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**Modal Parameter Extraction Methods**

Modal Analysis and Testing S. Ziaei-Rad

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**Type of Modal Analysis In this course By domain By Frequency range**

Frequency domain (FRFs) Time domain (IRFs or response history) By Frequency range SDOF method MDOF method In this course Single-FRF methods Multi-FRF methods

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**Preliminary Checks of FRF Data**

Visual checks Low-frequency asymptotes High-frequency asymptotes Incidence of anti-resonances Overall shape of FRF skeleton Nyquist plot inspection

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Basic Skeleton Theory IF IS ASYMPTOTIC TO IS ALSO ASYMPTOTIC TO ?

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Mobility Skeleton

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Skeleton Geometry

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**Skeleton Geometry Mass-dominated characteristics**

Stiffness-dominated characteristics Abnormal characteristics

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**Assessment of Multiple-FRF Data**

Principle Response Function (PRF)

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**Mode Indicator Functions (MIFs)**

The technique is used to determine the number of modes present in a given frequency range, to identify repeated natural frequencies and to pre-process the FRF data prior to modal analysis. Consider a set of FRF data from multiple excitation measurements or from multi-reference impact tests typically consists of an matrix where: n number of measurement DOFs p number of excitation or reference DOFs

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**Complex Mode Indicator Function (CMIF)**

The CMIF is the squares of the singular values and are usually plotted as a function of frequency in logarithmic form. Natural frequencies are indicated by large values of the first CMIF. Double modes by large values of second CMIF.

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Other MIFs MMIF: * results from the eigenvalue solution equation (*) for each frequency And these values are plotted as a function of frequency. The MMIF takes a value between 0 and 1, with the resonance frequencies now identified by minimum values of MMIF instead of Maximum values for the CMIF. RMIF: In this version, natural frequencies are identified by zero crossing of the RMIF values.

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**MIFs Complex Mode Indicator Multivariate Mode Indicator**

Function (CMIF) Multivariate Mode Indicator Function (MMIF)

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**Modal Analysis Method Curve Fit Analysis: 1- SDOF Methods**

2- MDOF Methods

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**BEST ESTIMATES FOR THE MODAL PARAMETERS**

Modal Analysis GIVEN: MEASURED FRF DATA: MODEL: DETERMINE: BEST ESTIMATES FOR THE MODAL PARAMETERS

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SDOF Curve-fit Method Im Re

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SDOF Modal Analysis (1) (2) (3)

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Complete FRF

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Peak Amplitude Method 1- First, individual resonance peak are detected on the FRF plot and the frequency of one of the maximum responses taken as natural frequency of that mode . 2- The local maximum value of the FRF is noted and the frequency bandwidth of the function for a response level is determined ( ). The two points are thus identified as (Half-power point) 3- The damping of the mode can now be estimated from one of the following formulae. 4- The modal constant can be found from:

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Peak Amplitude Method

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Peak Amplitude Method CASE (a) CASE (b)

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**Limitation of Peak Amplitude Method**

The estimates of both damping and modal constant depend heavily on the accuracy of maximum FRF level, while it is not possible to measure this quantity with great accuracy. Most of the errors in measurement are around the resonance region particularly for the lightly damped structures. Only real part of the modal constant can be calculated. The single mode assumption is not completely correct. Even with clearly separated modes, it is often found that the neighboring modes do contribute a noticeable amount to the total response. A more general method called circle-fit method will introduce in next section.

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**Circle-Fit Method Properties of Modal Circle**

Here, we consider a system with the structural damping. Thus, we shall use the receptance form of FRF. As we said earlier, it is this parameter that produce an exact circle in a Nyquist plot. If the structure possesses the viscous damping, then the mobility type FRF should be used. Although, this later need a different general approach, most of the following analysis and comment apply equally to that case Some modal analysis packages, offer the choice between the two types of damping and simply take the mobility or receptance data for the circle-fitting according to the selection.

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**Properties of Modal Circle**

The effect of modal constant is to scale and rotate the circle

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**Properties of Modal Circle**

Consider two points

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**Circle-Fit Analysis Procedure**

1- Select point to be used 2- Fit circle, calculate quality of fit 3- Locate natural frequency, obtain damping estimate 4- Calculate multiple damping estimates and scatter 5- Determine modal constant modulus and argument

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**Circle-Fit Analysis Procedure (Step 1)**

Select point to be used Can be automatic selection or by the operator judgment The selected point should not be influenced by neighboring modes The circle arc should be around 270 degree (if the second rule is not violated) Not less than six points should be used SELECT DATA POINTS

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**Circle-Fit Analysis Procedure (Step 2)**

Fit circle, calculate quality of fit Different routins can be used to fit the circle (e.g. least-square deviation) At the end of this process, the centre and radius of the circle are specified. An example of the process is shown in next slide. 12 13 15 14

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**Circle-Fit Analysis Procedure (Step 3)**

Locate natural frequency, obtain damping estimate The radial lines from the circle centre to the point around the resonace are drawn The sweep rate the can be calculated, then natural frequency and damping ratio The frequency of maximum response The frequency of maximum imaginary receptance The frequency of zero real receptance 12 13 15 (ii) 14 (i) (iii)

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**Circle-Fit Analysis Procedure (Step 3) Estimation of Natural Frequency**

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**Circle-Fit Analysis Procedure (Step 4) (Damping Estimate)**

Using different points (one below and one after resonance), a set of damping ratio will be calculated. Ideally they should all be identical If deviation is less than 4 to 5 percent, then we did a good analysis If the scatter is 20 to 30 percent, there is something unsatisfactory. If the variation of damping is random, is probably due to random noise If the variation is systematic, it is due to systematic errors (set-up, effect of near modes, non-linearity) 12 13 15 14

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**Circle-Fit Analysis Procedure (Step 4) (Damping Estimate)**

b c d e a- linear data b- random noise c- error in the data d- modal analysis error e- non-linearity

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**Inverse or Line-fit Method**

Standard FRF plot format Inverse FRF plot format

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**SDOF Modal Analysis Using Inverse FRF Data**

GENERAL SDOF ASSUMPTION: RESIDUAL EFFECTS OF OTHER MODES DEFINE: ONE OF THE VALUES OF NEAR AND AN ‘INVERSE’ FRF PARAMETER

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**SDOF Modal Analysis Using Inverse FRF Data**

WHERE

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**Analysis Step One From measured FRF near , fix one point ( at ) and**

Calculate for all other points. Plot and fit: Slopes of best-fit lines for

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Analysis Step One Note Where So

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**ANALYSIS STEP TWO - Plot - Repeat step one for all values of (Compute**

- Fit best straight line - Find Hence From Plot:step One SLOPE = SLOPE = INTERCEPT = INTERCEPT =

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**Line Fit Modal Analysis**

Plot of real and imaginary Line fit modal analysis a- Plot of Real and Imaginary b- Slope from a

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**Regenerated FRFs Measured and regenerated without Residual effect**

residuals

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Residuals LOW-FREQUENCY MODES HIGH-FREQUENCY MODES RESIDUALS

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**Representation of Residuals as Linear Functions**

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**High Frequency Residual**

Residuals Low Frequency Residual (L.F. Residual) High Frequency Residual (H.F. Residual)

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**L.F. Residuals (Rigid Body Modes)**

Z

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**ALL TERMS +VE ADDITIVE**

H.F. Residuals ALL TERMS +VE ADDITIVE SOME TERMS +VE, SOME -VE TENDENCY TO CANCEL

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**Modal Analysis Methods**

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**Modal Analysis in Frequency Domain**

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MDOF Curve-fit Method H

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**Curve - Fitting In General (Nonlinear Least-Squares)**

MEASURED FRF DATA: THEORETICAL MODEL FOR FRF DATA:

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**Modal Analysis Using Rational Fractions**

USE ALTERNATIVE FORMAT FOR FRF: INSTEAD OF PARTIAL FRACTION

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**Rational Fraction Curve Fits**

LET AND GIVEN SEVERAL VALUES OF FIND TO MINIMISE

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**Rational Fraction Curve Fits**

When L such equations are combined: Solution will be found, by minimizing the error function J This leads to:

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**Rational Fraction Curve Fits**

SETS UP EQUATION OF FORM: CONTAIN: VALUES VALUES EQUATIONS ARE OVERDETERMINED

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**Rational Fraction Approach**

- CURVE - FIT FORMULA TO MEASURED DATA TO FIND (REAL) COEFFICIENTS - THEN, SOLVE THE TWO POLYNOMIALS TO DETERMINE EQUIVALENT MODAL PARAMETERS: Measuring difference between original and regenerated FRFs using the derived modal properties. Measuring consistency of the various modal parameters for different model order choice and eliminating those which vary from run to run.

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Example

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Caution

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**MDOF Curve-fits: Light Damping**

It is found that some structures are very well respond to the above modal analysis procedures. This is mainly due to the difficulties in acquiring good measurements near resonances. This problem is in lightly-damped structures. In such structures, the damping is not very important, and the structure is modeled as an undamped one. The aim is to find natural frequencies and modal constants only by using data measured away from the resonance regions.

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**MDOF Curve-fits: Light Damping**

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**MDOF Curve-fits: Light Damping**

1- Measure FRF over frequency range of interest. 2- Locate the resonances and find the corresponding natural frequencies. 3- Select some data points away from the resonances. (No. of Points=No. of Modes+2) 4- Using the selected data and compute the modal constants. 5- Construct a regenerated curve and compare with the measured FRFs.

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**Selection of Response Data for Identification 1- Complete Modal Presentation**

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**Measured and Regenerated FRFs**

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**SO, CAN USE CURVE-FITTING OF TO FIND ESTIMATES OF & FROM SET OF FRFs.**

Global frequency Methods in the Frequency domain (Multiple Curve Fitting) SO, CAN USE CURVE-FITTING OF TO FIND ESTIMATES OF & FROM SET OF FRFs.

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**SDOF and MDOF Testing and Analysis**

MODAL TESTING ANALYSIS - DIFFERENT VALUES FOR r; r (MUST AVERAGE) + SINGLE VALUE (AVERAGED) FOR r; r + SINGLE VALUE FOR r; r {}r + SINGLE VALUE FOR {}r + MULTI-VARIATE MODE INDICATOR - MUST REPEAT FOR/ ALL FRFs + MODE INDICATOR FUNCTION + DOUBLE ROOTS - CANNOT DETECT DOUBLE MODES - CONSISTENT DATA - EXPENSIVE

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**Modal Analysis Strategies**

MULTIPLE ESTIMATES ROW/COL FRFs r; r; * (i.e. n FRFs) (i.e. n) {}r SINGLE ESTIMATES MULTIPLE ESTIMATES ROW/COLS FRFs r; r; (i.e. n p FRFs) * (i.e. n p) {}r MULTIPLE ESTIMATES (i.e. p) SINGLE ESTIMATES r ; r ; rAjk * ONE FRF

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**Mode Indicator Functions**

HOW TO IDENTIFY ‘GENUINE’ MODES? HOW TO DETECT ‘REPEATED’ MODES? HOW TO ESTIMATE MODAL FORCING? ORDINARY MODE INDICATOR FUNCTION (FROM ONE ROW/COLUMN OF [H] ) MULTIVARIATE MODE INDICATOR FUNCTION (FROM SEVERAL ROWS/COLUMNS OF [H] )

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**Ordinary Mode Indicator Functions**

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**Multivariate Mode Indicator**

CMIF RMIF MMIF GIVEN: N ROWS P COLUMNS

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**Global frequency Methods**

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**Global frequency Methods**

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**Global frequency Methods**

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**SO, CAN USE CURVE-FITTING OF TO FIND ESTIMATES OF & FROM SET OF FRFs.**

Global frequency Methods in the Frequency domain (Multiple Curve Fitting) SO, CAN USE CURVE-FITTING OF TO FIND ESTIMATES OF & FROM SET OF FRFs.

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**1-Global Rational Fraction Polynomial Method (GRFP)**

1- The basic of FRP was described for single FRF. 2- The method can be applied to multi-FRF data. 3- The fact is if we take several FRFs from the same structure, then the denominator will be the same for all FRFs. 4- For one FRF we had 2(2m+1) unknowns. If we analyze N FRFs separately, then we have to calculate 2N(2m+1) unknowns. 5- The number of coefficient for GRFP method is (N+1)(2m-1)

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Global SVD Method (1) where

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Global SVD Method

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**Global SVD Method Let’s consider a column of FRFs (p FRFs), then:**

where

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**NOW TAKE TWO NEARBY FREQUENCIES:**

Global SVD Method IF THEN ALSO NOW TAKE TWO NEARBY FREQUENCIES:

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Global SVD Method Assume that the effect of out of range modes is constant over the frequency range. or In a same way THESE EXPRESSIONS RELATE TO THE RECEPTANCE & MOBILITY TERMS FOR ONE i .

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**Global SVD Method (4) NOW TAKE SEVERAL (L) FREQUENCIES I=1,2,3,…..L**

ELIMINATE THIS LEADS TO AN EIGENPROBLEM: (4) WHERE

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**Global SVD Method Matrix is calculated directly from measured FRFs.**

The mobility matrix as:

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**NEXT TO FIND MODE SHAPES, RETURN TO 2**

Global SVD Method SOLVE 4 USING SVD IN ORDER TO DETERMINE RANK OF [], [’] AND THUS THE CORRECT NUMBER OF MODES (n) Sr ; r=1,2,….n NEXT TO FIND MODE SHAPES, RETURN TO 2 AND FIND FROM

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Example

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