A free vibration test is conducted on an aircraft landing gear strut modeled as an underdamped single-degree-of-freedom system. The displacement amplitude decreases from $x_0 = 12.0\text{ mm}$ to $x_5 = 1.5\text{ mm}$ after $n = 5$ complete successive cycles. The damping ratio $\zeta$ of the landing gear strut is closest to:

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### 1. Strategy & Theoretical Principles In underdamped free vibration, the logarithmic decrement $\delta$ is defined as the natural logarithm of the ratio of any two successive peak amplitudes separated by 1 period, or over $n$ periods: $$\delta = \frac{1}{n} \ln\left(\frac{x_0}{x_n}\right)$$ The exact relationship between logarithmic decrement $\delta$ and damping ratio $\zeta$ is: $$\delta = \frac{2\pi \zeta}{\sqrt{1 - \zeta^2}} \approx 2\pi \zeta \quad (\text{for } \zeta \ll 1)$$ ### 2. Analytical Formulation Given $n = 5$, $x_0 = 12.0\text{ mm}$, and $x_5 = 1.5\text{ mm}$: $$\frac{x_0}{x_5} = \frac{12.0}{1.5} = 8.0$$ $$\delta = \frac{1}{5} \ln(8.0) = \frac{1}{5} (2.07944) = 0.41589$$ ### 3. Step-by-Step Calculation From the exact damping relation: $$\frac{\delta^2}{4\pi^2} = \frac{\zeta^2}{1 - \zeta^2} \implies \zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}}$$ Calculate denominator: $$4\pi^2 = 4 \times (3.14159)^2 = 39.4784$$ $$\delta^2 = (0.41589)^2 = 0.17296$$ $$\sqrt{4\pi^2 + \delta^2} = \sqrt{39.4784 + 0.17296} = \sqrt{39.6514} \approx 6.2969$$ Evaluating $\zeta$: $$\zeta = \frac{0.41589}{6.2969} \approx 0.06605 \approx 0.066$$ Using the small damping approximation $\zeta \approx \frac{\delta}{2\pi} = \frac{0.41589}{6.28318} = 0.06619$, which differs by less than $0.2\%$. ### 4. Distractor Forensics - Forgetting to divide by $n=5$ gives $\zeta \approx 0.33$, a classic oversight. - Using $\log_{10}$ instead of natural logarithm yields $\zeta \approx 0.029$. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Always verify whether the amplitude decay is given across 1 cycle or across $n$ cycles. - **Standard Reference:** Rao, S.S., *Mechanical Vibrations*, 6th Edition, Pearson, Chapter 2 (Free Vibration of Single-Degree-of-Freedom Systems).

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