In under-damped vibrating system, if x₁ and x₂ are successive values of the amplitude on the same side of the mean position, then the logarithmic decrement is equal to

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HPCL Engineer Mechanical 12 Aug 2021 Official Paper (Shift 1)

Answer 1

Explanation:

Logarithmic decrement:

17.12.218.218

Logarithmic decrement is defined as the natural logarithm of the ratio of any two successive amplitudes on the same side of the mean line.

\(\delta = \ln \frac{{{x_0}}}{{{x_1}}} = \ln \frac{{{x_1}}}{{{x_2}}}\)

If the system executes n cycles:

\(\delta = \frac{1}{n}\ln \frac{{{x_o}}}{{{x_n}}} = \frac{{2\pi \xi }}{{\sqrt {1 - {\xi ^2}} \;}}\)

\(\begin{array}{l} \delta = \frac{1}{n}\ln \frac{{{x_o}}}{{{x_{n }}}} \end{array}\)