This problem investigates the number of cycles \(n\) re-

quired for a structural oscillation amplitude to decay to \(x \%\) of its maximum.

(a) What is the expression for the logarithmic decrement in terms of \(n\) and \(x\) ?

(b) Solve for \(n\) in terms of \(\zeta\) and \(x\).

(c) Plot \(n\) as a function of \(\zeta\) for (i) \(x=70 \%\), (ii) \(x=50 \%\), (iii) \(x=20 \%\) using the exact formula, and then using the approximate formula for \(\zeta=0.1\) and \(\zeta=0.5\).

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Table 3.1: Cases for tmax in Problem 9. Case (0) (in) (0) (in/s) w (rad/s) 1 2 0 5 0.1 2 5 0 0.1 2 3 5 5 0.1 2 4 5 5 0.5 2 5 5 5 1.1 2 6 5 7 5 5 0.1 4 5 0.1 10