What is the deflection in the center of the string?

Question

I took a strand of piano wire, 30 ft in length, and I stretched it along a building in the veritcal direction. It was pinned at both end such that it was free to rotate, but not to translate. The wind came howling by one night blowing with gusts around 60 mph. During that time the piano string bowed in the direction of the wind due to aerodynamic forces. I couldn't get up to the center to measure the actual deflection, but I wanted to get an approximation for it. My question is, how far off the center line was maximum deflection point of the wire?

Answer 1

No you didn't, but very interesting multi-disciplinary problem anyway!

Start by assuming the building is rigid. Neglect contributions to wire stretch of the wire weight. Assume sea level STP air properties. Temperature of wire is constant.

Calculate aerodynamic drag from the (assumed uniform over wire length) wind speed on a cylinder to determine the force/unit length.

Use the catenary (chain) equations to determine the tension and deflection, using the drag in lieu of weight. The solution should be more iterative than usual, as you should adjust the length of the wire for the tension using simple mechanics {E = stress/strain; E = (F/A)/(DeltaL/L)}.

Answer 2

suppose x-axis is the central line; if you can measure a deflection at say x=1 foot, then you may calculate amplitude of the wire using one of the equations as a good approximation: 1)chain function, 2)cosine, 3) or just parabola y=k*x^2;