Vibration tests are normally used in modal analysis, a process in which we describe a structure in terms of its dynamic properties like natural frequency, damping and mode shapes. To introduce modal analysis clearly, let’s consider a simple example. Imagine there is a freely supported flat plate, and we apply a force that varies according to sinusoidal pattern. So the plate will start to vibrate, and we measure the response of the plate with an accelerometer attached to one corner of the plate.

Now if we keep the amplitude of the exciting force the same, but change its frequency, we would notice that the amplitude of the response will also change. This is actually reasonable: the response is amplified more when the excitation frequency is closer to the natural frequency of the structure, and the response will be the maximum when the excitation frequency is the same as the natural frequency.

To do some further analysis, I would like to introduce a function which represents the relationship between input and output signal: frequency response function or FRF. To get the FRF of a system, we measure both the applied force and the response of the structure at the same time. Then the measured data is transferred from the time domain to the frequency domain using Fourier Transform algorithm. If we calculate the FRF of the flat plate, we will notice that the peaks in the FRF are at the natural frequencies of the system.

So we can see using the vibration tests on one structure we can get the FRF, and then we can find the natural frequencies according to the peaks of FRF.

Then why do we need the natural frequencies? And what’s its relation with THA? First, the natural frequencies are related to the stiffness of the structure: the higher the stiffness is, the higher the natural frequencies are. Second, the stiffness of the structure is related to the fixation between the implant and the bone: the better the fixation is, the higher the stiffness is. So we can use the vibration tests to get the natural frequencies of the implant-bone structure, then we can use the value of the natural frequencies to evaluate the fixation of the implant during THA.

Source: Peter Avitabile, *Experimental Modal Analysis*, 2001

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