Question

An under-damped single-degree-of-freedom system is freely oscillating with an initial amplitude \(A\). The initial velocity is zero. After five cycles of oscillation, the amplitude reduces to \(A/2\). Then the damping ratio of the system is ________________ % (rounded off to one decimal place) of critical damping.

Hint:

Amplitude decay over several cycles is an efficient way to estimate damping ratio without requiring natural frequency measurement.

Answer 1

Correct Answer: 2.1

For an under-damped SDOF system: \[ \frac{A_5}{A_0} = e^{-\zeta \omega_n t} \] But after \(n\) cycles: \[ t = nT_d = n \frac{2\pi}{\omega_d} \] and \[ \omega_d = \omega_n \sqrt{1 - \zeta^2} \] The decay ratio becomes: \[ \frac{A_5}{A_0} = e^{ -\frac{2\pi n \zeta}{\sqrt{1-\zeta^2}} } \] Given: \[ \frac{A_5}{A_0} = \frac{1}{2}, \qquad n = 5 \] \[ \ln(2) = \frac{10\pi \zeta}{\sqrt{1-\zeta^2}} \] Solve for \(\zeta\): \[ \zeta \approx 0.022 \] Thus, \[ \zeta_{%} = 100\zeta \approx 2.2% \] This lies in the expected range: \[ \boxed{2.1% \text{ to } 2.3%} \]
Final Answer: 2.1–2.3 %