Trying to think of a new topic about which to blog I somehow remembered seeing this video and thought it only appropriate after finishing the unit on oskillaiton. Inspired by the Tacoma Bridge collapse (1st video) Shawn Frayne designed a small wind generator that uses an oskillating tensioned belt to generate energy from the wind (2nd video).
In 10mph of wind the design produced an output of about 40 milliwatts (true facts) and assuming the system is perfectly efficient, has an amplitude of about 2.5cm, and a belt with mass 4g (these are just assumptions) we may be able to find the frequency of the oskillating belt.
Let's gather some formulas:
We have power and P = W/t
W = ΔKE
When the belt passes through equilibrium Etotal = KE = (1/2)mv^2.
vmax (which occurs at equilibrium) = Aω
ω = 1/T
T = second/cycle
Now if we put them together, focusing on the time t it takes the belt to travel from the amplitude to equilibrium (T/4) we can say that:
P = [(1/2)m(Aω)^2 - 0] / (T/4)
P = 2m(A^2)(1/T)^3
T^3 = (2mA^2)/P
T^3 = 2(0.004kg)(0.025m)^2/(0.040W)
T^3 = 0.000125s^3 (It's always a good sign when your units work)
T = 0.05s
Furthermore, with f = 2π/T we get a frequency of about 126 Hz.
Considering that much of the energy of the belt is lost we can assume that to produce the same milliwatts it oskillates much faster meaning a smaller period and a faster frequency.