The equation of motion for damped forced oscillation in a single-degree-of-freedom system is, if the external force is f(t):
mx'' + cx' + kx = f(t) (' '' indicates the first and second derivatives)
So, what kinds of external forces do we need to consider?
As for external forces:
(a) Repetitive forces that are applied repeatedly and steadily, such as motors.
(b) an instantaneous impact force, such as when being struck with a hammer.
These can all be represented using Fourier series.
f(t)＝a0＋a1coswt＋b1sinwt＋a2cos2wt＋b2sin2wt…
=f0+f1cos(wt+Φ1)+f2cos(2wt+Φ2)...
|f1|^2 = a1^2 + b1^2, tanΦ1 = a1/b1 (^ indicates exponentiation)
Solving the equations of motion involves describing the behavior of x mathematically. In solving this, we consider that x(t) is related to (synchronous with) f(t) once the initial transient state is over and a constant state is reached. Therefore, x(t) also takes the fundamental frequency w of f(t) and expresses it as a Fourier series;
X(t)＝ p0＋p1coswt＋q1sinwt＋p2cos2wt＋q2sin2wt…
＝X0＋X1cos（wt＋θ1）＋X2cos（2wt＋θ2）＋…
This is what is expected. Comparing the equations, we can see that f0, f1, f2… correspond to X0, X1, X2… respectively. As a representative of f(t), let the frequency of the nth term be v, and take a new Fcos(vt) as the steady-state response for the excitation force of a single harmonic oscillation;
mx''＋cx'＋kx＝Fcos(vt)
The solution;
x＝Acosvt＋Bsinvt＝Xcos(vt＋γ)
For now, the basic example of calculating X/F is introduced in the literature, but it seems like it's already a formula.
The graphs showing how vt is varied (wt, 2wt, etc.) are then used to explain the characteristics.
Using an FFT analyzer to detect forced displacement, force, and response vibration (displacement and acceleration α) with sensors, and experimentally measuring and graphing α/F and X/F with the FFT analyzer, is essentially the same as experimentally finding the solution.
The FFT analyzer
X-axis: frequency w, 2w…, Y-axis: X1/f1, X2/f2… and θ1, θ2…
This is how it is displayed. When there are excitation forces for multiple harmonic oscillations to which f1 and f2 are applied simultaneously, we can simply add up the responses of the excitation forces f1 and f2 of a single harmonic oscillation, and we can see that we can anticipate the response of a complex signal.
x＝X1cos（wt＋θ1）＋X2cos（2wt＋θ2）
Just understanding the concept of how to solve the equations of motion makes me feel like I can understand Fourier series, FFT analyzers that use them as their foundational theory, and how to interpret the data. Please refer to Ono Sokki Technical Report, "About FFT Analyzers."
(Excerpt from the email newsletter issued on June 21, 2002)
