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- 1. MAE 501 INDEPENDENT STUDYModal Analysis of a Rectangular Plate<br />By<br />Sasi BhushanBeera<br />#35763829<br />
- 2. Problem Details<br />Problem statement:<br />To determine the lowest six non-zero frequencies and associated mode shapes for a rectangular plate for three different thickness of the plate.<br />Geometry of the plate:<br /> Rectangular Plate with length 2a, width 2b and thickness h.<br /> γ = a/b=2<br /> ξ = h/b = 1/4, 1/16, 1/64<br /> a = 20m, b = 10 mm, h = 2.5mm, 0.625mm, 0.15625m<br /> Thus, the dimensions of the plate are as follows:<br /> Length of the plate = 40mm<br /> Width of the plate = 20mm<br /> Thickness of the plate = 2.5mm, 0.625mm, 0.15625mm<br />
- 3. Problem Details<br />Material Properties of the plate:<br /><ul><li>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</li></ul>Boundary Conditions:<br /><ul><li>The two adjacent edges of the rectangular plate are fixed while the other two are free.</li></ul>Element Details:<br /><ul><li>The modal analysis is performed using two types of elements i.e. solid and shell.</li></li></ul><li>Meshing:<br /><ul><li>A 3-D model of the plate was created and the plate was meshed using SOLID45 element.</li></ul> Solver used for Modal Analysis:<br /> The modal analysis was performed using the PCG Lanczos solver.<br /> Applying boundary conditions:<br /> The two adjacent sides of the plate are fixed in x and y directions and all the four sides are constrained in z direction.<br /> C0nvergence Criteria:<br /> Natural Frequency: For the problem to converge, the variation of all the six natural frequencies between two iterations should be less that 1%.<br />Modal Analysis using Solid45<br />
- 7. Case 1: h = 2.5 mm<br />Results<br />Case 2: h = 0.625 mm<br />
- 8. Results<br /><ul><li>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:</li></ul>Case 3: h = 0.15625 mm<br /> -As the thickness of the plate decreases the frequency values go on decreasing as the mass and the dimensions of the plate are decreased.<br /> -The lower frequencies converge quickly as compared to the higher <br /> frequencies.<br />
- 14. Case 1: h = 2.5 mm Case 2: h = 0.625 mm Case 3: h = 0.15625 mm<br />Convergence<br />
- 15. Meshing:<br /><ul><li>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.</li></ul> Solver used for Modal Analysis:<br /> The modal analysis was performed using the PCG Lanczos solver.<br /> Applying boundary conditions:<br /> The two adjacent sides of the plate are fixed in x and y directions and all the four sides are constrained in z direction.<br /> C0nvergence Criteria:<br /> Natural Frequency: For the problem to converge, the variation of all the six natural frequencies between two iterations should be less that 0.03%.<br />Modal Analysis using Shell63<br />
- 17. Case 1: h = 2.5 mm<br />Results<br />Case 2: h = 0.625 mm<br />
- 18. Results<br /><ul><li>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:</li></ul>Case 3: h = 0.15625 mm<br /> -As the thickness of the plate decreases the frequency values go on decreasing.<br /> -Lower frequencies converge quickly as compared to higher frequencies.<br />
- 24. Convergence<br />Case 1: h = 2.5 mm Case 2: h = 0.625 mm Case 3: h = 0.15625 mm<br />
- 25. Case 1: h = 2.5 mmCase 2: h = 0.625 mm Case 2: h = 0.15625 mm<br />Comparison of Results<br />
- 26. Comparison of Modal Shapes<br /><ul><li>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.</li></ul>Case 3: h = 0.15625 mm<br />Mode Shape for the 1st Natural Frequency:<br />Solid Elements Shell Elements<br />The mode shape obtained here is the same for both SOLID45 and SHELL63.<br />
- 28. Mode Shape for the 2nd Natural Frequency:<br />Solid Elements Shell Elements<br />Comparison of Modal Shapes<br />The mode shape obtained here is the same for both SOLID45 and SHELL63.<br />
- 29. Mode Shape for the 3rd Natural Frequency:<br />Solid Elements Shell Elements<br />Comparison of Modal Shapes<br />The mode shape obtained here is the same for both SOLID45 and SHELL63.<br />
- 30. Mode Shape for the 4th Natural Frequency:<br />Solid Elements Shell Elements<br />Comparison of Modal Shapes<br />The mode shape obtained here is the same with deformations in different directions<br />
- 31. Mode Shape for the 5th Natural Frequency:<br />Solid Elements Shell Elements<br />Comparison of Modal Shapes<br />Different mode shapes are obtained <br />
- 32. Mode Shape for the 6th Natural Frequency:<br />Solid Elements Shell Elements<br />Comparison of Modal Shapes<br />Different mode shapes are obtained <br />
- 33. <ul><li>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.</li></ul>Conclusion<br />

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