Forced Oscillations and Resonance — Question 2

Question 2

A displacement resonance needs sufficiently weak damping

For a driven oscillator with fixed force amplitude $F_0$, the steady displacement amplitude is $X=F_0/\sqrt{(k-m\Omega^2)^2+b^2\Omega^2}$. Define $\omega_0^2=k/m$ and $\beta=b/(2m)$.

1. Find the positive drive frequency that maximizes $X$, and state when it exists.

2. For $m=1\,\mathrm{kg}$, $k=25\,\mathrm{N/m}$ and $b=4\,\mathrm{kg/s}$, compare this frequency with $\omega_0$ and the free damped frequency. What if $b^2\ge2mk$?

Question 2 - Solution

A displacement resonance needs sufficiently weak damping

(a) Derivation and calculation

Maximizing $X$ is equivalent to minimizing$$D=(k-m\Omega^2)^2+b^2\Omega^2.$$$$D^{\prime}=2\Omega[-2m(k-m\Omega^2)+b^2].$$$$\boxed{\Omega_r=\sqrt{\frac km-\frac{b^2}{2m^2}}=\sqrt{\omega_0^2-2\beta^2}}.$$$$\boxed{b^2<2mk}\quad\text{is required for a positive-frequency peak}.$$

(b) Result and interpretation

$$m=1,\ k=25,\ b=4:\quad\beta=2,\quad\boxed{\Omega_r=\sqrt{17}=4.12311\,\mathrm{rad/s}}.$$$$\omega_d=\sqrt{21}=4.58258\,\mathrm{rad/s},\qquad\omega_0=5.00\,\mathrm{rad/s}.$$$$b^2\ge2mk\Rightarrow D^{\prime}>0\ (\Omega>0),\quad\boxed{X_{\max}=X(0)=F_0/k}.$$

Physical check

Displacement resonance, free damped oscillation, and undamped natural frequency are distinct measurements.