Question 9 4.4: Estimating Damping and Stiffness Measurement of the free res...
Estimating Damping and Stiffness
Measurement of the free response of a certain system whose mass is 500 kg shows that after six cycles the amplitude of the displacement is 10% of the first amplitude. Also, the time for these six cycles to occur was measured to be 30 s. Estimate the system’s damping c and stiffness k.
From the given data, n = 6 and B_{7}/B_{1} = 0.1. Thus, from (9.4.14),
δ = \frac{1}{n} \ln \frac{B_{1}}{B_{n+1}} (9.4.14)
δ = \frac{1}{6} \ln \left( \frac{B_{1}}{B_{7}} \right) = \frac{1}{6} \ln 10 = \frac{2.302}{6} = 0.384
From (9.4.13),
ζ = \frac{δ}{\sqrt{4\pi^{2} + δ^{2}}} (9.4.13)
ζ = \frac{0.384}{\sqrt{4 \pi^{2} + (0.384)^{2}}} = 0.066
Because the measured time for six cycles was 30 s, the period P is P = 30/6 = 5 s. Thus ω_{d} = 2 \pi/P = 2 \pi/5. The damped frequency is related to the undamped frequency as
ω_{d} = \frac{2 \pi}{5} = ω_{n} \sqrt{1 − ζ^{2}} = ω_{n} \sqrt{1 − (0.066)^{2}}
Thus, ω_{n} = 1.26 and
k = mω^{2}_{n} = 500(1.26)^{2} = 794 N/m
The damping constant is calculated as follows:
c = 2ζ \sqrt{mk} = 2(0.066) \sqrt{500(794)} = 83.2 N · s/m