Analysis Modal : Why is only one row or column of the FRF matrix needed

Analysis Modal Dr.Hashamdar

Why is only one row or column of the FRF matrix needed?

It is very important for us to understand how we arrive at mode shapes from the various measurements that are available in the frequency response function matrix. Without getting mathematical. let’s discuss this Let’s just take a look at the third row of the frequency response function matrix and concentrate on the first mode. If I look at the peak amplitude of the imaginary part of the frequency response function, I can easily see that the first mode shape for mode 1 can be seen (Fig 8a). So it seems fairly straightforward to extract the mode shape from measured data. A quick and dirty approach is just to measure the peak amplitude of the frequency response function for a number of different measurement points.

Now look at the second row of the frequency response function matrix and concentrate on the first mode (Fig 8b). If I look at the peak amplitude of the imaginary part of the frequency response function, I can easily see that the first mode shape for mode 1 can be seen from this row also.

We could also look at the first row of the frequency response function matrix and see the same shape. This is a very simple pictorial representation of what the theory indicates. We can use any row to describe the mode shape of the system. So it is very obvious that the measurements contain information pertaining to the mode shapes of the system.

Let’s now take a look at the third row again and concentrate on mode 2 now (Fig 8c). Again if I look at the peak amplitude of the imaginary part of the frequency response function, I can easily see the second mode shape for mode 2 can be seen.

And if I look at the second row of the frequency response function matrix and concentrate on the second mode, I will be a little surprised because there is no amplitude for the second mode (Fig 8d). I wasn’t expecting this but if we look at the mode shape for the second mode then we can quickly see that this is a node point for mode 2. The reference point is located at the node of the mode.

So this points out one very important aspect of modal analysis and experimental measurements. The reference point cannot be located at the node of a mode otherwise that mode will not be seen in the frequency response function measurements and the mode cannot be obtained.

Now we have only used 3 measurement points to describe the modes for this simple beam. If we add more input-output measurement locations then the mode shapes can be seen more clearly as shown in Figure 9. The figure shows 15 measured frequency response functions and the 3 measurement points used in the discussion above are highlighted. This figure shows the 15 frequency response functions in a waterfall style plot. Using this type of plot, it is much easier to see that the mode shapes can be determined by looking at the peaks of the imaginary part of the frequency response function.