MAE 501 INDEPENDENT STUDY
Modal Analysis of a Rectangular Plate
BySasi Bhushan Beera
#35763829
Problem DetailsProblem 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 = 40mmWidth of the plate = 20mmThickness of the plate = 2.5mm, 0.625mm, 0.15625mm
Problem DetailsMaterial 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.
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
Case 1: h = 2.5 mm
Results
It.NoMesh-Size(mm)
Freq-1 (Hz)
%E-Freq
Freq-2 (Hz)
%E-Freq
Freq-3 (Hz)
%E-Freq
Freq-4 (Hz)
%E-Freq
Freq-5 (Hz)
%E-FreqFreq-6(Hz)
%E-Freq
1 2.5 21.687 30.076 39.952 45.057 56.733 63.824
2 2.2521.51
-0.81616 29.777
-0.99415 38.48
-3.68442 44.431
-1.38935 53.049 -6.49358 62.317 -2.36118
3 221.41 -0.4649 29.633
-0.48359 38.417
-0.16372 44.125
-0.68871 52.995 -0.10179 61.48 -1.34313
4 1.7521.279
-0.61186 29.453
-0.60743 38.327
-0.23427 43.774
-0.79547 52.918 -0.1453 60.396 -1.76318
5 1.521.194
-0.39945 29.322
-0.44478 38.25 -0.2009 43.495
-0.63736 52.857 -0.11527 59.728 -1.10603
6 1.2521.134 -0.2831 29.224
-0.33422 38.184
-0.17255 43.278
-0.49891 52.808 -0.0927 59.275 -0.75844
Case 2: h = 0.625 mm
It.NoMesh-Size(mm)
Freq-1 (Hz)
%E-Freq
Freq-2 (Hz)
%E-Freq
Freq-3 (Hz)
%E-Freq
Freq-4 (Hz)
%E-Freq
Freq-5 (Hz)
%E-FreqFreq-6(Hz)
%E-Freq
1 2.5 21.687 30.076 39.952 45.057 56.733 63.824
2 2.2521.51
-0.81616 29.777
-0.99415 38.48
-3.68442 44.431
-1.38935 53.049 -6.49358 62.317 -2.36118
3 221.41 -0.4649 29.633
-0.48359 38.417
-0.16372 44.125
-0.68871 52.995 -0.10179 61.48 -1.34313
4 1.7521.279
-0.61186 29.453
-0.60743 38.327
-0.23427 43.774
-0.79547 52.918 -0.1453 60.396 -1.76318
5 1.521.194
-0.39945 29.322
-0.44478 38.25 -0.2009 43.495
-0.63736 52.857 -0.11527 59.728 -1.10603
6 1.2521.134 -0.2831 29.224
-0.33422 38.184
-0.17255 43.278
-0.49891 52.808 -0.0927 59.275 -0.75844
Case 3: h = 0.15625 mm
Results
It.NoMesh-Size(mm)
Freq-1 (Hz)
%E-Freq
Freq-2 (Hz)
%E-Freq
Freq-3 (Hz)
%E-Freq
Freq-4 (Hz)
%E-Freq
Freq-5 (Hz)
%E-FreqFreq-6(Hz)
%E-Freq
1 1.25 1.4241 2.0588 3.149 4.269 4.7007 4.9974
21.125 1.4153
-0.61793 2.0313
-1.33573 3.0922
-1.80375 4.2264
-0.99789 4.6007 -2.12734 4.895 -2.04907
31 1.4099
-0.38154 2.0152 -0.7926 3.0585
-1.08984 4.1991
-0.64594 4.5408 -1.30198 4.8333 -1.26047
40.875 1.4052
-0.33336 2.0015
-0.67983 3.0298
-0.93837 4.1735
-0.60965 4.4889 -1.14297 4.7794 -1.11518
50.75 1.4019
-0.23484 1.9923
-0.45966 3.0106
-0.63371 4.154
-0.46723 4.4533 -0.79307 4.7419 -0.78462
•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:
•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:
-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.
Case 1: h = 2.5 mm Case 2: h = 0.625 mm Case 3: h = 0.15625 mm
Convergence
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.
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
Case 1: h = 2.5 mm
Results
Case 2: h = 0.625 mm
It.NoNo of divisions
Freq-1 (Hz)
%E-Freq
Freq-2 (Hz)
%E-Freq
Freq-3 (Hz)
%E-Freq
Freq-4 (Hz)
%E-Freq
Freq-5 (Hz)
%E-Freq
Freq-6 (Hz)
%E-Freq
1 2.5 15.456 24.593 39.838 39.975 52.277 56.704
22.25 15.501
0.291149 24.766
0.703452 39.908
0.175712 40.207
0.580363 52.628
0.671423 56.637 -0.11816
32 15.51
0.058061 24.799
0.133247 39.894
-0.03508 40.282
0.186535 52.699
0.134909 56.624 -0.02295
41.75 15.513
0.019342 24.811
0.048389 39.899
0.012533 40.309
0.067027 52.724
0.047439 56.62 -0.00706
51.5 15.514
0.006446 24.817
0.024183 39.899 0 40.321 0.02977 52.736 0.02276 56.618 -0.00353
It.NoNo of divisions
Freq-1 (Hz)
%E-Freq
Freq-2 (Hz)
%E-Freq
Freq-3 (Hz)
%E-Freq
Freq-4 (Hz)
%E-Freq
Freq-5 (Hz)
%E-Freq
Freq-6 (Hz)
%E-Freq
1 2.5 3.864 6.1483 9.9596 13.069 15.303 15.354
22.25 3.8753
0.292443 6.1914
0.701007 10.052
0.927748 13.157
0.673349 15.456
0.999804 15.468 0.742478
31.75 3.8774
0.054189 6.1998
0.135672 10.071
0.189017 13.175
0.136809 15.489
0.213509 15.494 0.168089
41.5 3.8782
0.020632 6.2028
0.048389 10.077
0.059577 13.181
0.045541 15.501
0.077474 15.504 0.064541
51.25 3.8785
0.007736 6.2042 0.02257 10.08
0.029771 13.184 0.02276 15.505
0.025805 15.508 0.0258
Case 3: h = 0.15625 mm
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.•The lower frequencies converge quickly as compared to the higher frequencies.•The convergence details of each case are plotted graphically below:
•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:
-As the thickness of the plate decreases the frequency values go on decreasing.
-Lower frequencies converge quickly as compared to higher frequencies.
It.NoNo of divisions
Freq-1 (Hz)
%E-Freq
Freq-2 (Hz)
%E-Freq
Freq-3 (Hz)
%E-Freq
Freq-4 (Hz)
%E-Freq
Freq-5 (Hz)
%E-Freq
Freq-6 (Hz)
%E-Freq
1 2.5 3.864 6.1483 9.9596 13.069 15.303 15.354
22.25 3.8753 0.292443 6.1914 0.701007 10.052 0.927748 13.157 0.673349 15.456 0.999804 15.468 0.742478
31.75 3.8774 0.054189 6.1998 0.135672 10.071 0.189017 13.175 0.136809 15.489 0.213509 15.494 0.168089
41.5 3.8782 0.020632 6.2028 0.048389 10.077 0.059577 13.181 0.045541 15.501 0.077474 15.504 0.064541
51.25 3.8785 0.007736 6.2042 0.02257 10.08 0.029771 13.184 0.02276 15.505 0.025805 15.508 0.0258
ConvergenceCase 1: h = 2.5 mm Case 2: h = 0.625 mm Case
3: h = 0.15625 mm
Case 1: h = 2.5 mm Case 2: h = 0.625 mm Case 2: h = 0.15625 mm
Comparison of Results
Case 3: h = 0.15625 mm
Mode Shape for the 1st Natural Frequency:
Solid Elements Shell Elements
Comparison of Modal Shapes
•The mode shape obtained here is the same for both SOLID45 and SHELL63.•The mode shape obtained here is the same for both SOLID45 and SHELL63.
The mode shape obtained here is the same for both SOLID45 and SHELL63.
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.
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.
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
Mode Shape for the 5th Natural Frequency:
Solid Elements Shell Elements
Comparison of Modal Shapes
Different mode shapes are obtained
Mode Shape for the 6th Natural Frequency:
Solid Elements Shell Elements
Comparison of Modal Shapes
Different mode shapes are obtained
• 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.
Conclusion