Frequency Response Function (FRF)
Dr Michael Sek, 2016
FREQUENCY RESPONSE FUNCTION (FRF)
The concept of Frequency Response Function (Figure 1) is at
the foundation of modern experimental system analysis. A
linear system such as an SDOF or an MDOF, when subjected
to sinusoidal excitation, will respond sinusoidally at the same
frequency and at specific amplitude that is characteristic to the
frequency of excitation. The phase of the response, in general
case, will be different than that of the excitation. The phase
difference between the response and the excitation will vary
with frequency. The system does not need to be excited at one
frequency at the time. The same applies if the system is
subjected to a broadband excitation comprising a blend of
many sinusoids at any given time, such as in the white noise
(Gaussian random excitation) or an impulse. It is obvious that,
in order to find how the system responds at various
frequencies, the excitation and the response signals must be
subjected to the DFT.
The characteristics of a system that describe its response to
excitation as the function of frequency is the Frequency
Response Function H(f) defined as the ratio of the complex
spectrum of the response to the complex spectrum of the
excitation. The spectra are raw (unfolded two-sided).
H( f ) 
X( f )
F( f )
The H(f) is a spectrum whose magnitude |H| is the ratio of
|X| / |F| and the phase H = X - F .
Figure 2 shows an example of experimental setup.
Figure 1 Concept of Frequency Response Function
(Brüel&Kjær "Structural Testing")
Figure 2 Car body undergoing testing to acquire its FRFs (Brüel&Kjær "Structural Testing")
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Frequency Response Function (FRF)
Dr Michael Sek, 2016
VIRTUAL EXPERIMENT TO MEASURE THE FREQUENCY RESPONSE FUNCTION
Let's find the FRF of a system in a virtual experiment. For this purpose let's use excited by a force SDOF system (m=100kg,
c=1000N/(m/s), k=1e6N/m) discussed in "ODEs. Vibration of SDOF system. Transfer Function".
The model of SDOF system will be used as a virtual system, whose characteristics will be determined from analysis of its
response to excitation signals. The system will be treated as a black box, as illustrated by the faded section of Simulink®
model in Figure 3, labelled VIRTUAL SYSTEM. For comparison, the system is modelled with two alternative approaches
(ODE and transfer function) and the results can be viewed in the Scope. Only one model is required for the rest of this project.
As in a real experiment, we will obtain the "experimental" data from the "digital scope "Scope1". One needs to enable the
scope's storage feature. Since the Scope1 has multiple inputs connected to it we choose the data format "Structure with time" as
shown in Figure 4. After running the model once we can check field names in the structure ScopeData1.
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VIRTUAL SYSTEM
Figure 3 Virtual experimental setup to acquire the data for the FRF of a system
>> ScopeData1
ScopeData1 =
time: [2048x1 double]
signals: [1x4 struct]
blockName: 'SDOF/Scope1'
>> ScopeData1.signals
ans =
1x4 struct array with fields:
values
dimensions
label
title
plotStyle
Figure 4 Data logging settings for Scope1 and field names in the structure ScopeData1
It is obvious that, following the order of connections to Scope1, the essential "measurements" are accessible in the structure
ScopeData1 as shown in Table 1.
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Frequency Response Function (FRF)
Dr Michael Sek, 2016
Table 1 ScopeData1 structure fields associations.
Time
ScopeData1.time
Excitation Force
ScopeData1.signals(1).values
Response Acceleration
ScopeData1.signals(2).values
Response Velocity
ScopeData1.signals(3).values
Response Displacement ScopeData1.signals(4).values
In the Model Configuration Parameters Data Import/Export tab "Save to workspace" items can be un-ticked since the data is
returned via Scope1. Save options must be changed as shown in Figure 5 in order to ensure that the data is returned at
appropriate equispaced time intervals. The variables dt and tmax control the sampling interval (or reciprocal of sample
frequency) and the duration of the virtual experiment. These are the same decisions one needs to make in a real experiment.
Under Solver tab it is advised to change the Max step size from auto to dt.
Figure 5 Save options settings of Simulation Configuration Parameters Data Import/Export.
Random Gaussian Excitation
Random Gaussian signal is a broadband signal. Settings for Random Number block suitable for using as a broadband
excitation are shown in Figure 6.
Variable Fmax controls the maximum excitation. Variance in the
Random Number block refers to the squared standard deviation  , also
termed the Root Means Square or RMS. The normal random signal only
rarely exceeds 3. The entered expression for the variance will cause the
maximum instantaneous force to be approximately equal Fmax.
Figure 6 Settings of Random Number block
The FRFs between the response and excitation force can be found for any parameter that describes the response of the system,
i.e. acceleration, velocity and displacement. Figure 7 shows the results for random excitation obtained with the code shown in
the Appendix. The resonance is near 15 Hz. The curves look noisy. Random excitation requires longer sample time and
averaging. Note that the folding scaling and multiplication by 2 are not required for the folding of H(f) since H(f) is the
ratio.
In practice, in most cases only one type of response parameter is measured and typically it is acceleration using accelerometers.
Accelerance FRF is calculated directly from measured signals. Mobility and receptance FRFs are obtained by integration, i.e.
by dividing the accelerance by the corresponding values of j and (j)2, respectively.
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Frequency Response Function (FRF)
Dr Michael Sek, 2016
Figure 7 Random excitation and responses of the system under test and the corresponding FRFs
Impulse Excitation
Better results are obtained with an impulse excitation (see Figure 9). Such excitation in the form of a square impulse can be
produced with the Pulse Generator set up as shown in Figure 8.
Figure 8 Settings of Pulse Generator block to produce an impulse excitation.
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Frequency Response Function (FRF)
Dr Michael Sek, 2016
Figure 9 Excitation and responses of the system under test (impact excitation) and the corresponding FRFs
Identification (Recovery) of System's Parameters from its FRF
FRFs allow to recover the "unknown" parameters of the system



The magnitude of accelerance Ha at large frequencies approaches equals 1/mass.
The magnitude of receptance Hx at near-zero frequency approaches 1/stiffness coefficient.
The width of magnitude of mobility Hv is related to the damping.
Using the results in Figure :
 mass m= 1 / 0.01 = 100 kg
 stiffness coefficient k= 1 / 0.1e-5 = 1e6 N/m
The results match the values used for the simulation.
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Frequency Response Function (FRF)
Dr Michael Sek, 2016
APPENDIX
AN EXAMPLE OF A FUNCTION USED TO GENERATE A HARMONIC SIGNAL
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Frequency Response Function (FRF)
Dr Michael Sek, 2016
IMPLEMENTATION OF THE FOLDING ALGORITHM
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Frequency Response Function (FRF)
Dr Michael Sek, 2016
The code used to obtain the FRFs of the system in Figure 3 and produce Figure 7 and
Figure 9.
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Frequency Response Function (FRF)
Dr Michael Sek, 2016
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