Question 2.7: A displacement–time plot for a free vibration analysis of a ...
A displacement–time plot for a free vibration analysis of a SDOF system gives a displacement of 7.5 mm at time t_{1} and 1 mm after 7 cycles. Determine logarithmic decrement and damping ratio.
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Logarithmic decrement is given by
\pmb{\delta} =\frac{1}{j} \log _{e}\frac{x_{i}}{x_{i+j}} =2j \pi \xi \frac{\omega }{\omega _{D}}
\pmb{\delta} = \frac{1}{7} \log _{e}\frac{7.5}{1.0} =0.288
∴ Damping ratio x = δ/2π = 0.288/2 × π = 0.046 or 4.6%
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