Small oscillations and engineering implications

A good mathematical challenge is to understand how the unusual dynamics identified in the introduction arise. Before we look at those problems we will first look at a simpler problem, namely, how does the bridge behave under small oscillations.

After the collapse of the bridge, an engineering analysis determined that the section of the bridge formed by the roadway and the large thick supports on its side, called stiffening-plate girders, did not absorb the turbulence of wind gusts; at the same time, the narrow, two-lane roadway gave the span a high degree of flexibility.

Mathematically, modeling the bridge is very complicated if you account for the position of the bridge as a function of space and time. However, if you are only interested in the amplitude of the vertical motion and horizontal deflection, then ordinary differential equations can be used to model the motion. This is what we will do. The figure to the right schematically shows the vertical deflection and the horizontal deflection. The simplest equation that is an appropriate model of the bridge assumes that the supports of the bridge act like Hookes law springs.

The model uses Newton's second law and assumes the cables supporting the road surface behave like Hooke's law springs. The ordinary differential equation that models how the bridge oscillates torsionally is

where c>0 is the coefficient of viscous damping divided by the mass, k is the spring constant of the cables divided by the mass, and f(t) is the force of the bridge due to the wind. Notice that it has been assumed that the viscous damping is like friction and its assumed to be proportional to x'.

The vertical displacement is also governed follows by a second order constant coefficient linear differential equation, and gravity acts as a constant forcing term.

The equation should look familiar to you. It also models the mass-spring-daphpot systems, the pendulum for small swings, and electrical circuits. Your assignment in this section of the project is to study the effects of c and forcing for a given k. To get the Maple worksheet, click here or to view a pdf version, click here.

The engineers recognized that in this equation, if c were large enough, the viscous damping would prevent oscillations from growing very large. To implement this in bridges, the engineers replaced the stiffening-plate girders with web trusses in future bridges. Notice the difference in construction below, the old bridge is on the left and the new bridge is on the right.

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