VES App Note #14
9/5/2000
ME’scopeVES Application Note #14
Multiple Reference Curve Fitting To Find Repeated Roots
INTRODUCTION
DISPLAYING SHAPES IN ANIMATION
In this note, the capabilities of the Complex Mode Indicator
Function (CMIF) and Multi-Variate Mode Indicator Function
(MMIF) in the Curve Fitting panel of Visual Modal Pro are
used to find and estimate the modal parameters of repeated
roots in an axi-symmetric structure.
To display the mode shapes of the circular disk in animation,
To create a structure with repeated roots, a finite element
model of a circular plate was constructed using NASTRAN
For Windows, and its modal parameters (eigenvalues & eigenvectors) were found using the FEA program. Both the
structure model and its modes were then exported to
ME’scopeVES.
Figure 1 shows an animated display of two of the modes of
the circular disk model. Notice that the two modes have the
same frequency, but different mode shapes. The mode
shapes are in fact very similar. They are merely “rotated” by
45 degrees from one another, as evidenced by their nodal
lines.
REPEATED ROOTS: 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 the File | Project | Open command, and open
the App Note #14 ….PRJ from the Modal Analysis subdirectory under the ME’scopeVES directory.
•
Press the Animate button in the Structure window to
initiate animation.
•
Press the Arrange Windows button in the
ME’scopeVES window to display the Structure and
Shape Table windows together in the Work Area.
•
Select the Circular Disk- Z DOFs Only.SHP file as the
animation source.
•
To display a different mode shape, display the Shape
Headers spreadsheet in the Shape Table and depress its
shape button.
Notice that pairs of repeated roots are lis ted in the Shape
Table, starting at 108 Hz up to 1442 Hz, with the exception
of the single mode (non-repeated root) at 201 Hz. (The first
three modes are rigid body modes, verified by displaying
their mode shapes.)
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 App Note, a set of multiple reference FRFs will be
synthesized using several repeated root pairs, and a single
non-repeated root. These FRFs will then be processed using
both CMIF and MMIF to recover the modal parameters of all
of the modes. Then, the curve fitting answers will be compared with the original parameters.
Figure 1. Disk Showing Repeated Root Mode Shapes.
Page 1 of 8
VES App Note #14
9/5/2000
Figure 2. Shape Table Showing Repeated Root Pairs.
Page 2 of 8
VES App Note #14
9/5/2000
FRF SYNTHESIS
To synthesis the FRFs,
To exercise the multiple reference curve fitting capabilities in
the Visual Modal option, a set of FRFs will be synthesized
using 7 modes from the Shape Table, (three pairs of repeated
roots and the 201 Hz non-repeated root).
•
Click on OK. Answer Yes and OK in the following dialogs.
With the Shape Headers spreadsheet displayed,
After the FRFs are synthesized, a Data Block window open
displaying them. Notice from the Title bar that there are 320
FRFs in the Data Block.
•
•
Select shapes 4 through 10. (Depress Shape button 4,
hold down the Shift key and depress Shape button 10.)
This will select the 7 modes from 108 Hz to 397 Hz. To determine two suitable references for synthesizing FRFs,
•
Drag the vertical blue splitter bar to the left to display
the Traces spreadsheet.
Notice that each pair of Traces includes a Roving DOF and
each of the Reference DOFs (5Z and 15Z).
With animation stopped, execute the Display | Point
Labels command in the Structure window.
We will choose Points 5 and 15 for the two references. They
are close together, yet distinct enough to test the multiple
reference capabilities of ME’scopeVES. To synthesize a set
of FRFs using these two references,
•
Execute the Tools | FRF Synthesis command 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 two Reference DOFs (5Z and 15Z),
•
Hold down the Ctrl key, click on 5Z to select it, scroll to
15Z and click on it to select it.
Figure 4. Data Block Window Showing Synthesized FRF.
MODE INDICATOR FUNCTIONS
To initiate curve fitting,
•
Press the Modes | Modal Parameters button on the Data
Block Toolbar, or execute the command from the Modes
menu.
This will display the Curve Fitting Panel on the right of the
window, with the FRFs in the upper left and the Mode Indicator graph in the lower left of the window. To compute and
display the Modal Peaks Function (MPF),
Figure 3. FRF Synthesis Dialog Box
Showing DOF Selections.
•
Make sure that the Curve Fit | Mode Indicator | Use
Imaginary command button is depressed.
•
Make sure that the MPF radio button is selected in the
Curve Fitting Panel, spread the band cursors to enclose
all of the peaks in the FRF, and press the Count Peaks
button.
•
Lower the peak counter threshold line (horizontal dotted line on the MPF) by using the scroll bar to the right
of the MPF graph, so that red dots are displayed on all
of the peaks.
Page 3 of 8
VES App Note #14
9/5/2000
Red dots should be displayed on 4 peaks, and the Modes box
on the Curve Fitting Panel should have a 4 in it. The Modal
Peaks Function only shows evidence of 4 modes in the cursor band. It doesn’t show evidence of the 3 repeated roots
and the single non-repeated root, for a total of 7 modes. To
see this, either the CMIF or the MMIF method must be used.
The 2 CMIF curves clearly indicate the presence of 3 repeated roots. (Both CMIF curves have 3 peaks at the same
frequency.) Also, one CMIF curve has an additional peak at
about 200 Hz, indicating a non-repeated root.
CMIF Method
Another matrix resulting from the SVD calculation is a matrix
of Modal Participation Factors. These Participation Factors
indicate how strongly each mode “participates in” (or is represented by) the FRFs for each Reference.
The CMIF (Complex Mode Indicator Function) performs a
singular value decomposition (SVD) of the multiple reference
FRF data. The SVD decomposes the FRF matrix into the
product of three matrices for each frequency in the cursor
band. In this case the FRF matrix has 160 rows and 2 columns, (or 160 Roving DOFs and 2 Reference DOFs ).
One of the matrices in the SVD is a diagonal matrix of singular values. These are plotted as the CMIF curves in the
Mode Indicator graph area. Since there are 2 References,
there will also be 2 CMIF curves. Each peak on a CMIF
curve is as indicator of a mode of vibration. To use the
CMIF method,
•
Select the CMIF indicator on the Curve Fitting Panel.
•
Press the Count Peaks button. A dialog box will open,
telling you how many CMIFs will be calculated. Press
the Yes button to proceed with the calculation.
ESTIMATING FREQUENCY & DAMPING
These Modal Participation Factors are kept in memory with
the CMIFs but are not displayed. (They can be displayed by
using the Modes | Copy to New File | CMIFs command when
not curve fitting.)
The Modal Participation Factors are used to “weight” the
FRFs during both the F & D and Residues curve fitting
steps. To estimate frequency & damping for the 7 modes in
the cursor band,
•
Press the F & D button. A dialog box will open,
summarizing the parameters to be used for curve fitting.
Click on Yes to initiate curve fitting.
After they are calculated, the 2 CMIF curves are displayed
together in an overlaid format in the Mode Indicator graph
area on the lower left side of the window. Each CMIF curve
is a different color. Also, all of the peaks above the threshold line are displayed as red dots.
•
Adjust the threshold line downward using the scroll bar
to the right of the CMIFs until 7 red dots are displayed
and the Modes box contains a 7, as shown in Figure 5.
When the curve fitting is completed, the frequencies &
damping of the 7 modes will be listed in the spreadsheet in
the Curve Fitting Panel.
Figure 5. CMIF Curves Indicating 7 Modes.
Figure 6. Listing of Modal Frequencies & Damping.
Page 4 of 8
VES App Note #14
9/5/2000
Notice that the frequency estimates agree precisely with the
values in Figure 2. Notice also that the parameter estimates
are ordered in the spreadsheet. The estimates corresponding to the peaks in the first CMIF curve are followed by the
repeated root estimates corresponding to the peaks in the
second CMIF curve .
Notice also that all of the mo des have 1% critical damping,
which agrees with the values in Figure 2. (Since the original
modes where FEA modes (with no damping), this amount of
damping was artificially added to all of the analytical modes.)
This completes the second curve fitting step. In this case,
multiple reference curve fitting was performed to estimate
modal frequency & damping, using the modal participation
factors obtained from the first step.
NOTE: When either the CMIF or MMIF method is used for
multiple reference curve fitting, the order of the parameter
estimates in the spreadsheet should not be changed until
Residues have been estimated for all modes of interest.
ESTIMATING RESIDUES
Once the modal frequencies & damping are estimated, modal
residues (magnitudes & phases) can be estimated by a third
curve fitting step. To estimate residues,
•
Press the Residues button on the Curve Fitting panel. A
dialog box will open, summarizing the parameters to be
used for curve fitting. Click on Yes to initiate curve fitting.
•
Residue estimates and Fit Functions can be viewed by
scrolling through the FRFs using the scroll bar to the
right of the measurements graph.
•
Scroll through the measurements until the Synthesized
FRF <21Z:5Z> is displayed, as shown in Figure 7.
Notice that there is no peak at 108 Hz in this measurement.
From inspection of the magnitudes & phases of the 108 Hz
repeated root, it is clear that the two modes are equal in
magnitude by 180 degrees different in phase. In other
words these two modes are summing to zero, or canceling
each other at this DOF.
This completes the third curve fitting step. Again, multiple
reference curve fitting was performed to estimate modal residues, using the modal participation factors obtained from the
first step.
MMIF Method
The MMIF (Multi-Variate Mode Indicator Function) is used
in the same manner as the CMIF method in curve fitting, but
it is based on different assumptions and yields slightly different results.
•
Select the MMIF indicator on the Curve Fitting Panel.
•
Press the Count Peaks button. A dialog box will open,
telling you how many MMIFs will be calculated. Press
the Yes button to proceed with the calculation.
Two MMIF curves will be displayed as shown in Figure 8.
When the residues have been estimated for the 7 modes and
all 320 FRFs, the residue estimates for each FRF are displayed in the spreadsheet. Also, a red Fit Function is overlaid on each FRF in the upper left corner of the window.
Figure 8. 2 MMIFs Showing 7 Peaks.
•
Figure 7. FRF and Curve Fit Showing No Peak At 108 Hz.
Page 5 of 8
Adjust the threshold line using the scroll bar to the right
of the MMIF graph, until 7 red dots are displayed and
the Modes box contains a 7.
VES App Note #14
9/5/2000
To estimate frequency & damping for the 7 modes,
•
SAVING SHAPES
Press the F & D button. A dialog box will open, summarizing the parameters to be used for curve fitting. Click
on Yes to initiate curve fitting.
When the curve fitting is completed, the frequencies &
damping of the 7 modes will be listed and selected in the
spreadsheet in the Curve Fitting Panel. The first 7 unselected
modes in the spreadsheet were estimated with the CMIF
indicator.
To clear out the Fit Functions remaining from the CMIF
based curve fitting,
•
Execute the Curve Fit | Clear Fit functions command or
press its button on the Toolbar.
When curve fitting is completed, mode shapes can be saved
in a Shape Table from the multiple reference FRFs.
•
To unselect all of the modes, double click on the Mode
column heading in the modal parameter spreadsheet.
•
Press the Save Shapes button on the Curve Fitting
Panel. A dialog box will open as shown in Figure 10.
This dialog contains a listing all of the modal parameter estimates, and “normalized” modal participation factors (relative
strengths) for each mode and each reference. The strongest
reference for each mode (largest participation factor) is already selected.
To estimate residues for the selected modes,
•
Press the Residues button on the Curve Fitting panel.
Click on Yes to initiate residue curve fitting.
When the residues have been estimated for the selected 7
modes and all 320 FRFs, the residue estimates for each FRF
are displayed in the modal parameter spreadsheet, and a new
red Fit Function is overlaid on each FRF in the upper left
corner of the window. This is shown in Figure 9.
Figure 10. Save Shapes Dialog Box.
Figure 9. Curve Fitting Results Using MMIF.
Multiple Reference curve fitting has now been performed
using both the CMIF and MMIF mode indicators.
NOTE: The RFP (Rational Fraction Polynomial) curve fitting method is automatically selected when either the CMIF
or MMIF mode indicator is used. The RFP method performs
multiple reference curve fitting utilizing the Modal Participation Factors generated by these two mode indicators.
•
Press OK to save the shapes for the selected References into a Shape Table.
•
Press the New File button in the Shape Table Selection
dialog box, and click on OK to save the shapes.
The new Shape Table window will open displaying the frequency & damping of the 14 modes identified by the two
methods, CMIF and MMIF.
Comparing the 108 Hz Modes
To compare the MAC values of the repeated root mode
shapes at 108 Hz.,
•
Page 6 of 8
Press the Shape button of modes 1, 5, 8, & 12 to select
those modes in the Shape Table window.
VES App Note #14
•
9/5/2000
Execute the Display | MAC | Table command. The MAC
values of the 4 modes will be displayed as shown in
Figure 11.
•
Press the Animate button in the Structure window to
initiate animation.
•
Select the Shape Table used to display the MAC values
in Figure 12 (Shape Table 2.SHP) as the animation
source.
•
Depress the Animate | Contours | Lines button on the
Toolbar.
•
Display only the Top View.
•
Execute Animate | Comparison | Side by Side. The display should look like that shown in Figure 13.
•
Make sure that the same Shape Table (Shape Table
2.SHP) is selected in both Animation Source list boxes
on the Toolbars.
•
Display Shape 1 on the left, and Shape 2 on the right.
From the MAC values, it is clear that shapes 1 & 12 are
identical, and shapes 5 & 8 are also identical.
Figure 12. MAC Values of 108 Hz Shapes.
Now, let’s compare these shapes with the original 108 Hz
repeated root shapes.
•
Display the Shape Headers in the original Shape Table,
Circular Disk- Z DOFs Only.SHP.
•
Press the Shape buttons of shapes 4 & 5 (the 108 Hz
modes) to select them.
•
Execute Edit | Copy to New File, and click on OK to store
the two shapes into a new Shape Table.
•
From the new Shape Table, execute Edit | Paste Shape
Table. In the dialog box that opens, select the Shape
Table with the curve fitting results in it, and click on OK.
•
Select all of the 108 Hz shapes in the Shape Table.
(Shapes 1& 2 are the original shapes. Shapes 3, 7, 10,
& 14 are the curve fitting results.
•
Execute Display | MAC | Table again to display the
MAC values shown in Figure 12.
Figure 13. Top View of Shapes 1 & 2.
Figure 12. MAC Values of Original & Curve Fit Shapes.
Notice that the original shapes (1 & 2) are orthogonal to one
another (their MAC = 0.0). Likewise, the CMIF mode shapes
(3 & 7), and the MMIF shapes (10 & 14) are also orthogonal to one another. However, neither the CMIF nor the
MMIF shapes are the same as (MAC = 1.0) or orthogonal to
(MAC = 0.0) the original shapes.
•
Now display Shape 3 in on the right side.
•
Execute Animate | Comparison | Overlay. The overlaid
shapes should like those shown in Figure 14.
Notice that the nodal lines of these two shapes are slightly
different. However, from the contours it is clear that if one
shape were rotated slightly, it would match the other shape.
To understand this apparent discrepancy, the shapes must
be displayed in animation.
Page 7 of 8
VES App Note #14
9/5/2000
Figure 14. Top View of Shapes 1 & 3 Overlaid.
CONCLUSIONS
Two different mode indicators (CMIF and MMIF) were used
for finding repeated roots in this Application Note. Both
indicators clearly indicated three sets of repeated roots and a
single non-repeated root.
Both indicators also provided modal participation factors
that were utilized during the F & D and Residues curve fitting steps. Both of these multiple reference curve fitting
steps correctly identified the modal parameters of all seven
modes.
The MAC values of the 108 Hz repeated root were then compared, first between the CMIF and the MMIF results, and
then between the curve fitting results and the original mode
shapes.
The CMIF and MMIF results agreed perfectly with one another. Each pair of repeated root shapes found from each
mode indicator were orthogonal to one another, and identical
to the shapes of the other indicator.
Non-Unique Solution
The MAC values between the curve fit results and the original shapes indicated that the curve fitting shapes were not
the same as the original mode shapes. However, when the
shapes were displayed in animation, it became clear that the
curve fit shapes were “slightly rotated” from the original
shapes.
These non-unique solutions of mode shapes (eigenvectors)
are typical of repeated root problems. As further evidence of
this phenomenon, the curve Fit Functions matched the FRFs
perfectly during curve fitting, yet the Fit Functions were
synthesized using different mode shapes than those used to
synthesize the original FRFs.
Page 8 of 8