Getting Started with Modal Curvefitting

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In experimental modal analysis, after collecting a set of Frequency Response Function (FRF) data on a structure, the next step is to extract a meaningful set of modes and their associated modal parameters including:

  • natural frequencies
  • damping values
  • mode shapes

This process of extracting this information from the FRFs is referred to a modal curvefitting or modal parameter estimation (Figure 1).

 

Modal_Curvefit_Process2.pngFigure 1: FRFs are curvefit to determine modal parameters including frequency, damping, and mode shape.

This article is an introduction on how to use Simcenter Testlab for modal curvefitting.  It has several parts:

  1. Background and Considerations
  2. XY Dataset Analogy
  3. Simcenter Testlab Modal Analysis software:
    • FRF and Bandwidth Selection
    • Stabilization Diagram
    • Shape calculation
    • Modal Validation
  4. Repeated Roots: Two Modes at the Same Frequency

Background and Considerations

 

Modal curvefitting measured FRFs is a tall task!  There are several considerations (shown in Figure 2):

  • How can the correct number of natural frequencies in the structure be determined from the FRFs? How to tell if too many modes are estimated?  How to know if too few modes are estimated?
  • How is it possible to tell if a peak in the FRF data is measurement noise, or an actual resonance?
  • What if modes are so close together that they look like a single peak? How can these adjacent modes be mathematically separated from each other?

 

Modal_Parameter_Estimation_Challenges.pngFigure 2: Modal Parameter Estimation challenges – Right: Determine number of modes, Middle: Noise on FRFs, Left: Closely spaced modes.

To help with modal curvefitting, software such as Simcenter Testlab Modal Analysis can assist in the process. Using a Stabilization Diagram as a guide, the best mathematical modal model (frequencies, damping, etc.) that describes the FRF data set can be determined.

 

Before detailing the Simcenter Testlab software, an analogy might help to better understand curvefitting.

 

XY Dataset Analogy

 

To illustrate the curvefitting process, this analogy uses a set of X and Y data as shown in Figure 3.

 

XYdata.pngFigure 3: Set of X and Y data that follows a mathematical trend with but has some random variation in values.

The goal is to calculate a curve that best describes the X and Y data.

 

Viewing the data points, they appear to have an underlying mathematical relationship, which looks to be a third order polynomial. But not all the data points fall exactly along the polynomial. 

 

In fact, when the XY data was generated, the data points of a third order polynomial were perturbed to create some variation.  This variation is like noise on acquired experimental data.

 

What process can be used to find the best mathematical representation of the data? One approach would be:

  1. Try different order polynomials. For example, first fit the data with a first order polynomial curve, then a second order polynomial, then a third order polynomial, etc.
  2. Find the order number and coefficients for the polynomials that minimizes the difference (or error) between the mathematical equation and data points.

This process is animated for orders one through ten in Figure 4.

 

stab_gif3.gifFigure 4: Various Order Polynomial Fits to XY Data.

Consider the first order polynomial curve to the XY data shown in Figure 5.

 

First_Order_Poly.pngFigure 5: A first order polynomial is used to describe the data.

The first order polynomial follows the general trend of the XY data, but does not make a perfect fit to the data points. The curve parameters (slope and offset) were calculated to minimize the difference between the curve and the data points. 

 

To a trained observer, it is obvious that a higher order polynomial is needed to fit the data points better. Increasing the polynomial order to three, a better fit to the data is obtained as shown in Figure 6.

 

Third_Order_Poly.pngFigure 6: Fit to data with 3rd order polynomial.

That is pretty good fit!  Of course, there is some random variation in the XY data, making it impossible for the polynomial curve to pass through all the data points.

 

Even a fourth or fifth order polynomial will fit the data well, with low error, as shown in Figure 4.  Without any prior knowledge, how would a user know if a 3rd, 4th, or 5th order polynomial is correct?  Engineering judgement from the user may ultimately be required.

 

What happens if the order of the polynomial is increased even more?  A 10th order polynomial fit is shown in Figure 7

 

Tenth_Order_Poly.pngFigure 7: High order polynomial, while passing through more data points, is probably a poorer mathematical representation of the XY data.

Using a tenth order polynomial, the small variations in data points cause the polynomial to over and under shoot in order to try and pass through each data point.

 

The calculated error, based on the distance between data points and the curve, of the tenth order might be better than lower orders: the curve passes through more of the data points.  However, it is clear that it is not fitting the underlying data pattern well - there are some large excursions in the curve which deviate from the original third order polynomial shape.

 

What is there to conclude from this analogy about fitting different order polynomials to a set of XY data?:

  • Simply increasing the mathematical order in the curvefit does guarantee correct results.
  • Any curvefitting routine will need to ignore noise and other variation in the source data.
  • An analyst will be required to make a judgement about the correct order and math model to accept.

 

Key_Curvefit.png

 

Let’s expand this polynomial analogy to curvefitting of actual modal FRF data using Simcenter Testlab Modal Analysis.

 

Modal Curvefitting with Simcenter Testlab Modal Analysis

 

The process of estimating a good set of modal parameters from measured FRFs has similarities to the XY data analogy:

  • The structural modes of the test object dictate the underlying mathematical relationship in the FRFs.
  • There is some measurement noise/variation on top of the measured FRF data due to equipment noise floor, small structural non-linearities, mass shifting due to accelerometer placement, etc.

But there are additional considerations, as illustrated in Figure 8:

  • Multiple functions or FRFs from different locations on structure must be considered.
  • Multiple modes (natural frequency , damping, mode shape) over the frequency range must be taken into account.

 

Modal_Curvefit.pngFigure 8: Multiple modes over multiple locations need to be considered in a modal analysis curvefit.

There are several unknowns to solve for (modes, etc) and many FRF data points to consider.  The modal curvefitter software and its algorithms should:

  • Find the correct number of modes (i.e., order), over a given frequency range for the test object.
  • Determine the natural frequencies, damping, and other modal information that gives the best fit to the measured FRFs.
  • Minimize the error between calculated modes and measured FRF data.
  • Ignore noise or other random variations in the data.
  • Make sure any extraneous or improper modes are not included in the final modal model.

In the end, the user will need to manually inspect the results and make sure that all of the above is done properly.

 

Key_Engineering_Judgement.png

 

The Process

 

The steps in performing a modal curvefit in Simcenter Testlab Modal Analysis are shown in Figure 9.

 

Simcenter_Testlab_Modal_Analysis_Flow_Chart.pngFigure 9: Steps in Modal Curvefitting process.

This process may need to be repeated until a satisfactory curvefit is achieved.  The first three steps in the process are all done in the Polymax or Time MDOF worksheet of Simcenter Testlab Modal Analysis.  The validation steps have separate workbooks called Modal Synthesis and Modal Validation respectively.

 

Open the Simcenter Modal Analysis software (Tools -> Add-ins -> Modal Analysis and Polymax Modal Analysis) to get started. It is highly recommended to use the Polymax algorithm for the modal curvefitting process, see the upcoming section on Modal Assurance Criterion for an explanation.

 

Select FRFs and Frequency Range

 

The first step in doing a modal curvefit is to select the measured FRFs to be analyzed and set the frequency range of analysis as shown in Figure 10.

 

Simcenter_Testlab_Modal_Frequency_Selection.pngFigure 10: The FRF and Bandwidth selection step of the Simcenter Testlab Modal Analysis. The display curve is the sum of all the FRFs being used in the Modal Curvefit.

Some key features of the Bandwidth selection minor worksheet:

  • FRF Selection - The FRF selection is done in the upper left. By default, all measurement runs of the current project section are used for analysis.  If FRFs of the same name exist, only the latest measurement (based on the data time stamp) is used.  The assumption is that if a bad FRF measurement was acquired (with poor coherence for example) the latest measurement would represent the proper FRF after the problem was discovered and data reacquired.
  • Bandwidth Selection - The frequency range for analysis is selected using a double cursor. Typically, it is best practice to position the cursors at areas of low response, or anti-resonance, so that that no modal information is truncated.
  • Data Visualization Overlay - In the lower left of the worksheet, the Display section allows the user to select the FRF functions to be shown on the screen while curvefitting. The choice “Selected Function” allows individual measurements to be shown, while “Sum” is an averaged FRF of the entire data set. 

This sum FRF data shown in the display is for visualization to help select the bandwidth for analysis. All the FRFs will be used in the analysis.  The complete list of FRFs being analyzed is shown on the left side of the screen.

 

Stabilization Diagram

 

Inside the Stabilization minor worksheet (the top right of the screen), the Simcenter Testlab software creates a visual guide, called a stabilization diagram, for selecting potential modes as shown in Figure 11.

 

Simcenter_Testlab_Modal_Stabilization_Diagram.pngFigure 11: The Stabilization Diagram step of the Simcenter Testlab Modal Analysis.

The stabilization diagram has many rows, each containing letters:

  • Row Number - The number on the right side represents the number of potential modes (like the order of the polynomial from the previous analogy) that the software algorithm uses to describe the FRF data.
  • Letters - The letters in each row represent potential modes the software determines by analyzing the measured FRFs. The type of letter indicates the quality of the modal estimate.

The model size, in the lower left corner, is normally set two to three times higher than the number of modes expected in the data to allow for enough potential mode selections.  The model size is the number of modes the Simcenter Testlab software algorithm fits to the FRF data.

 

Stabilization Diagram Rows

 

The number of letters, or potential modes, in each row will never exceed the row number (Figure 12).  It can be less than the row number because the Simcenter Testlab will not display meaningless mode selections.  For example, if a potential mode was identified by the curvefitting algorithm with negative frequency or negative damping, it would not be displayed.

 

Stabilization_Rows.pngFigure 12: The row number on the right side of the stabilization diagram indicates how many modes the Testlab software used to try and describe the FRFs.

By hovering the mouse over a letter in the stabilization diagram, the potential frequency and damping value is shown below.  The letter (which represents a potential mode) can be selected for the curvefit analysis by clicking on it.

 

Stabilization Diagram Letters

 

The letters indicate the repeatability of a particular potential modal solution.  This is done by comparing the potential solutions (frequencies, dampings, etc) from the current row to the potential solutions in the previous row as shown in Figure 13.

 

Stabilization_Letters.pngFigure 13: The letters on the stabilization diagram indicate the repeatability of a potential modal solution.

The letters indicate the stability of the solutions, with the letter s being the most repeatable.

  • o – New: This solution was not present on the previous row.
  • f – Frequency: If the frequency of the potential mode is stabilized with respect to the previous row, than the letter f is used.
  • d – Damping: Both frequency and damping values are stabilized relative to previous row.
  • v – Vector: Frequency and modal participation vector are both stabilized.
  • s – Stable: Modal solution is completely stabilized in frequency, damping, and modal participation vector.

The modal participation vector is related to the mode shape component of the driving point FRF measurements.

 

Key_Say_Yes_to_S.png

 

Ideally, columns of letter s indicate the presence of modes (Note: a very lightly damped mode at 50 or 60 Hz, even with a strong column of letter s, is most probably electrical interference and not a mode of the test object).

 

Key_Only_One_S.png

 

It is only necessary to pick one letter s from any column of letters since the solution is repeatable over the rows/model order.

 

Stabilization Diagram Tolerances

 

How repeatable?  In the lower left of the stabilization diagram screen is a button labelled “Tolerances…”.  By pressing on it, the tolerance percentages used for the letter designations are shown (Figure 14). 

 

Tolerances.pngFigure 14: Tolerances for determining the letter assigned to possible modal solutions are set in the lower left corner of the Stabilization Diagram.

Tolerances can be adjusted.  For example, if it is desired to know the frequency of the mode with less than 1% variability in the frequency, the tolerance could be tightened to 0.1%. 

 

Calculate Shapes

 

After selecting the best set of potential modes, the next step is to calculate mode shapes as shown in Figure 15. Click on the “Shapes” minor worksheet at the top left of the screen.

 

Simcenter_Testlab_Modal_Shape_Calculation.pngFigure 15: The Mode Shape calculation step of the Simcenter Testlab Modal Analysis.

Press the “Calculate” button on the middle of the left side to determine the shapes.  The modes with shapes will be listed in the lower left. View the modes and scroll through them using the arrows on the lower right side.

 

Modal Model Validation

 

With a complete set of modes and mode shapes, now the results can be double-checked through some validation steps.  It is important that no modes were missed, and that not too many modes were included.

Two commonly used validation methods are: Modal Synthesis and Modal Assurance Criterion. These methods are available in two separate workbooks. 

 

Modal Synthesis: Checking for Missed Modes

 

Using the calculated mode set, FRFs can be synthesized and compared against the originally measured FRFs.

 

In doing this, it is often possible to catch mistakes where modes were left out of the analysis as shown in Figure 16.

 

Synthesized_FRF.pngFigure 16: Comparison of synthesized FRFs (green) with measured FRFs (red). When no modes are missed, the correlation percentage is high, and error is low (left graph). When modes are missed, the synthesized and measured FRFs do not overlay, and the correlation is lower while error is higher.

Each measured FRF can be checked in this manner.  For each FRF, a correlation and error percentage are calculated and display.  It is desirable to have the correlation percentage as high as possible, close to 100%.  The error percentage should be low, as close to zero percent as possible.

 

Key_Modal_Synthesis.png

 

If modes where missing in the original analysis, go back to the Stabilization step and add them.

 

Modal Assurance Criterion: Checking for Too Many Modes

 

It is possible to have a good modal synthesis but have selected too many modes.  The Modal Assurance Criterion can help identify this situation.

 

The Modal Assurance Criterion (MAC) compares two mode shapes.  If the modes are similar in shape, the value of the MAC will be close to 100%.  Two shapes that are different will have a MAC close to 0%.

Ideally, each mode identified during modal parameter estimation should have its own unique mode shape (like no two snowflakes have the same shape).

 

How could too many modes be selected?  Consider Figure 17, which shows the Stabilization Diagram for two different modal curvefitting algorithms: Time MDOF and Polymax.

 

Stabilization_Comparison.pngFigure 17: The chances of selecting too many modes can be increased by algorithm used to create Stabilization diagram.

It would be easy for a user to select a duplicate or extra mode on the Time MDOF stabilization diagram (left side of Figure 17).  It is less likely that a duplicate mode would be selected from the stabilization diagram on the right side, which used the Polymax algorithm. The Polymax curvefitting algorithm does a better job of rejecting potential modes that are not possibly real.

 

When an extra or duplicate mode is selected, the shape of the modes will be very similar to modes nearby in frequency. Similar shapes are physically impossible: each mode shape should be unique. This mode-shape-similarity will be flagged as a high “off-diagonal” MAC term as shown in Figure 18.

 

MAC_Matrix.pngFigure 18: High off-diagonal MAC terms close in frequency indicate too many modes were selected.

The figure above shows that the 304 Hz mode and 311 Hz mode are likely the same mode. One of the modes should be deselected. Like the XY dataset analogy, the error between the modal model and FRF data might be smaller, but the underlying modal model is wrong.  Extra modes should be removed.

 

Key_MAC.png

 

See the ‘Modal Assurance Criterion Knowledge Base’ article for more information.  In addition to selecting too many modes, the MAC has other uses including indicating if too few measurement points were used in the modal survey.

 

Repeated Roots: Two Modes at Same Frequency

 

In many structures, it is possible to have two modes occur at the same frequency.  This happens often in symmetric structures (tubes, disks, etc).

 

To determine if a structure has a repeated root, use the Mode Indicator Function (MIF). The MIF will dip in the presence of a mode. If there are two modes at close frequencies, and there are two MIFs, each MIF will dip indicating the presence of two modes at the same frequency as shown in Figure 19.

 

Simcenter_Testlab_MIF.pngFigure 19: Mode Indicator Function indicates two modes at same frequency.

The Mode Indicator Function “dips down” at frequencies where there are two modes. In Figure 19, both the primary mode indicator function (in green) and the secondary mode indicator function (in blue) dip at the same frequency.  This “double dip” indicates the presence of two modes at these frequencies. 

 

Key_MIF.png

 

Mathematically, it is necessary to have two references to have both a primary and secondary mode indicator function.

 

As a final touch, consider using the Maximum Likelihood estimation of a Modal Model (MLMM). The MLMM algorithm makes small, automated adjustments to the frequencies and dampings for a better match to the FRF data. 

 

Conclusions

 

Hope this help!  Questions?  Post a reply, email peter.schaldenbrand@siemens.com, or contact Siemens PLM GTAC support.

 

Modal Data Acquisition

Modal Analysis and Operational Deflection Shapes

Comments
Siemens Experimenter Siemens Experimenter
Siemens Experimenter

I believe that a little background on the theory of modal analysis might aid in the understanding of modal curve fitting.

 

Modal analysis starts with the equation of motion of a structure. For a structure with a single degree of freedom this is:

 eqn of motion.pngEquation 1: Equation of Motion of a SDOF System

After performing a Fourier Transform and manipulating terms, that equation can be expressed in the form of a transfer function (Equation 2).

 

 

transfer function.pngEquation 2: Transfer Function of a SDOF System

 

 

The left side of this equation, H(w), is the measured frequency response function. The terms on the right side contain information about the mode shapes, modal scaling factors, natural frequencies, and damping. Some of these relationships are illustrated below in Figure 1.

 

modal parameters.pngFigure 1: Transfer Function Terms and their Associated Modal Parameters

 

One of the goals of a modal curve fit is to find the polynomial that best matches the measured FRF. Then, the mode shapes, natural frequencies, and damping can be extracted from the polynomial.

 

Almost any structure in the real world will be a multiple degree of freedom system. The transfer function for a multiple degree of freedom system is similar to Equation 2, but now there is a summation (Equation 3). The transfer function is equal to the sum of all of the modes of the system.

 

other transfer function.pngEquation 3: Transfer Function of a MDOF System

 

Consider a structure that has three modes. The FRF is pictured below in Figure 2. The transfer function will be equal to the summation of the three modes of the system, and is written below.

 other transfer function.pngEquation 4: 3 Mode Transfer Function

 

 summation of modes.pngFigure 2: A FRF showing three modes. The FRF is equal to the linear superposition of all three modes.

 

Because H(w) is known, a curve fit can be performed, and then frequency, damping, and mode shapes can be extracted for each mode.

 

If you are interested in learning more about the theory behind modal parameter estimation check out the Search Manuals, Theory, FAQ tool which can be found in the desktop shortcut for Simcenter Testlab.

 

 

Search FAQs.pngFigure 3: Use the Search Manuals, Theory, FAQ to find additional theory documents in Simcenter Testlab