Discussion
The logarithmic decrement measured for a viscously damped single degree of freedom system is 0.125. The value of the damping factor in % is closest to
Answer: D
The damping factor is closest to 2.0%.
Logarithmic decrement \(\delta\) and damping factor \(\zeta\) are related by
\[ \begin{aligned} \delta &= \frac{2\pi\zeta}{\sqrt{1 - \zeta^{2}}} \\ \zeta &\approx \frac{\delta}{2\pi} = \frac{0.125}{6.283} \\ &= 0.0199 \approx 2\% \end{aligned} \]
The exact form \(\zeta = \delta/\sqrt{4\pi^{2} + \delta^{2}}\) gives the same 1.99%. 1.0% is what dividing by \(4\pi\) instead of \(2\pi\) gives, and 0.5% and 1.5% are smaller and larger than any reading of the formula allows.
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