Question

A single degree of freedom system is undergoing free oscillation. The natural frequency and damping ratio of the system are $1$ rad/s and $0.01$ respectively. The reduction in peak amplitude over three cycles is __________% (rounded off to one decimal place).

Hint:

For lightly damped systems ($\zeta \ll 1$), \[ \delta \approx 2\pi\zeta \] This approximation greatly simplifies GATE numerical problems on vibrations.

Answer 1

Correct Answer: 17.2

Step 1: Recall logarithmic decrement.
For an underdamped single degree of freedom system, the logarithmic decrement \( \delta \) is given by: \[ \delta = \frac{2\pi \zeta}{\sqrt{1-\zeta^2}} \] where \( \zeta \) is the damping ratio.
Step 2: Substitute the given values.
Given: \[ \zeta = 0.01 \] \[ \delta = \frac{2\pi \times 0.01}{\sqrt{1-(0.01)^2}} \approx 2\pi \times 0.01 = 0.0628 \]
Step 3: Reduction in amplitude over three cycles.
The ratio of amplitudes after \( n \) cycles is: \[ \frac{x_n}{x_0} = e^{-n\delta} \] For three cycles: \[ \frac{x_3}{x_0} = e^{-3\delta} = e^{-3 \times 0.0628} = e^{-0.1884} \approx 0.828 \]
Step 4: Calculate percentage reduction in amplitude.
\[ \text{Reduction} = (1 - 0.828) \times 100 = 17.2 \]
Step 5: Conclusion.
The reduction in peak amplitude over three cycles is: \[ \boxed{17.2\%} \]