Question

A 5.0 kg object oscillates on a spring with a spring constant of 180 N/m. The damping constant is 0.20 kg/s. The system is driven by a sinusoidal force of maximum value 50 N and an angular frequency of 20 rad/s.
i) Determine the amplitude and phase lag of the oscillations.
ii) If the driving frequency is varied, at what frequency will resonance occur?
iii) Determine the maximum amplitude the system could possibly have.

Answer 1

$\omega_0=\sqrt{\frac{k}{m}}=\sqrt{\frac{180}{5}}=6(rad/s)$

(i) $A=\frac{F_0}{m\sqrt{(\omega_0^2-\omega^2)^2+4\beta^2\omega^2}}=\frac{50}{5\cdot\sqrt{(6^2-20^2)^2+4\cdot0.2^2\cdot20^2}}=0.027(m)$

$\tan\phi=\frac{2\beta\omega}{\omega^2-\omega_0^2}=\frac{2\cdot0.2\cdot20}{20^2-6^2}=0.022\to$ $\phi=1.26°$

(ii) $\omega_{res}=\sqrt{\omega_0^2-2\beta^2}=\sqrt{6^2-2\cdot0.2^2}=5.99(rad/s)$

(iii) $A_{res}=\frac{F_0/m}{2\beta\sqrt{\omega_0^2-\beta^2}}=\frac{50/5}{2\cdot0.2\sqrt{6^2-0.2^2}}=4.17(m)$