In a modal-based analysis, the problem is reduced by representing the dynamics of the structure as a combination of small number of its most significant eigenmodes. This is very efficient when the frequency content of the loads applied to the structure is limited, so that only a small number of modes will be excited.
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You solve the same problem as in the previous section. Just as for the direct solution, it is important that you set the absolute tolerance small enough.
Results and Discussion
The plot in Figure 9 below shows the same x-displacement as in the previous section but with results from both the full and the modal based time-dependent analysis. The correspondence between the solutions is good even though only the six first eigenmodes are used.
Figure 9: x-displacement at a point on the loaded surface for full and modal analyses.
Figure 10 below shows the von Mises stress in the bracket at 0.0036 s. The maximum value is 237 MPa, which can be compared with the 252 MPa computed using the direct solution above. In general, more modes than what is needed to compute accurate displacements are required to obtain good stress solutions.
Solved with COMSOL Multiphysics 4.2a
Figure 10: von Mises stress for modal solution.
Notes About the COMSOL Implementation
When you create a new study, it is possible to directly select “Time-Dependent Modal.” Such a study, however, generates a complete solver sequence including the eigenvalue computation step. Because the eigenvalues are already available, you create an “Empty Study” and add the study steps manually.
The undamped eigenmodes are used as the base, and the damping is provided by the material.
In the modal procedure, all loads must have the same variation in time, specified in the study step. This means that you should not enter any time-dependent loads (that is, loads with an explicit dependency on the time using the time variable t).
Modeling Instructions
M O D E L W I Z A R D
1 In the Model Builder window, right-click the root node and choose Add Study. 2 Go to the Model Wizard window.
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3 Find the Studies subsection. In the tree, select Custom Studies>Empty Study. 4 Click Finish.
S O L I D M E C H A N I C S
Time-Dependent Load
In the Model Builder window, right-click Model 1>Solid Mechanics>Time-Dependent Load and choose Disable.
Boundary Load 3
1 Right-click Solid Mechanics and choose Boundary Load.
2 In the Model Builder window, right-click Boundary Load 3 and choose Rename. 3 Go to the Rename Boundary Load dialog box and type Modal Time-Dependent
Load in the New name edit field.
4 Click OK.
5 Select Boundary 29 only.
6 Go to the Settings window for Boundary Load.
7 Locate the Force section. Specify the FA vector as
S T U D Y 5
1 In the Model Builder window, right-click Study 5 and choose Rename.
2 Go to the Rename Study dialog box and type Study 5 (Modal Time-Dependent) in the New name edit field.
3 Click OK.
4 Right-click Study 5 and choose Time-Dependent Modal. S T U D Y 5 ( M O D A L T I M E - D E P E N D E N T )
Step 1: Time-Dependent Modal
1 Go to the Settings window for Time-Dependent Modal.
2 Locate the Study Settings section. In the Times edit field, type range(0,2e-4,10e-3).
1.5[MPa] X
0 Y
0 Z
Solved with COMSOL Multiphysics 4.2a
Solver 5
1 In the Model Builder window, right-click Study 5 (Modal Time-Dependent) and choose Show Default Solver.
2 Expand the Study 5 (Modal Time-Dependent)>Solver Configurations node.
3 Right-click Study 5 (Modal Time-Dependent)>Solver Configurations>Solver 5 and choose Rename.
4 Go to the Rename Solver dialog box and type Modal Time-Dependent Sequence in the New name edit field.
5 Click OK.
Modal Time-Dependent Sequence
1 In the Model Builder window, expand the Study 5 (Modal Time-Dependent)>Solver Configurations>Modal Time-Dependent Sequence node, then click Modal Solver 1. 2 Go to the Settings window for Modal Solver.
3 Locate the General section. Find the Tolerance subsection. In the Absolute global tolerance edit field, type 1e-5.
4 Locate the Eigenpairs section. From the Solution list, choose Eigenfrequency Sequence. 5 Click to expand the Advanced section.
6 In the Load factor edit field, type 1+sin(2*pi*500*t-pi/2).
7 In the Model Builder window, right-click Study 5 (Modal Time-Dependent) and choose Compute.
R E S U L T S
Data Sets
1 In the Model Builder window, right-click Results>Data Sets>Solution 5 and choose Rename.
2 Go to the Rename Solution dialog box and type Modal Time-Dependent Solution in the New name edit field.
3 Click OK. Stress (solid)
Reproduce the plot in Figure 10 by following these instructions:
1 In the Model Builder window, click Stress (solid). 2 Go to the Settings window for 3D Plot Group.
3 Locate the Data section. From the Time list, choose 0.0036.
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4 Click the Plot button.
5 In the Model Builder window, right-click Stress (solid) and choose Rename. 6 Go to the Rename 3D Plot Group dialog box and type Modal Time-Dependent
Stress Contour in the New name edit field.
7 Click OK.
Add the results from this study to the graph of the direct time dependent results, so that the methods can be compared.
Time-Dependent Displacement Graphs
1 In the Model Builder window, right-click Results>Time-Dependent Displacement Graphs and choose Point Graph.
2 Select Point 30 only.
3 Go to the Settings window for Point Graph.
4 In the upper-right corner of the y-Axis Data section, click Replace Expression. 5 From the menu, choose Displacement field, X component (u).
6 Locate the Data section. From the Data set list, choose Modal Time-Dependent Solution.
7 Locate the y-Axis Data section. From the Unit list, choose mm. 8 Click to expand the Coloring and Style section.
9 Find the Line style subsection. From the Color list, choose Green. 10Click the Plot button.