Modal Analysis of a Rectangular Plate - PDF

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Modal Analysis of a Rectangular Plate - PDF

  1. 1. MAE 501: PROJECT 02 Modal Analysis of Rectangular Plate Project Report Sasi Bhushan Beera Person# 35763829
  2. 2. Table of Contents Problem Details: ...................................................................................................................................... 4 Problem statement: .............................................................................................................................. 4 Geometry of the plate: ......................................................................................................................... 4 Material Properties of the plate: ........................................................................................................... 4 Boundary Conditions: .......................................................................................................................... 4 Element Details: .................................................................................................................................. 4 Modal Analysis using SOLID45: ............................................................................................................. 5 Meshing: ............................................................................................................................................. 5 Solver used for Modal Analysis: .......................................................................................................... 5 Applying boundary conditions: ............................................................................................................ 5 Convergence Criteria: .......................................................................................................................... 5 Results: ............................................................................................................................................... 5 Observations: ...................................................................................................................................... 8 Modal Analysis using SHELL 63: ........................................................................................................... 9 Meshing: ............................................................................................................................................. 9 Solver used for the analysis: .............................................................................................................. 10 Applying boundary conditions: .......................................................................................................... 10 Convergence Criteria: ........................................................................................................................ 10 Results: ............................................................................................................................................. 10 Observations: .................................................................................................................................... 12 Comparison of modal analysis using SOLID45 and SHELL63:.............................................................. 13 Case 1 i.e. h = 2.5mm: ....................................................................................................................... 13 Case 2 i.e. h = 0.625mm: ................................................................................................................... 14 Case3 i.e. h = 0.15625mm: ................................................................................................................ 15 Modal Shapes: ....................................................................................................................................... 17 Mode Shape for the 1st Natural Frequency: ........................................................................................ 17 Mode Shape for the 2nd Natural Frequency: ........................................................................................ 17 Mode Shape for the 3rd Natural Frequency: ........................................................................................ 18 2|P ag e
  3. 3. Mode Shape for 4th Natural Frequency: ............................................................................................. 18 Mode Shape for 5th Natural Frequency: ............................................................................................ 189 Conclusion: ........................................................................................................................................... 19 3|P ag e
  4. 4. 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 = 20 mm, b = 10 mm, h = 2.5mm, 0.625mm, 0.15625mm - Thus, the dimensions of the plate are as follows: o Length of the plate = 40mm o Width of the plate = 20mm o Thickness of the plate = 2.5mm, 0.625mm, 0.15625mm Material Properties of the plate: - ν = Poisson’s ratio = 0.05 - Material of the plate : Cast Iron - E = Modulus of Elasticity = 139.7GPa - 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. Now, we will go through the details of the modal analysis of the plate in case of the different thickness values as obtained on running the problem using ANSYS 12. 4|P ag e
  5. 5. Modal Analysis using SOLID45: Meshing: - A 3-D model of the plate was created and the plate was meshed using SOLID45 element. - Properties of SOLID45: o SOLID45 is used for the 3-D modeling of solid structures. The element is defined by eight nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions. o The element has plasticity, creep, swelling, stress stiffening, large deflection, and large strain capabilities. - The mesh was refined per iteration using manual size control. Thus, the mesh was refined along all the three dimensions of the plate. 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. Convergence Criteria: - Natural Frequency: For the problem to converge, the variation of all the six natural frequencies between two iterations should be less that 1%. Results: - Case 1 i.e. h = 2.5 mm: o It was observed that the frequency values go on decreasing as the mesh size is refined, thus, converging to the lowest frequency values. o The problem was converged in six iterations and their details of are as follows: 5|P ag e
  6. 6. Mesh- It. Freq-1 %E- Freq-2 %E- Freq-3 %E- Freq-4 %E- Freq-5 %E- Freq-6 %E- Size( No (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq mm) 21.687 30.076 39.952 45.057 56.733 63.824 1 2.5 - - - - - - 2 2.25 0.816 0.994 3.684 1.389 6.493 2.361 21.51 16 29.777 15 38.48 42 44.431 35 53.049 58 62.317 18 - - - - - - 0.464 0.483 0.163 0.688 0.101 1.343 3 2 21.41 9 29.633 59 38.417 72 44.125 71 52.995 79 61.48 13 - - - - - - 0.611 0.607 0.234 0.795 0.145 1.763 4 1.75 21.279 86 29.453 43 38.327 27 43.774 47 52.918 3 60.396 18 - - - - - - 0.399 0.444 0.200 0.637 0.115 1.106 5 1.5 21.194 45 29.322 78 38.25 9 43.495 36 52.857 27 59.728 03 - - - - - - 0.283 0.334 0.172 0.498 0.092 0.758 6 1.25 21.134 1 29.224 22 38.184 55 43.278 91 52.808 7 59.275 44 - Case 2 i.e. h = 0.625mm: o This problem converged in six iterations and their details are as follows: 6|P ag e
  7. 7. Mesh- It. Freq-1 %E- Freq-2 %E- Freq-3 %E- Freq-4 %E- Freq-5 %E- Freq-6 %E- Size( No (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq mm) 2.5 5.7937 8.3503 12.901 18.249 19.684 21.133 1 - - - - - - 2 1.061 1.750 2.464 2.586 3.276 3.435 2.25 5.7322 5 8.2041 84 12.583 93 17.777 44 19.039 77 20.407 39 - - - - - - 0.696 1.097 1.557 1.771 2.127 2.249 3 2 5.6923 07 8.1141 01 12.387 66 17.462 95 18.634 21 19.948 23 - - - - - - 0.799 1.127 1.550 2.170 2.109 2.536 4 1.75 5.6468 33 8.0226 67 12.195 01 17.083 43 18.241 05 19.442 6 - - - - - - 0.469 0.676 1.000 1.281 1.463 1.476 5 1.5 5.6203 29 7.9683 84 12.073 41 16.864 98 17.974 74 19.155 19 - - - - - - 0.295 0.421 0.654 0.812 0.990 0.908 6 1.25 5.6037 36 7.9347 67 11.994 35 16.727 38 17.796 32 18.981 38 - Case 3 i.e. h = 0.15625mm: o Initially, the model is meshed with a much finer mesh of 1.25 mm. o Thus, the problem converged in five iterations and their details are as follows: 7|P ag e
  8. 8. Mesh- It. Freq-1 %E- Freq-2 %E- Freq-3 %E- Freq-4 %E- Freq-5 %E- Freq-6 %E- Size( No (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq mm) 1.25 1.4241 2.0588 3.149 4.269 4.7007 4.9974 1 - - - - - - 2 0.617 1.335 1.803 0.997 2.127 2.049 1.125 1.4153 93 2.0313 73 3.0922 75 4.2264 89 4.6007 34 4.895 07 - - - - - - 0.381 0.792 1.089 0.645 1.301 1.260 3 1 1.4099 54 2.0152 6 3.0585 84 4.1991 94 4.5408 98 4.8333 47 - - - - - - 0.333 0.679 0.938 0.609 1.142 1.115 4 0.875 1.4052 36 2.0015 83 3.0298 37 4.1735 65 4.4889 97 4.7794 18 - - - - - - 0.234 0.459 0.633 0.467 0.793 0.784 5 0.75 1.4019 84 1.9923 66 3.0106 71 4.154 23 4.4533 07 4.7419 62 Observations: - 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. - The convergence details of each case are plotted graphically below: - Case 1 i.e. h =2.5mm: 8|P ag e
  9. 9. - Case 2 i.e. h = 0.625mm: - Case 3 i.e. h = 0.15625mm: Modal Analysis using SHELL 63: Meshing: - A 2D-model of the plate was created and the plate was meshed using SHELL63 element. - The thickness of the plate was entered as a real constant of the SHELL63 element. - Properties of SHELL63: o SHELL63 has both bending and membrane capabilities. o Both in-plane and normal loads are permitted. 9|P ag e
  10. 10. o The element has six degrees of freedom at each node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z-axes. Stress stiffening and large deflection capabilities are included. - The mesh was refined per iteration using the manual size control. The mesh size was refined along the two dimensions of the plate Solver used for the analysis: - The PCG Lanczos solver was used to perform the modal analysis. Applying boundary conditions: - The fixed boundary conditions are applied to the two adjacent edges and all the edges are constrained in z direction. Convergence Criteria: - It was observed that SHELL63 gave much better convergence that SOLID45. Thus, a lower convergence criterion was decided for SHELL63 element. - Natural Frequency: For the problem to converge, the variation of all the six natural frequencies between two iterations should be less that 0.03%. Results: - Case 1 i.e. h = 2.5mm: As the mesh size is refined, the frequency values increased. o The problem converged in five iterations and their details are as follows: No of It. Freq-1 %E- Freq-2 %E- Freq-3 %E- Freq-4 %E- Freq-5 %E- Freq-6 %E- divisi No (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq ons 2.5 15.456 24.593 39.838 39.975 52.277 56.704 1 - 2 0.291 0.703 0.175 0.580 0.671 0.118 2.25 15.501 149 24.766 452 39.908 712 40.207 363 52.628 423 56.637 16 - - 3 0.058 0.133 0.035 0.186 0.134 0.022 2 15.51 061 24.799 247 39.894 08 40.282 535 52.699 909 56.624 95 10 | P a g e
  11. 11. - 0.019 0.048 0.012 0.067 0.047 0.007 4 1.75 15.513 342 24.811 389 39.899 533 40.309 027 52.724 439 56.62 06 - 0.006 0.024 0.029 0.022 0.003 5 1.5 15.514 446 24.817 183 39.899 0 40.321 77 52.736 76 56.618 53 - Case 2 i.e. h = 0.625mm: o The problem converged in five iterations and their details are as follows: No of It. Freq-1 %E- Freq-2 %E- Freq-3 %E- Freq-4 %E- Freq-5 %E- Freq-6 %E- divisi No (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq ons 2.5 3.864 6.1483 9.9596 13.069 15.303 15.354 1 0.292 0.701 0.927 0.673 0.999 0.742 2 2.25 3.8753 443 6.1914 007 10.052 748 13.157 349 15.456 804 15.468 478 0.054 0.135 0.189 0.136 0.213 0.168 3 1.75 3.8774 189 6.1998 672 10.071 017 13.175 809 15.489 509 15.494 089 0.020 0.048 0.059 0.045 0.077 0.064 4 1.5 3.8782 632 6.2028 389 10.077 577 13.181 541 15.501 474 15.504 541 0.007 0.022 0.029 0.022 0.025 0.025 5 1.25 3.8785 736 6.2042 57 10.08 771 13.184 76 15.505 805 15.508 8 - Case3 i.e. h = 0.15625mm: o The problem converged in six iterations and their details are as follows: No of It. Freq-1 %E- Freq-2 %E- Freq-3 %E- Freq-4 %E- Freq-5 %E- Freq-6 %E- divisi No (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Freq ons 2.5 0.966 1.5371 2.4899 3.2673 3.8257 3.8384 1 0.9688 0.291 0.702 0.927 0.670 1.001 0.745 2 2.25 2 925 1.5479 622 2.513 748 3.2892 278 3.864 124 3.867 102 0.9693 0.055 0.135 0.183 0.136 0.214 0.170 3 2.00 6 738 1.55 668 2.5176 048 3.2937 811 3.8723 803 3.8736 675 11 | P a g e
  12. 12. 0.9695 0.019 0.045 0.067 0.045 0.077 0.061 4 1.75 5 601 1.5507 161 2.5193 525 3.2952 541 3.8753 473 3.876 958 0.9696 0.009 0.025 0.031 0.024 0.036 0.028 5 1.5 4 283 1.5511 795 2.5201 755 3.296 278 3.8767 126 3.8771 38 0.9696 0.004 0.006 0.015 0.012 0.018 0.015 6 1.25 8 125 1.5512 447 2.5205 872 3.2964 136 3.8774 057 3.8777 475 Observations: - As the thickness of the plate decreases the frequency values go on decreasing. - Lower frequencies converge quickly as compared to higher frequencies. - The convergence details of each case are plotted graphically below: - Case 1 i.e. h = 2.5mm: 12 | P a g e
  13. 13. - Case 2 i.e. h = 0.625mm: - Case 3 i.e. h = 0.15625mm: Comparison of modal analysis using SOLID45 and SHELL63: Case 1 i.e. h = 02.5 mm: - The converged frequency values obtained using SOLID45 and SHELL63 are plotted in the graph below: 13 | P a g e
  14. 14. - It can be observed that the natural frequencies values obtained using SOLID45 are higher than those obtained using SHELL63. Case 2 i.e. h = 0.625mm: - The converged frequency values obtained using SOLID45 and SHELL63 are plotted in the graph below: 14 | P a g e
  15. 15. - The difference in the frequency values of SOLID45 and SHELL63 obtained in Case 2 is lower as compared to Case 1. Case3 i.e. h = 0.15625mm: - The converged frequency values obtained using SOLID45 and SHELL63 are plotted in the graph below: 15 | P a g e
  16. 16. - It can be observed that the natural frequency values obtained from SHELL63 and SOLID45 are almost the same. - Thus, it can be concluded that, the shell elements show better performance for lower shell thickness. 16 | P a g e
  17. 17. Modal Shapes: The mode shapes obtained for Case 3 (i.e. h = 0.15625m) for both SHELL63 and SOLID45 are given below: Mode Shape for the 1st Natural Frequency: Solid Elements: Shell Elements: - The mode shape obtained here is the same for both SOLID45 and SHELL63. Mode Shape for the 2nd Natural Frequency: Solid Elements: Shell Elements: - The mode shape for both the elements is same. 17 | P a g e
  18. 18. Mode Shape for the 3rd Natural Frequency: Solid Elements: Shell Elements: - The mode shape in both the cases is the same. Mode Shape for 4th Natural Frequency: Solid Elements: Shell Elements - Same mode shape is obtained in both the cases with deformations in different directions. Mode Shape for 5th Natural Frequency: Solid Elements: Shell Elements: 18 | P a g e
  19. 19. - Different mode shapes are obtained. Mode shape for 6th Natural Frequency: Solid Elements: Shell Elements: - Different mode shapes are obtained. Conclusion: - 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. - Meshing with SHELL elements is easier as compared to SOLID mesh in case of complex structures. - SHELL elements give better performance as the shell thickness go on decreasing. 19 | P a g e
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