In a physics book I came across the following solution to a differential equation modeling forced, damped, oscillating motion: $$x(t)=A\cos(\omega t+\phi)$$ where $$A=\frac{F_0}{\sqrt{m^2\left(\omega^2-\omega_0^2\right)^2+b^2\omega^2}}$$ The differential equation which prompted this solution is $$-kx-b\dot{x}+F_0\sin(\omega t)=m\ddot{x}$$
This doesn't feel quite intuitively right, however. When the angular frequency $\omega$ of the thing which is causing the forced oscillating is equal to the natural angular frequency $\omega_0$ of the system it is causing an oscillation in, this seems to work intuitively. But when this isn't the case, it seems like $x(t)$ cannot be modeled as a single trig function, but rather a combination of multiple.
For example, in this video at 1:26, the frequency $\omega$ of the vibration generator is set to 2Hz, which is different than the natural frequency $\omega_0$ of the system (this turns out to be $\approx3.33$Hz) and the motion of the mass does not appear to resemble a single trig function, but rather multiple.
Here is my best attempt to graph the seeming motion at 1:26:
$$f(x)=\cos(x)-\cos(2x)$$
My question: Is the formula incorrect/a simplified version? If so, what is the correct one? If not, where am I going wrong here?
Best Answer
The solution of a forced and damped harmonic oscillator $$m\ddot{x}=-kx - b\dot{x} +F_0 \cos\omega_{\text{d}} t$$ is actually the sum of steady-state part $x_{\text{s}}$ and a transient part $x_{\text{t}}$ $$x = x_{\text{s}} + x_{\text{t}}$$ where $$x_{\text{s}} \sim A\cos(\omega t+\phi) \\ x_{\text{t}} \sim A' e^{-\gamma t} \sin (\omega' t+\phi')$$
The transient part depends on the initial conditions and both parts are present at the start. As time goes on, the transient part decays exponentially, leaving only the steady-state solution, which is the one you quoted. The full equations can be found here. Therefore, at the beginning, the motion will not be a single pure trig function but rather a sum of two trig functions.