App Note #14
www.vibetech.com
25-Feb-14
ME’scope Application Note #14 Multi-Reference Curve Fitting To Find Repeated Roots INTRODUCTION
DISPLAYING SHAPES IN ANIMATION
In this note, the VES-4500 Multi-Reference Modal Analysis Option will be used to estimate the modal parameters of the repeated roots in an axi-symmetric structure.
To display the mode shapes of the circular disk in animation:
Repeated Root: A structure is said to have a repeated root if two or more of its modes have the same frequency but different mode shapes.
Execute Draw | Animate Shapes in the STR: Circular Disk Structure window.
To create a structure with repeated roots, a finite element model of a circular plate was constructed using NASTRAN, and its modal parameters (natural frequencies & mode shapes) were found as the eigensolution to the equations of motion for the structure. The natural frequencies are called eigenvalues and the mode shapes are called eigenvectors. Both the structure model and its modal parameters were then imported into ME’scope.
Open App Note 14 - Repeated Roots.VTprj from the ME’scope\Application Notes folder.
Select SHP: Circular Disk Z DOFs Only in the animation source list on the Structure window toolbar. Click on a Select Shape button in the Shape Table to display that mode shape in animation. Notice that pairs of repeated roots are listed in the Shape Table, starting at 108 Hz up to 1442 Hz, with the exception of the single mode (a non-repeated root) at 201 Hz.
Figure 1. Disk Showing Repeated Root Mode Shapes. Figure 1 shows an animated display of two of the modes of the circular disk. Notice that the two modes have the same frequency (108.68 Hz), but different mode shapes. The mode shapes are very similar, but are merely “rotated” by 45 degrees from one another, as evidenced by their nodal lines. This structure has a number of repeated roots, indicated by the pairs of identical frequencies in the Shape Table shown in Figure 2. In this Application Note, a set of multiple reference FRFs will be synthesized using the repeated root pairs. These FRFs will then be processed using multi-reference curve fitting to recover the modal parameters of the modes. Finally, the curve fitting estimates will be compared with the original modal parameters. Figure 2. Shape Table Showing Repeated Root Pairs.
Page 1 of 5
App Note #14
www.vibetech.com
Notice also that all of the modes have 1% critical damping. Normally, modes extracted from FEA models have no damping. The 1% critical damping was artificially added to the modes so that they could be used to synthesize FRFs.
25-Feb-14
Click on OK in the next dialog box that opens, enter “5Z 15Z 25Z” into the next dialog box, and click on OK.
FRF SYNTHESIS A set of FRFs will be synthesized using the modes from the Shape Table. Three reference DOFs will be chosen and FRFs will be synthesized between all of the mode shapes DOFs and the three reference DOFs. To determine two suitable references for synthesizing FRFs: Execute Display | Points | Point Labels in the Structure window.
Figure 4. FRF Synthesis Dialog Box. After the FRFs are synthesized, a Data Block window will open. Notice from the Title bar that 480 FRFs were synthesized, 160 Roving DOFs paired with each of the 3 Reference DOFs. Scan through all of the FRFs using the vertical scroll bar between the graphics and the Traces spreadsheet. Figure 3. Structure Model Showing Point Labels. We will choose Points 5, 15 & 25 for the three references. To synthesize a set of FRFs using these three references: Execute Tools | Synthesize FRFs in the Shape Table window. The FRF Synthesis dialog box will open. To select all of the Roving DOFs in the dialog box: Click on the first Roving DOF (1Z) to select it. Hold down the Shift key, scroll to the end of the Roving DOF list and click on the last DOF (160Z). To select the Reference DOFs (5Z, 15Z, & 25 Z): Hold down the Ctrl key, click on 5Z to select it, scroll to 15Z and click on it to select it, scroll to 25Z and click on it to select it. Enter a Block Size of 2000. Enter an Ending Frequency of 1500 Hz. To synthesis the FRFs, click on OK.
Figure 5. Data Block Containing 480 Synthesized FRFs. Differentiation of FRFs Notice that the units of the FRFs are (acceleration / force units). Notice also that the mode shapes have units of (displacement / force - seconds).
Page 2 of 5
App Note #14
www.vibetech.com
25-Feb-14
Since acceleration responses were selected in one of synthesis dialog boxes, the FRFs were first synthesized in (displacement / force) units, and then double differentiated to yield FRFs in (acceleration / force) units.
Use the scroll bar on the right side of the Mode Indicator to move the noise threshold line upward until only 19 peaks are counted.
MULT-REFERENCE CURVE FITTING To initiate curve fitting,
Make sure that Multi-Reference Polynomial is selected in the Method box on the Frequency & Damping tab.
Execute Modes | Modal Parameters in the Data Block window.
Press the Frequency & Damping button, and click on Yes in the dialog box that opens..
This will display the FRFs in the upper-left, the Mode Indicator graph in the lower-left, the curve fitting tabs in the upper-right, and the modal parameters spreadsheet in the lower-right of the window.
When curve fitting is completed, the frequency & damping of 19 modes will be listed in the Modal Parameters spreadsheet on the lower-right side of the window.
Frequency & Damping
Notice that modal frequencies of 9 pairs of repeated roots have been correctly identified and the single mode at 202 Hz was also correctly identified. Notice also that all of the modal damping valued are indentified as 1% critical damping. Modal Participation Factors Right click on the column heading area in the Modal Parameters spreadsheet, and select Show/Hide Columns from the menu. The File | Data Block Options dialog box will open. Check Participation in the dialog box, and click on OK.
Figure 6. Data Block Window During Curve Fitting. Complex Mode Indicator Function (CMIF) The CMIF function is calculated by performing a Singular Value Decomposition of all of the FRFs. One CMIF curve is created for each reference in the FRFs. In this case, three CMIFs are created. Select Multi-Reference CMIFs from the selection box on the Mode Indicator tab. Press the Count Peaks button on the Mode Indicator tab. Each of the CMIFs is colored with a different color, which is chosen on the Contour Colors tab in the File | Data Block Options box. Also, the peaks on each CMIF curve are counted, and the total number of peaks on all three curves is displayed in the Peaks box on the Mode Indicator tab. Each peak that is counted is indicated with a red dot on the CMIF. Figure 6 shows that 21 modes were counted. We already know that only 19 modes were used to synthesize the FRFs. The extra two peaks are due to “crossovers” between the CMIF curves, and were counted in error.
Figure 7. Frequency, Damping and Participation Factors. Modal participation factors are “weights” (with values from 0.0 to 1.0) that indicate the participation of each mode in each reference of FRFs. A participation factor = 1.0 means that a mode is strongly represented in a particular reference of data. In other words, its resonance peaks in that reference of data are larger than the peaks of other modes. Furthermore, its mode
Page 3 of 5
App Note #14
www.vibetech.com
25-Feb-14
shape component corresponding to the Reference DOF in the FRF is close to an anti-node of the shape.
ing of all of the modes were recovered by curve fitting the FRFs. But, were the mode shapes recovered?
Likewise, a participation factor = 0.0 means that a mode is not represented in a particular reference of data, (has no resonance peaks). Also, its mode shape component is close to a node (zero amplitude) of the mode shape.
The shape comparison display and the Modal Assurance Criterion (MAC) will be used to compare the original FEA mode shapes with the mode shapes obtained by curve fitting the synthesized FRFs.
Residues
Press the Save Shapes button on the Residues & Save Shapes tab.
Make sure that Multi-Reference Polynomial is selected in the Method box on the Residues & Save Shapes tab. Press the Residues button, and click on Yes in the dialog box that opens. When curve fitting is completed, the residues of 19 modes will be listed in the Modal Parameters spreadsheet on the lower-right side of the window. In addition, a synthesized red curve fit function is overlaid on each FRF. Since the FRFs were originally synthesized from mode shapes, each fit function overlays its corresponding FRF.
Click on New File button in the dialog box that opens, and click on OK. Enter “CMIF” into the dialog box that opens, and click on OK. A new Shape Table will open with 19 mode shapes in it.
Use the scroll bar next to the Trace graphics to scroll through the FRFs and overlaid fit functions. When you release the mouse button, the residues corresponding to each FRF are listed in the Modal Parameters spreadsheet. Modal frequency & damping are global estimates, and therefore they will not change as you scroll the display of the FRFs. Modal residues are local to each FRF and they will be different for each FRF.
Figure 9. Shape Table Containing 19 Residue Mode Shapes.
COMPARING MODE SHAPES
Scroll through the DOFs in the lower spreadsheet to examine the mode shapes.
Nine pairs of repeated roots, plus a single non-repeated root were found from a multi-reference set of FRFs using MultiReference curve fitting. It is evident that the frequency & damp-
Notice that there are 460 M#s (or DOFs) in each shape. Each of the 160 Roving DOFs is paired with one of the 3 Reference DOFs.
Figure 8. Modal Parameters Following Residue Curve Fitting.
Page 4 of 5
App Note #14
www.vibetech.com
New Animation Equations The mode shapes in the SHP: Circular Disk Z DOFs Only had 160 M#s in them. In order to display mode shapes from the SHP: CMIF Shape Table, new animation equations must be created.
25-Feb-14
Execute Animate | Compare Shapes in the Structure window. Now, shapes from SHP: CMIF and SHP: Circular Disk Z DOFs Only should be displayed together in animation, as shown in Figure 10 below.
Execute Tools | Create Animation Equations in the SHP: CMIF Shape Table, and click of OK in the dialog box that opens. The following dialog box will open, indicating that 460 new animation equations have been created.
In fact, only 160 equations were created, but each equation has three terms in it, one for each M# in the SHP: CMIF Table that has the same Roving DOF. NOTE: Only the Roving DOFs of the shape M#s are considered when animation equations are created by matching shape DOFs to structure model DOFs. Execute Tools | Animate Shapes in the SHP: CMIF Table. The following dialog box will open, reminding you that you must select a reference from which to display mode shapes in animation.
Figure 10. Comparison of FEA and Multi-Ref Curve Fit Shapes Select different references in SHP: CMIF. Notice that the shape does not change when a different Reference DOF is selected. NOTE: When the same shape is displayed for different Reference DOFs, this indicates that the shape is a mode shape. Mode shapes are inherent properties of a structure, and are independent of the Reference DOF. Notice also that even though the 108.68 Hz repeated roots from the two sources have similar shapes, the shapes from one source are “rotated” (about 25 degrees) from the other. The MAC value equals 0.5 for all 108.68 Hz pairs, indicating that the shapes in each pair are not the same. Select Shape #3 (201.61 Hz) in both Shape Tables. Not only do these two shapes look the same, but their MAC value equals 1.0, confirming that multi-reference curve fitting has recovered the same mode shape using multi-reference curve fitting on the synthesized FRFs.
Click on OK to open the DOF selection box. NOTE: The DOF selection box should be left open during animation so that shapes can be displayed for one Reference DOF at a time. Execute Animate | Contours | Contour Colors. Execute Animate | Deformation | Undeformed. Execute File | Structure Options to open the Structure window Options box. On the Animation tab, check Display MAC. Close the Data Block window BLK: 5Z 15Z 25Z.
Page 5 of 5