A Frequency Response Function (or FRF), in experimental modal analysis:
In a Frequency Response Function measurement the following can be observed:
In Experimental Modal Analysis
Many types of input excitations and response outputs can be used to calculate an experimental FRF. Some examples:
For a experimental modal analysis on a mechanical structure, typically the input is force and output is acceleration, velocity or displacement.
Forces can be applied and measured via:
Responses can be measured by:
Generally, the input force spectrum (X) should be flat versus frequency, exciting all frequencies uniformly. When viewing the response (Y), the peaks in the response indicate the natural/resonant frequencies of the structure under test.
Because the FRF response is "normalized" to the input , the peaks in the resulting FRF function are resonant frequencies of the test object.
Imaginary FRFs and Mode Shapes
A FRF is a complex function which contains both an amplitude (the ratio of the input force to the response, for example: g/N) and phase (expressed in degrees, which indicates whether the response moves in and out of phase with the input).
Any function that has amplitude and phase can also be transformed to real and imaginary terms, as described the equation below:
After transforming the FRF from Amplitude & Phase to Real & Imaginary, some interesting things happen:
If several FRFs are acquired at different locations on the structure, and they are all phased with respect to a common reference, the imaginary part of the FRFs can be used to plot the mode shape.
In the example below, six FRF measurements were taken on a simple metal plate hung in freefree boundary conditions. The six FRFs are located as follows:
When plotting the imaginary portion of the FRF, and looking at 532 Hz:
When plotting the imaginary values at 532 Hz on a stick figure model of the plate, it can be seen that the plate is in torsion.
Who knew that viewing “imaginary” FRFs could be so useful?!
Digital Signal Processing Terminology
In nomenclature, a FRF is typically represented by the single capital letter H. The input is X and output is Y. H, X and Y are all functions versus frequency.
The FRF is the crosspower (Sxy) of the input (x) and output (y) divided by the autopower (Sxx) of input.
The autopower Sxx is the complex conjugate of the input spectrum to itself, which becomes an all real function, containing no phase. The crosspower Sxy is the complex conjugate of the output spectrum and the input spectrum, and contains both amplitude and phase.
Averaging FRF Measurements: Coherence & Estimators
It is common practice to measure the FRF measurement several times to ensure that a reliable estimate of the structures transfer function is being measured. The repeatability of the individual FRFs is checked by estimating a coherence function, while the average is calculated using different estimator methods, depending on the desired end result.
Coherence
Coherence is function versus frequency that indicates how much of the output is due to the input in the FRF. It can be indicator of the quality of the FRF. It evaluates the consistency of the FRF from measurement to repeat of the same measurement.
The value of a coherence function ranges between 0 and 1:
When the amplitude of a FRF is very high, for example at a resonant frequency, the coherence will have a value close to 1.
When the amplitude of the FRF is very low, for example at an antiresonance, the coherence will have a value closer to 0. This is because the signals are so low, their repeatability is made inconsistent by the noise floor of the instrumentation. This is acceptable/normal. When the coherence is closer to 0 than 1 at a resonant frequency, or across the entire frequency range, this indicates a problem with the measurement.
Problems could include:
Note that if only one measurement is performed, the coherence will be a value of 1! The value will be one across the entire frequency range – giving the appearance of a “perfect” measurement. This is because at least two FRF measurements need to be take and compared to start to calculate a meaningful coherence function. Don’t be fooled!
Estimators
When measuring a Frequency Response Function on a structure by inputting a 30 Hz forcing frequency. Using three different force levels, the following happens:
Why the variation between measurements? Unlike generating a FRF from a Finite Element Model, measuring a FRF may not return the same value every time a measurement is taken: structures are not completely linear and there can be small amounts of instrumentation noise in the measurement.
The three measurements had similar results: 5.0, 5.1 and 4.9 g/N. Which one is correct?
To determine the “correct” value, estimators are used for calculating the amplitude ratio (H) of the input to output of FRFs. There are three main FRF estimators in use today: H1, H2 and HV estimators.
When trying to characterize a structure the following table of data was gathered over 5 individual FRF measurements at three different frequencies:
The following is a simplified example for learning purposes. In a single FRF measurement, when looking at 3 different frequencies, the following may be observed over 5 individual measurements:
These X and Y pairs are plotted, a line is fit to the data. The slope of the line (typically g/N) will determine the amplitude of the FRF. The estimators affect how the data is fit and how much each data point is adjusted to create the best fit line.
H1 Estimator
The most commonly used estimator is the H1estimator, which assumes that there is no noise on the input and consequently that all the X measurements (the input) are accurate. All noise (N) is assumed to be on the output Y.
This estimator tends to give an underestimate of the FRF if there is noise on the input. H1 estimates the antiresonances better than the resonances. Best results are obtained with this estimator when the inputs are uncorrelated.
H2 Estimator
Alternatively, the H2 estimator can be used. This assumes that there is no noise on the output and consequently that all the Y measurements are accurate. Noise (M) is assumed to be only on input X.
This estimator tends to give an overestimate of the FRF if there is noise on the output. This estimator estimates the resonances better than the antiresonances. Notice the corrections are bigger for the 245 Hz antiresonance frequency than for the 133 Hz resonance frequency.
Hv Estimator
The Hv estimator provides the best overall estimate of the frequency function. It approximates to the H2 estimator at the resonances and the H1 estimator at the antiresonances. It does however require more computational time than the other two, which is not an issue for today's computers. The Hv estimator assumes noise (M and N) is on both the X input and Y output.
Conclusion
Frequency Response Functions (FRFs) are used to measure and characterize the dynamic behavior of a structure.
FRFs contain information about:
When creating an average FRF, coherence functions can give indications of FRF quality, while estimation methods are used to account for noise on the measurements.
Questions? Feel free to email peter.schaldenbrand@siemens.com
Related Structural Dynamics Links:
Digital Signal Processing:
Hi admin.
Could you sent to me data of FRF for this example?
Thank you so much.
Hello Dungle,
I'm not sure why you want these particular FRFs but some very similar ones are available in the Modal Analysis tutorial located at https://support.industrysoftware.automation.siemens.com/docs/lms/test.lab/.
The difference is that the article shows the FRFs from a 6 point impact test on a flat plate. The Modal Analysis tutorial shows the data from a shaker test on a 15 point geometry of the same flat plate. The FRFs should be the same since they are system characteristics of the plate.
Hi friends.
Respect this topic i woud like to know about use FRF or CrossPower Calculated from other device in EMA or OMA, in the Test.Lab. Can I Enter the processed data directly to calculate modal parameters? How I do it? or maybe I need to modify the header from my processed data (FRF or Crosspower) and then entered it to the "operational data selection" mode (in OMA) without use "operational data collection" mode? or in "Modal data selection" mode (for EMA)?.
Greetings
Jair
Hello Jair,
FRF's can be used in Modal Analysis. Crosspowers (or time histories to calculate Crosspowers from) can be used in Operational Modal Analysis. You can bring the data into Testlab from any data format that Testlab can import like Excel, Universal file, SDF, etc. For Excel you need a data header and you can a) Copy similar data to excel to get the required headers or b) ask your local support office to provide some example Excel files.
In rev 17 and earlier we can import these file types. Some require extra licenses and addins need to be turned ON. In rev 18 and later, you don't need to turn on extra addins and extra licenses are not needed.
Data type 
Extension 
Additional Licenses needed (Addins) 

ASAM Transport Format (XML) 
.atfx 
ASAM ODS Data Driver 

CadaX Project DB 
.ix0, .ix1, .dat, .idx, .mtx 
None 

CadaX TRDS 
.trds 
None 

Cadax TDF database 
.tdf 
None 

SIMCENTER XS / Recorder Data 
.xtrp / .trp 
Standalone Recording 

Excel files 
.xls, .xlsx 
Excel Data Driver 

Matlab 
.mat 
None 

Nastran 
.nas 
Nastran Data Driver 

nCode DAC 
.dac 
Durability Drivers Pack 

RoadRunner or Pimento 
Frequency Data Files Order Data Files Octave Data Files Time Data Files 
.fdf .odf .zdf .tdf 
None 
Standard Data File 
.sdf 
None 

Stereolithography File 
.stl 
None 

Throughput Files 
TDF LDSF 
.ix0, .ix1, .dat, .idx, .mtx .ldsf 
None 
Teac TaffMAT (Teac data Acquisition File Format) 
.dat, .hdr 
None 

Testlab 
.lms 
None 

Test.Xpress Data File (Simcenter Extended Data Format) 
.xdf 
None 

Universal Data File 
.unv 
None 

Virtual.Lab Data File 
.vl2tl 
None 

Windows Audio File 
.wav 
None 

Head 
.hdf 
Head Data Drivers 

AMESim 
.ame 
None 

NI Diadem 
.dat 
Durability Drivers Pack 

Somat SIE 
.sie 
Durability Drivers Pack 

Lexade 
.mes 
Durability Drivers Pack 

MDF 3.0 
.mdf 
Durability Drivers Pack 

Datx 
.datx 
DATX Data Driver 

ASC II 
.asc 
None 

MTS IDEAS 
.ati / .afu 
None 
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