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MAE 501: PROJECT 02 Modal Analysis of Rectangular Plate Project Report Sasi Bhushan Beera Person# 35763829
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MAE 501: PROJECT 02
Modal Analysis of Rectangular Plate
Project Report
Sasi Bhushan Beera Person# 35763829
2 | P a g e
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
3 | P a g e
Mode Shape for 4th Natural Frequency: ............................................................................................. 18
Mode Shape for 5th Natural Frequency: ............................................................................................ 189
Conclusion: ........................................................................................................................................... 19
4 | P a g e
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.
5 | P a g e
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:
6 | P a g e
It.
No
Mesh-
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-
Freq
Freq-6
(Hz)
%E-
Freq
1 2.5 21.687
30.076
39.952
45.057
56.733
63.824
2 2.25 21.51
-
0.816
16 29.777
-
0.994
15 38.48
-
3.684
42 44.431
-
1.389
35 53.049
-
6.493
58 62.317
-
2.361
18
3 2 21.41
-
0.464
9 29.633
-
0.483
59 38.417
-
0.163
72 44.125
-
0.688
71 52.995
-
0.101
79 61.48
-
1.343
13
4 1.75 21.279
-
0.611
86 29.453
-
0.607
43 38.327
-
0.234
27 43.774
-
0.795
47 52.918
-
0.145
3 60.396
-
1.763
18
5 1.5 21.194
-
0.399
45 29.322
-
0.444
78 38.25
-
0.200
9 43.495
-
0.637
36 52.857
-
0.115
27 59.728
-
1.106
03
6 1.25 21.134
-
0.283
1 29.224
-
0.334
22 38.184
-
0.172
55 43.278
-
0.498
91 52.808
-
0.092
7 59.275
-
0.758
44
- Case 2 i.e. h = 0.625mm:
o This problem converged in six iterations and their details are as follows:
7 | P a g e
It.
No
Mesh-
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-
Freq
Freq-6
(Hz)
%E-
Freq
1 2.5 5.7937
8.3503
12.901
18.249
19.684
21.133
2 2.25 5.7322
-
1.061
5 8.2041
-
1.750
84 12.583
-
2.464
93 17.777
-
2.586
44 19.039
-
3.276
77 20.407
-
3.435
39
3 2 5.6923
-
0.696
07 8.1141
-
1.097
01 12.387
-
1.557
66 17.462
-
1.771
95 18.634
-
2.127
21 19.948
-
2.249
23
4 1.75 5.6468
-
0.799
33 8.0226
-
1.127
67 12.195
-
1.550
01 17.083
-
2.170
43 18.241
-
2.109
05 19.442
-
2.536
6
5 1.5 5.6203
-
0.469
29 7.9683
-
0.676
84 12.073
-
1.000
41 16.864
-
1.281
98 17.974
-
1.463
74 19.155
-
1.476
19
6 1.25 5.6037
-
0.295
36 7.9347
-
0.421
67 11.994
-
0.654
35 16.727
-
0.812
38 17.796
-
0.990
32 18.981
-
0.908
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:
8 | P a g e
It.
No
Mesh-
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-
Freq
Freq-6
(Hz)
%E-
Freq
1 1.25 1.4241
2.0588
3.149
4.269
4.7007
4.9974
2 1.125 1.4153
-
0.617
93 2.0313
-
1.335
73 3.0922
-
1.803
75 4.2264
-
0.997
89 4.6007
-
2.127
34 4.895
-
2.049
07
3 1 1.4099
-
0.381
54 2.0152
-
0.792
6 3.0585
-
1.089
84 4.1991
-
0.645
94 4.5408
-
1.301
98 4.8333
-
1.260
47
4 0.875 1.4052
-
0.333
36 2.0015
-
0.679
83 3.0298
-
0.938
37 4.1735
-
0.609
65 4.4889
-
1.142
97 4.7794
-
1.115
18
5 0.75 1.4019
-
0.234
84 1.9923
-
0.459
66 3.0106
-
0.633
71 4.154
-
0.467
23 4.4533
-
0.793
07 4.7419
-
0.784
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:
9 | P a g e
- 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.
10 | P a g e
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:
It.
No
No of
divisi
ons
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
2 2.25 15.501
0.291
149 24.766
0.703
452 39.908
0.175
712 40.207
0.580
363 52.628
0.671
423 56.637
-
0.118
16
3 2 15.51
0.058
061 24.799
0.133
247 39.894
-
0.035
08 40.282
0.186
535 52.699
0.134
909 56.624
-
0.022
95
11 | P a g e
4 1.75 15.513
0.019
342 24.811
0.048
389 39.899
0.012
533 40.309
0.067
027 52.724
0.047
439 56.62
-
0.007
06
5 1.5 15.514
0.006
446 24.817
0.024
183 39.899 0 40.321
0.029
77 52.736
0.022
76 56.618
-
0.003
53
- Case 2 i.e. h = 0.625mm:
o The problem converged in five iterations and their details are as follows:
It.
No
No of
divisi
ons
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
2 2.25 3.8753
0.292
443 6.1914
0.701
007 10.052
0.927
748 13.157
0.673
349 15.456
0.999
804 15.468
0.742
478
3 1.75 3.8774
0.054
189 6.1998
0.135
672 10.071
0.189
017 13.175
0.136
809 15.489
0.213
509 15.494
0.168
089
4 1.5 3.8782
0.020
632 6.2028
0.048
389 10.077
0.059
577 13.181
0.045
541 15.501
0.077
474 15.504
0.064
541
5 1.25 3.8785
0.007
736 6.2042
0.022
57 10.08
0.029
771 13.184
0.022
76 15.505
0.025
805 15.508
0.025
8
- Case3 i.e. h = 0.15625mm:
o The problem converged in six iterations and their details are as follows:
It.
No
No of
divisi
ons
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 0.966
1.5371
2.4899
3.2673
3.8257
3.8384
2 2.25
0.9688
2
0.291
925 1.5479
0.702
622 2.513
0.927
748 3.2892
0.670
278 3.864
1.001
124 3.867
0.745
102
3 2.00
0.9693
6
0.055
738 1.55
0.135
668 2.5176
0.183
048 3.2937
0.136
811 3.8723
0.214
803 3.8736
0.170
675
12 | P a g e
4 1.75
0.9695
5
0.019
601 1.5507
0.045
161 2.5193
0.067
525 3.2952
0.045
541 3.8753
0.077
473 3.876
0.061
958
5 1.5
0.9696
4
0.009
283 1.5511
0.025
795 2.5201
0.031
755 3.296
0.024
278 3.8767
0.036
126 3.8771
0.028
38
6 1.25
0.9696
8
0.004
125 1.5512
0.006
447 2.5205
0.015
872 3.2964
0.012
136 3.8774
0.018
057 3.8777
0.015
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:
13 | P a g e
- 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:
14 | P a g e
- 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:
15 | P a g e
- 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:
16 | P a g e
- 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.
17 | P a g e
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.
18 | P a g e
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:
19 | P a g e
- 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.
20 | P a g e
