Download - Modal Analysis of a Rectangular Plate - PPT

Transcript
Page 1: Modal Analysis of a Rectangular Plate - PPT

MAE 501 INDEPENDENT STUDY

Modal Analysis of a Rectangular Plate

BySasi Bhushan Beera

#35763829

Page 2: Modal Analysis of a Rectangular Plate - PPT

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

Page 3: Modal Analysis of a Rectangular Plate - PPT

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.

Page 4: Modal Analysis of a Rectangular Plate - PPT

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

Page 5: Modal Analysis of a Rectangular Plate - PPT

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

Page 6: Modal Analysis of a Rectangular Plate - PPT

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.

Page 7: Modal Analysis of a Rectangular Plate - PPT

Case 1: h = 2.5 mm Case 2: h = 0.625 mm Case 3: h = 0.15625 mm

Convergence

Page 8: Modal Analysis of a Rectangular Plate - PPT

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

Page 9: Modal Analysis of a Rectangular Plate - PPT

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

Page 10: Modal Analysis of a Rectangular Plate - PPT

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

Page 11: Modal Analysis of a Rectangular Plate - PPT

ConvergenceCase 1: h = 2.5 mm Case 2: h = 0.625 mm Case

3: h = 0.15625 mm

Page 12: Modal Analysis of a Rectangular Plate - PPT

Case 1: h = 2.5 mm Case 2: h = 0.625 mm Case 2: h = 0.15625 mm

Comparison of Results

Page 13: Modal Analysis of a Rectangular Plate - PPT

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.

Page 14: Modal Analysis of a Rectangular Plate - PPT

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.

Page 15: Modal Analysis of a Rectangular Plate - PPT

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.

Page 16: Modal Analysis of a Rectangular Plate - PPT

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

Page 17: Modal Analysis of a Rectangular Plate - PPT

Mode Shape for the 5th Natural Frequency:

Solid Elements Shell Elements

Comparison of Modal Shapes

Different mode shapes are obtained

Page 18: Modal Analysis of a Rectangular Plate - PPT

Mode Shape for the 6th Natural Frequency:

Solid Elements Shell Elements

Comparison of Modal Shapes

Different mode shapes are obtained

Page 19: Modal Analysis of a Rectangular Plate - PPT

• 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