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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?
- the logarithmic decrement is 0.5
- the angular frequency of oscillation is exactly sqrt(161/4) rad/s
- the time period of oscillation is exactly 1 s
- 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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