App Note #6 www.vibetech.com 2/25/14

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ME’scope Application Note #6

Single versus Multiple Reference Curve Fitting

INTRODUCTION

NOTE: Steps in this Application Note can be duplicated using any Package that includes the VES-3000 Signal Pro-cessing and VES-4000 Modal Analysis options.

In ME’scope, experimental modal parameters are typically estimated from a set of Frequency Response Functions (FRFs) measured from a structure. Each FRF is an estimate of the dynamic characteristics between two degrees of free-dom (DOFs) of the structure. Each DOF contains both a point and a direction of measurement on the structure.

In this note, curve fitting will be used on sets of synthesized FRFs to demonstrate the drawbacks and benefits of both single reference and multiple reference curve fitting.

MIMO Model

All of the FRF measurements that can possibly be made between pairs of DOFs on a structure can be assembled into an FRF matrix. This FRF matrix is part of a Multiple Input Multiple Output (MIMO) model. A MIMO model can then be used to calculate any structural response (or output) due to any combination of excitation forces (or inputs). The MIMO model is represented mathematically as,

{ ( )} [ ( )]{ ( )}X j H j F j (1)

where: [ ( )]H j = FRF matrix (N by N)

{ )}X(j = Fourier transforms of

displacement responses, or outputs (N-vector)

{ ( )}F j = Fourier transforms of

excitation forces, or inputs (N-vector)

N = number of DOFs

Figure 1 depicts a set of FRF measurements being taken on a structure. The input is fixed at point 7 in the vertical di-rection, and the varying outputs are at points 1, 2, 3, etc., each in the vertical direction. Each FRF is measured be-tween the fixed input DOF and a different output DOF.

SINGLE REFERENCE TEST: In Figure 1, because the single excitation force is fixed, this is a single reference test.

Figure 1. Single Reference FRF Measurements.

All of these FRF measurements are put into one column of the FRF matrix, corresponding to the input at point 7 in the vertical direction. Each FRF is also placed in the row corre-sponding to its output point & direction.

NOTE: Each column of the FRF matrix corresponds to an input DOF, and each row corresponds to an output DOF.

ROVING IMPACT TEST

In a single reference test, a single (fixed) input or a single (fixed) output DOF is used during FRF measurement. In the example in Figure 1, a single fixed input was used, and a single column of FRFs in the FRF matrix was measured.

In a roving impact test, an accelerometer is used to measure a single (fixed) output, and a roving hammer is used to pro-vide input at multiple input DOFs. This measurement pro-cess is shown in Figure 2.

The FRFs resulting from a roving hammer test will fill in a single row of the FRF matrix. In this case, the row corre-sponds to the single (fixed) response or output, and each FRF corresponds to a different force or input. A set of FRFs, each one with the same output and a different input, occupy a single row of the FRF matrix.

To summarize, a single reference set of FRFs occupies a single row or single column of the FRF matrix.

In the majority of cases, a single row or single column of FRFs from the FRF matrix in a MIMO model is sufficient to estimate all of the modal parameters of the structure, in the frequency band of the FRFs.

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Figure 2. Single Reference Impact Test

MULTIPLE REFERENCE TESTING

Multiple reference modal testing means that two or more (fixed) output DOFs or two or more (fixed) input DOFs are used during FRF measurement. This results in the meas-urement of two or more rows or two or more columns of FRFs in the FRF matrix.

If curve fitting one row or one column of FRFs is usually sufficient for identifying the modes of a structure, then why measure and curve fit multiple rows or columns? There are several good reasons.

1. If the reference DOF corresponds to a nodal point (a DOF with zero value) of the mode shape of a particular mode, then evidence of that mode (its resonance peak), will not appear in any FRF. Hence, the mode will be missed.

2. Certain references (rows or columns) of FRFs may pro-vide larger resonance peaks for some modes, hence better modal parameter estimates than if only a single reference (row or column) is used.

3. If the structure is large or non-linear, multiple shakers (inputs) may be needed to achieve sufficient signal lev-els for making effectively linear FRF measurements.

4. The structure may contain closely coupled modes or repeated roots. These curve fitting cases can only be resolved by using a multi-reference curve fitting meth-od. (These two cases are addressed in App Notes #14 & #15.)

Multiple reference FRF measurement requires that all of the fixed reference signals (multiple inputs or multiple outputs) be simultaneously acquired with at least one roving signal. This requires multiple transducers and a multi-channel ac-quisition system that can simultaneously acquire all chan-nels of data. Furthermore, if multiple fixed inputs are used, the input signals must be uncorrelated with one another.

As a minimum, the same signal cannot be used for multiple inputs

MODAL PARAMETERS

We will start with a set of known modal parameters so that the curve fitting results can be checked. The modal parame-ters are listed in Figure 3. This modal data has already been entered into a Shape Table in ME’scope.

Mode Number 1 2 3

Frequency (Hz) 50.0 52.0 55.0 Damping () 3.0 2.5 3.0

DOFs --------- Residues ---------

1Z:5Z 1.0 0.0 -1.0

2Z:5Z 0.5 -0.25 -0.6

3Z:5Z 0.0 -0.5 0.6

4Z:5Z -0.5 -0.75 -0.3

5Z:5Z 1.0 1.0 0.8

Figure 3. Modal Parameters (3 Modes, 5 FRFs).

Open the App Note 06-Single versus Multiple Refer-ence Curve Fitting Project from the ME’copeVES\Application Notes folder.

Double click on SHP: 3 Close Modes in the Project panel to open the Shape Table with the modal parame-ters in it.

Figure 4. Shape Table with Modal Parameters.

Notice that the modal damping is expressed both in Hz units as well as a percent of critical damping. Notice also that each mode shape component (or M#) is a Measurement Type of Residue Mode Shape with Units of in/lbf-sec.

Each component of a Residue Mode Shape corresponds to the FRF from which it was obtained by curve fitting. (In this case, we are starting with Residue Mode Shapes that have Roving DOFs from 1Z to 5Z, and a reference DOF of 5Z.

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NOTE: FRFs have DOFs in the following format;

FRF DOFs = Roving DOF : Reference DOF

Finally, notice that two modes have nodal points; that is, points were their mode shape components are “0”. Shape 1 has a nodal point at DOF 3Z, and Shape 2 has a nodal point at DOF 1Z.

FRF SYNTHESIS

To synthesize a single reference set of FRFs using the mod-al parameters in SHP: 3 Close Modes

Right click in the spreadsheet area in SHP: 3 Close Modes, and execute Tools | Synthesize FRFs.

The Synthesize FRFs dialog box will open, as shown below.

Change the parameters in dialog box, and click on OK.

Select Displacement in dialog box that opens, and click on OK.

In the next dialog box that opens, enter “Single Refer-ence 5Z”, and click on OK.

Execute Format | Rows/Columns and select (5, 1) to display all of the FRFs, as shown in Figure 5.

Figure 5. Synthesized FRFs

SINGLE REFERENCE CURVE FITTING

To curve fit these FRFs,

Right click in the graphics area of BLK: Single Refer-ence 5Z, and execute Modes | Modal Parameters.

This will display the curve fitting panel with several tabs on the right side of the Data Block window.

Mode Indicator

The first step in curve fitting is to determine how many mode peaks (resonance curves) are present in a frequency band. The Mode Indicator assists you in making this de-termination.

Press the Count Peaks button on the Mode Indicator tab.

Click on OK in the dialog box that opens.

The Indicator curve on the lower left shows three resonance peaks (with red dots on them), indicating three modes. Notice that the title above the Mode Indicator says CMIF Using Imaginary Part.

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Notice also that 3 Peaks is listed on the Mode Indicator tab. This is the number of modes, or the model size, that will be used for curve fitting.

Figure 6. Mode Indicator Showing Three Peaks.

Rules for Calculating Mode Indicators

If the FRFs have acceleration or displacement respons-es, or numerators, the Imaginary Part of the FRF data should be used to calculate the Mode Indicator.

If the FRFs have velocity responses, or numerators, the Real Part of the FRF data should be used.

In all other cases, the Magnitude of the FRF data should be used.

Estimating Frequency & Damping

The next curve fitting step is to estimate frequency & damp-ing. Before executing this step, the Modes box on the Fre-quency & Damping tab must be equal to at least the num-ber of resonance peaks counted on the Mode Indicator.

If number of modes is set higher than the number of peaks, parameters for extra modes will also be estimated during curve fitting. These (computational) modes must then be deleted from the Modal Parameters spreadsheet.

In this case, we already know that there are three modes in the FRFs since they were synthesized using three modes. In a real world situation, the number of modes in a frequency band may not be very apparent.

Press the F & D button, and click on YES to estimate the modal frequencies & damping.

Press the Frequency & Damping button on the Fre-quency & Damping tab.

Click on Yes in the dialog box that opens.

Figure 7. Frequency & Damping Estimates.

The Modal Parameters spreadsheet in Figure 7 lists the fre-quency & damping estimates obtained by the Global Poly-nomial curve fitting method, (which was used when the Frequency & Damping button was pressed). These esti-mates closely match the original modal frequency & damp-ing values that were used to synthesize the FRFs

Estimating Residues

Once modal frequency & damping have defined in the Modal Parameter spreadsheet for one or more modes, modal residues can be estimated by a second curve fitting step.

Press the Residues button on the Residues & Save Shapes tab.

Click on Yes in the dialog box that opens.

After residue curve fitting is completed, a red fit function is displayed on top of each FRF, and Residue estimates for each mode and each FRF are listed in the Modal Parame-ters spreadsheet. The Residues & Save Shapes tab, (shown in Figure 8), indicates that the Polynomial curve fitting method was used when the Residues button was pressed.

Scroll though the FRF Traces

NOTE: When the mouse button is released during scroll-ing, the Residues for the current FRF Trace are displayed in the Modal Parameters spreadsheet.

Notice that all of the Residue estimates closely match the original residue values that were used to synthesize the FRFs

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Figure 8. Residue Estimates.

Saving Shapes

Once residues have been estimated for each mode and each FRF, they can be saved in a Shape Table as Residue Mode Shapes.

Press the Save Shapes button on the Residues & Save Shapes tab.

A dialog box will open, containing a list of current Shape Tables.

Press the New File button, enter “Single Ref Fit” into the dialog box that opens, and click on OK.

Figure 9. Single Reference Curve fitting Results.

Notice that Residue Mode Shapes have been saved in the Shape Table (shown in Figure 9), that the curve fitting method used for each frequency & damping estimate is shown in the Label column of the frequency & damping spreadsheet, and that the curve fitting method used for each residue estimate is shown in the Label column of the shapes spreadsheet

Comparing Shapes

A pair of mode shapes can be compared in two ways;

1. Graphically, by using the Shape Comparison animated display

2. Numerically, by calculating a MAC (Modal Assurance Criterion) value for the pair.

Comparison animation requires that a 3D model of the structure be available for displaying shapes spatially. Since none is available in this Project, MAC values will be used to compare shapes. The original shapes in SHP: 3 Close Modes will be compared with the shapes in SHP: Single Ref Fit.

MAC Comparison

Each MAC value is calculated between a pair of shapes. The following rules apply when using MAC;

MAC values range between 0 and 1. If the MAC value is equal to 1.0, the two shapes are

identical. If the MAC value is equal to or greater than 0.9, the

two shapes are very similar. If the MAC value is less than 0.9, the two shapes are

different.

To compare the shapes in SHP: Single Ref Fit with the orig-inal shapes in SHP: 3 Close Modes,

Execute Display | MAC in the SHP: Single Ref Fit window.

The Shape Table file selection box will open

Select the SHP: 3 Close Modes in the dialog box that opens, and click on OK.

A Bar Chart of MAC values will open, as shown in Figure 10.

Figure 10. MAC Bar Chart

Hover the mouse pointer over each of the vertical bars to display its value.

Each bar is a MAC value between a mode shape in SHP: Single Ref Fit and a mode shape in SHP: 3 Close Modes.

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The large red bars each have a MAC value of 1.0, indicat-ing that each mode shape in SHP: Single Ref Fit is identi-cal to a mode shape in SHP: 3 Close Modes.

QUICK FIT

The three curve fitting steps that were previously carried out can also be done by executing the Curve Fit | Quick Fit command.

Position the mouse pointer on the Mode Indicator graph in the BLK: Single Reference 5Z window, and execute Quick Fit in the drop down menu

Notice that a second set of modal parameter estimates has been added to the Modal Parameters spreadsheet, and that these values are identical to the first set of parameters. The-se are the results of the Quick Fit.

Figure 11. Quick Fit Results.

MULTIPLE REFERENCE CURVE FITTING

The Residue Mode Shapes in SHP: 3 Close Modes are only defined for a single reference DOF, 5Z.

NOTE: Each Residue Mode Shape component has both Roving and Reference DOFs, indicating the FRF from which it was estimated.

Residue Mode Shapes can be re-scaled into UMM Mode Shapes. UMM Mode Shapes are scaled so as to provide Unit Modal Masses, the details of which are discussed in the ME’scope operating manual.

NOTE: Each UMM Mode Shape component has only a Roving DOF, indicating the Point & direction on the struc-ture for which it is defined.

Each UMM Mode Shape component also contains units similar to Residue Mode Shape components. UMM Mode Shape units are always (displacement / (force - seconds)).

Modal Models

A set of Residue Mode Shapes that was obtained by curve fitting a set of calibrated FRFs is called a Modal Model. A Modal Model contains all of the mass, stiffness, and damp-ing properties of the real structure from which the calibrated FRFs were measured. A set of UMM Mode Shapes is also called a Modal Model.

A multi-reference set of FRFs can only be synthesized from a multi-reference set of Residue Mode Shapes. The follow-ing two steps will be carried out in order to synthesize mul-ti-reference FRFs

A set of single-reference Residue Mode Shapes can be re-scaled into a set of UMM Mode Shapes.

A set of UMM Mode Shapes can be re-scaled into a set of multi-reference Residue Mode Shapes.

To create UMM Mode Shapes from the mode shapes in SHP: 3 Close Modes

Position the mouse pointer over the spreadsheet in the SHP: 3 Close Modes window, and execute Tools | Scaling | Residue to UMM Shapes in the drop down menu

Press the New File button in the dialog box that opens, enter “UMM Mode Shapes” into the next dialog box, and click on OK.

Figure 12. UMM Mode Shapes

Notice that the UMM Mode Shape components have differ-ent values because of the re-scaling. Notice also that the mode shape components have no Reference DOFs, only Roving DOFs.

Position the mouse pointer over the spreadsheet in the UMM Mode Shapes window, and execute Tools | Scaling | UMM to Residue Shapes in the drop down menu

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Press the New File button in the dialog box that opens, enter “Multi-Ref Residue Shapes” into the next dialog box, and click on OK.

Figure 13. Multi-Ref Residue Mode Shapes

Notice in Figure 13 that the mode shapes have a total of 25 components, 5 components with each of the 5 Reference DOFs, 1Z to 5Z. Notice also that the Residue mode shape components with Reference DOF 5Z are the same as the original mode shapes in SHP: 3 Close Modes. This is a multi-reference set of Residue Mode Shapes, from which a set of multi-reference FRFs can be synthesized,

Multi-Reference FRF Synthesis

To synthesize a multiple reference set of FRFs using the modal parameters in SHP: Multi-Ref Residue Modes

Right click in the spreadsheet area in SHP: Multi-Ref Residue Modes, and execute Tools | Synthesize FRFs.

The Synthesize FRFs dialog box will open, as shown below.

Change the parameters in dialog box, and click on OK.

Select Displacement in dialog box that opens, and click on OK.

In the next dialog box that opens, enter “Multi-Ref FRFs”, and click on OK.

A new Data Block window will open with 25 synthesized FRFs in it. To curve fit these FRFs using Quick Fit,

Position the mouse pointer over the trace graphics in the BLK: Multi-Ref FRFs window, and execute Mod-al Parameters in the drop down menu

The curve fitting tabs will be displayed in the Data block window.

Position the mouse pointer on the Mode Indicator graph, and execute Quick Fit in the drop down menu

The Quick Fit results shown in Figure 14, are obtained by curve fitting all 25 multi-reference FRFs. A residue was estimated for each mode and each FRF, the same as with a single reference set of FRFs.

Figure 14. Quick Fit of Multi-Reference FRFs

Comparing Multi-Reference Shapes

Press the Save Shapes button on the Residues & Save Shapes tab.

A dialog box will open, containing a list of current Shape Tables.

Press the New File button, enter “Multi Ref Fit” into the dialog box that opens, and click on OK.

A new Shape Table with multi-reference Residue Mode Shapes in it will open. Notice in Figure 15 that all of the mode shape components for Shape 1 with Reference DOF 3Z have essentially a “0” value. This is because Shape 1 had a nodal point at DOF 3Z. The same is true for Shape 2 and Reference DOF 1Z.

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Figure 15. Multi-Reference Curve Fitting Modes

Multi-Reference MAC

To compare the shapes in SHP: Multi Ref Fit with the orig-inal shapes in SHP: 3 Close Modes,

Execute Display | MAC in the SHP: Multi Ref Fit window.

The Shape Table file selection box will open

Select the SHP: 3 Close Modes in the dialog box that opens, and click on OK.

Because the shapes in SHP: Multi Ref Fit are multi-reference shapes, a message box will open informing you that you must select a reference first. Then the DOF Selec-tion dialog box will open with a Reference DOF selected.

Each reference of Residue Mode Shapes contains the same mode shape, except at those references where the mode shape has a nodal point. Shape 1 has a nodal point at DOF 3Z, and Shape 2 has a nodal point at DOF 1Z.

Select Reference DOF 1Z and then select Reference 3Z.

Figure 16. Low MAC Value for Reference 1Z

For Reference 1Z and 3Z, one of the mode shape pairs does not correlate well (it has a low MAC value). Figure 16 shows a low MAC value for shape pair 2 when Reference DOF 1Z is selected.

For all other Reference DOFs except 1Z and 3Z, all three mode shape pairs are identical (they have MAC values of 1.0)