Given a non-periodic function f(t), the Fourier Transform is defined by the equation
24 See, for example, Fourier Series and Boundary Value Problems, R.V.Churchill and J.W.Brown
A Designer’s Guide To Finite Element Analysis Mechanics, Mathematics, And Other Theory Here we assume that the non-periodic function is a function with an infinite
period. Note that the independent variable in the “original” function, f(t), was “t”, while the independent variable in the “transformed” function is ω. It is customary to interpret ω as the angular frequency (radians / second). This is natural, since it is the frequency of the sine and cosine terms in the Fourier Series.
We say that the transform has taken us from the time domain to the frequency domain. The inverse transform takes us back from the frequency to the time domain.
Of the several properties that the Fourier Transform has, one in particular is interesting: the Fourier Transform of f’(t) is given by iωF(ω). This means differential equations can be transformed into algebraic equations25. The transform has a host of other interesting properties, covering areas like time shifting, modulation, etc. which are very useful in signal processing.
Theory guarantees us that the two representations are entirely equivalent:
there is no loss of information going from one domain to the other. Which form we choose to use depends purely on convenience.
It's just a question of which is easier to understand and comprehend.
Compare the problem to that of listening to an orchestra and trying to pick out instruments. A trained ear can do it, but for the average ear extracting such info from the time-domain-signal is hard. If, instead, you use the Fourier Transform to break the signal into a spectrum, you can clearly see signatures of different instruments, since each has a different frequency range.
To illustrate this, consider the 3-degree-of-freedom system shown below26. Models such as this are not just instructive – they are also used quite widely in practice. Called lumped mass models, they are used extensively in a wide variety of applications. One such example, the simulation of vehicle crash, is included for reference at the end of this book.
25 Recall Laplace transforms – they serve a similar purpose.
26 These figures are from The Fundamentals Of Modal Testing, Application Note 243-3, Agilent Technologies
Since the system has 3 degrees of freedom, it will have 3 modes of vibration. These are the natural frequencies - ω1, ω2 and ω3 - and the corresponding mode shapes.
Our design goal is to control the response of the system. That is, given any source of excitation, we want to first predict, then alter, the response. The first step in this is to understand how the modes of vibration contribute to the response. Since the modes depend on k and m, we can tune the response by making appropriate changes to the structure.
The time-response, shown below, doesn’t provide much insight. It does tell us how the response changes with time, but tells us nothing of the relative importance of the 3 modes.
But now let’s look at the Fourier Transform – the frequency domain plot.
A Designer’s Guide To Finite Element Analysis Mechanics, Mathematics, And Other Theory
Note that the peaks in the plot correspond to ω1, ω2 and ω3, the natural frequencies of the system. The contribution of ω1 is more than that of ω2, so we’ll probably be better off tuning the structure to change ω1.
In fact, as the figure below shows, the frequency domain plot can be obtained as the superposition of the plots for each individual mode!
Remember this, since we will discuss the Modal Superposition Method later on in our study.
Frequency domain plots can intimidate a beginner, but there’s no reason to be intimidated. While a physical interpretation of the frequency domain plot is not always possible (in fields like electrical engineering, for instance), mechanical vibrations are kinder. As shown in the figure, the frequency-domain and time-frequency-domain plots of the modes of vibration of a cantilever beam are very easy to correlate with the physical behavior of the vibrating beam.
While the “actual” beam, being a continuous body, has infinite degrees of freedom, our physical measurement is “reducing” it to a 3-degree of freedom body if we measure just the first 3 modes. The plot along the
“frequency axis” is the frequency-domain plot we’ve seen above. Obviously the plot will change depending on the measurement point, just as the time response changes at each measurement point – recall the FRF we saw earlier.
Figuring out where to measure (i.e. the choice of the measurement points) is one of the many challenges test-engineers face, and is one reason why the FE method goes so well with testing in the design of vibrating
equipment. FE is often used to suggest measurement points.
It is harder to visualize the equivalence between the time-domain and frequency-domain forms of a function for complicated structures, but the principle is the same.
The Fourier Transform is an invaluable tool in the study of vibrations.
Several times, it is implemented in real-time. That is, even as a signal is acquired, its Transform is calculated.