The small oscillation model that you studied in the first section of the assignment is not accurate at larger oscillations because it makes geometric approximations that are wrong for large oscillations.
When the geometry is properly accounted for, the new ordinary differential equation that models the bridge is
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. This equation should also look familiar. If the cosine term is ignored, this equation is the equation of a damped pendulum that is being forced. The extra cosine term accounts for the changing angle between the vertical suspending cables and the torsionally oscillating bridge surface. If you are interested in the derivation of the equation, go here.
Complimenting this equation is another ordinary differential equation for the vertical displacement
where g is the gravitational coefficient. This behavior of solutions of this equation should be familiar to you. We have studied this equation as a forced mass-spring-dashpot model. The complimentary solutions of this equation all tend to zero as time tends to infinity, and the particular solution is yp = 3g/k.
The torsional oscillation equation cannot be solved with analytic methods, and so numerical methods must be used. In the final section of your assignment, you will study numerical solutions of these equations. To get the Maple worksheet, click here or to see a pdf version, click here.
What you will find in doing this section of the project is that accounting for the geometry of the bridge in the model gives solutions that have large oscillations that persist for a long time.