Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final...
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![Page 1: Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d2b5503460f949ffeee/html5/thumbnails/1.jpg)
Modal Analysis of Rectangular Simply-Supported Functionally
Graded PlatesBy Wes Saunders
Final Report
![Page 2: Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d2b5503460f949ffeee/html5/thumbnails/2.jpg)
Purpose
• To use Finite Element Analysis (FEA) to perform a modal analysis on functionally graded materials (FGM) to determine modes and mode shapes.
![Page 3: Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d2b5503460f949ffeee/html5/thumbnails/3.jpg)
Background• FGMs
– FGMs are 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.
![Page 4: Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d2b5503460f949ffeee/html5/thumbnails/4.jpg)
Problem Description• 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
![Page 5: Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d2b5503460f949ffeee/html5/thumbnails/5.jpg)
Methodology
• FEA
• Modal analysis performed by COMSOL eigenfrequency module.
![Page 6: Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d2b5503460f949ffeee/html5/thumbnails/6.jpg)
Methodology (cont.)• 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
![Page 7: Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d2b5503460f949ffeee/html5/thumbnails/7.jpg)
Results - Isotropic
• Isotropic results matched with theory
• Reasons for isotropic case– Verify FEA model– Check plate thickness
limit– Have baseline for
comparison to FGM
![Page 8: Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d2b5503460f949ffeee/html5/thumbnails/8.jpg)
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
![Page 9: Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d2b5503460f949ffeee/html5/thumbnails/9.jpg)
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
![Page 10: Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d2b5503460f949ffeee/html5/thumbnails/10.jpg)
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
![Page 11: Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d2b5503460f949ffeee/html5/thumbnails/11.jpg)
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
![Page 12: Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d2b5503460f949ffeee/html5/thumbnails/12.jpg)
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.