Question

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

Hint:

Use logarithmic decrement \( \delta = 2\pi\zeta \) to estimate damping effects. For a system oscillating over \(n\) cycles: \[ A_n = A_0 e^{-n\delta} \Rightarrow {Reduction} = \left(1 - \frac{A_n}{A_0} \right) \times 100 \] Always apply exponential decay logic for amplitude in damped vibrations.

Answer 1

Step 1: Use the logarithmic decrement formula. The logarithmic decrement \( \delta \) for lightly damped systems is: \[ \delta = 2\pi \zeta = 2\pi (0.01) = 0.0628 \] Step 2: Compute amplitude ratio after 3 cycles. Let \(A_0\) be the initial amplitude. Then the amplitude after 3 cycles is: \[ A_3 = A_0 e^{-3\delta} = A_0 e^{-3 \times 0.0628} = A_0 e^{-0.1884} \approx A_0 \times 0.84 \] Step 3: Find percentage reduction. \[ {Reduction} = \left(1 - \frac{A_3}{A_0} \right) \times 100 = (1 - 0.84) \times 100 = 16% \]