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# Modal Analysis of a Rectangular Plate - PPT

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• 1. MAE 501 INDEPENDENT STUDYModal Analysis of a Rectangular Plate
By
Sasi BhushanBeera
#35763829
• 2. Problem Details
Problem statement:
To determine the lowest six non-zero frequencies and associated mode shapes for a rectangular plate for three different thickness of the plate.
Geometry of the plate:
Rectangular Plate with length 2a, width 2b and thickness h.
γ = a/b=2
ξ = h/b = 1/4, 1/16, 1/64
a = 20m, b = 10 mm, h = 2.5mm, 0.625mm, 0.15625m
Thus, the dimensions of the plate are as follows:
Length of the plate = 40mm
Width of the plate = 20mm
Thickness of the plate = 2.5mm, 0.625mm, 0.15625mm
• 3. Problem Details
Material Properties of the plate:
• ν = Poisson’s ratio = 0.05
• 4. Material of the plate : Cast Iron
• 5. E = Modulus of Elasticity = 139.7GPa
• 6. ρ = Density = 7300 kg/m3
Boundary Conditions:
• The two adjacent edges of the rectangular plate are fixed while the other two are free.
Element Details:
• The modal analysis is performed using two types of elements i.e. solid and shell.
• Meshing:
• A 3-D model of the plate was created and the plate was meshed using SOLID45 element.
Solver used for Modal Analysis:
The modal analysis was performed using the PCG Lanczos solver.
Applying boundary conditions:
The two adjacent sides of the plate are fixed in x and y directions and all the four sides are constrained in z direction.
C0nvergence Criteria:
Natural Frequency: For the problem to converge, the variation of all the six natural frequencies between two iterations should be less that 1%.
Modal Analysis using Solid45
• 7. Case 1: h = 2.5 mm
Results
Case 2: h = 0.625 mm
• 8. Results
• As the thickness of the plate decreases the frequency values go on decreasing as the mass and the dimensions of the plate are decreased.
• 9. The lower frequencies converge quickly as compared to the higher frequencies.
• 10. The convergence details of each case are plotted graphically below:
• 11. As the thickness of the plate decreases the frequency values go on decreasing as the mass and the dimensions of the plate are decreased.
• 12. The lower frequencies converge quickly as compared to the higher frequencies.
• 13. The convergence details of each case are plotted graphically below:
Case 3: h = 0.15625 mm
-As the thickness of the plate decreases the frequency values go on decreasing as the mass and the dimensions of the plate are decreased.
-The lower frequencies converge quickly as compared to the higher
frequencies.
• 14. Case 1: h = 2.5 mm Case 2: h = 0.625 mm Case 3: h = 0.15625 mm
Convergence
• 15. Meshing:
• A 2D-model of the plate was created and the plate was meshed using SHELL63 element.
• 16. The thickness of the plate was entered as a real constant of the SHELL63 element.
Solver used for Modal Analysis:
The modal analysis was performed using the PCG Lanczos solver.
Applying boundary conditions:
The two adjacent sides of the plate are fixed in x and y directions and all the four sides are constrained in z direction.
C0nvergence Criteria:
Natural Frequency: For the problem to converge, the variation of all the six natural frequencies between two iterations should be less that 0.03%.
Modal Analysis using Shell63
• 17. Case 1: h = 2.5 mm
Results
Case 2: h = 0.625 mm
• 18. Results
• As the thickness of the plate decreases the frequency values go on decreasing as the mass and the dimensions of the plate are decreased.
• 19. The lower frequencies converge quickly as compared to the higher frequencies.
• 20. The convergence details of each case are plotted graphically below:
• 21. As the thickness of the plate decreases the frequency values go on decreasing as the mass and the dimensions of the plate are decreased.
• 22. The lower frequencies converge quickly as compared to the higher frequencies.
• 23. The convergence details of each case are plotted graphically below:
Case 3: h = 0.15625 mm
-As the thickness of the plate decreases the frequency values go on decreasing.
-Lower frequencies converge quickly as compared to higher frequencies.
• 24. Convergence
Case 1: h = 2.5 mm Case 2: h = 0.625 mm Case 3: h = 0.15625 mm
• 25. Case 1: h = 2.5 mmCase 2: h = 0.625 mm Case 2: h = 0.15625 mm
Comparison of Results
• 26. Comparison of Modal Shapes
• The mode shape obtained here is the same for both SOLID45 and SHELL63.
• 27. The mode shape obtained here is the same for both SOLID45 and SHELL63.
Case 3: h = 0.15625 mm
Mode Shape for the 1st Natural Frequency:
Solid Elements Shell Elements
The mode shape obtained here is the same for both SOLID45 and SHELL63.
• 28. Mode Shape for the 2nd Natural Frequency:
Solid Elements Shell Elements
Comparison of Modal Shapes
The mode shape obtained here is the same for both SOLID45 and SHELL63.
• 29. Mode Shape for the 3rd Natural Frequency:
Solid Elements Shell Elements
Comparison of Modal Shapes
The mode shape obtained here is the same for both SOLID45 and SHELL63.
• 30. Mode Shape for the 4th Natural Frequency:
Solid Elements Shell Elements
Comparison of Modal Shapes
The mode shape obtained here is the same with deformations in different directions
• 31. Mode Shape for the 5th Natural Frequency:
Solid Elements Shell Elements
Comparison of Modal Shapes
Different mode shapes are obtained
• 32. Mode Shape for the 6th Natural Frequency:
Solid Elements Shell Elements
Comparison of Modal Shapes
Different mode shapes are obtained
• 33.
• The natural frequency values of any structure depend on its dimensions and boundary conditions. In this case, the frequency values decrease with decrease in thickness of the plate.
• 34. Meshing with SHELL elements is easier as compared to SOLID mesh in case of complex structures.
• 35. SHELL elements give better performance as the shell thickness go on decreasing.
Conclusion