# Force-dependency of frequency response of driven harmonic oscillator with damping

For a driven harmonic oscillator with damping of the form $$$$\ddot{x} + 2\xi\omega_0\dot{x} + \omega_0^2x = \frac{F_0}{m}cos(wt)$$$$ with damping ratio $$\xi$$ and natural resonance frequency $$\omega_0$$, the resonance frequency "for a given $$F_0$$" is $$$$w_r = w_0\sqrt{1-2\xi^2}$$$$ according to Wikipedia. The way this is phrased suggests some dependency on the applied Force $$F_0$$, but I can't gather what it might be from the presented maths. Alternatively, it is stated that for this type of oscillator, $$$$x(t) = \frac{F_0}{mZ\omega}sin(wt)$$$$ i.e. the max. amplitude $$x_0$$ depends linearly on $$F_0$$. What, however, is the relationship between $$x_0$$ and $$w_r$$, the resonance frequency?

Background

I conducted a student experiment with an oscillating string driven via an electro magnet controlled by a Voltage $$V_{drive}$$. The oscillator is considered to be governed by Duffing-type equations of motion, $$$$\ddot{x} + \Gamma{x}+ \eta\dot{x}x^2 + \alpha x^3 + \Omega^2(t)x = \frac{F_0}{M}cos(wt)$$$$ which for large oscillation amplitudes has frequency response $$$$w_{res} = w_0 + \frac{3}{8}\frac{\alpha}{M\omega_0}x_0^2.$$$$ I've recorded the frequency response for different driving voltages (i.e. different oscillation amplitudes), and at low amplitudes, I'm not getting the quadratic trend stated above, which is expected, since this relationship only holds when large amplitudes make the $$x^3$$-term dominate. However, I did wonder what this frequency response would look like at low amplitudes, when nonlinear terms can be dropped, so I can come up with a good explanation for what I'm seeing at low amplitudes.

"For a particular driving frequency called the resonance, or resonant frequency $$\omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}$$, the amplitude (for a given $$F_{0}$$) is maximal."
The intended meaning is not that the resonant frequency depends on the loading but that of the ways to load the system, namely, (1) at different loading magnitudes $$F_0$$ and/or (2) different frequencies $$\omega$$, only the latter is being considered.
In other words, one could also increase the amplitude by increasing $$F_0$$, but that's not being considered here: $$F_0$$ is taken as constant.
Thus, we could write the amplitude for a given $$\boldsymbol{F_0}$$ as $$\sim\frac{1}{mZ_m}$$, where the impedance $$Z_m$$ goes to zero at the resonant frequency as the damping ratio $$\zeta$$ is reduced to zero: