The mass of a single-degree damped vibrating system is 7.5 kg and it makes 24 free oscillation in 14 s when disturbed from its equilibrium position. The amplitude of vibration reduces to 0.25 of its initial value after five oscillations. Then the logarithmic decrement will be

Concept:

Logarithmic decrement is the log of successive decrement ratio:

Logarithmic decrement 

\(\delta =\ln\left| {\frac{{{X_{n}}}}{{{X_{n+1 }}}}} \right|= \frac{1}{n}\ln\left| {\frac{{{X_{1}}}}{{{X_{n+1 }}}}} \right|\)

Where n is the number of oscillations made to reduce the amplitude from X₁ to X_(n+1)

Calculation:

Given X₅ = 0.25 X₀

\(\frac{{{X_5}}}{{{X_0}}} = 0.25\;\)

\(\frac{{{X_5}}}{{{X_0}}} = \frac{X_5}{X_4}\times\frac{X_4}{X_3}\times\frac{X_3}{X_2}\times\frac{X_2}{X_1}\times\frac{X_1}{X_0}\)

Logarithmic decrement 

\(\delta = \frac{1}{5}\ln\left| {\frac{{{X_0}}}{{{X_5}}}} \right|\)

\(\delta = \frac{1}{5}\ln\left| 4 \right|\)