Forced Oscillations and Resonance — Question 5

Question 5

Static response and phase identify all three oscillator parameters

An unknown linear oscillator has fixed $m,k,b>0$. A constant force $2.0\,\mathrm{N}$ displaces its equilibrium by $.040\,\mathrm{m}$. With a sinusoidal force of amplitude $2.0\,\mathrm{N}$, the steady displacement lags by $90^\circ$ at $\Omega=10\,\mathrm{rad/s}$ and has amplitude $.10\,\mathrm{m}$ there.

1. Infer $k,m,b$ from these three observations.

2. Predict the frequency of maximum displacement. Explain why identifying the $90^\circ$ point is different from searching for the largest displacement.

Question 5 - Solution

Static response and phase identify all three oscillator parameters

(a) Derivation and calculation

Static equilibrium determines stiffness:$$\boxed{k=F/x_e=2.00/.0400=50.0\,\mathrm{N/m}}.$$At a $90^\circ$ displacement lag,$$k-m\Omega^2=0\Rightarrow\boxed{m=k/\Omega^2=50/100=.500\,\mathrm{kg}}.$$$$X=\frac{F_0}{b\Omega}\Rightarrow\boxed{b=\frac{2.00}{10(.100)}=2.00\,\mathrm{kg/s}}.$$

(b) Result and interpretation

$$\beta=b/(2m)=2.00\,\mathrm{s^{-1}},\quad\omega_0=10.0\,\mathrm{rad/s}.$$$$\boxed{\Omega_r=\sqrt{\omega_0^2-2\beta^2}=\sqrt{92}=9.59166\,\mathrm{rad/s}}.$$The phase condition locates $\omega_0$ while the displacement maximum is shifted downward by damping.

Physical check

The inference assumes settled response and frequency-independent linear $m,k,b$.