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App Note #4

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25-Feb-14

ME’scope Application Note #4 Using Structural Dynamics Modification for Sub-Structuring INTRODUCTION NOTE: The steps in this Application Note can be duplicated using any Package that includes the VES-5000 SDM option. In this application note, two substructures will be joined together using the SDM (Structural Dynamics Modification) option in ME’scope. A simple two degree-offreedom (2-DOF) substructure will be mounted on a flat plate substructure to model the response of the combined substructures. WHAT IS SUBSTRUCTURING? In brief, sub-structuring is the process of modeling the attachment of two or more structures together using FEA elements. Each of the structures is termed a substructure and it is dynamically described by a set of its modes, called a modal model. The objective is to calculate the new modes of the combined structure, starting with only the modes of each substructure and the attachment FEA elements. The modes of the substructures may be either EMA modes (obtained experimentally), or FEA modes (obtained from an FEA model).

Figure 1. Plate-on-Springs SubStructure. This constrained structure is represented dynamically by a set of 12 mode shapes. Each mode shape has 63 DOFs, describing Z-axis (vertical) motion at 63 Points in a 7 by 9 grid of Points. The 2-DOF substructure shown in Figure 2 consists of two point masses connected together with a single spring. The modal model for this 2-DOF spring-mass substructure will be obtained by solving for the modes of its FEA model.

Each substructure modal model needs to be a complete description of the substructure’s dynamics, including its boundary conditions. If a substructure is modeled in the free-free condition, its modal model must include its rigidbody modes. If a substructure is attached to ground, no rigid-body modes are required. NOTE: If one substructure is to be mounted onto another and is not attached to ground, the rigid-body modes of the mounted structure must be included in its modal model. To perform sub-structuring, the following is needed: 1. Properly scaled mode shapes (i.e. a modal model) of each substructure. 2. Properly scaled 3D model of each substructure, with FEA elements added between the substructures where they are to be connected together. THE SUBSTRUCTURES The plate-on-springs substructure is an (8 x 10 x ¼ inch) aluminum plate supported vertically by four springs of 100 lb/in stiffness and restrained in-plane by four more springs as shown in Figure 1.

Figure 2. 2-DOF SubStructure. A lower spring shown in Figure 2 will be used during substructuring to “mount” the 2-DOF substructure on the Plateon-Springs substructure.

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App Note #4

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2/25/2014

 In the dialog box that opens, make the selections shown below, and click on OK.

GETTING STARTED  Open the App Note 04-Using SDM for SubStructuring Project from the ME’copeVES\Application Notes folder. Modes of the Plate-On-Springs  Open STR: Plate on Springs and SHP: Plate on Springs  Right click in the graphics area of the STR: Plate on Springs window, and execute Shapes from the menu

Draw | Animate

 Click on each Shape button in the SHP: Plate on Springs window to display its shape Notice that the first three modes (25.99, 29.93 & 33.27 Hz), are rigid body modes, where the plate simply moves vertically on the support springs and does not flex.  Select Displacement in the next dialog box, and click on OK  Press the New File button in the next dialog box, enter “Driving Point FRFs” into the next dialog box, and click on OK.  Execute Format | Overlaid in the Data Block window that opens.

Figure 3. First Torsional Mode of the Plate-on-Springs Notice also that the plate center is located along node lines for all torsional modes. A 2-DOF substructure will be attached to the center of the plate. VERIFYING THE MODAL MODEL One way to verify the plate modes and also determine which modes are participating in the response at the center of the plate-on-springs is to synthesize Driving Point FRFs from the modal model. NOTE: A Driving Point FRF is an FRF that has a Roving DOF equal to its Reference DOF. In order to synthesize FRFs using the mode shapes in SHP: Plate on Springs, they will require modal damping.  Double click on the Damping column heading in the SHP: Plate on Springs window  Enter “1” into the dialog box that opens, and click on OK.

Figure 4. 63 Overlaid Driving Point FRFs.  Right click, select Horizontal Axis from the command menu, and select Log Frequency in the Show Axis As list in the dialog box that opens.  The overlaid plot of Driving Point FRFs in Figure 4 clearly shows peaks for all 12 modes.  Execute Format | Rows Columns, and select 1,1

 Execute Tools | Synthesize FRFs

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App Note #4

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2/25/2014

Figure 5. Driving Point FRF 32Z:32Z.  Scroll the Trace display to display FRF 32Z:32Z, as shown in Figure 5 Only four resonance peaks (25.99, 476.0, 772.6 & 1731 Hz) appear in FRF 32Z:32Z.  Zoom the display around the highest frequency peak.

The cursor value is .0025344 in/lb. This flexibility value is equal to the inverse of the stiffness, or 394.5 lb/in, which is close to the stiffness of the four supporting springs. The mass of the plate (5.78 lb) can be estimated from the mass line which is higher in frequency than the rigid body mode resonance peak. The mass line should be a horizontal line in an (acceleration/force) FRF, which can be obtained by double differentiating the (displacement/force) FRF. 

Execute Tools | Differentiate twice in the Data Block window.

 Double click on the Units column in the Traces spreadsheet.  In the dialog box that opens, type “g/lb”, and click on OK.  Click on Yes to re-scale the Traces to g/lb units. Figure 6. Zoomed FRF with Peaks at 1691 &1731 Hz, The Zoomed display in Figure 6 verifies that both the 1691 Hz & 1732 Hz modes participate in the response at Point 32, the center of plate-on-springs.

 Position the Line cursor on the mass line above the rigid body mode, as shown in Figure 7.

Point 32 is a nodal point for the other eight modes (29.93, 33.27, 387.4, 903.1, 1028, 1403, 1692 and 2116 Hz). It is expected that attaching a substructure to this DOF will not affect these modes, but will only affect the modes that have peaks in the Driving Point FRF 32Z:32Z. Mass & Stiffness Lines The 32Z:32Z FRF has units of (displacement/force), and clearly shows the participation of the 25.99 Hz rigid-body mode. The sum of the stiffnesses of the four springs supporting the plate (4 x 100 lb/in = 400 lb/in) can be estimated from the frequency stiffness line below this rigid body resonance.  Position the Line cursor on the stiffness line, as shown below.

Figure 7. Cursor on Mass Line of Rigid Body Mode. The cursor value is 0.1727 g/lb for the nearly horizontal mass line. The inverse of this value is 5.79 lb/g, which is a close estimate of the mass of the plate in a 1g gravitational field, or the weight of the plate. By examining properties of synthesized FRFs for the plate, two of its key dynamic properties have been verified. This proves that its modal model is a valid representation of its dynamics, and therefore can be confidently used with SDM for sub-structuring.

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App Note #4

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2/25/2014

MODES OF THE 2-DOF SUBSTRUCTURE

 Click on the Masses tab

The modes of the 2-DOF substructure will be calculated from its FEA model, consisting of two FEA Point masses connected together by an FEA linear spring.

 Execute Edit | Add in the FEA Properties window  Click on the Springs tab

Adding Two FEA Masses  Execute Display | Surfaces | Transparent in the STR: 2-DOF SubStructure window.  Right click in the graphics area, and execute Object List |

 Enter “0.25” into the Mass (lbm) column.  Execute Edit | Add in the FEA Properties window  Enter “5000” into the Stiffness (lb/in) column. Assigning Properties and Constraining Motions

FEA Masses from the menu

 Right click in the graphics area in the STR: 2-DOF Sub-

 Right click in the graphics area, and execute Add Object from the menu  Click near the Point in the center of the Top Mass cube (with Point label 100) to add an FEA mass to the Point.  Execute Draw | Substructures | Add Selected Objects to Substructure.  In the dialog box that opens, select Top Mass, and click on Add Objects.  Click near the Point in the center of the Bottom Mass cube (with Point label 101) to add an FEA mass to the Point.  Execute Draw | Substructures | Add Selected Objects to Substructure.  In the dialog box that opens, select Bottom Mass, and click on Add Objects. Adding an FEA Spring between the Masses

Structure window, and execute Object List | Masses from the menu

FEA

 Double click on the FEA Property column heading in the mass properties spreadsheet, select Mass 1 from the list, and click on OK.  Double click on the Direction column heading, and select Z from the list in the dialog box.  Right click in the graphics area, and execute Object List | FEA Springs from the menu  Double click on the FEA Property column heading in the spring properties spreadsheet, select Spring 1 from the list, and click on OK.  Double click on the Point 1 Direction column heading, and select Z from the list in the dialog box.  Double click on the Point 2 Direction column heading, and select Z from the list in the dialog box. Solving for the 2-DOF Substructure Modes

 Right click in the graphics area, and execute Object List | FEA Springs from the menu  Right click in the graphics area, and execute Add Object from the menu  Click near the Point in the center of the Top Mass (with Point label 100) and then on the Point in the center of the Bottom Mass (with Point label 101) to add the FEA spring.

The 2-DOF FEA model now consists of 2 FEA Masses connected together with an FEA Spring. To solve for the two modes of this model,  Execute FEA | Calculate FEA Modes The following dialog box will open.

 Right click in the graphics area, and execute Add Object again to disable the Add operation.  Execute Draw | Substructures | Add Selected Objects to Substructure.  In the dialog box that opens, select Top Spring, and click on Add Objects Adding FEA Properties To create FEA Properties for the two masses and the spring,

 Click on Yes, set up the next dialog box as shown below, and click on OK

 Execute FEA | FEA Properties to open the FEA Properties window

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App Note #4

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2/25/2014

 Enter “2-DOF Substructure Modes” into the dialog box that opens, and click on OK. The Shape Table shown below will open with two modes in it. Notice that the first mode at 0.0 Hz is a rigid body mode, with equal mode shape components. The second mode at 625.45 Hz is a flexible body mode, with equal and opposite mode shape components.

Figure 9. 2-DOF & Plate-on-Springs Substructures. Merging the Two Shape Tables To merge together the mode shapes of the two substructures,  Execute Edit | Copy Shapes to File in the SHP: PlateOn-Springs window.  Click on New File in the dialog box that opens, and enter “Merged Mode Shapes” into the next dialog box.  Execute Edit | Paste Shapes from File in the SHP: Merged Mode Shapes window.  Select SHP: 2-DOF Substructure Modes from the list in the dialog box that opens, and click on OK.

Figure 8. 2-DOF Substructure Modes.

Block Diagonal Format

MERGING THE TWO MODELS Sub-structuring requires that the substructure models be merged together in the same Structure window, and also that the mode shapes of the two substructures be merged together in the same Shape Table window. Merging the Substructure Models The two substructures will be copied into a new Structure window.  Execute Edit | Copy Objects to File in the STR: PlateOn-Springs window.  Click on New File in the dialog box that opens, and enter “Merged Substructures” into the next dialog box.  Execute Edit | Paste Objects from File in the STR: Merged Substructures window.

The two modes of the 2-DOF substructure have now been merged with the modes of the plate-on-springs substructure. Because the DOFs of the 2-DOF substructure are different than the DOFs of the plate-on-springs substructure, the shapes have been merged together in a “block diagonal” format. In block diagonal format (shown in Figure 10), all shapes share the same DOFs, and the shape components of each shape are zero for DOFs where that shape was not originally defined. Notice also in Figure 10 that the M#s were also copied from the two Shape Tables, and that the list of M#s no longer contains unique M#s. To give all of the shape components unique M#s,  Execute Tools | Renumber M#s in SHP: Merged Mode Shapes to renumber the shape M#s.  Execute Tools | Create Animation Equations in SHP: Merged Mode Shapes to create new animation equations

 Select STR: 2-DOF Substructure in the dialog that opens, and click on Paste. The two substructures will be displayed together in the Merged Substructures window, as shown in Figure 9.

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App Note #4

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2/25/2014

 Right click in the graphics area, and execute Add Object again to disable the Add operation.  Double click on the FEA Property column heading in the spring properties spreadsheet, select Spring 1 from the list, and click on OK.  Double click on the Point 1 Direction column heading, and select Z from the list in the dialog box.  Double click on the Point 2 Direction column heading, and select Z from the list in the dialog box. INTERPOLATION OF UN-MEASURED POINTS The corner Points of the Mass substructures, and all of the interior Points on the two Spring substructures are unmeasured Points. NOTE: An un-measured Point is also called an Interpolated Point because it requires an Interpolated animation equation in order to be animated. Interpolating the Mass Cube Points  Select the Top Mass and Bottom Mass substructures. Figure 10. Substructure Modes in Block Diagonal Format.

 Execute Draw | Animation Equations | Create Interpolated.

ADDING THE ATTACHMENT FEA SPRING

 Click through the dialog boxes that open.

An FEA Spring Object will be used to mount the 2-DOF substructure on the plate-on-springs substructure, by adding an FEA Spring between the Bottom Mass and the center of the plate.

Interpolating the Spring Points

 Right click in the graphics area, and execute Object List | FEA Springs from the menu in the STR: Plateon-Springs window

 Hold down the Ctrl key and click near the bottom Point of FEA Spring 2 and the center Point (Point 32) of the plate-on-springs to select them, as shown in Figure 11.

 Right click in the graphics area, and execute Add Object from the menu

 Execute Draw | Animation Equations | Create Interpolated

 Click on the Point in the center of the Bottom Mass cube (with Point label 101) and then on the Point at the bottom of the Bottom Spring substructure to add the FEA spring as shown below.

 Click through several dialog boxes until the following box opens indicating that one Interpolated equation has been created for the un-measured spring end Point.

 Right click in the graphics area, and execute Object List | FEA Springs from the menu

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App Note #4

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2/25/2014

 The FEA Spring 2 object that connects the two substructures together in the STR: Merged Substructures window Checking the Animation Equations SDM uses the animation equations on the structure model for its computations. Therefore, the structure should be animated using the shapes of the un-modified structure before using SDM to calculate new modes.  Right click in the graphics area in the STR: Merged Substructures window, and execute Animate Shapes When a plate mode is selected, the plate should be deflected. When a 2_DOF mode is selected the masses and spring should be deflected.

Figure 11. Spring Point and Center Point Selected. The un-measured end Point at the bottom of Spring 2 now has the same animation equation as Point 32 on the plate, but it is an Interpolated equation. This equation will be converted from an Interpolated to a Measured equation so that the rest of the Points on the Spring substructure can be Interpolated from its two Measured end Points.  Execute Draw | Animation Equations | Interpolated to Measured To create Interpolated equations for all of the Spring substructures,  Select all of the Spring substructures in the STR: Merged Substructures window

Figure 12. Animation of the 625.45 Hz mode. When you are satisfied that both the plate modes and the 2DOF modes are animating correctly,  Execute SDM | Calculate New Modes

 Execute Draw | Animation Equations | Create Interpolated

 Select SHP: Merged Mode Shapes in the dialog box that opens, and click on OK

 Make sure that the following dialog box has a “2” in it, and click on OK

 If the following dialog box opens, click on No, make the two FEA Masses and the FEA Spring 1 invisible, and execute SDM | Calculate New Modes again.

SUB-STRUCTURING WITH SDM SDM will use the following to calculate new modes for the combined substructures;  Shapes in the SHP: Merged Mode Shapes window

When the SDM calculation has completed, a dialog box will open for saving the new mode shapes.

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App Note #4

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2/25/2014

 Enter “New Mode Shapes” into the dialog box, and click on OK.

phase with the plate center and the lower mass is nearly stationary.

The new mode shapes can now be displayed in animation of the model in STR: Merged Substructures.

Modes 7 & 8 show interaction between the 2-DOF substructure and the 772.6 Hz mode of the plate substructure. In mode 7 (704.2 Hz), the masses move out-of-phase with one another, and the lower mass moves in-phase with the plate center. The motion of the plate is small, again indicating that the 2DOF substructure is absorbing energy from the plate. In mode 8 (797.3 Hz), the plate is quite active. The lower mass moves out-of-phase with both the plate and the upper mass.

Right click in the graphics area in the STR: Merged Substructures window, and execute Animate Shapes

Note that the first mode (24.92 Hz) is a rigid-body translation and that the springs of the 2-DOF substructure do not deform. The attached 2-DOF is acting like a rigid mass (of 0.5 lb) attached to the center of the plate. The deformation of the plate is of the same as that of its original 25.99 Hz mode shape before the 2-DOF substructure was attached to it. The frequency of the new mode can be calculated by the simple SDOF relationship of a mass on a spring:

SUMMARY Two substructures were dynamically coupled together using SDM. In this application note you: 1.

Added FEA Masses and FEA Springs to 3D model of a 2-DOF structure.

2.

Solved for the modes of 2-DOF FEA model.

3.

 Right click in the graphics area in the STR: Merged Substructures window, and execute Compare Shapes

Merged the modes of this model with the modes of a second plate-on-springs structure.

4.

Connected the two substructures together with an FEA Spring.

 Right click in the graphics area, execute Structure Options, and check Display MAC on the Animation tab

5.

Solved for the new modes of the coupled substructures using SDM.

f Final  f Plate

MassPlate 5.78  25.99  24.93 Hz 5.78  0.5 MassFinal

COMPARING SHAPES

 Right click in the graphics area, and execute Compare Shapes | Synchronize Shapes The first modes of the modified and unmodified structure should be displayed as shown below. The MAC value indicates that these mode shapes are 95% alike.

Modes 2 & 3 remain unchanged because DOF 32Z was a node point for these shapes in the plate-on-springs substructure. Modes 5 and 9 through 14 also remain unchanged because all of these modes have a node point at DOF 32Z. Modes 4 and 6 reflect interaction between the 2-DOF substructure and the 476.0 Hz mode of the plate substructure. With mode 4 (276.8 Hz), the masses move in-phase with one another and tend to suppress or absorb motion from the plate. In mode 6 (481.2 Hz), the upper mass moves out-of-

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ME'scopeVES Application Note #04 - Using Structural Dynamics Modification for Sub-Structuring  
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