Transcript Final Presentation.ppt

Modal Analysis of Rectangular
Simply-Supported Functionally
Graded Plates
By Wes Saunders
Final Report
Purpose
• To use Finite Element Analysis (FEA) to
perform a modal analysis on functionally
graded materials (FGM) to determine modes
and mode shapes.
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.
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
Methodology
• FEA
• Modal analysis performed by COMSOL
eigenfrequency module.
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
Results - Isotropic
• Isotropic results
matched with
theory
• Reasons for
isotropic case
– Verify FEA model
– Check plate thickness
limit
– Have baseline for
comparison to FGM
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
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
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
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
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.