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# Modal Analysis : Just what are these measurement that are called FRFs

Dr.Hashamdar : The frequency response function is very simply the ratio of the output response of a structure due to an applied force. We measure both the applied force and the response of the structure due to the applied force simultaneously. (The response can be measured as displacement, velocity or acceleration.) Now the measured time data is transformed from the time domain to the frequency domain using a Fast Fourier Transform algorithm found in any signal processing analyzer and computer software packages. Due to this transformation, the functions end up being complex

valued numbers; the functions contain real and imaginary components or magnitude and phase components to describe the function. So let’s take a look at what some of the functions might look like and try to determine how modal data can be extracted from these measured functions.

Let’s first evaluate a simple beam with only 3 measurement locations (Fig 6). We see the beam below with 3 measurement locations and 3 mode shapes. There are 3 possible places forces can be applied and 3 possible places where the response can be measured. This means that there are a total of 9 possible complex valued frequency response functions that could be acquired; the frequency response functions are usually described with subscripts to denote the input and output locations as hout,in (or with respect to typical matrix notation this would be hrow,column) The figure shows the magnitude, phase, real and imaginary parts of the frequency response function matrix. (Of course, I am assuming that we remember that a complex number is made  up of a real and imaginary part which can be easily converted to magnitude and phase. Since the frequency response is a complex number, we can look at any and all of the parts that can describe the frequency response function.) Now let’s take a look at each of the measurements and make  some remarks on some of the individual measurements that could be made.

First let’s drive the beam with a force from an impact at the tip of the beam at point 3 and measure the response of the beam at the same location (Fig 7). This measurement is referred to as h33. This is a special measurement referred to as a drive point measurement. Some important characteristics of a drive point measurement are all resonances (peaks) are separated by anti-resonances the phase looses 180 degrees of phase as we pass over a resonance and gains 180 degrees of phase as we pass over an anti-resonance the peaks in the imaginary part of the frequency response function must all point in the same direction So as I continue and take a measurement by moving the impact force to point 2 and measuring the response at point 3 and then moving the impact force on to point 1 to acquire two more measurements as shown. (And of course I could continue on to collect any or all of the additional input-output combinations.) So now we have some idea about the measurements that we could possibly acquire. One important item to note is that the frequencyresponse function matrix is symmetric. This is due to the fact that the mass, damping and stiffness matrices that describe the system are symmetric. So we can see that hij = h ji – this is called reciprocity. So we don’t need to actually measure all the terms of the frequency response function matrix.

One question that always seems to arise is whether or not it is necessary to measure all of the possible input-output combinations and why is it possibly to obtain mode shapes from only one row or column of the frequency response function matrix. 