App Note 04 www.vibetech.com 2-Aug-18 ME scope Application Note 04 Using SDM for Sub-Structuring NOTE: The steps in this Application Note can be duplicated using any Package that includes the VES-5000 SDM option. Click here to download the ME scope Demo Project file for this App Note. WHAT IS SUBSTRUCTURING? In this application note, two substructures will be joined together using the SDM (Structural Dynamics Modification) option in ME scope. A simple two degree-of-freedom (2-DOF) substructure will be mounted on a flat plate substructure to model the response of the two combined substructures. 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). 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 six rigid-body 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. Unit Modal Mass (UMM) are required by SDM 2. A 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 the figure below. Plate-on-Springs SubStructure. Page 1 of 15
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 the figure below 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. 2-DOF Spring-Mass SubStructure. During sub-structuring, the lower spring will be attached to the Plate-on-Springs substructure to mount the 2-DOF substructure on the Plate-on-Springs substructure. GETTING STARTED Open the Project file for this App Note. Modes of the Plate-On-Springs Open the STR: Plate on Springs and SHP: Plate on Springs windows Right click in the graphics area of the STR: Plate on Springs window, and execute Draw Animate Shapes from the menu Click on each Shape button in the SHP: Plate on Springs window to display its shape Notice that the first three modes (25, 29 & 33 Hz), are rigid body modes, where the plate simply moves vertically on the support springs and does not deform. First Torsional Mode of the Plate-on-Springs Page 2 of 15
Notice also that the plate center is located along node lines for all torsional modes. The 2-DOF Spring-Mass substructure will be attached to the center of the plate. VERIFYING THE MODAL MODEL One way to verify the plate modes and 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 between two DOFs where its Roving DOF equals 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 (Hz) column heading in the SHP: Plate on Springs window Enter 1 into the dialog box that opens, and click on OK. This addd 1 Hz of damping to all of the modes Execute Tools Synthesize FRFs In the dialog box that opens, make the selections shown below, 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. 63 Overlaid Driving Point FRFs. Page 3 of 15
Double click on the horizontal axis (X-Axis) and select Log Frequency in the dialog box that opens. The overlaid plot of Driving Point FRFs above clearly shows peaks for 12 modes. Execute Format Rows Columns, and select 1,1 Scroll the M# display to display FRF 32Z:32Z, as shown in the figure below Only four resonance peaks appear in FRF 32Z:32Z. Driving Point FRF 32Z:32Z. Zoom the display around the highest frequency peak. Zoomed FRF with two Peaks at near 1700 Hz, The Zoomed display verifies that near 1700 Hz, only two modes participate in the response at Point 32, the center of the Plate-on-Springs substructure. Point 32 is a nodal point for the other modes. Therefore, attaching a substructure to DOF 32Z 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), The un-zoomed display of FRF 32Z:32Z clearly shows the participation of the 26 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 stiffness line at a frequency below this rigid body resonant frequency. Position the Line cursor on the stiffness line, as shown below. Page 4 of 15
The cursor value is.0025454 in/lb. This flexibility value is equal to the inverse of the stiffness, or 393 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 BLK: Driving Point FRFs window Double click on the Units column in the M#s spreadsheet In the dialog box that opens, type g/lb, and click on OK Click on Yes to re-scale the M#s to g/lb units Position the Line cursor on the mass line above the rigid body mode in M#32, as shown below Cursor on Mass Line of Rigid Body Mode. The cursor value is 0.1726 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 (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 the dynamics of a structure, and therefore can be confidently used with SDM to model structural modifications. MODES OF THE 2-DOF SUBSTRUCTURE 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. Page 5 of 15
Adding Two FEA Masses Execute Display Surfaces Transparent in the STR: 2-DOF SubStructure window. Right click in the graphics area, and execute Edit Current Objects FEA Masses 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 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 Right click in the graphics area, and execute Edit Current Objects 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 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, Execute FEA FEA Properties to open the FEA Properties window Click on the Masses tab Execute Edit Add in the FEA Properties window Enter 0.25 into the Mass (lbm) column Click on the Springs tab Execute Edit Add in the FEA Properties window Enter 5000 into the Translational Stiffness (lb/in) column Page 6 of 15
Assigning Properties and Constraining Motions Right click in the graphics area in the STR: 2-DOF SubStructure window, and execute Edit Current Objects FEA Masses from the menu 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 Edit Current Objects 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 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 Click on Yes, set up the next dialog box as shown below, and click on OK Page 7 of 15
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 essentially 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. NOTE: Only the Translational mode shape components are non-zero. All of the Rotational mode shape components are zero. MERGING THE TWO MODELS Sub-structuring has two requirements; 2-DOF Substructure Modes. 1. The substructure models have to be merged together in the same Structure window 2. The mode shapes of the substructures have to be merged together in the same Shape Table Merging the Substructure Models The two substructures will be copied into a new Structure window. Execute Edit Copy Objects to File in the STR: Plate-On-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 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 below. Page 8 of 15
Merging the Two Shape Tables To merge the mode shapes of the two substructures, 2-DOF & Plate-on-Springs Substructures. Execute Shapes Copy to File in the SHP: Plate-On-Springs window. Click on New File in the dialog box that opens, and enter Merged Mode Shapes into the next dialog box Execute Shapes Paste 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 Shape Table Block Diagonal Format The two modes of the 2-DOF Substructure have now been merged with the modes of the Plate-on-Springs substructure. Because the M# DOFs of the 2-DOF Substructure are different from the M# DOFs of the Plate-on-Springs substructure, the shapes have been merged together in a block diagonal format. In block diagonal format, all shapes share the same M# DOFs, and the shape components of each shape are zero valued for M# DOFs where that shape is not defined. Substructure Modes in Block Diagonal Format. Page 9 of 15
Creating More M# Links The M#s of the Plate-On-Springs are already linked to their substructure. However, because the M#s of the mode shapes of the 2-DOF Substructure have been added to the M#s of the Plate-On-Springs, they have to be linked to the model in the STR: Merged Substructures window. Select the Top Mass and Bottom Mass in the STR: Merged Substructures window. Execute Animate Link M#s in the SHP: Merged Mode Shapes window and link this Shape Table to the Merged Substructures window After the Links have been created the following dialog box will open. Only the M#s DOFs of the two masses have been linked to the model. SORTING THE POINTS BY THEIR LABELS It is useful to sort the Points spreadsheet by their labels so that it is easier to identify and select them. This brings the numbered Points to the top of the Points spreadsheet Execute Edit Edit Current Objects Points Execute Edit Sort Objects by Label ADDING THE FEA SPRING BETWEEN THE SUBSTRUCTURES An FEA Spring Object will be used to mount the 2-DOF Substructure on the Plate-on-Springs substructure. The two substructures will be attached together by adding an FEA Spring between the Bottom Mass and the center of the plate. Right click in the graphics area, and execute Edit Current Objects FEA Springs from the menu in the STR: Plate-on-Springs window Right click in the graphics area, and execute Add Object from the menu Click on the Point in the center of the Bottom Mass cube (with Point label 101) and then on Point 32 on the Plate-on-Springs model to add the FEA spring as shown below. Page 10 of 15
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. INTERPOLATED LINKS FOR THE UN-MEASURED POINTS The corner Points of the Mass cubes, and all of the interior Points on the two springs (on the 2-DOF Substructure), are un-measured Points. NOTE: An un-measured Point is also called an Interpolated Point because it requires an Interpolated M# Link in order to be animated. Creating Interpolated M# Links for the Mass Cube Points Select the Top Mass and Bottom Mass substructures Execute M# Links Create Interpolated M# Links Click through the dialog boxes, and enter 1 in the following dialog box When the links are created, the following dialog box will open Interpolating the Top Spring Points Right click in the graphics area, and execute Edit Current Objects FEA Springs from the menu Select Spring 1 in the Objects spreadsheet Right click in the graphics area, and execute Edit Current Objects Substructures from the menu Select Top Spring in the Objects spreadsheet, as shown below Page 11 of 15
Execute M# Links Create Interpolated M# Links Click through several dialog boxes and enter 2 in the following dialog box. After the M# Links have been created, the following dialog box will open. Interpolating the Bottom Spring Points Right click in the graphics area, and execute Edit Current Objects FEA Springs from the menu Select Spring 2 in the Objects spreadsheet Right click in the graphics area, and execute Edit Current Objects Substructures from the menu Select Bottom Spring in the Objects spreadsheet, as shown below Execute M# Links Create Interpolated M# Links Click through several dialog boxes and enter 2 in the following dialog box. After the M# Links have been created, the following dialog box will open. Page 12 of 15
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 The FEA Spring 2 object that connects the two substructures together in the STR: Merged Substructures window Checking the M# Links SDM uses the M# Links on the structure model for its computations. To check the M# Links, 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 Draw Animate Shapes When a plate mode is selected, the plate should be deflected. When a mode of the 2-DOF Substructure is selected, the masses and spring should be deflected. Animation of the 625.45 Hz mode. When you are satisfied that both the plate modes and the 2-DOF modes are animating correctly, Execute SDM Calculate New Modes Select SHP: Merged Mode Shapes in the dialog box that opens, and click on OK If the following dialog box opens, click on No Page 13 of 15
Make the Top Mass, Bottom Mass and Top Spring invisible, as shown above Execute SDM Calculate New Modes again When the SDM calculation has completed, a dialog box will open for saving the new mode shapes. Enter New Mode Shapes into the dialog box, and click on OK. The new mode shapes can now be displayed in animation of the model in STR: Merged Substructures. Right click in the graphics area in the STR: Merged Substructures window, and execute Draw 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 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: MassPlate 5.78 f Final f Plate 25.99 24.93 Hz Mass 5.78 0.5 Final COMPARING SHAPES Right click in the graphics area in the STR: Merged Substructures window, and execute Draw Compare Shapes Right click in the graphics area, execute Structure Options, and check Display MAC on the Animation tab Right click in the graphics area, and execute Animate Compare Shapes Maximum MAC The first mode of the modified structure should compare closely with the first mode of the unmodified sub structures, as shown below. The MAC value indicates that these two mode shapes are 97% alike. Modes 2 & 3 remained unchanged after the substructures were connected together because DOF 32Z was a nodal point for those shapes in the Plate-on-Springs substructure. Many of the higher frequency modes were also unchanged because all of those modes have a nodal point at DOF 32Z. Modes 4 and 6 reflect interaction between the 2-DOF substructure and the 476.0 Hz first bending mode of the Plate-on-Springs substructure. Mode 4 (276.8 Hz) of the connected substructures, with the two masses moving in-phase with one another, suppressed the motion of the first bending mode of the plate. On the other hand, mode 6 (481.2 Hz) of the connected substructures, with the two masses moving out-of-phase with one another, did not suppress the first bending mode of the plate. Page 14 of 15
Modes 7 & 8 of the connected substructures exhibit similar influence on the 772.6 Hz mode of the Plate-on- Springs substructure. Mode 7 (704.2 Hz), suppresses the motion of the 772.6 Hz mode while mode 8 does not. SUMMARY SDM was used to dynamically couple two substructures together using a single FEA spring element. In this App Note the following steps were carried out: 1. FEA Masses and an FEA Spring were used to create a 2-DOF structure 2. The two modes of 2-DOF FEA model were solved for 3. The modes of the 2-DOF FEA model were merged with the modes of a Plate-on-Springs structure in block diagonal format 4. The two substructures were connected together with using a single FEA Spring 5. SDM was used to solve for the new modes of the coupled substructures Page 15 of 15