App Note #24
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2/25/2014
ME’scope Application Note #24 Choosing Reference DOFs for a Modal Test INTRODUCTION Finite Element Analysis (FEA) can provide excellent guidance for conducting an experimental modal analysis. A well thought out FEA-based test plan reduces testing time and leads to better experimental results. In this note we will use FEA mode shapes to plan the excitation of a physical structure. We will answer important questions such as:
How many shakers (or Reference accelerometers for a roving impact test) will be required to accurately identify the modes of the structure?
Where should these References be located?
Steps in the application note can be duplicated using VT-550 Visual Modal Pro or any package that includes option VES-350 Advanced Signal Processing.
FEA Mode shapes of pinned-pinned beam.
Contour Lines (left) identify node-lines, while Color Contours (right) locate anti-nodes.
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App Note #24
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To select a mode for animation:
THE FEA MODEL The modeled test article is a small-scale bridge model with a span of 27 feet and a width of 3 feet. Its ends are pinned to rigid attachment supports.
Click on the desired mode Shape in the Model Bridge Modes.SHP Spreadsheet.
Viewing Contour Lines The finite element model contains six modes below 50 Hz and 175 Z-direction degrees-of-freedom (DOFs). All of the modes have modest damping of less than 3%. Note: These FEA modes were calculated using the VES550 Advanced Modification option contained in the Visual SDM Pro package.
Execute: Animate | Contours | Lines. Colored contour lines are now added to the animated display.
Execute: Animate | Step. The animation will stop at the maximum deformation of the mode shape.
OPENING THE PROJECT Open ME’scope. Execute: File | Project | Open.
Double-click within the 3D-View of the mode shape display. The window will change to the Quad View.
Select Model Bridge from the More Examples /Application Notes/#24 subdirectory.
The Model Bridge.STR Structure and Model Bridge Modes.SHP Shape Table windows will open. Animating Mode Shapes Execute: Window | Arrange | For Animation. To initiate animation: Execute: Draw | Animate in the Structure window.
Double-click within the Top (+Z) pane to replace the Quad View with the Top View.
Select each of the six Shapes and observe its (black) node lines.
Note that the four bending modes at 4.35, 11.97, 34.46 and 38.78 Hz all have parallel node lines that run across the width of the bridge (in the Y-direction). The 4.35 Hz mode has 2 node lines at the end-points of the bridge. The 11.97 Hz mode has 3 node lines, including one across the center span. The 34.46 Hz mode has 4 node lines including two near 1/3 and 2/3 of the span. The 38.78 Hz mode has 5 node lines, including one at the span center. The two torsion modes at 22.84 and 46.22 Hz have a node line running down the center of the bridge (in the Xdirection) and Y-direction node lines at the ends of the bridge. The 46.22 Hz mode has a third Y-direction node line running across the center of the bridge. Viewing Color Contours The appearance of Color Contours is determined by the selected palette of Contour Colors. To select colors for the contours: Execute: File | Options in the Model Bridge.STR window.
Animating Model Bridge modes.
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Selecting Contour Colors.
Select the Contour Colors tab.
Press the 7 Band Default button.
Uncheck Scale Between Limits.
Check Interpolate Colors.
Press the OK button.
|Magnitude| color scaling versus signed magnitude.
Select each of the six Shapes and compare the results with those shown on page 1.
If the color Contour Values range between negative and positive numbers, signed magnitude scaling is in effect, as shown on the right in the prior figure. To select |magnitude| scaling, execute Animate | Vector Direction | X axis, Animate | Vector Direction | Y axis and Animate | Vector Direction | Z axis.
Bright (near white) spots are the anti-nodes (locations of maximum response) for each mode. These are highly desirable Reference locations from which to identify a mode experimentally.
Note: when only a single Vector Direction is enabled, signed magnitude scaling is in effect. When two or more Vector Directions are turned on, |magnitude| color scaling is used.
Execute: Animate | Contours | Fill. The contours are filled with color.
Magnitude versus Signed Magnitude SELECTING REFERENCE DOFs To view the color “thermometer” showing the magnitude associated with each color: Execute: Animate | Contours | Values. The resulting thermometer should span 0 (black) to the largest |magnitude| (white) in the Mode Shape, as shown in the following figure.
Most modal tests are done using a single reference testing method where either a single shaker or a single fixed reference accelerometer is used. Single reference testing provides the most precise results when the reference DOF is highly responsive to all of the modes of interest. Hence, we seek a Reference DOF that has a driving-point FRF with a large peak for each resonance. Finding such a point using multiple trial measurements can be time consuming. Determining it with FEA mode shapes is a much more efficient process.
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The Shape Product dialog opens, indicating that 6 Mode Shapes and 175 DOFs will be used in the calculation.
Multiple References One possible reason for Multiple Reference testing and curve-fitting is to find repeated roots (see Application Note #14). Since there are no repeated roots among the FEA modes, there is no preliminary evidence that multiple reference testing will be required.
Press the Animate Shape Product button. The Shape Product is displayed.
However, the possibility of closely coupled modes does exist, because the 3rd and 4th (distinct root) modes are quite close in frequency. Nevertheless, we will first determine whether these two modes (and the others) can be identified from a single reference test. Using a Single Reference All modal testing is aided by recalling some important facts about FRFs:
In any FRF, the magnitude of each resonance peak (i.e. its Residue) is proportional to the product of two mode shape components, one corresponding to the response DOF and the other corresponding to the force DOF. In a driving point FRF, the magnitude of each resonance peak is proportional to the square of the mode shape component for the driving point DOF. If a DOF is on a node line of a mode, its corresponding shape component is equal to zero.
We will use these facts plus the Shape Product command in ME’scope to locate ideal Reference DOFs on the bridge. Shape Product The Tools | Shape Product command in the Shape Table window animates a shape formed by multiplying all (selected) mode shapes together. If any mode shape has a shape component equal to zero, the Shape Product is zero. Thus, only those DOFs with non-zero shape components for all of the modes will form a bright spot in a Color Contour display of a Shape Product. The “bright spots” in a Shape Product display of all FEA modes represent the best candidates for a Reference DOF. To calculate and display the Shape Product: Execute: Tools | Shape Product in the Model Bridge Modes.SHP window.
Shape Product of all six modes indicates four equally attractive Reference DOFs and 16 secondary choices. To identify the DOF label of a point in the display: Execute: Animate | Interactive | Point Label.
Click on any displayed point. The Point Label will be displayed next to the point.
To select multiple points, hold down the Ctrl key and click on the desired points.
As shown by the red arrows in the figure above, four DOFs (36Z, 40Z, 136Z and 140Z) have bright points indicating maximum values of the Shape Product. These are the four “primary” choices for the single Reference DOF. Sixteen other DOFs (including 55Z, 60Z, 70Z and 75Z) are bright points of lower intensity. We will also investigate these “secondary” DOFs. Synthesizing the Driving-Point FRFs We can verify the correctness of this finding by synthesizing the driving-point FRFs and examining their resonance peaks.
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Press the Close button in the Shape Product dialog.
App Note #24
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Execute: Tools | Synthesize FRFs in the Model Bridge Modes.SHP window. The Synthesize FRFs dialog will open.
Enter All driving-point FRFs as the new file name and click on the OK button. The All driving-point FRFs.BLK window will open.
Minimize the Model Bridge.STR and Model Bridge Modes.SHP windows.
In the All driving-point FRFs.BLK Spreadsheet:
Click on M#36 (36Z:36Z), M#40 (40Z:40Z), M#136 (136Z:136Z) and M#140 (140Z:140Z) in the Select column.
Enter 2000 as the Block Size.
Enter 50 as the Frequency Span.
Execute: Format | Overlay Selected Traces.
Check the Driving Points Only box.
Execute: Display | CoQuad | Upper/Lower.
Click on the first DOF in the Roving DOF list. Hold down the Shift key and click on the last DOF to select all of the 175 DOFs in the Roving DOF list.
Press the OK button. The FRF Synthesis dialog will open.
Press the OK button to proceed with the synthesis of the 175 FRFs. The FRF Response dialog will open.
Overlay of “primary” DOF driving-point FRFs. Note that the four FRFs selected are exact matches of one another. These four DOFs are symmetrically located on the model bridge, so identical FRFs should be expected.
Select Acceleration and press OK. When the calculation is complete, the New File dialog will open.
However, note that only five resonance peaks appear in the (lower) Imaginary part of each FRF. Each imaginary peak is also bounded by minimum and maximum peaks in the (upper) Real part. These peaks are characteristic of a resonance.
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To expand the display around 23 Hz where two modes were expected: Execute: Display | Zoom.
Move the cursor into the plot area where it will change to a Zoom cursor ().
Move the Zoom cursor to about 20 Hz, hold down the left mouse button and drag the Zoom cursor to about 26 Hz. Release the left mouse button. The display will Zoom between the limits of the Zoom cursor.
Overlay of “secondary”DOF FRFs with “primary” FRF.
Double-click in the Color field of M#40 (40Z:40Z) and change its color to red.
Click on M#36 (36Z:36Z), M#136 (136Z:136Z) and M#140 (140Z:140Z) to unselect them.
Click on M#55 (55Z:55Z), M#60 (60Z:60Z), M#70 (70Z:70Z) and M#75 (75Z:75Z) in the Select column. Execute: Display | Imaginary.
Zoom of prior plot centered on 3rd and 4th resonances. In this zoomed display we can see that there are actually two resonance peaks. However, the half-power bandwidths of these two modes overlap one another, indicating that these are closely coupled modes.
Note in the prior figure that some of the “secondary” FRFs exhibit higher peaks than FRF 40Z:40Z for some of the modes. However, none of them gives any indication of the 23.46 Hz 3rd mode as shown in the zoomed display below.
To restore the full span of the display: Hold down the shift key and execute: Display | Mooz. Now let’s see if any of the “secondary” driving point FRFs (55Z: 55Z, 60Z: 60Z, 70Z: 70Z or 75Z: 75Z) indicate all 6 modes.
Zoom of prior plot centered on 3rd and 4th resonances.
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Curve Fitting the Single Reference Data Set We will now curve-fit a “primary” DOF driving-point FRF. If a single reference test is possible, all six modes should be accurately identified.
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Click on M#36 (36Z:36Z), M#136 (136Z:136Z) and M#140 (140Z:140Z) to select them in addition to M#40 (40Z:40Z). Execute: Edit | Select Traces | Invert to select all of the FRFs except the four “primary” DOF FRFs.
Select (only) M#40 (40Z:40Z) in the All drivingpoint FRFs.BLK Spreadsheet.
Execute: Curve Fit | Quick Fit.
Execute: Modes | Modal Parameters. Execute: Curve Fit | Quick Fit. When curve fitting is completed, modal Frequencies, Damping and Residues will be listed in the modal parameters spreadsheet and a red fit function overlaid on the FRF.
Quick Fit of “non-primary” FRFs. Note that only five of the six modes are identified from these 171 driving point FRFs. The 3rd mode (22.84 Hz) was found, but the 4th mode (23.46 Hz) is missed.
“Quick Fit” results from 40Z:40Z FRF. As shown above, the Quick Fit results match the six FEA natural frequencies and damping (inset) very closely. This indicates that a single Reference test of this structure is possible. However, the peaks of the “secondary” DOF FRFs indicated the 4th mode is not easily identified. In fact, this closely coupled mode can only be identified from the four “primary” DOFs. This will be demonstrated by curvefitting all 171 “non-primary” driving point FRFs. To clear the curve fit data and fit all but the “primary” DOF driving-point FRFs:
In summary, while a single reference test of the Model Bridge structure is possible, the Reference DOF selection is too critical. All six of the modes can only be identified if one of the “primary” DOFs :36Z, :40Z, :136Z or :140Z is used as the Reference DOF. The closely coupled 4th mode will be missed by using any other DOF as the Reference. A multiple reference test is better for identifying closely coupled modes.
MULTIPLE REFERENCE TESTING To successfully test the Model Bridge, at least two Reference DOFs that clearly indicate each of the closely coupled 3rd and 4th modes are needed. We will start by finding a Reference DOF from which the 4th mode (23.46 Hz) can be clearly identified.
Execute: Curve Fit | Clear Fit Functions.
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As shown in the Color Contours of the following figure, choosing 88Z at the geometric center of the bridge will accomplish this. This DOF lies on node lines for the 2nd, 3rd, 5th and 6th modes, which means that force applied to this DOF will only excite the 1st, and 4th modes. Secondly, we need a Reference that will excite the remaining 2nd, 3rd, 5th and 6th modes effectively. The Shape Product can be used to identify potential DOFs.
Shape Product suggests 65Z as a 2nd Reference. The figure below compares the 3rd and 4th Mode Shapes using a Contour Lines display. Note that the first Reference DOF (88Z) is on a node line of the 3rd mode, while the second Reference DOF (65Z) is on a node line of the 4th mode. This reference pair will provide the best possible separation of these closely coupled modes while providing good excitation of all of the other modes.
88Z Reference will excite the 1st and 4th modes. In the Model Bridge.SHP Spreadsheet: ď‚
Select (only) Shapes 2, 3, 5 and 6. Execute: Tools | Shape Product.
ď‚
Animate the Shape Product and turn on Color Contours.
The following Shape Product Color Contour shows that four DOFs (61Z, 65Z, 111Z and 115 Z) are equally attractive as the Reference from which to excite the selected modes. We will choose 65Z.
65Z is on 4th mode node; 88Z is on 3rd mode node.
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Comparing Single and Mulitple Reference DrivingPoints Synthesize the 40Z:40Z, 65Z:65Z and 88Z:88Z driving point FRFs and display them as shown in the following figure. Note from the FRF peak amplitudes, that the Multiple Reference FRFs (black Traces) indicate excellent excitation of both of the closely coupled 3rd and 4th modes. Also, the Multiple Reference FRFs indicate more excitation of the 1st mode than the Single Reference FRFs (red Traces). The 2nd, 5th and 6th modes will receive about equal excitation from either test.
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Click on 65Z in the Reference DOF list. Then hold down the Ctrl key and click on 88Z to select these two DOFs in the Reference DOFs list.
Press the OK button. The FRF Synthesis dialog will open.
Press the OK button. The FRF Response dialog will open.
Select Acceleration and press OK. The New File dialog will open.
Enter Dual Reference FRFs as the new file name and click on the OK button. The Dual Reference FRFs.BLK window will open.
MULTIPLE REFERENCE CURVE FITTING To curve fit the Multiple Reference set of FRFs: Execute: Modes | Modal Parameters in the Dual Reference FRFs.BLK window.
Select CMIF and Multiple Reference as the Methods on the Mode Indicator tab. Execute: Modes | Quick Fit.
Single Reference compared to Multiple Reference FRFs. When curve fitting is completed, modal Frequencies, Damping and Residues are listed in the modal parameters spreadsheet and a red fit function is overlaid on each FRF.
SYNTHESIZING MULTIPLE REFERENCE FRFs A Multiple Reference curve fit requires a rectangular matrix of FRF measurements between all response DOFs and all (two in this case) reference DOFs. To synthesize the required FRFs, in the Model Bridge Modes.SHP Spreadsheet: Execute: Tools | Synthesize FRFs. The Synthesize FRFs dialog will open.
Enter 2000 as the Block Size.
Enter 50 as the Frequency Span.
Uncheck the Driving Points Only box.
Click on the first DOF in the Roving DOF list. Hold down the Shift key and click on the last DOF to select all of the DOFs in the Roving DOF list. Multiple Reference Quick Fit results.
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Note that the resulting Frequency and Damping values for all six modes agree well with the modal parameters of Model Bridge Modes.SHP (inset) in the figure above.
COMPARING MODE SHAPES Lastly, to further validate the “experimental” mode shapes, we will save them into a Shape Table and compare them with the FEA shapes.
Press the Save Shapes button on the Residues & Mode Shapes tab. The Save Shapes from Multiple References dialog will open.
Enter Dual Reference Modes as the new file name and press the OK button. The Dual Reference Modes.SHP window will open.
Close all other windows.
Open the Model Bridge Modes.SHP window.
In the Dual Reference Modes.SHP window: Execute: Display | Cross MAC. The Shape Table Selection dialog will open.
This dialog confirms that the 1st and 4th modes were identified from the 88Z Reference while the 2nd, 3rd, 5th and 6th were identified from 65Z, as planned.
Press the OK button. The Shape Table Selection dialog will open.
Select Model Bridge Modes.SHP and click on OK. The Modal Assurance Criterion (MAC) Table will open.
MAC Table between curve-fit and FEA modes. The MAC matrix measures the similarity between mode shapes. Each row in the Table represents a curve-fit mode shape while each column represents an FEA mode shape. The numbers in the Table are the MAC values.
Press the New File button. The New File dialog will open.
A MAC value of 1 indicates that the shapes in the row and column are identical. A value of 0.9 indicates strong similarity between the shapes. A value of 0 means the two shapes are completely different.
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Notice that the diagonal values are all 1, indicating a perfect match between the curve-fit and FEA mode shapes. Also notice that all of the off-diagonal values are 0, indicating that each mode shape is unique from all the others.
Press the Plot button. The MAC Table will be replaced by a plot of the MAC values.
MAC Plot between curve-fit and FEA mode shapes. The MAC Plot summarizes the shape findings concisely. You can see at a glance that curve fitting the Multiple Reference set of FRFs correctly identified all of the mode shapes in the FEA model.
SUMMARY In this note you have used the Shape Product of FEA modes to:
Locate 4 primary Reference DOFs, any one of which could be used in a single reference modal test to identify all 6 modes of a bridge structure.
Locate two Reference DOFs with which to perform a multiple reference test, which more clearly identifies the closely-coupled 3rd and 4th modes of a model bridge structure.
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