Simcenter Testing Solutions What is a Frequency Response Function (FRF)?

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A Frequency Response Function (or FRF) is:

  • is a frequency based 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
An experimentally measured Frequency Response Function (or FRF) is shown in Figure 1:

Figure 1: Bode Plot of Amplitude and Phase of a FRF function. Amplitude has peaks corresponding to natural frequencies/resonances of test object. Phase has shift at resonant frequency.

The following can be learned about a structure from a FRF:

  • 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

Article Contents:
1. Background
   1.1 Single Degree of Freedom (SDOF) Response
   1.2. Mass, Stiffness, and Damping Regions of a FRF
2. Experimental FRF Measurement Considerations
   2.1 Measurement Types and Equipment
   2.2 Input Force
   2.3 Crosspower, Spectrum, and Autopower Formulation
   2.4  Imaginary FRFs and Mode Shapes
   2.5. Averaging FRF Measurements: Coherence 
   2.6 FRF Estimators
      2.6.1  H1 Estimator
      2.6.2 H2 Estimator
      2.6.3 Hv Estimator
3. Conclusion

1. Background

There are several ways to understand a Frequency Response Function (FRF):

1.1 Single Degree of Freedom System (SDOF) Response

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A single degree of freedom (SDOF) system, consisting of a mass-spring-damper, is often used to understand dynamic systems (Figure 2).  The response and input to such a system is described in a FRF function.


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Figure 2: Mass-spring-damper single degree of freedom system.

In the mass-spring-damper system:

  • m – mass of the system [kg]
  • c – damping of system [Ns/m]
  • k – stiffness [N/m]
  • f – force applied to system [N]
  • x – displacement [m]

As a function of frequency, this mass-spring-damper system has a peak response (x) due to the force (f) at the natural frequency (ωn). The natural frequency of the mass spring system is equal to the square root of the stiffness over the mass as given in Equation 1.

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Equation 1: Natural frequency of a mass-spring-damper system is the square root of the stiffness divided by the mass.

The FRF of such a system is shown in Figure 3.

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Figure 3: A Frequency Response Function (FRF) of a Single-Degree-of-Freedom (SDOF) system in terms of displacement over force.

When exciting all frequencies with same amount of force, the natural frequency creates a peak in the response.

Real world objects are more complicated than a single degree of freedom system.  Real world systems have multiple natural frequencies because they are composed of a distributed set of masses and stiffnesses.

More about natural frequencies in the following knowledge article: Natural Frequency and Resonance.

1.2. Mass, Stiffness, and Damping Regions of a FRF

Plotting the Frequency Response Function of a single degree of freedom system, there are three distinct regions (Figure 4) below, at, and above the natural frequency of the system.

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    Figure 4: Responses below the natural frequency are governed by stiffness, at the natural frequency damping governs the response, above the natural frequency mass governs the response.

    The frequency response function of the system can be broken into three regions:

    • Stiffness – Below the natural frequency, the stiffness governs the response of the SDOF system
    • Damping – At the natural frequency, the damping governs the response of the SDOF system
    • Mass – Above the natural frequency, the mass governs the response of the SDOF system

    Structures in the real world are more complicated than a single degree of freedom system.  Real world systems have multiple natural frequencies because they are composed of a distributed set of mass and stiffness.

    More about the mass, stiffness, and damping regions of FRFs in the following knowledge articles: 

    2. Experimental FRF Measurement Considerations

    There are several considerations when measuring the Frequency Response Function of a real world structure.

    2.1 Measurement Types and Equipment

    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 outputs 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 are typically applied and measured using either modal impact hammers or shakers (Figure 5):

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    Figure 5: Left - Impact hammers for modal excitation, Middle/Right: Electrodynamic shakers for modal excitation.

    Responses can be measured by:

    • Accelerometers: acceleration vibration
    • Lasers: surface velocity
    • String Pots, Photogrammetry: displacement
    More knowledge articles on measurement excitation techniques:

    2.2 Input Force

    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. This is illustrated in Figure 6.

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    Figure 6: A force with flat frequency response is applied to a structure to identify resonant frequencies in the response.

    Because the FRF response is "normalized" to the input , the peaks in the resulting FRF function are resonant frequencies of the test object.

    When using a modal impact hammer, the stiffness and mass of the modal hammer tip are important for getting a flat input force: What Modal Impact Hammer Tip Should I Use?

    2.3 Crosspower, Spectrum, and Autopower Formulation

    To calculate an experimental FRF, the input force autopower is used in conjunction with the crosspower of the response/input.

    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 as shown in Figure 7.


    Figure 7: H represents the FRF between input X and output Y

    The FRF is the crosspower (Sxy) of the input (x) and output (y) divided by the autopower (Sxx) of input as shown in Equation 2 below.

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    Equation 2: The FRF (H) is a crosspower divided by an autopower.

    The autopower Sxx is the complex conjugate (a-ib) of the input spectrum multiplied by itself (a+ib), which becomes an all real function, containing no phase. The crosspower Sxy is the complex conjugate of the output spectrum multiplied by the input spectrum, and contains both amplitude and phase.

    Related knowledge articles:

    2.4 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 3 below:

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    Equation 3: Relationship between Amplitude, Phase, Real, and Imaginary.

    After transforming the FRF from Amplitude & Phase to Real & Imaginary, some interesting things happen (Figure 8):

    • 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

    Figure 8: Left - FRF expressed in Amplitude and Phase, Right - FRF expressed in Real and Imaginary

    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 (Figure 9). 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

    Figure 9: Plot of the imaginary portion of six FRFs on simple plate


    When plotting the imaginary portion of the FRF, and looking at 532 Hz (Figure 10):

    • 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)


    Figure 10: The amplitude of the imaginary portion of six FRFs plotted on plate geometry.

    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?!

    In real world structures, when modes are close together in frequency, reading the imaginary amplitude of the FRF is not enough to determine the mode shape.  The modes must be separated using techniques like modal curvefitting. 

    More in these knowledge articles:

    2.5 Averaging FRF Measurements: Coherence 

    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 (Figure 11) 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.

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    Figure 11: Green: Coherence, Red: FRF

    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!

    2.6 FRF Estimators

    Consider measuring a FRF at a single frequency on a structure. 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 as shown in Table 1.

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    Table 1: FRF data at three different frequencies from five different 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.

    2.6.1  H1 Estimator

    commonly used estimator is the H1-estimator (Figure 12). It 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.

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    Figure 12: H1 Estimator for FRF measurement

    This estimator tends to give an underestimate of the FRF if there is noise on the input as illustrated in Figure 13.

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    Figure 13: Left - FRF function, Right - Graph of X and Y values from 5 separate measurements, and Y-only corrections for average

    The H1 method estimates the anti-resonances better than the resonances. Best results are obtained with this estimator when the inputs are uncorrelated.

    2.6.2  H2 Estimator

    Alternatively, the H2 estimator (Figure 14) 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.

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    Figure 14: H2 Estimator for FRF measurement

    This estimator tends to give an overestimate of the FRF if there is noise on the output. Notice the corrections are bigger for the 245 Hz anti-resonance frequency than for the 133 Hz resonance frequency as shown in Figure 15.

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    Figure 15: Left - FRF function, Right - Graph of X and Y values from 5 separate measurements, and X-only corrections for average

    The H2 estimator estimates the resonances better than the anti-resonances.

    2.6.3  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 considers noise (N and M) on both the input X and output Y (Figure 16).

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    Figure 16: Hv Estimator for FRF measurement

    It does however require more computational time than the other two estimator methods (H1 and H2), which is not an issue for today's computers. The estimator effects are shown in Figure 17.

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    Figure 17: Left - FRF function, Right - Graph of X and Y values from 5 separate measurements, and corrections in both X and Y for average

    The Hv Estimator is commonly used when the source of noise in the measurements is not fully known.

    3. 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 email

    Related Structural Dynamics Links:

    Digital Signal Processing:

    KB Article ID# KB000043927_EN_US



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