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 free-free 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 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 anti-resonance, 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!
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.
The most commonly used estimator is the H1-estimator, 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 anti-resonances better than the resonances. Best results are obtained with this estimator when the inputs are uncorrelated.
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 anti-resonances. Notice the corrections are bigger for the 245 Hz anti-resonance frequency than for the 133 Hz resonance frequency.
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 anti-resonances. 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.
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.
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