Harmonic Motion - Fourier Approximation What does this mean below?

Question

There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of fourier expansions in the differential equation of motion, such as:

$m \ddot{u} + c\dot{u}+ku + g_{NL}=-z_{0}sin(wt)$

where $u$, $\dot{u}$, and $\ddot{u}$ are response displacement, velocity and acceleration, respectively. And $c$ and $k$ are damping and stiffness terms; and $u_{NL}$ is the nonlinear (hysteretic) restoring force term. $z_0$ and $w$ are amplitude and frequency of the applied force; and $t$ is time.

There is a method developed based on the fourier expansion of functions, so called "hybrid frequency–time domain method (HFT)" and there is this explanation of it at the paper attached in the link below. There is FFT and IFFT used in this method and it is explained in this part of the paper on 2nd page:

"... to use an initial guess function that is introduced in the time domain, obtaining its Fourier coefficients ~uN via the FFT. Likewise, the nonlinear forces are evaluated in the time domain and their Fourier coefficients ~gN obtained via FFT. Once a new approximation for ~uN is found, its time domain representation is obtained via the inverse fast Fourier transform in order to calculate the nonlinear forces, and then, this procedure repeats. Evaluating nonlinear forces in the time domain allows higher accuracy and simplicity in the formulation, which are key features of the HFT method."

http://pressurevesseltech.asmedigitalcollection.asme.org/article.aspx?articleID=2085509

Can anyone please explain what this means explanation above means? Why there is a need for FFT and then IFFT?

If there is initial guess of uN and gN, why can't we just use matlab's fourier fit for example:

fit(t, u,'fourier3')

Similarly for gN too? I have no idea about where IFFT be done!

I appreciate if anyone can help to understand it better.

Answer 1

It looks like I need to proceed a little bit about what I understand from this problem:

1. I understand the use of fft to calculate the coeff of displacement in fourier series from time domain to freq domain, which results in sines and cosines function approximation.

2. And using the IFFT to calculate the response in time domain makes sense too, but what happens then? Compare the response in freq domain or compare in time domain to the initial guess or what?

Answer 2

I understood the FFT and IFFT parts of my question. I am still not sure about very first part part: defining nonlinear forces in the time domain. If it is a harmonic motion and is nonlinear restoring forces are a hysteresis, and then I can directly convert this force-deformation over one period of the applied force (T=2*pi/w). Then I can obtain the fourier series approximation of both assumed force and deformation.

The problem with this paragraph in the paper is that it is very vaguely written. They are not clearly saying what is being repeated for example. They didn't explain how nonlinear forces are evaluated in time domain. Since I didn't get their procedure exactly, I still don't see any meaning to use IFFT too. That's probably even unnecessary.

If there is anyone who understand the procedure, and wanna try to explain?