In this chapter we study special non autonomous second-order equations and non autonomous systems. Sections 5.1--5.3 cover the forced harmonic oscillator and resonance. The approach in Sections 5.1 and 5.2 is fairly traditional, relying on "guess and test" (the method of undetermined coefficients) to obtain general solutions of nonhomogeneous second-order equations. These two sections suffice for a standard treatment of forced oscillators. Section 5.3 gives a more qualitative description of forcing and resonance.

Sections 5.4 and 5.5 are independent of each other and of the rest of the text. They can be covered in any order and at anytime after chapter 3.

Section 5.4 discusses the collapse of the Tacoma Narrows Bridge. Recent work by A. C. Lazer and P. J. McKenna indicates that this sites the studies at the time of the collapse which concluded that resonance due to the forcing caused by vortex shedding was {\bf not} the cause of the collapse. (It is extremely unlikely that vortex shedding could have occurred at sufficiently regular intervals for traditional resonance to have been the cause.) Their work indicates that a more likely explanation is that the collapse was due to nonlinear effects occurring during large oscillations.

In Section 5.5 we discuss other periodically forced systems and introduce return maps. This section is long and considerably more challenging than the previous sections.

5.1 Forced Harmonic Oscillators

In this section we introduce the forced harmonic oscillator equations and outline the Method of Undetermined Coefficients. Only exponential forcing terms are used in the text. Exponential and polynomial forcing are covered in the exercises. Trigonometric forcing is covered in Section 5.2. Our approach here is traditional except that we are honest about this being a guess and test technique. In finding the general solution of the homogeneous equation, students will probably revert to using systems. This is another good opportunity to reinforce the relationship between systems and second-order equations.

A somewhat more difficult (but more satisfying) approach to second- guessing of particular solutions can be made by choosing the second guess equal to

is the solution of the homogeneous equation, then solving for z(t).

The section ends with an RLC circuit example. The coefficients of this example are realistic values, so the time scales on Figures 5.2 and 5.3 are typical.

Comments on selected exercises

Exercise 1 applies the Method of undetermined coefficients to first-order, constant coefficient equations. Many of the applications from Section 1.8 can be solved using this method.

Exercise 2 emphasizes that the Extended Linearity Principle for nonhomogeneous equations is different from the Linearity Principle for homogeneous equations.

Exercises 3--15 are standard nonhomogeneous equations with exponential forcing (as in the text).

Exercises 16--27 extend the Method of Undetermined Coefficients to polynomial forcing terms and to sums of polynomials and exponentials.

5.2 Sinusoidal Forcing and Resonance

Harmonic oscillators with periodic (i.e., trigonometric) forcing is covered in this section. The approach is again fairly standard except for the fact that complex exponentials are used in the guess and test step of the Method of Undetermined Coefficients. This simplifies the guessing procedure and reinforces the use of complex numbers begun in Chapter 3. It also works well with the notion of phasors which are standard in Electric Circuit Theory.

Resonance is introduced by example and the qualitative analysis is done from the formulas. That resonant effects require fairly precise tuning of the forcing frequency (Figure 5.9) should be emphasized (see Section 5.4).

Comments on selected exercises

Exercises 1--6 are standard Method of Undetermined Coefficients problems.

Exercise 7 touches on the notions of natural response and forced response. These are covered in more detail in Section 5.3.

Exercises 8 and 9 are essay problems on resonance. We believe the story in Exercise 9 is true, but can't remember the name of the University. If you happen to know more about this story, we would love to hear about it. Send comments to odes@math.bu.edu

Exercises 10--12 discuss "phasors", a notation and technique which is standard in electrical engineering.

5.3 Qualitative Analysis of the Forced Harmonic Oscillator

In this section we give a qualitative analysis of the forced harmonic oscillator equation by converting it to a system and studying the change in the energy. Familiarity with at least the beginning of Section 4.2 is almost a requirement for understanding this section. This section also helps to set up Section 5.5 on return maps. It can be safely skipped (in particular, it is independent of Section 5.4).

Comments on selected exercises

Exercises 1--4 are essay problems on the qualitative effect of resonance while Exercises 5--7 involve computation in the change in the energy for a forced harmonic oscillator, while exercises 5 to 7 involve computation in the change in the energy for a forced harmonic oscillator. In Exercise 7, the energy function should be

5.4 The Tacoma Narrows Bridge

The official investigation into the collapse of the Tacoma Narrows bridge concluded that resonant forcing was not the cause of the collapse. Periodic forcing caused by the shedding of vortices during high winds could not reasonably be expected to maintain a sufficiently precise period for a sufficient time to cause the collapse.

In this section, we discuss recent work of A.C. Lazer and P.J. McKenna (see SIAM Reviews, Vol. 32, No. 4, 1990, p. 537-578 and the references cited there). They model the motion of light flexible suspension bridges and observe that there is a significant nonlinearity that occurs when the bridge oscillators with moderately high amplitude. When the bridge is below equilibrium, the stretched cables act as springs pulling the bridge up and the linear harmonic oscillator model is reasonable. On the other hand, when the bridge is significantly above its equilibrium position, the cables are slack and do not push down. Only the constant force of gravity pulls down on the bridge. They show that a simple nonlinear system with this sort of (discontinuous) nonlinearity can exhibit stable, large amplitude oscillations in the presence periodic forcing over a range of frequencies. Based on this idea, they also develop more accurate partial differential equations models for the full motion of the bridge.

This section develops the Lazer, McKenna system and reports on the results of their numerical and analytical investigations. While the numerical work is fairly delicate and the theorems moderately difficult, the results are well within the reach of the typical differential equations student.

There are many good examples where resonance leads to dramatic behavior in physical systems. That this is not the case for the Tacoma Narrows bridge and that the real culprit was the use of a linear model when a nonlinear model was required is an important lesson.

Comments on selected exercises

Exercises 1--4 are modifications of the Lazer, McKenna system for suspension bridges. Exercises 5--8 discuss another application given by Lazer and McKenna to objects bobbing up and down in water which can lift out of the water.

5.5 Periodic Forcing of Nonlinear Systems and Chaos

This section looks at periodically forced nonlinear systems. An attempt is made to introduce Poincare return maps. This is a long and difficult section, requiring considerable mathematical maturity. A more accessible introduction to recent results in Dynamical Systems (e.g., chaos) is found in Chapter 6.

Comments on selected exercises

One problem with trying to present this material at this level is coming up with exercises which are doable without very fancy technology. Exercises 1--9 try to reinforce the relationship between the graphs of components of solutions and the return map pictures.

Comments on the Labs

Lab 5.1: A Periodically Forced RLC Circuit

The equations modeling RLC circuits, which are frequently given as a main motivation why second-order equations must be studied instead of first-order systems, are actually first developed as a first-order system. The dependent variables are voltage over the capacitor and the current. This system is then converted into a second-order equation. This lab is meant to emphasize this point. It can be assigned after Section 5.2

Lab 5.2: A Predator-Prey System with Periodic Immigration

This lab adds a periodic forcing term to the familiar predator-prey model of Section 2.1 and provides opportunity to explore behavior numerically. The quality of the technology available will probably be a large determining factor on the success of this lab.

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Version 1.1. May, 1996.