Measuring damping ratio (20pts):
ak-m-c system is under harmonic excitation of frequency \omega , it will reach a steady
state. We can measure the amplitude of the steady state displacement at different
frequency to get the relation between the two x(\omega ). Then we plot the steady state
amplitude x versus force frequency \omega . We find the curve has a peak at \omega _(0)=(50)/(s), and
near the peak the curve can be fitted by a simplified equation: x(\omega )=10(1)/(\sqrt(1+400(1-(\omega )/(\omega _(0)))^(2)))cm.
Determine the half power bandwidth and estimate \zeta . (5pts)
cx(\omega ) has a peak of 1 cm at certain
frequency. The amplitude of the force is fixed at 1 N and spring constant is 1000(N)/(m).
Estimate \zeta \zeta is smallF_(0). We find that: the maximum displacement is 9 cm ; the displacement falls within 5cm+-
0.1 cm after 5 s ; and approaches 5 cm when we wait for long enough time. The vibrating mass
in the system m=1kg. Determine: Damping ratio, natural frequency \zeta , and spring constant kM_(P)=(x_(max)-x_(ss))/(x_(ss)),\zeta =-(ln(M_(p)))/(\sqrt(\pi ^(2)+ln^(2)M_(p))), settling time t_(s)=