The following article will attempt to explain the basic theory of the frequency response function. This basic theory will then be used to calculate the frequency response function between two points on a structure using an accelerometer to measure the response and a force gauge hammer to measure the excitation.
Fundamentally a frequency response function is a mathematical representation of the relationship between the input and the output of a system.
So for example the frequency response function between two points on a structure. It would be possible to attach an accelerometer at a particular point and excite the structure at another point with a force gauge instrumented hammer. Then by measuring the excitation force and the response acceleration the resulting frequency response function would describe as a function of frequency the relationship between those two points on the structure.
The basic formula for a frequency response function is
![\[ H(f) = \frac{Y(f)}{X(f)} \]](http://blog.prosig.com/wp-content/ql-cache/quicklatex.com-117472ddaefda02951a3cd11b149cf0d_l3.png "Rendered by QuickLaTeX.com")
Where 
is the frequency response function.
And 
is the output of the system in the frequency domain.
And where 
is the input to the system in the frequency domain.
Frequency response functions are most commonly used for single input and single output analysis, normally for the calculation of the 
or 
frequency response functions. These are used extensively for hammer impact analysis or resonance analysis.
The frequency response function is used in situations where the output to the system is expected to be noisy when compared to the input.
The frequency response function is used in situations where the input to the system is expected to be noisy when compared to the output.
Additionally there are other possibilities, but they are outside of the scope of this article.
or can be used for resonance analysis or hammer impact analysis. is most commonly used with random excitation.
The breakdown of is as follows,
![\[ H1(f) = \frac{S_{xy}(f)}{S_{xx}(f)} \]](http://blog.prosig.com/wp-content/ql-cache/quicklatex.com-d8525a8a335e2424a70ca5a5baf3f74f_l3.png "Rendered by QuickLaTeX.com")
Where is the frequency response function.
And 
is the Cross Spectral Density in the frequency domain of 
and 
And where 
is the Auto Spectral Density in the frequency domain of .
In very basic terms the frequency response function can be described as
![\[ H1(f) = \frac{Cross\,Spectral\,Density\,of\,the\,Input\,and\,Output}{Auto\,Spectral\,Density\,of\,the\,Input} \]](http://blog.prosig.com/wp-content/ql-cache/quicklatex.com-47251a95c61e8a8e06b256c418a176db_l3.png "Rendered by QuickLaTeX.com")
The breakdown of therefore is as follows,
![\[ H2(f) = \frac{S_{yy}(f)}{S_{yx}(f)} \]](http://blog.prosig.com/wp-content/ql-cache/quicklatex.com-4f49e7494df6f5d3845e1cb3fa7f4ba5_l3.png "Rendered by QuickLaTeX.com")
Where is the frequency response function.
And 
is the Cross Spectral Density in the frequency domain of and
And where 
is the Auto Spectral Density in the frequency domain of
In very basic terms the frequency response function can be described as
![\[ H2(f) = \frac{Auto\,Spectral\,Density\,of\,the\,Output}{Cross\,Spectral\,Density\,of\,the\,Input\,and\,Output} \]](http://blog.prosig.com/wp-content/ql-cache/quicklatex.com-56447c0edb9b82f048dbab23ae9e16f8_l3.png "Rendered by QuickLaTeX.com")
In the following example we will discuss and show the calculation of the frequency response function.
The excitation or input would be the force gauge instrumented hammer, as shown in Figure 1 as a time history.
Figure 1: X(t)
In this case the response or output would be the accelerometer, as shown in Figure 2.
Figure 2: Y(t)
However as discussed earlier the frequency response function is a frequency domain analysis, therefore the input and the output to the system must also be frequency spectra. So the force and acceleration must be first converted into spectra.
The first part of the analysis requires the Cross Spectral Density of the input and output, this is . This is calculated using the response as the first input and the excitation as the second input to the Cross Spectral Density Analysis in DATS the result is shown in Figure 3. Were 
being calculated for use with for example, then the excitation would be the first input and the response the second input to the Cross Spectral Density Analysis in DATS.
Figure 3: Sxy(f)
Next the Auto Spectral Density of the input, or excitation signal is required. This is calculated using the Auto Spectral Density Analysis in DATS, this analysis is sometimes known as Auto Power, the result of which is shown in Figure 4, this is .
Figure 4: Sxx(f)
The Cross Spectrum is then divided by the Auto Spectrum and the resulting frequency response function is shown in Figure 5.
Figure 5: H1(f)
The entire analysis as used in DATS.toolbox is shown in Figure 6, the data flow from the original input and output, force and response, can be seen through to the frequency response function. The DATS software does, of course, provide a single step transfer function analysis. We have deliberately used the long-hand form below to illustrate the steps in this article.
Figure 6: Complete DATS worksheet (Click to expand)
It is necessary to understand that for the purposes of understanding and clarity in this article some important steps have been glossed over, windowing of the input for example, to allow the basic understanding of what makes up the frequency response function.
Hello,
Normally people don’t expect to see an Autospectrum, or Power Spectral Density on a linear scale, and it looks so noisy, that I suspect the impact was recorded using a Hanning weighting – a common mistake.
The end result, – the transfer function is also odd, to my eyes.
I would have expected a bode plot.
I am aware that linear display is useful for fatigue and stress purposes, but the frequency range is too high for this to be the application.
Hi Ken,
Thanks for posting on our blog.
I’d like to respond to your points if I may.
I understand your point of view, the Power Spectral Density is normally shown on a logarithmic scale (Y axis) as you suggest.
However for the purposes of this article it is not, we have shown it as linear. Here in this article we are trying to explain and discuss the basics in a straight forward simple fashion.
With regards to your points about noise and windowing, we apply a window to the force signal. We also apply a window to the response if it is very noisy, but we prefer not to do this.
The window is an exponential window, like a transient window for example. It is not a hanning window or similar.
The structure in question is a very simple structure, if a little noisy, and you’re correct it’s not being used for fatigue analysis.
With regards to the final output, our DATS software can quickly switch between Modulus & Phase and Real & Imaginary. So it is possible show in the Bode format you mention.
what can be infered from figure 5,please explain..
Hello Eapen,
I’m not sure anything can be inferred from Figure 5, it simply shows the Frequency Response Function for this particular test.
how can v obtain auto spectral density of velocity from auto spectral density of the displacement
Hi Kiran,
Thanks for asking a question on our blog.
This conversion is one of the fundamental laws of Newton’s physics.
It is possible to convert from Acceleration to Velocity to Displacement using calculus, specifically integration.
Our Prosig software performs this conversion in an advanced fashion to take account of the constant and remove this error from the results.
You should keep in mind that the original time series is required for this conversion.
If you would like to discuss this feature further please feel free to contact us directly.
Hi
Just say you have several accelerometers on a complex vibrating structure. Each accelerometer has a slightly different frequency spectrum. Lets also pretend that you have a mic at some distance from this vibrating structure, and that you are trying to locate the particular component or part of the structure that is responsible for a radiating a particular tonal frequency. Would the cross spectrum be valuable in identifying which accelerometer is the culprit of this offending sound?
Many thanks
Hello Paul,
Thanks for asking a question on our blog.
We can see that you have a good understanding of what you are trying to do.
In our opinion the Cross Spectrum is a step in the correct direction, but you should go one step further and calculate the Coherence between each vibration source and the microphone response, you can do this very easily with a software package like DATS.
To actually rank each vibration with respect to the microphone you should consider something like the Source Contribution Package, again as part of DATS.
The Source Contribution Analysis uses a method called Singular Value Decomposition. The Singular Value Decomposition computation produces an eigenvector matrix, this matrix is used to derive the cross spectra between the vibration references and the measured sound response. These cross spectra are then used to calculate the Reference Related Auto spectra at the response position. Each Reference Related Auto spectrum is related to the coherent contributions from the particular references and source input.
If you would like to discuss this further please feel free to contact us directly.
Going back to Paul’s hypothetical situation. Suppose that the accellerometers are measuring the start of an event that will be producing the noise that is being measured, but that there is no real reason to expect there to be any frequency correlation between them, would any conventional analysis work then?
For example, if you have 5 drums and a lightbeam sensor for each drum that triggered just before the drumstick hit each one, and that the drums are being hit in a regular sequence. How might one analyse the data (one mic channel and 5 drumstick sensors) to determine which drum was loudest/highest-pitch etc?
Hi Andy,
Thanks for asking a question on our blog.
I think there is some difference between your question and Pauls.
In Paul’s case the vibration responses are all related.
In your case they are not.
Therefore the same analysis would not apply.
I think you are trying to find which drum gives a certain frequency response when hit. The easiest thing to do would be to use a microphone, mounted about the drums and hit each drum in sequence. Starting and stopping a data capture for each drum. Then simply frequency analyse each of these data captures. You might need to do it several times to build up an average for each drum as you may hit it differently each time.
This will give you the full frequency spectrum for each drum.
You wouldn’t need any accelerometers at all.
Sorry, perhaps I took my analagous situation too far. This is actually exactly the same question as Paul’s, but we have a slightly different conception of the nature of the data. I should perhaps have chosen a better analogy, as the trigger pulses occur at 20-150Hz which would be a pretty inhuman rate of drumstick operation.
Hi Andy,
Thanks for the additional information, but I still don’t understand your application and can’t comment on it unless you can explain what your trying to do. Perhaps you could contact us directly to discuss?
I am looking for the information about the engine order, used for frequency response analysis.
We are working on frequency response Analysis for the exhaust systems, there we use different engine orders, i.e. 1st, 1.5, 2nd, 2.5 etc orders. Kindly let me know what is the actual meaning of this engine order. All I know is, its the disturbances created per rotation of the crank shaft. I need information.
- Ashujc
Hello Ashujc,
Thanks for asking a question on our blog.
An engine order is really two separate words, ‘engine’ and ‘order’
Engine is obvious but ‘order’ not so.
You could have an order of anything that rotates, not just an engine. For example wind mill blades have their own orders.
An order is the speed that something happens at.
So if a shaft is rotating at 100 times per second you would have a fundamental frequency of 100Hz. If there were two blades on opposite sides of the shaft somewhere long it’s length, this shaft they would be causing an excitation or noise at 200Hz because there is two of them. You could say the noise from the blades is a 2nd order noise of the main shaft.
The same goes for any other number. An order is the relationship to the main fundamental frequency that occurs in multiples of the fundamental.
James,
Thanks for a clear example.
Would be more useful if you could publish the parameters used by DATS.
e.g.
Sampling rate:
Anti-aliasing filter:
Number data in each FFT:
Windowing:
Ya
Hello Ya Huang,
Thank you for your feedback.
We wanted to keep the article as simple as possible, so we have kept away from any specific numbers and data, just the main principles.
If you would like to discuss in further detail, please feel free to contact us directly.
Hi James,
Your example is very instructive. However as a new comer in FRF, I could not figure out how you obtain the phase for FRF from the steps mentioned in figure 6. Obviously the division can only give a real number, which is the amplitude of FRF. Thanks.
Hello Steve,
Thank you for asking a question on our blog.
You have closely studied figure-6, which is good to read.
In figure-6 the CSD (Cross Spectral Density) is divided by the ASD (Auto Spectral Density). The CSD is a complex number, the ASD in a real number.
Any mathematics that involve a complex number will result in a complex answer, so the answer will have both a real and imaginary part or expressed differently a modulus and phase.
For example.
Where the CSD is represented by A+ iB, where A is the real part and iB the imaginary part.
And where the ASD is represented by C, where C is real.
The formula in figure-6 would be,
(A+iB) / C
Which is exactly equal to
A/C + i(B/C)
If you have any further questions, please feel free to ask.
I’m reading this because I’ve just been sent a newsletter that points to the page.
Are you sure you should really be advising use of H1 and H2 on single tap test recordings?
You see I agree with your first equation H(f)=Y(f)/X(f) but the subsequent equations are unhelpful if you end up using segment averaging (inherent in the calculation of the power and cross spectral densities) on a single tap. The windowed data for the force time history segments will all be zero except for the one window that contains the actual impact. Moreover the response in the subequent windows has nothing to do with the corresponding force window leading to potentially significant bias errors in the transfer function.
If you must use segment averaging to obtain transfer functions for tap test data then I think you need to make it very clear that this is for multiple taps and that the window length used must exactly correspond to each tap and response time history (i.e. there must be no segmentation within each time history pair) and the window length must be sufficient to capture the full decay of the vibration following the tap. Your readers should also know that the highlighted inappropriateness of segment avergaing for tap test data cannot be overcome by multiple taps at random intervals – that only compounds the errors introducing rippling in the estimated transfer functions because of the [Fourier transform properties of the] inherent similarity between one tap and the next.
Hello Stuart,
Thank you once again for your comments.
First of all I can only comment on the software we produce at Prosig and the methods we would recommend.
This article is intended to give a basic understanding of the concept of what we call Hammer Impact tests or you refer to as Tap tests.
For Hammer Impact Analysis we do not use or suggest overlapping segments, we would indeed suggest this is an incorrect method.
So I agree with your points, it is just you have assumed we use a method we do not use or recommend.
Our Hammer Impact software uses a Wizard to set-up the sample rates and durations to match exactly. The data is then processed as one entire block, including a pre-trigger. A force block and response block, having had force and response windows applied.
Thanks again for your comments.