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: Use the expression for logarithmic decrement. 
For an underdamped single degree of freedom system, the logarithmic decrement is related to the damping ratio by:

\[ \delta = \frac{2\pi \zeta}{\sqrt{1-\zeta^2}} \]

Here, $\zeta$ represents the damping ratio of the system.

Step 2: Insert the given damping ratio.
The damping ratio is specified as:

\[ \zeta = 0.01 \]

Since the damping is very small, the denominator is approximately equal to 1, giving:

\[ \delta \approx 2\pi \times 0.01 = 0.0628 \]

Step 3: Determine amplitude change after three cycles.
The ratio of the amplitude after $n$ cycles to the initial amplitude is given by:

\[ \frac{x_n}{x_0} = e^{-n\delta} \]

For three successive cycles:

\[ \frac{x_3}{x_0} = e^{-3 \times 0.0628} = e^{-0.1884} \approx 0.828 \]

Step 4: Compute the percentage decrease in amplitude.
The percentage reduction in peak amplitude is:

\[ (1 - 0.828) \times 100 = 17.2\% \]

Step 5: Final result.
 

\[ \boxed{17.2\%} \]