App Note #15 www.vibetech.com 25-Feb-14

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ME’scope Application Note #15

Multi-Reference Curve Fitting To Find Closely Coupled Modes INTRODUCTION

In this note, the VES-4500 Multi-Reference Modal Analysis Option will be used to estimate the modal parameters of two closely coupled modes in a plate structure.

Closely Coupled Modes: A structure is said to have two or more closely coupled modes is they are represented by only one resonance peak in a set of FRF or spectral data.

In this note, the mode shapes of two closely coupled modes of a rectangular plate will be identified from a multi-reference set of FRF measurements. The plate, shown in the Figures below, is made of Poly Vinyl Chloride (PVC). The two Figures show the operating deflection shape (ODS) of the plate at 190 Hz, when displayed from two different Ref-erence DOFs, 7Z and 35Z. These two ODS’s are clearly different!

Figure 1. 190 Hz ODS From Reference 7Z.

Figure 2. 190 Hz ODS From Reference 35Z.

A 2-reference set of FRFs can be used to identify 2 closely coupled modes, if there are any. But, the first question to ask is, “How do we know if the structure has any closely coupled modes?” This question can be answered by dis-playing the operating deflection shapes (ODS’s) from each reference DOF in animation. The animated shape capability of ME’scope will be used to discover the first evidence of closely coupled modes.

Property of Mode Shapes: All mode shapes are inherent properties of a structure. This means that they do not de-pend on the Reference DOFs used to acquire a multi-reference set of FRFs.

ME’scope displays in animation the ODS data at the cursor position in a Data Block of FRFs. If the structure is “lightly damped”, the ODS is a close approximation of a mode shape when the cursor is placed at a resonance peak. Using the above property of mode shapes, the ODS should not change if displayed from one Reference DOF versus anoth-er in a multi-reference set of FRFs.

If the ODS at a resonance peak does not change when dis-played from different references in a multi-reference set of FRFs, this is a strong indication of a single mode. If on the other hand, the ODS is different when displayed from dif-ferent references, this is an indication of closely coupled modes.

Figure 1 shows the ODS at 190 Hz from the FRFs from Reference 7Z. Figure 2 shows the ODS at 190 Hz from the FRFs from Reference 35Z. Clearly, these are different shapes, indicating the ODS is not dominated by a single mode, hence it must the summation of contributions from closely coupled modes.

Another clue that the ODS is not dominated by a single mode is that 190 Hz is the lowest frequency resonance on the structure. In almost all cases, the fundamental (lowest frequency) mode of a structure is the first bending or first torsional mode. These two modes have very simple mode shapes, unlike the ODS shown in Figure 1 & 2.

Open App Note 15 - Closely Coupled Modes.VTprj from the ME’scope\Application Notes folder.

This Project contains a Structure file with the plate model, and a Data Block with 70 FRFs that were acquired from the plate.

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FRF DOFs

Slide the vertical blue splitter bar to the left in the BLK: PVC Plate window to expose the DOFs column in the Traces spreadsheet.

Scroll through the Trace properties spreadsheet, and notice that there are 70 FRF, 35 with reference 7Z and 35 with reference 35Z.

Figure 3. Traces Spreadsheet Showing 2 Reference DOFs.

NOTE: Trace DOFs for cross-channel measurements such as FRFs have the following format;

Trace DOF = Roving DOF : Reference DOF

This multi-reference set of FRFs was acquired during a rov-ing impact test of the plate. Two accelerometers were at-tached to the plate at Points 7 & 35, and the plate was im-pacted at each of the 35 Points (labeled on its top side in Figures 1 & 2) using an instrumented hammer.

The accelerometers sensed motion in the vertical (Z) direc-tion at Points 7 & 35. Hence the two Reference DOFs are 7Z & 35Z. The plate was impacted in the vertical (Z) direc-tion at each of the 35 Points. Hence the Roving DOFs range from 1Z to 35Z.

Each FRF was calculated from a pair of signals that were acquired between one of the 35 roving impact points and one of the 2 fixed reference accelerometers. Therefore, each Trace DOF contains a Roving DOF and a Reference DOF, indicating the two Points & directions between which the FRF was calculated.

CREATING ANIMATION EQUATIONS

In order to display the ODS’s of the PVC Plate in animation directly from the FRFs, animation equations must be created which assign each FRF measurement (M#) to the Point & direction on the plate model from which is was measured.

NOTE: When Animation equations are created, only the Roving DOF of the M#s in the current Animation source is considered.

Execute Draw | Animation Equations | Create Meas-ured (Assign M#s) in the Structure window.

Click on OK in the dialog box that opens.

The following dialog box will open, confirming that 35 equations have been created.

Press OK to complete the equation creation.

Measurements were made at the 35 Points on the top of the Plate, so a single Measured equation was created to animate each Point in the Z direction. To examine the animation equations,

Execute Draw | Animation Equations | Equation Edi-tor.

Click on the Animation Equations tab.

Click near a Point to display its animation equation on the tab.

Measured Animation Equation for Point 1.

There are two Traces (M#s) with the same Roving DOF for each Point, (each Roving DOF is paired with Reference DOF 7Z & DOF 35Z). Therefore, each animation equation has two terms in it, each term containing the M# corre-sponding to one of the References.

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To initiate animation from the BLK: PVC Plate window,

Execute Draw | Animate Shapes in the Structure win-dow.

The following dialog box will open.

This dialog opened because there are Points on the structure model that don’t have an animation equation. In this case, all of the 35 (un-numbered) Points on the bottom side of the Plate were un-measured, hence they have no animation equation. Animation equations for un-measured Points are called Interpolated equations.

Interpolated equations are created from nearby Measured animation equations. The Measured equations at Points closest to each un-measured Point are weighted more heavi-ly than those farther away.

Click on Yes (in the dialog box shown above) to create Interpolated equations. The following dialog box will open.

This dialog allows you to choose the number of nearest Measured Points to be used to create each Interpolated equa-tion.

Enter “1” into the dialog box above, and click on OK.

This will create an Interpolated equation for each Point on the bottom of the plate using the Measured equation direct-ly above it on the top of the plate.

Because shape animation is going to occur from an Anima-tion Source with multiple Reference Traces in it, the follow-ing dialog box will open next.

Click on OK.

The Select Traces By dialog box will open with Select Traces By Reference DOF chosen from its list of selection options.

Animation of the shape data at the cursor position in the BLK: PVC Plate window will now begin.

Resize the Structure window to half of the Work Area.

Execute Window | Arrange Windows | For Animation in the ME’scope window, to arrange the windows as shown below.

Drag the cursor to the lowest frequency peak (190 Hz), as shown below.

Execute Animate | Step | Stop/Continue to stop the animation.

The ODS at 190 Hz, using the M#s from Reference 7Z, should be displayed. The ODS should look like the shape shown below. If indeed a single mode is excited at 190 Hz, this ODS should be dominated by its mode shape.

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ODS from Reference 7Z at 190 Hz

Select Reference 35Z in the Select Traces By dialog box.

Notice how the ODS changes to a completely different shape (shown below) when a different Reference is selected.

If a single mode exists at 190 Hz, the ODS should be the same, no matter which reference it is displayed from. Clear-ly, this is evidence that more than one mode exists at 190 Hz.

ODS from Reference 35Z at 190 Hz.

Drag the cursor to the next higher frequency peak (about 427 Hz).

Execute Animate | Step | Stop/Continue again to re-sume the animation.

Select Reference DOF 7Z and then 35Z in the Select Traces By dialog box.

Now notice that the ODS does not change when a different reference is selected. This is clear evidence that a single mode exists at 427 Hz.

MULTIPLE REFERENCE CURVE FITTING

Multiple reference curve fitting uses the extra information in a set of multiple reference FRFs to extract the parameters of closely coupled modes.

Execute Modes | Modal Parameters command in the BLK: PVC Plate window to initiate curve fitting.

Turn the cursors OFF.

Execute Curve Fit | Clear All Fit Data to clear all cur-rent curve fitting data from the Data Block.

On the Mode Indicator tab, select Multi-Reference CMIF from the methods list.

Press the Count Peaks button, and click on OK in the dialog box that opens.

Two CMIF curves will be calculated and displayed as shown below. Red dots are placed on each peak counted on the two curves, and the number of peaks counted in dis-played in the “peaks” box on the tab.

Use the scroll bar on the right side of the CMIFs to slide the noise threshold line upward until only 7 peaks are counted, as shown above.

NOTE: The lowest frequency peaks are rigid body modes of the plate, which will be ignored.

Notice that red dots appear on two peaks near 190 Hz on the CMIFs. This is an indication that two modes exist near 190 Hz. The red dots on the other five peaks indicate dis-tinct modes.

Frequency & Damping Curve Fitting

On the Frequency & Damping tab, press the Frequen-cy & Damping button, and click on Yes in the dialog box that opens.

Notice that Multi-Reference Polynomial was selected on the method list. Notice also that frequency & damping are

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now listed for 7 modes in the Modal Parameters spread-sheet, as shown below.

Notice that a mode was found at 187.8 Hz, and the second one at 189.3 Hz. These are the two closely coupled modes that cannot be found using a Single Reference curve fitting method.

Residue Curve Fitting

On the Residues & Save Shapes tab, press the Residues button, and click on Yes in the dialog box that opens.

The magnitudes & phases of complex residues now appear in the Modal Parameters spreadsheet for the 7 modes, as shown below.

When residue curve fitting has been completed, the Modal Parameters spreadsheet contains different residues for each FRF Trace, and a red Fit Function is overlaid on each Trace, as shown above.

NOTE: The Fit Function is synthesized from the parame-ters in the Modal Parameters spreadsheet.

Notice also that the Multi-Reference Polynomial was se-lected from the methods list. This method was used for res-idue curve fitting.

Use the scroll bar on the right of the Traces display to scroll through the FRF Traces, displaying their modal parameters and Fit Functions.

Press the Save Shapes button to save the modal parame-ters into a Shape Table.

Press the New File button in the dialog box that opens, and click on OK in the next dialog box.

A new Shape Table window will open with the mode shapes in it, as shown below.

Notice that the Measurement Type of the mode shapes in the lower spreadsheet is Residue Mode Shape. This is be-cause the residues obtained from curve fitting each FRF Trace were saved as mode shape components, in the same order as the Traces in the Data Block. This is re-emphasized in the following note.

NOTE: Each component of a Residue Mode Shape has the same M# and DOFs as the Trace from which it was ob-tained by Residue curve fitting.

DISPLAYING THE MODE SHAPES

Since Residue Mode Shapes have the same M#s and DOFs are the Traces from which they were obtained by curve fit-ting, the animation equations that were previously created to animate ODS’s from the BLK: PVC Plate can also be used to animate the mode shapes.

Close the BLK: PVC Plate window.

Resize the Structure window to half of the Work Area.

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Execute Window | Arrange Windows | For Animation in the ME’scope window, to arrange the windows as shown below.

Execute Tools | Animate Shapes in the SHP: Shape Table 1 window to initiate animation from the Shape Table.

The following dialog box will open, reminding you that the Shape Table contains multiple reference mode shapes, and that you can only animate shapes from one reference at a time.

Click on OK.

Animation will begin showing Shape 1, the 187.8 Hz mode shape, as shown below. This mode is the recognized by its mode shape as the first bending mode of the plate.

Select Reference DOF 7Z and then 35Z in the Select Traces By dialog box.

First Bending Mode Shape at 187.8 Hz.

Notice that the mode shape does not change when the M#s for a different reference DOF are selected.

Press the Shape 2 button in the Shape Table to display the 189.8 Hz mode shape.

First Torsion Mode Shape at 189.3 Hz.

Its animated mode shape makes it clear that this is the first torsion mode of the plate.

You can display the other higher frequency mode shapes and verify that they do not change when displayed from a different Reference DOF.

CONCLUSIONS

In this Application Note, the animation display of ODS’s directly from a set of multiple reference FRFs was used to obtain the first indication of closely coupled modes. For lightly damped structures such as this PVC plate, the ODS is typically dominated by a single mode at or near a resonant frequency. Therefore, the ODS at a resonant frequency should look like the dominant mode shape, no matter which reference is selected during the ODS display.

As shown in Figures 1 & 2, the ODS at the 190 Hz reso-nance peak when reference 7Z was selected was distinctly different from the ODS when reference 35Z was selected. On the other hand, the ODS did not change when the cursor was placed on one of the higher frequency peaks, and a dif-ferent reference DOF was selected.

To correctly estimate the modal parameters of closely cou-pled modes, a multiple reference set of FRFs, together with Multi-Reference curve fitting methods are necessary.

Multi-Reference curve fitting correctly estimated the modal parameters of two closely coupled modes, at 187 & 189 Hz. The 187 Hz mode was the first bending mode, and the 189 Hz mode was the first torsional mode of the plate. A first bending and first torsional mode can be expected at low frequencies in virtually every structure that is tested