x'' + c x' + k sin(x) cos(x) = f(t)
When x is small, cos(x) is nearly one and sin(x) is nearly x so this last equation is similar to the small oscillation equation. The extra cosine term accounts for the changing angle between the vertical suspending cables and the torsionally oscillating bridge surface. The derivation of the equation appears here.
The ordinary differential equation for the vertical displacement y(t) is
where g is the gravitational coefficient. The homogeneous solution is yh = e-ct/2(A cos wt + B sin wt), 2w=(c2-4k/3)1/2 where A and B are constants. This solution tends to zero as t tends to infinity. A particular solution is yp = 3g/k. The general solution is the sum of the two solutions: y(t) = yh(t) + yp(t).
Including the geometry of the bridge in the model gives solutions x(t) with large amplitude oscillations that persist for a long time. The torsional oscillation equation has no known explicit solution, and so numerical methods must be used. In the final section of your assignment, you will study numerical solutions of these equations.
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