We are learning about oscillations and second-order dynamic systems. We observe a crane at a construction site on campus and notice that as soon as the crane operator releases the load, the whole crane oscillates back and forth several times as the vibration damps out – just like the second-order dynamic systems we have been studying in class. A cell phone video clip is used to analyze this. Using some video processing software, we measure the following data from his video clip: With the load in place, the initial deflection of the crane (relative to its zero-load resting state) is 8.50 degrees at time t = 0. At t = 13.2 s, exactly three full periods later, the peak deflection is 0.50 degrees.
[The final deflection is zero degrees after the load has been released and the oscillations have died out.]
a) Calculate the damped (actual) natural frequency (in Hz).
b) Use the log-decrement method to calculate the damping ratio, showing all your work.
c) If there were no damping (no friction in the system), at what frequency (in Hz) would the crane oscillate?