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A mass-spring-damper system has k = 161 N/m, m = 4 kg, and viscous damping coefficient b. The oscillation amplitude is found to fall to 1/e of its initial value after two complete cycles. Which of the following is correct for this (lightly damped) system?

1. the logarithmic decrement is 0.5
2. the angular frequency of oscillation is exactly sqrt(161/4) rad/s
3. the time period of oscillation is exactly 1 s
4. b = 4 kg/s

Correct answer: the logarithmic decrement is 0.5

Solution

Logarithmic decrement per cycle delta = (1/n) ln(A0/An). Here ln(A0/An) = ln(e/1) = 1 over 2 cycles, so delta = 1/2 = 0.5. The natural angular frequency sqrt(k/m) is only approximate for damped motion, the period is not exactly 1 s, and b is not exactly 4; only the log decrement statement is exact.

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