Modal Analysis of Rectangular Simply-Supported Functionally Graded PlatesBy Wes SaundersFinal Report

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

BackgroundFGMsFGMs are defined as an anisotropic material whose physical properties vary throughout the volume, either randomly or strategically, to achieve desired characteristics or functionalityFGMs 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 AnalysisModal analysis involves imposing an excitation into the structure and finding when the structure resonates, and returns multiple frequencies, each with an accompanying displacement field.

Problem DescriptionEach CaseFrequencies (4)Mode Shapes (4)Plates 1m x 1m, 0.025m and 0.05m thickCase ACompare to theoretical valuesCase B-DCompare to isotropicSelect CasesCompare to Efraim formula

MethodologyFEA

Modal analysis performed by COMSOL eigenfrequency module.

Methodology (cont.)Mori-Tanaka estimation of material properties for FGMsGives accurate depiction of material properties at certain point in the thickness, dependent on volume fractions and material properties of the constituent materialsGet density (), shear (K) and bulk () moduliUse elasticity to get expressions for E and

Results - IsotropicIsotropic results matched with theoryReasons for isotropic caseVerify FEA modelCheck plate thickness limitHave baseline for comparison to FGM

Results LinearRepresents, on average,a 50/50 metal-ceramic FGMh=0.05m frequencies were bounded by their constituent materialsMode shapes 2a and 2b swapped from where they were in isotropic cases

Results Power Law n=2Represents, on average, a 67/33 metal-ceramic FGMFrequencies are bounded by their constituent materialsMode shapes are changed by addition of ceramic

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

Comparison to EfraimEfraim formula is a simple method of determining FGM frequencies by knowing the isotropic frequencies of the constituent materialsMethod showed results that were 6-11% lower than the FEA results

ConclusionsUsing 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 ceramicA 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 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.