli Use elasticity to get expressions for E and ν Methodology (cont.)

7. Isotropic results matched with theory Reasons for isotropic case Verify FEA model Check plate thickness limit Have baseline for comparison to FGM Results - Isotropic

8. Results – Linear • Represents, on average, a 50/50 metal-ceramic FGM • h=0.05m frequencies were bounded by their constituent materials • Mode shapes 2a and 2b swapped from where they were in isotropic cases

9. Results – Power Law n=2 • Represents, on average, a 67/33 metal-ceramic FGM • Frequencies are bounded by their constituent materials • Mode shapes are changed by addition of ceramic

10. Results – Power Law n=10 • Represents, on average, a 91/9 metal-ceramic FGM, or a metal plate with a thin ceramic coating • Frequencies were very close to isotropic metal frequencies • Mode shapes 2a and 2b highly distorted due to presence of ceramic

11. Comparison to Efraim • Efraim formula is a simple method of determining FGM frequencies by knowing the isotropic frequencies of the constituent materials • Method showed results that were 6-11% lower than the FEA results

12. Conclusions • Using the isotropic and MT checks, a great level of confidence was achieved in the results. • When considering a FGM that is metal and ceramic, the frequency seems to follow the metal while the mode shape seems to the follow the ceramic • A FGM should be thick enough so that enough data is able to be extracted from the cross-section of the plate. • The Mori-Tanaka estimate is heavy computationally, as demonstrated in Appendix C. For future use, the use of µ and K should stand to simplify the calculations. • The FEA results were found to be within 6-11% of the computed values from Efraim. Efraim formula can be used as a starting point reference or sanity check on more complex FGM FEA models.