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

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,

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

 The breakdown of  therefore is as follows,

 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

 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 response function would normally be shown in modulus & phase form as shown in Figure 6.

Figure 6: H1(f) shown as modulus & phase

The entire analysis as used in DATS.toolbox is shown in Figure 7, 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 7: 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.