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A Frequency Response Function (or FRF), in experimental modal analysis:

- is a frequency based measurement function
- used to identify the resonant frequencies, damping and mode shapes of a physical structure
- sometimes referred to a “transfer function” between the input and output
- expresses the frequency domain relationship between an input (x) and output (y) of a linear, time-invariant system

In a Frequency Response Function measurement the following can be observed:

- Resonances - Peaks indicate the presence of the natural frequencies of the structure under test
- Damping - Damping is proportional to the width of the peaks. The wider the peak, the heavier the damping
- Mode Shape – The amplitude and phase of multiple FRFs acquired to a common reference on a structure are used to determine the mode shape

**In Experimental Modal Analysis**

Many types of input excitations and response outputs can be used to calculate an experimental FRF. Some examples:

- Mechanical Systems: Inputs in force (Newtons) and outputs in Acceleration (g's), Velocity (m/s) or Displacement (meter)
- Acoustical Systems: Inputs in Q (Volume Acceleration) and outputs in Sound Pressure (Pascals)
- Combined Acoustic and Mechanical systems: Inputs in force (either Q or Newtons) and ouputs in Sound Pressure (Pa), Acceleration (g's), etc
- Rotational Mechanical Systems: Inputs in Torque (Nm) and output in Rotational Displacement (degrees)

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:

- Impact Hammers
- Electrodynamic Shakers

Responses can be measured by:

- Accelerometers: measure acceleration vibration
- Lasers: measure surface velocity
- String Pots, Photogrammetry: displacement

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:

- The real part of the FRF will equal zero at natural/resonant frequencies
- The imaginary will have “peaks” either above or below zero which indicate resonant frequencies. The direction of the peaks can be used to determine the mode shape associated with the natural/resonant frequency

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:

- Measurement points 1 and 3 are on one end of plate
- Measurement points 7 and 9 are in the center
- Measurement points 13 and 15 are on the other end, opposite points 1 and 3

When plotting the imaginary portion of the FRF, and looking at 532 Hz:

- Measurement points 1 and 13 move in phase, and are opposite corners of the plate (red and cyan)
- Measurement points 7 and 9 have low amplitude (blue and magenta)
- Measurement points 3 and 15 are in phase on opposite corners of the plate (brown and green)

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:

- A value of 1 at a particular frequency indicates that the FRF amplitude and phase are very repeatable from measurement to measurement.
- A value of 0 indicates that opposite – the measurements are not repeatable, which is a possible “warning flag” that there is an error in the measurement setup.

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:

- Instrumentation error – For example, ICP power is not being supplied to transducer that requires ICP power
- Inconsistent excitation – Structure is not being hit by an impact hammer consistent (for example, operator is tired and striking structure at different angles between impacts)
- Insufficient force – The structure is not being excited. For example, a very small hammer (example: size of pencil) on a large object (example: size of a bridge) with a large distance between excitation and response measurement

*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:

- Measurement #1 – Two Newtons of input force results in 10 g’s of acceleration response: Ratio of response to input is 5.0 g/N
- Measurement #2 – One Newton of input force results in 5.1 g’s of acceleration response: Ratio of response to input is 5.1 g/N
- Measurement #3 – Three Newtons of input force at 30 Hz result in 14.7 g’s: Ratio of response to input is 4.9 g/N

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 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.

**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 anti-resonances. Notice the corrections are bigger for the 245 Hz anti-resonance 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 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.

**Conclusion**

Frequency Response Functions (FRFs) are used to measure and characterize the dynamic behavior of a structure.

FRFs contain information about:

- Resonant frequencies
- Damping
- Mode shape

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 contact us!

**Related Structural Dynamics Links:**

- Siemens Structural Dynamics Solutions
- What modal impact hammer tip should I use?
- Modal testing: roving hammer or roving accelerometer?
- How to calculate damping from a FRF?
- LMS Test.Lab setup tricks for triaxial accelerometers
- What is a Power Spectral Density (PSD)?
- Modal Assurance Criterion (MAC)
- LMS Test.Lab Impact Testing

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