In moderate winds, high-tension power lines sometimes \"sing\" or hum as a resul
ID: 2003063 • Letter: I
Question
In moderate winds, high-tension power lines sometimes "sing" or hum as a result of transverse waves on the lines that winds can generate. The cables that carry high-voltage electricity are made of aluminum with a steel core added for strength. The cross-sectional areas of the steel and aluminum in the line are 39 mm2 and 220 mm2, respectively. The cable is 1.8 cm in diameter and is under a tension of 4.4 104 N. Typically, the towers are placed about 0.19 km apart. (See the figure below.)
(a) What is the fundamental frequency of such a power line? (The densities of steel and aluminum are given in this table.) Hz
(b) The wind can induce vibrations when flowing around a cylinder. The frequency of the vibrations induced this way is modeled well by the equation fwind = 1 5 vwind d where vwind is the speed of the wind and d is the diameter of the cylinder. The wind will induce vibration this way only if the frequency of the induced oscillation matches a natural frequency of the power line. It is found that the power line hums when the wind blows at a steady 6.8 m/s. What harmonic of the line is this induced vibration exciting?
(c) If the line is humming at the 153rd harmonic, how fast is the wind blowing?
Explanation / Answer
(a)
the linear mass density of the cable is,
u = usteel + ualuminum =[Area*density] + [Area*density]
= [39*10-6 m2 * 7800] + [220*10-6 m2 * 2700]
= 0.8982 kg/m3
thus, the frequency is,
f = [1 / 2L] sqrt[T/u] = [1 /2*190] sqrt[4.4x104 / 0.8982 ] = 0.582 Hz
(b)
the frequency of vibration is,
f' = [1/5][v/d] = [1 / 5][6.8 / (0.018)] =75.56 Hz
the number of harmonic is,
n = f'/f = 75.56 / 0.582 = 129.8 = 130
(c)
the speed of wind is,
v = 5nfd = 5*153*0.582*0.018 = 8.01 m/s
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