Chapter 5, Problem 32RE

Galloping Gertie Bridges are good examples of vibrating mechanical systems that are constantly subjected to external forces, from cars driving on them, water pushing against their foundation, and wind blowing across their superstructure. On November 7, 1940, only four months after its grand opening, the Tacoma Narrows Suspension Bridge at Puget Sound in the state of Washington collapsed during a windstorm. See Figure 5.R.6. The crash came as no surprise since “Galloping Gertie,” as the bridge was called by local residents, was famous for a vertical undulating motion of its roadway which gave many motorists a very exciting crossing. For many years it was conjectured that the poorly designed superstructure of the bridge caused the wind blowing across it to swirl in a periodic manner and that when the frequency of this force approached the natural frequency of the bridge, large upheavals of the lightweight roadway resulted. In other words, it was thought the bridge was a victim of mechanical resonance. But as we have seen on page 207, resonance is a linear phenomenon which can occur only in the complete absence of damping. In recent years the resonance theory has been replaced with mathematical models that can describe large oscillations even in the presence of damping. In his project, The Collapse of the Tacoma Narrows Suspension Bridge, that appeared in the last edition of this text, Gilbert N. Lewis examines simple piecewise-defined models describing the driven oscillations of a mass (a portion of the roadway) attached to a spring (a vertical support cable) for which the amplitudes of oscillation increase over time. In this problem you are guided through the solution of one of the models discussed in that project.

The differential equation with a piecewise-defined restoring force,

$\frac{d^{2}x}{d{t}^2}+F(x)=\sin 4t,F(x)=\{\begin{array}{c}4x,x\ge 0\\ x,x<0\end{array}$

is a model for the displacement x(t) of a unit mass in a driven spring/mass system. As in Section 5.1 we assume that the motion takes place along a vertical line, the equilibrium position is x = 0, and the positive direction is downward. The restoring force acts opposite to the direction of motion: a restoring force 4x when the mass is below (x > 0) the equilibrium position and a restoring force x when the mass is above (x < 0) the equilibrium position.

1. (a) Solve the initial-value problem

   $\frac{d^{2}x}{d{t}^2}+4x=\sin 4t,x(0)=0,x\text{'}(0)=v_0>0.$ (2)

   The initial conditions indicate that the mass is released from the equilibrium position with a downward velocity. Use the solution to determine the first time t₁ > 0 when x(t) = 0, that is, the first time that the mass returns to the equilibrium position after release. The solution of (2) is defined on the interval [0, t₁]. [Hint: The double-angle formula sin 4t = 2 sin 2t cos 2t will be helpful.]

2. (b) For a time interval on which t > t₁ the mass is above the equilibrium position and so we must now solve the new differential equation

   $\frac{d^{2}x}{d{t}^2}+x=\sin 4t$ (3)

   One initial condition is x(t₁) = 0. Find x'(t₁) using the solution of (2) in part (a). Find a solution of equation (3) subject to these new initial conditions. Use the solution to determine the second time t₂ > t₁ when x(t) = 0. The solution of (3) is defined on the interval [t₁, t₂]. [Hint: Use the double-angle formula for the sine function twice.]

3. (c) Construct and solve another initial-value problem to find x(t) defined on the interval [t₂, t₃], where t₃ > t₂ is the third time when x(t) = 0.
4. (d) Construct and solve another initial-value problem to find x(t) defined on the interval [t₃, t₄], where t₄ > t₃ is the fourth time when x(t) = 0.
5. (e) Because of the assumption that v₀ > 0 one down-up cycle of the mass is completed on the intervals [0, t₂], [t₂, t₄], [t₄, t₆], and so on. Explain why the amplitudes of oscillation of the mass must increase over time. [Hint: Examine the velocity of the mass at the beginning of each cycle.]
6. (f) Assume in (2) that v₀ = 0.01. Use the four solutions on the intervals in parts (a), (b), (c), and (d) to construct a continuous piecewise-defined function x(t) defined on the interval [0, t₄]. Use a graphing utility to obtain a graph of x(t) on [0, t₄].

FIGURE 5.R.6 Collapse of the Tacoma Narrows Suspension Bridge