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App Note #3

www.vibetech.com

25-Feb-14

ME’scope Application Note #3 Modeling a Rib Stiffener with SDM INTRODUCTION NOTE: The steps in this Application Note can be duplicated using any Package that includes the VES-5000 SDM (Structural Dynamics Modification) Option. In this note, the VES-5000 SDM (Structural Dynamics Modification) option will be used to model the addition of a rib stiffener to a flat aluminum plate. This study will use the mode shapes of the unmodified flat plate plus finite elements to model the rib. Both bar (beam) and quadrilateral plate elements will be used as alternate ways to model the rib. Then the Modal Assurance Criteria (MAC) will be used to compare the SDM results with the modes of finite element model of the ribbed plate.

of the same material as the plate, and is attached along the centerline of the 25 inch length of the plate with cap screws. FEA PLATE AND RIB MODELS The flat plate was modeled using the VES-8000 Experimental FEA Option in ME’scope. The unmodified plate (without the rib) was modeled with 80 FEA Quad plate elements, and the plate with rib was modeled with 100 FEA Quad elements. Each Quad element was 3/8-inches thick, and was defined between 4 nodes in a 2.5 inch square grid of nodes on the plate. The rib was modeled with 2 rows of FEA Quads, each 1.5 inches high by 2.5 inches wide by 3/8-inches thick. The 11 nodes on the bottom of the rib coincide with nodes on the centerline of the plate. The FEA Quad elements were given the following material properties: Modulus of Elasticity: 107 lb/in2 Poisson’s Ratio: 0.33 Density: 0.101 lb/in3 FEA Data Two sets of FEA mode shapes were generated; one set for the plate without the rib, and one set for the plate with the rib attached to its centerline.

Figure 1A. Unmodified Plate (80 FEA Quads).

The FEA modes were edited to make them more like typical experimental results.  The six rigid body (zero frequency) modes were deleted.  All rotational degrees-of-freedom (DOFs) in the mode shapes were also deleted.  Finally, all X and Y direction DOFs were deleted, leaving mode shapes describing dynamic behavior solely in the vertical (Z) direction. MODES OF THE UNMODIFIED STRUCTURE  Open App Note 03 - Modeling a Rib Stiffener with SDM.VTprj from the ME’scope \Application Notes folder.

Figure 1B. Plate with Rib (100 FEA Quads). The test article is a 3/8-inch thick 20 by 25 inch rectangular plate constructed of 6061-T6511 aluminum. The stiffening rib is 25 by 3 inches and is also 3/8-inches thick. It is made

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App Note #3

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25-Feb-14

Figure 2. Project Panel. Notice from the Project Panel that the Project contains two Structures and two Shape Tables.  Double-click on SHP: Aluminum Plate in the Project Panel to open its window. This Shape Table contains 24 mode shapes ranging in frequency from 92.4 Hz to 1421.4 Hz, as shown in Figure 4. Notice that all of the damping values are 0.0. This is typical of FEA models, which don’t include damping. Animating the Shapes To view the mode shapes in animation,  Double-click on STR: Aluminum Plate under Structures in the Project panel to open a Structure window with the plate model in it.  Re-size the Structure window to fill about 2/3’s of the Work Area, as shown below.

Figure 4. FEA Modes of the Plate Without Rib.  Execute Animate | Deformation | Undeformed. The mode shape of Shape 1 will be selected in the Shape Table and its mode shape displayed in animation in the Structure window.  Click on each Shape button in the SHP: Aluminum Plate window to display its shape. Figure 3. 92.5 Hz Torsional Mode.  Execute Window | Arrange Windows | For Animation in the ME’scope window.  Execute Draw | Animate Shapes in the Structure window to start the shape animation.

 Execute Animate | Contours | Node Lines to display the shape node lines.  Double click on the 3D View to display the Quad View.  Double click on the Z View (Top View) to display the shape node lines more clearly.

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App Note #3

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25-Feb-14

 Verify the following units on the Units tab, and click on OK.

Figure 5. Node Lines of Shape 2. The mode shapes with a node line on the centerline of the plate (such as Shape 1) will not be affected by a rib stiffener attached to the centerline. The mode shapes without a node line on the centerline of the plate (such as Shape 2), will be significantly affected by the rib stiffener. MODELING THE RIB WITH BAR ELEMENTS An FEA Bar is simply a beam of fixed cross-section. Each FEA Bar is described by;  its length

NOTE: If you change units, make sure that both the Point coordinates in the Structure window and the shapes in the Shape Table are re-scaled to match the new units. Adding FEA Bars to the Model FEA Bars will be added between all points on the centerline of the plate.  In the STR: Aluminum Plate window, execute Animate | Draw Structure to stop the animation.  Double click on the Z-View to display the Quad View.  Double click on the 3D View.

 its material properties

 its cross-sectional properties. An FEA Bar is attached to a structure model at its two endpoints. At each end-point, the FEA Bar applies stiffness and inertial restraint to the structure. In this case, only Zdirection translations are defined by the mode shapes, so the FEA Bar will only affect the Z-direction deformation of the plate. Checking the Units

Execute Edit | Object List | FEA Bars. Execute Edit | Add Object to enable the Add Objects operation.

 Beginning at one end of the centerline, click on each Point pair along the centerline, to add an FEA Bar between each pair, as shown below.  Execute Edit | Add Object again to disable the Add Objects operation.

Before performing any structural modifications, it is important to make sure that the units of the model in the Structure window match the shape units of the Shape Table. NOTE: Before using SDM, the units in the Structure Options dialog box must match the shape units in the Shape Table. Notice that the units of the mode shapes in the SHP: Aluminum Plate window are in/lb-sec. The structure model in the STR: Aluminum Plate window must also have length units of inches (or in), and force units of lbf (or lb). In addition, the mass units of the structure model must be lbm.  Execute File | Structure Options in the STR: Aluminum Plate window, and select the Units tab.

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Figure 6. Plate Showing 10 FEA Bars.


App Note #3

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25-Feb-14

All of the FEA Bars have the same width b (3/8 inch) and height h (3 inches). Therefore; Area = (3/8) x (3) = 1.125 in2 Ixx = (1/12) x (3/8)3 x (3) = 0.01318 in4 Iyy = (1/3) x (3/8) x (3)3 = 3.375 in4 Ixy = 0.0 in4 J = 0.01318 + 3.375 = 3.388 in4 To enter the calculated cross-section properties into the FEA Properties window,  Execute FEA | FEA Properties in the Structure window.  Select the Bars tab.  Execute Edit | Add.  Select Aluminum for the drop down list in the Materials column.  Enter 1.125 into the cell in the Area column.  Enter 0.01318 into the cell in the Ixx column.  Enter 3.375 into the cell in the Iyy column.

Figure 7. Cross-section of a rectangular FEA Bar. FEA Bar Cross-Section Properties

 Enter 0.0 into the cell in the Ixy column.

The FEA Bar cross-section is described by its Area and four area moments calculated with respect to the attachment point at the bottom center of the cross-section.

 Enter 3.388 into the cell in the J column.  Close the FEA Properties window.

The area moments (Ixx, Iyy, Ixy and J) are computed with respect to the local cross-section axes, shown in figure 7 above. For a rectangular cross-section, they are calculated with the following equations; b

h

2

Area   dA  b  dx  h  dy  bh b

0

b

I xx   y 2 dA  h

2

 y dy  2

b

FEA Bar properties. (1)

2

 Double click on the FEA Properties column header in the FEA Bars spreadsheet in the Structure window.

3

bh 12

(2)

2

 Select Bar 1 from the drop down list in the dialog box that opens, and click on OK. Now the newly defined Bar 1 property has been assigned to all 10 FEA Bars on the structure model.

h

I yy   x 2 dA  b  x 2 dx  0

bh 3 3

  b2   I xy   xydA   x  ydy dx  0  0  b  2

(3)

h

J   x 2  y 2 dA  I zz  I xx  I yy

(4)

Bar Orientation Note from Figure 7 above that the cross-section properties are computed with respect to a set of local section axes. The orientation of these cross-section axes must be defined relative to the global axes. This is accomplished by choosing an Orientation Point for the cross-section of each FEA Bar. NOTE: An Orientation Point is any Point that is not in line with the attachment points of the FEA Bar. The Y-axis of the

(5)

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App Note #3

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cross-section is assumed to lie in the plane defined by the Orientation Point and the two FEA Bar attachment points. We will orient the cross-section Y-axis of all 10 FEA Bars by choosing Point 19 on the Plate as their Orientation Point. This Point is labeled in the Figure below.

25-Feb-14

After the modification has been completed, a dialog box will open asking you to choose a Shape Table for storing the new mode shapes.  Press the New File button, enter the name “FEA Bars” into the dialog box that opens, and click on OK. A new Shape Table window will open with the new mode shapes in it.  Execute Tools | Animate Shapes from SHP: FEA Bars to start the animated display of the new mode shapes Notice all of the first 20 modes have little or no motion along the centerline of the plate. All of these mode shapes are reflecting the strong influence of the rib stiffener. Truncated Modal Model

Figure 8. Model with Bar 1 Properties & Orient Point 19.  Double click on the Orient Point column heading in the FEA Bars spreadsheet.  Enter 19 into the dialog box, and click on OK. The 10 FEA Bars are now ready to model the rib stiffener. MAKING THE MODIFICATION WITH SDM Now the model is ready to compute the effect of the FEA Bar rib stiffener on the modes of the plate.  Execute SDM | Calculate New Modes.

The last 4 modes (21 through 24) have mode shapes that don’t reflect the influence of the rib stiffener. These are called computational modes, and result because the modal model of the unmodified structure was truncated. The real plate structure has an infinite number of modes, at least in principle. The truncated modal model used to represent the dynamics of the unmodified structure only had 24 modes in it. Because the dynamic model is truncated, the SDM solution includes some computational modes (with unrealistic mode shapes and higher frequencies), in order to compensate for the absence of the higher frequency modes in the modal model of the unmodified structure. COMPARING MODIFIED AND UNMODIFIED MODES Some of the natural frequencies in the SHP: FEA Bars Shape Table match exactly with frequencies in the Shape Table of unmodified structure. Comparison animation can be used to see how many of the modes were not changed by the rib stiffener.

 Select SHP: Aluminum Plate in the dialog box that opens. The following dialog box will open.

 Execute File | Structure Options, click on the Animation tab, and select Sources on Right, check Display MAC, and click on OK.  Execute Animate | Compare Shapes in the STR: Aluminum Plate window.  Execute Windows | Arrange Windows | For Animation in the ME’scope window.

 Verify that new modes will be calculated using only 10 FEA Bars, and click on Yes. The SDM algorithm uses the modes of the unmodified structure in SHP: Aluminum Plate plus the FEA Bars attached to the structure model in STR: Aluminum Plate to create a set of equations of motion for the modified structure. These equations are then solved for the new modes of the structure.

The SHP: FEA Bars mode shapes should be displayed on the left, and the SHP: Aluminum Plate shapes on the right, as shown below.  Right click in the structure graphics area, and execute Synchronize Shapes from the menu.

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App Note #3

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25-Feb-14

MAC Values As you click on each shape in either Shape Table to display its shape in animation, the corresponding shape in the other table that is nearest in frequency will also be displayed. NOTE: During comparison animation, if Animate | Compare Shapes | Synchronize Shapes is enabled, each time a new shape is displayed, the comparison shape nearest in frequency will also be displayed. Notice that the MAC value for each shape pair is also displayed in the structure window graphics area. MAC is a numerical measure of how much alike two shapes are.

A MAC bar chart is another way to compare all of the FEA modes with all of the SDM modes together.  Execute Animate | Compare Shapes again to stop the animation.  Execute Display | MAC in the SHP: Aluminum Plate with Rib window.  Select SHP: FEA Bars from the list, and click on OK.

MAC = 1.0 means the shapes are identical

The MAC window will open displaying a bar chart, as shown below.

MAC >= 0.9 means the shapes are very similar

Eighteen mode shape pairs have MAC >= 0.90

MAC < 0.9 means the shapes are different

This is significant agreement between the SDM and FEA results. Some factors that could account for differences in the two solutions are;

Eleven pairs of shapes have MAC values greater than 0.9. This means that the rib stiffener had no influence on eleven modes of the unmodified plate. The remaining 9 modes (of modes 1 through 20) in SHP: FEA Bars are brand new modes, with new mode shapes created by the influence of the rib stiffener. COMPARING SDM AND FEA MODES

The new modes calculated with the SDM method will now be compared with the FEA modes obtained from the STR: Aluminum Plate with Rib model.  Open the STR: Aluminum Plate with Rib window by clicking on it in the Project panel.

 FEA Bar elements were used to model the rib with SDM while FEA Quad plate elements were used to model the rib in the FEA model.  SDM used a truncated modal model (with only 24 modes) to represent the dynamics of the unmodified structure.  The FEA Bars were attached to the plate at only 11 Points.  The FEA model was heavily discretized, using only 121 nodes or 726 DOFs to model the plate and rib.

 Open the SHP: Aluminum Plate with Rib Shape Table window by clicking on it in the Project panel.  Execute Animate | Compare Shapes in the STR: Aluminum Plate window.  Execute Windows | Arrange Windows | For Animation in the ME’scope window. The SHP: Aluminum Plate with Rib mode shapes should be displayed on the left, and the SHP: FEA Bars shapes on the right, as shown below.

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App Note #3

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Figure 9. MAC Values between SDM & FEA modes.

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25-Feb-14


ME'scopeVES Application Note #03 - Modeling a Rib Stiffener with SDM